\(\int (d+e x)^m (b x+c x^2)^3 \, dx\) [440]

Optimal result

Integrand size = 19, antiderivative size = 267 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\frac {d^3 (c d-b e)^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{2+m}}{e^7 (2+m)}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^7 (7+m)} \]

[Out]

d^3*(-b*e+c*d)^3*(e*x+d)^(1+m)/e^7/(1+m)-3*d^2*(-b*e+c*d)^2*(-b*e+2*c*d)*(e*x+d)^(2+m)/e^7/(2+m)+3*d*(-b*e+c*d
)*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(3+m)/e^7/(3+m)-(-b*e+2*c*d)*(b^2*e^2-10*b*c*d*e+10*c^2*d^2)*(e*x+d)^(
4+m)/e^7/(4+m)+3*c*(b^2*e^2-5*b*c*d*e+5*c^2*d^2)*(e*x+d)^(5+m)/e^7/(5+m)-3*c^2*(-b*e+2*c*d)*(e*x+d)^(6+m)/e^7/
(6+m)+c^3*(e*x+d)^(7+m)/e^7/(7+m)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {712} \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\frac {3 d (c d-b e) \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac {(2 c d-b e) \left (b^2 e^2-10 b c d e+10 c^2 d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac {3 c \left (b^2 e^2-5 b c d e+5 c^2 d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{m+6}}{e^7 (m+6)}+\frac {d^3 (c d-b e)^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{m+2}}{e^7 (m+2)}+\frac {c^3 (d+e x)^{m+7}}{e^7 (m+7)} \]

[In]

Int[(d + e*x)^m*(b*x + c*x^2)^3,x]

[Out]

(d^3*(c*d - b*e)^3*(d + e*x)^(1 + m))/(e^7*(1 + m)) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x)^(2 + m))/(e
^7*(2 + m)) + (3*d*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^(3 + m))/(e^7*(3 + m)) - ((2*c*d -
b*e)*(10*c^2*d^2 - 10*b*c*d*e + b^2*e^2)*(d + e*x)^(4 + m))/(e^7*(4 + m)) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*
e^2)*(d + e*x)^(5 + m))/(e^7*(5 + m)) - (3*c^2*(2*c*d - b*e)*(d + e*x)^(6 + m))/(e^7*(6 + m)) + (c^3*(d + e*x)
^(7 + m))/(e^7*(7 + m))

Rule 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {d^3 (c d-b e)^3 (d+e x)^m}{e^6}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{1+m}}{e^6}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{2+m}}{e^6}+\frac {(2 c d-b e) \left (-10 c^2 d^2+10 b c d e-b^2 e^2\right ) (d+e x)^{3+m}}{e^6}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{4+m}}{e^6}-\frac {3 c^2 (2 c d-b e) (d+e x)^{5+m}}{e^6}+\frac {c^3 (d+e x)^{6+m}}{e^6}\right ) \, dx \\ & = \frac {d^3 (c d-b e)^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)^{2+m}}{e^7 (2+m)}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {3 c^2 (2 c d-b e) (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^7 (7+m)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.88 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\frac {(d+e x)^{1+m} \left (\frac {d^3 (c d-b e)^3}{1+m}-\frac {3 d^2 (c d-b e)^2 (2 c d-b e) (d+e x)}{2+m}+\frac {3 d (c d-b e) \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^2}{3+m}-\frac {(2 c d-b e) \left (10 c^2 d^2-10 b c d e+b^2 e^2\right ) (d+e x)^3}{4+m}+\frac {3 c \left (5 c^2 d^2-5 b c d e+b^2 e^2\right ) (d+e x)^4}{5+m}-\frac {3 c^2 (2 c d-b e) (d+e x)^5}{6+m}+\frac {c^3 (d+e x)^6}{7+m}\right )}{e^7} \]

[In]

Integrate[(d + e*x)^m*(b*x + c*x^2)^3,x]

[Out]

((d + e*x)^(1 + m)*((d^3*(c*d - b*e)^3)/(1 + m) - (3*d^2*(c*d - b*e)^2*(2*c*d - b*e)*(d + e*x))/(2 + m) + (3*d
*(c*d - b*e)*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^2)/(3 + m) - ((2*c*d - b*e)*(10*c^2*d^2 - 10*b*c*d*e
+ b^2*e^2)*(d + e*x)^3)/(4 + m) + (3*c*(5*c^2*d^2 - 5*b*c*d*e + b^2*e^2)*(d + e*x)^4)/(5 + m) - (3*c^2*(2*c*d
- b*e)*(d + e*x)^5)/(6 + m) + (c^3*(d + e*x)^6)/(7 + m)))/e^7

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(903\) vs. \(2(267)=534\).

Time = 2.11 (sec) , antiderivative size = 904, normalized size of antiderivative = 3.39

method	result	size
norman	\(\frac {c^{3} x^{7} {\mathrm e}^{m \ln \left (e x +d \right )}}{7+m}+\frac {\left (b^{3} e^{3} m^{3}+3 b^{2} c d \,e^{2} m^{3}+18 b^{3} e^{3} m^{2}+39 b^{2} c d \,e^{2} m^{2}-15 b \,c^{2} d^{2} e \,m^{2}+107 b^{3} e^{3} m +126 b^{2} c d \,e^{2} m -105 b \,c^{2} d^{2} e m +30 c^{3} d^{3} m +210 b^{3} e^{3}\right ) x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+22 m^{3}+179 m^{2}+638 m +840\right )}+\frac {c^{2} \left (3 b e m +c d m +21 b e \right ) x^{6} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+13 m +42\right )}+\frac {m d \left (b^{3} e^{3} m^{3}+18 b^{3} e^{3} m^{2}-12 b^{2} c d \,e^{2} m^{2}+107 b^{3} e^{3} m -156 b^{2} c d \,e^{2} m +60 b \,c^{2} d^{2} e m +210 b^{3} e^{3}-504 b^{2} d \,e^{2} c +420 b \,c^{2} d^{2} e -120 c^{3} d^{3}\right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{5}+25 m^{4}+245 m^{3}+1175 m^{2}+2754 m +2520\right )}-\frac {6 d^{4} \left (b^{3} e^{3} m^{3}+18 b^{3} e^{3} m^{2}-12 b^{2} c d \,e^{2} m^{2}+107 b^{3} e^{3} m -156 b^{2} c d \,e^{2} m +60 b \,c^{2} d^{2} e m +210 b^{3} e^{3}-504 b^{2} d \,e^{2} c +420 b \,c^{2} d^{2} e -120 c^{3} d^{3}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{7} \left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right )}+\frac {3 \left (b^{2} e^{2} m^{2}+b c d e \,m^{2}+13 b^{2} e^{2} m +7 b c d e m -2 c^{2} d^{2} m +42 b^{2} e^{2}\right ) c \,x^{5} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+18 m^{2}+107 m +210\right )}+\frac {6 m \,d^{3} \left (b^{3} e^{3} m^{3}+18 b^{3} e^{3} m^{2}-12 b^{2} c d \,e^{2} m^{2}+107 b^{3} e^{3} m -156 b^{2} c d \,e^{2} m +60 b \,c^{2} d^{2} e m +210 b^{3} e^{3}-504 b^{2} d \,e^{2} c +420 b \,c^{2} d^{2} e -120 c^{3} d^{3}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{6} \left (m^{7}+28 m^{6}+322 m^{5}+1960 m^{4}+6769 m^{3}+13132 m^{2}+13068 m +5040\right )}-\frac {3 \left (b^{3} e^{3} m^{3}+18 b^{3} e^{3} m^{2}-12 b^{2} c d \,e^{2} m^{2}+107 b^{3} e^{3} m -156 b^{2} c d \,e^{2} m +60 b \,c^{2} d^{2} e m +210 b^{3} e^{3}-504 b^{2} d \,e^{2} c +420 b \,c^{2} d^{2} e -120 c^{3} d^{3}\right ) d^{2} m \,x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{5} \left (m^{6}+27 m^{5}+295 m^{4}+1665 m^{3}+5104 m^{2}+8028 m +5040\right )}\)	\(904\)
gosper	\(\text {Expression too large to display}\)	\(1528\)
risch	\(\text {Expression too large to display}\)	\(1745\)
parallelrisch	\(\text {Expression too large to display}\)	\(2517\)

[In]

int((e*x+d)^m*(c*x^2+b*x)^3,x,method=_RETURNVERBOSE)

[Out]

c^3/(7+m)*x^7*exp(m*ln(e*x+d))+(b^3*e^3*m^3+3*b^2*c*d*e^2*m^3+18*b^3*e^3*m^2+39*b^2*c*d*e^2*m^2-15*b*c^2*d^2*e
*m^2+107*b^3*e^3*m+126*b^2*c*d*e^2*m-105*b*c^2*d^2*e*m+30*c^3*d^3*m+210*b^3*e^3)/e^3/(m^4+22*m^3+179*m^2+638*m
+840)*x^4*exp(m*ln(e*x+d))+c^2*(3*b*e*m+c*d*m+21*b*e)/e/(m^2+13*m+42)*x^6*exp(m*ln(e*x+d))+m*d*(b^3*e^3*m^3+18
*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+107*b^3*e^3*m-156*b^2*c*d*e^2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d*e^2+4
20*b*c^2*d^2*e-120*c^3*d^3)/e^4/(m^5+25*m^4+245*m^3+1175*m^2+2754*m+2520)*x^3*exp(m*ln(e*x+d))-6*d^4*(b^3*e^3*
m^3+18*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+107*b^3*e^3*m-156*b^2*c*d*e^2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d
*e^2+420*b*c^2*d^2*e-120*c^3*d^3)/e^7/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)*exp(m*ln(e
*x+d))+3*(b^2*e^2*m^2+b*c*d*e*m^2+13*b^2*e^2*m+7*b*c*d*e*m-2*c^2*d^2*m+42*b^2*e^2)*c/e^2/(m^3+18*m^2+107*m+210
)*x^5*exp(m*ln(e*x+d))+6/e^6*m*d^3*(b^3*e^3*m^3+18*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+107*b^3*e^3*m-156*b^2*c*d*e^
2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d*e^2+420*b*c^2*d^2*e-120*c^3*d^3)/(m^7+28*m^6+322*m^5+1960*m^4+676
9*m^3+13132*m^2+13068*m+5040)*x*exp(m*ln(e*x+d))-3*(b^3*e^3*m^3+18*b^3*e^3*m^2-12*b^2*c*d*e^2*m^2+107*b^3*e^3*
m-156*b^2*c*d*e^2*m+60*b*c^2*d^2*e*m+210*b^3*e^3-504*b^2*c*d*e^2+420*b*c^2*d^2*e-120*c^3*d^3)*d^2/e^5*m/(m^6+2
7*m^5+295*m^4+1665*m^3+5104*m^2+8028*m+5040)*x^2*exp(m*ln(e*x+d))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1449 vs. \(2 (267) = 534\).

Time = 0.44 (sec) , antiderivative size = 1449, normalized size of antiderivative = 5.43 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

-(6*b^3*d^4*e^3*m^3 - 720*c^3*d^7 + 2520*b*c^2*d^6*e - 3024*b^2*c*d^5*e^2 + 1260*b^3*d^4*e^3 - (c^3*e^7*m^6 +
21*c^3*e^7*m^5 + 175*c^3*e^7*m^4 + 735*c^3*e^7*m^3 + 1624*c^3*e^7*m^2 + 1764*c^3*e^7*m + 720*c^3*e^7)*x^7 - (2
520*b*c^2*e^7 + (c^3*d*e^6 + 3*b*c^2*e^7)*m^6 + 3*(5*c^3*d*e^6 + 22*b*c^2*e^7)*m^5 + 5*(17*c^3*d*e^6 + 114*b*c
^2*e^7)*m^4 + 15*(15*c^3*d*e^6 + 164*b*c^2*e^7)*m^3 + (274*c^3*d*e^6 + 5547*b*c^2*e^7)*m^2 + 6*(20*c^3*d*e^6 +
1019*b*c^2*e^7)*m)*x^6 - 3*(1008*b^2*c*e^7 + (b*c^2*d*e^6 + b^2*c*e^7)*m^6 - (2*c^3*d^2*e^5 - 17*b*c^2*d*e^6
- 23*b^2*c*e^7)*m^5 - (20*c^3*d^2*e^5 - 105*b*c^2*d*e^6 - 207*b^2*c*e^7)*m^4 - 5*(14*c^3*d^2*e^5 - 59*b*c^2*d*
e^6 - 185*b^2*c*e^7)*m^3 - 2*(50*c^3*d^2*e^5 - 187*b*c^2*d*e^6 - 1072*b^2*c*e^7)*m^2 - 12*(4*c^3*d^2*e^5 - 14*
b*c^2*d*e^6 - 201*b^2*c*e^7)*m)*x^5 - (1260*b^3*e^7 + (3*b^2*c*d*e^6 + b^3*e^7)*m^6 - 3*(5*b*c^2*d^2*e^5 - 19*
b^2*c*d*e^6 - 8*b^3*e^7)*m^5 + (30*c^3*d^3*e^4 - 195*b*c^2*d^2*e^5 + 393*b^2*c*d*e^6 + 226*b^3*e^7)*m^4 + 3*(6
0*c^3*d^3*e^4 - 265*b*c^2*d^2*e^5 + 401*b^2*c*d*e^6 + 352*b^3*e^7)*m^3 + 5*(66*c^3*d^3*e^4 - 249*b*c^2*d^2*e^5
+ 324*b^2*c*d*e^6 + 509*b^3*e^7)*m^2 + 18*(10*c^3*d^3*e^4 - 35*b*c^2*d^2*e^5 + 42*b^2*c*d*e^6 + 164*b^3*e^7)*
m)*x^4 - (b^3*d*e^6*m^6 - 3*(4*b^2*c*d^2*e^5 - 7*b^3*d*e^6)*m^5 + (60*b*c^2*d^3*e^4 - 192*b^2*c*d^2*e^5 + 163*
b^3*d*e^6)*m^4 - 3*(40*c^3*d^4*e^3 - 200*b*c^2*d^3*e^4 + 332*b^2*c*d^2*e^5 - 189*b^3*d*e^6)*m^3 - 4*(90*c^3*d^
4*e^3 - 345*b*c^2*d^3*e^4 + 456*b^2*c*d^2*e^5 - 211*b^3*d*e^6)*m^2 - 12*(20*c^3*d^4*e^3 - 70*b*c^2*d^3*e^4 + 8
4*b^2*c*d^2*e^5 - 35*b^3*d*e^6)*m)*x^3 - 36*(2*b^2*c*d^5*e^2 - 3*b^3*d^4*e^3)*m^2 + 3*(b^3*d^2*e^5*m^5 - (12*b
^2*c*d^3*e^4 - 19*b^3*d^2*e^5)*m^4 + (60*b*c^2*d^4*e^3 - 168*b^2*c*d^3*e^4 + 125*b^3*d^2*e^5)*m^3 - (120*c^3*d
^5*e^2 - 480*b*c^2*d^4*e^3 + 660*b^2*c*d^3*e^4 - 317*b^3*d^2*e^5)*m^2 - 6*(20*c^3*d^5*e^2 - 70*b*c^2*d^4*e^3 +
84*b^2*c*d^3*e^4 - 35*b^3*d^2*e^5)*m)*x^2 + 6*(60*b*c^2*d^6*e - 156*b^2*c*d^5*e^2 + 107*b^3*d^4*e^3)*m - 6*(b
^3*d^3*e^4*m^4 - 6*(2*b^2*c*d^4*e^3 - 3*b^3*d^3*e^4)*m^3 + (60*b*c^2*d^5*e^2 - 156*b^2*c*d^4*e^3 + 107*b^3*d^3
*e^4)*m^2 - 6*(20*c^3*d^6*e - 70*b*c^2*d^5*e^2 + 84*b^2*c*d^4*e^3 - 35*b^3*d^3*e^4)*m)*x)*(e*x + d)^m/(e^7*m^7
+ 28*e^7*m^6 + 322*e^7*m^5 + 1960*e^7*m^4 + 6769*e^7*m^3 + 13132*e^7*m^2 + 13068*e^7*m + 5040*e^7)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 21005 vs. \(2 (250) = 500\).

Time = 4.42 (sec) , antiderivative size = 21005, normalized size of antiderivative = 78.67 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)**m*(c*x**2+b*x)**3,x)

[Out]

Piecewise((d**m*(b**3*x**4/4 + 3*b**2*c*x**5/5 + b*c**2*x**6/2 + c**3*x**7/7), Eq(e, 0)), (-b**3*d**3*e**3/(60
*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x
**5 + 60*e**13*x**6) - 6*b**3*d**2*e**4*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**
10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 15*b**3*d*e**5*x**2/(60*d**6*e**7 + 360*d*
*5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6
) - 20*b**3*e**6*x**3/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e
**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 6*b**2*c*d**4*e**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*
e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 36*b**2*c*d**3*e*
*3*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d
*e**12*x**5 + 60*e**13*x**6) - 90*b**2*c*d**2*e**4*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 +
1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 120*b**2*c*d*e**5*x**3/(60*d
**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**
5 + 60*e**13*x**6) - 90*b**2*c*e**6*x**4/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**1
0*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 30*b*c**2*d**5*e/(60*d**6*e**7 + 360*d**5*e
**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) -
180*b*c**2*d**4*e**2*x/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*
e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 450*b*c**2*d**3*e**3*x**2/(60*d**6*e**7 + 360*d**5*e**8*x + 9
00*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 600*b*c**
2*d**2*e**4*x**3/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*
x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 450*b*c**2*d*e**5*x**4/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e
**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) - 180*b*c**2*e**6*x*
*5/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e
**12*x**5 + 60*e**13*x**6) + 60*c**3*d**6*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 +
1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 147*c**3*d**6/(60*d**6*e**7 +
360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**
13*x**6) + 360*c**3*d**5*e*x*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**
10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 822*c**3*d**5*e*x/(60*d**6*e**7 + 360*d**5
*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6)
+ 900*c**3*d**4*e**2*x**2*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*
x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 1875*c**3*d**4*e**2*x**2/(60*d**6*e**7 + 360*
d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x*
*6) + 1200*c**3*d**3*e**3*x**3*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e
**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 2200*c**3*d**3*e**3*x**3/(60*d**6*e**7 +
360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**
13*x**6) + 900*c**3*d**2*e**4*x**4*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d*
*3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 1350*c**3*d**2*e**4*x**4/(60*d**6*e*
*7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60
*e**13*x**6) + 360*c**3*d*e**5*x**5*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d
**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6) + 360*c**3*d*e**5*x**5/(60*d**6*e**7
+ 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e**10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e*
*13*x**6) + 60*c**3*e**6*x**6*log(d/e + x)/(60*d**6*e**7 + 360*d**5*e**8*x + 900*d**4*e**9*x**2 + 1200*d**3*e*
*10*x**3 + 900*d**2*e**11*x**4 + 360*d*e**12*x**5 + 60*e**13*x**6), Eq(m, -7)), (-b**3*d**3*e**3/(20*d**5*e**7
+ 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 5*b**3*d**
2*e**4*x/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e*
*12*x**5) - 10*b**3*d*e**5*x**2/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 1
00*d*e**11*x**4 + 20*e**12*x**5) - 10*b**3*e**6*x**3/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 20
0*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 12*b**2*c*d**4*e**2/(20*d**5*e**7 + 100*d**4*e**8*x +
200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 60*b**2*c*d**3*e**3*x/(20*d**5*
e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 120*b*
*2*c*d**2*e**4*x**2/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x
**4 + 20*e**12*x**5) - 120*b**2*c*d*e**5*x**3/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*
e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 60*b**2*c*e**6*x**4/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**
3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) + 60*b*c**2*d**5*e*log(d/e + x)/(20*d**5
*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) + 137*b
*c**2*d**5*e/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 2
0*e**12*x**5) + 300*b*c**2*d**4*e**2*x*log(d/e + x)/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200
*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) + 625*b*c**2*d**4*e**2*x/(20*d**5*e**7 + 100*d**4*e**8*x
+ 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) + 600*b*c**2*d**3*e**3*x**2*log
(d/e + x)/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e
**12*x**5) + 1100*b*c**2*d**3*e**3*x**2/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*
x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) + 600*b*c**2*d**2*e**4*x**3*log(d/e + x)/(20*d**5*e**7 + 100*d**4*e**
8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) + 900*b*c**2*d**2*e**4*x**3
/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5
) + 300*b*c**2*d*e**5*x**4*log(d/e + x)/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*
x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) + 300*b*c**2*d*e**5*x**4/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e
**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) + 60*b*c**2*e**6*x**5*log(d/e + x)/(20*d**5
*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 120*c
**3*d**6*log(d/e + x)/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11
*x**4 + 20*e**12*x**5) - 274*c**3*d**6/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x
**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 600*c**3*d**5*e*x*log(d/e + x)/(20*d**5*e**7 + 100*d**4*e**8*x + 200
*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 1250*c**3*d**5*e*x/(20*d**5*e**7 +
100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 1200*c**3*d*
*4*e**2*x**2*log(d/e + x)/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e
**11*x**4 + 20*e**12*x**5) - 2200*c**3*d**4*e**2*x**2/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 2
00*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 1200*c**3*d**3*e**3*x**3*log(d/e + x)/(20*d**5*e**7 +
100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 1800*c**3*d*
*3*e**3*x**3/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 2
0*e**12*x**5) - 600*c**3*d**2*e**4*x**4*log(d/e + x)/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 20
0*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 600*c**3*d**2*e**4*x**4/(20*d**5*e**7 + 100*d**4*e**8*
x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**12*x**5) - 120*c**3*d*e**5*x**5*log(d/
e + x)/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d*e**11*x**4 + 20*e**1
2*x**5) + 20*c**3*e**6*x**6/(20*d**5*e**7 + 100*d**4*e**8*x + 200*d**3*e**9*x**2 + 200*d**2*e**10*x**3 + 100*d
*e**11*x**4 + 20*e**12*x**5), Eq(m, -6)), (-b**3*d**3*e**3/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 +
16*d*e**10*x**3 + 4*e**11*x**4) - 4*b**3*d**2*e**4*x/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d
*e**10*x**3 + 4*e**11*x**4) - 6*b**3*d*e**5*x**2/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**1
0*x**3 + 4*e**11*x**4) - 4*b**3*e**6*x**3/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3
+ 4*e**11*x**4) + 12*b**2*c*d**4*e**2*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e*
*10*x**3 + 4*e**11*x**4) + 25*b**2*c*d**4*e**2/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*
x**3 + 4*e**11*x**4) + 48*b**2*c*d**3*e**3*x*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 +
16*d*e**10*x**3 + 4*e**11*x**4) + 88*b**2*c*d**3*e**3*x/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16
*d*e**10*x**3 + 4*e**11*x**4) + 72*b**2*c*d**2*e**4*x**2*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*
e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 108*b**2*c*d**2*e**4*x**2/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**
2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 48*b**2*c*d*e**5*x**3*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8
*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 48*b**2*c*d*e**5*x**3/(4*d**4*e**7 + 16*d**3*e**8*x
+ 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) + 12*b**2*c*e**6*x**4*log(d/e + x)/(4*d**4*e**7 + 16*d*
*3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 60*b*c**2*d**5*e*log(d/e + x)/(4*d**4*e**7 +
16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 125*b*c**2*d**5*e/(4*d**4*e**7 + 16*d*
*3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 240*b*c**2*d**4*e**2*x*log(d/e + x)/(4*d**4*
e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 440*b*c**2*d**4*e**2*x/(4*d**4*e
**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 360*b*c**2*d**3*e**3*x**2*log(d/e
+ x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 540*b*c**2*d**3*e*
*3*x**2/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 240*b*c**2*d**2*
e**4*x**3*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 2
40*b*c**2*d**2*e**4*x**3/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) -
60*b*c**2*d*e**5*x**4*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e*
*11*x**4) + 12*b*c**2*e**6*x**5/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*
x**4) + 60*c**3*d**6*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**1
1*x**4) + 125*c**3*d**6/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) +
240*c**3*d**5*e*x*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x
**4) + 440*c**3*d**5*e*x/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*e**11*x**4) +
360*c**3*d**4*e**2*x**2*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*
e**11*x**4) + 540*c**3*d**4*e**2*x**2/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e**10*x**3 + 4*
e**11*x**4) + 240*c**3*d**3*e**3*x**3*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e*
*10*x**3 + 4*e**11*x**4) + 240*c**3*d**3*e**3*x**3/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e*
*10*x**3 + 4*e**11*x**4) + 60*c**3*d**2*e**4*x**4*log(d/e + x)/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x*
*2 + 16*d*e**10*x**3 + 4*e**11*x**4) - 12*c**3*d*e**5*x**5/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 +
16*d*e**10*x**3 + 4*e**11*x**4) + 2*c**3*e**6*x**6/(4*d**4*e**7 + 16*d**3*e**8*x + 24*d**2*e**9*x**2 + 16*d*e
**10*x**3 + 4*e**11*x**4), Eq(m, -5)), (6*b**3*d**3*e**3*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**
9*x**2 + 6*e**10*x**3) + 11*b**3*d**3*e**3/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 18
*b**3*d**2*e**4*x*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 27*b**3*d**2*e
**4*x/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 18*b**3*d*e**5*x**2*log(d/e + x)/(6*d**
3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 18*b**3*d*e**5*x**2/(6*d**3*e**7 + 18*d**2*e**8*x +
18*d*e**9*x**2 + 6*e**10*x**3) + 6*b**3*e**6*x**3*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2
+ 6*e**10*x**3) - 72*b**2*c*d**4*e**2*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x
**3) - 132*b**2*c*d**4*e**2/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 216*b**2*c*d**3*e
**3*x*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 324*b**2*c*d**3*e**3*x/(6*
d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 216*b**2*c*d**2*e**4*x**2*log(d/e + x)/(6*d**3*e
**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 216*b**2*c*d**2*e**4*x**2/(6*d**3*e**7 + 18*d**2*e**8*
x + 18*d*e**9*x**2 + 6*e**10*x**3) - 72*b**2*c*d*e**5*x**3*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e
**9*x**2 + 6*e**10*x**3) + 18*b**2*c*e**6*x**4/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3)
+ 180*b*c**2*d**5*e*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 330*b*c**2*d
**5*e/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 540*b*c**2*d**4*e**2*x*log(d/e + x)/(6*
d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 810*b*c**2*d**4*e**2*x/(6*d**3*e**7 + 18*d**2*e*
*8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 540*b*c**2*d**3*e**3*x**2*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x +
18*d*e**9*x**2 + 6*e**10*x**3) + 540*b*c**2*d**3*e**3*x**2/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6
*e**10*x**3) + 180*b*c**2*d**2*e**4*x**3*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10
*x**3) - 45*b*c**2*d*e**5*x**4/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 9*b*c**2*e**6*
x**5/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 120*c**3*d**6*log(d/e + x)/(6*d**3*e**7
+ 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 220*c**3*d**6/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x*
*2 + 6*e**10*x**3) - 360*c**3*d**5*e*x*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x
**3) - 540*c**3*d**5*e*x/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 360*c**3*d**4*e**2*x
**2*log(d/e + x)/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 360*c**3*d**4*e**2*x**2/(6*d
**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) - 120*c**3*d**3*e**3*x**3*log(d/e + x)/(6*d**3*e**7
+ 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3) + 30*c**3*d**2*e**4*x**4/(6*d**3*e**7 + 18*d**2*e**8*x + 18
*d*e**9*x**2 + 6*e**10*x**3) - 6*c**3*d*e**5*x**5/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**
3) + 2*c**3*e**6*x**6/(6*d**3*e**7 + 18*d**2*e**8*x + 18*d*e**9*x**2 + 6*e**10*x**3), Eq(m, -4)), (-12*b**3*d*
*3*e**3*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 18*b**3*d**3*e**3/(4*d**2*e**7 + 8*d*e**8*x +
4*e**9*x**2) - 24*b**3*d**2*e**4*x*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 24*b**3*d**2*e**4*x
/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 12*b**3*d*e**5*x**2*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**
9*x**2) + 4*b**3*e**6*x**3/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 72*b**2*c*d**4*e**2*log(d/e + x)/(4*d**2
*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 108*b**2*c*d**4*e**2/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 144*b**2*c
*d**3*e**3*x*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 144*b**2*c*d**3*e**3*x/(4*d**2*e**7 + 8*d
*e**8*x + 4*e**9*x**2) + 72*b**2*c*d**2*e**4*x**2*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 24*b
**2*c*d*e**5*x**3/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 6*b**2*c*e**6*x**4/(4*d**2*e**7 + 8*d*e**8*x + 4*
e**9*x**2) - 120*b*c**2*d**5*e*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 180*b*c**2*d**5*e/(4*d*
*2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 240*b*c**2*d**4*e**2*x*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x
**2) - 240*b*c**2*d**4*e**2*x/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 120*b*c**2*d**3*e**3*x**2*log(d/e + x
)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 40*b*c**2*d**2*e**4*x**3/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2)
- 10*b*c**2*d*e**5*x**4/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 4*b*c**2*e**6*x**5/(4*d**2*e**7 + 8*d*e**8
*x + 4*e**9*x**2) + 60*c**3*d**6*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 90*c**3*d**6/(4*d**2*
e**7 + 8*d*e**8*x + 4*e**9*x**2) + 120*c**3*d**5*e*x*log(d/e + x)/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 1
20*c**3*d**5*e*x/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 60*c**3*d**4*e**2*x**2*log(d/e + x)/(4*d**2*e**7 +
8*d*e**8*x + 4*e**9*x**2) - 20*c**3*d**3*e**3*x**3/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) + 5*c**3*d**2*e**
4*x**4/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2) - 2*c**3*d*e**5*x**5/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2)
+ c**3*e**6*x**6/(4*d**2*e**7 + 8*d*e**8*x + 4*e**9*x**2), Eq(m, -3)), (60*b**3*d**3*e**3*log(d/e + x)/(20*d*e
**7 + 20*e**8*x) + 60*b**3*d**3*e**3/(20*d*e**7 + 20*e**8*x) + 60*b**3*d**2*e**4*x*log(d/e + x)/(20*d*e**7 + 2
0*e**8*x) - 30*b**3*d*e**5*x**2/(20*d*e**7 + 20*e**8*x) + 10*b**3*e**6*x**3/(20*d*e**7 + 20*e**8*x) - 240*b**2
*c*d**4*e**2*log(d/e + x)/(20*d*e**7 + 20*e**8*x) - 240*b**2*c*d**4*e**2/(20*d*e**7 + 20*e**8*x) - 240*b**2*c*
d**3*e**3*x*log(d/e + x)/(20*d*e**7 + 20*e**8*x) + 120*b**2*c*d**2*e**4*x**2/(20*d*e**7 + 20*e**8*x) - 40*b**2
*c*d*e**5*x**3/(20*d*e**7 + 20*e**8*x) + 20*b**2*c*e**6*x**4/(20*d*e**7 + 20*e**8*x) + 300*b*c**2*d**5*e*log(d
/e + x)/(20*d*e**7 + 20*e**8*x) + 300*b*c**2*d**5*e/(20*d*e**7 + 20*e**8*x) + 300*b*c**2*d**4*e**2*x*log(d/e +
x)/(20*d*e**7 + 20*e**8*x) - 150*b*c**2*d**3*e**3*x**2/(20*d*e**7 + 20*e**8*x) + 50*b*c**2*d**2*e**4*x**3/(20
*d*e**7 + 20*e**8*x) - 25*b*c**2*d*e**5*x**4/(20*d*e**7 + 20*e**8*x) + 15*b*c**2*e**6*x**5/(20*d*e**7 + 20*e**
8*x) - 120*c**3*d**6*log(d/e + x)/(20*d*e**7 + 20*e**8*x) - 120*c**3*d**6/(20*d*e**7 + 20*e**8*x) - 120*c**3*d
**5*e*x*log(d/e + x)/(20*d*e**7 + 20*e**8*x) + 60*c**3*d**4*e**2*x**2/(20*d*e**7 + 20*e**8*x) - 20*c**3*d**3*e
**3*x**3/(20*d*e**7 + 20*e**8*x) + 10*c**3*d**2*e**4*x**4/(20*d*e**7 + 20*e**8*x) - 6*c**3*d*e**5*x**5/(20*d*e
**7 + 20*e**8*x) + 4*c**3*e**6*x**6/(20*d*e**7 + 20*e**8*x), Eq(m, -2)), (-b**3*d**3*log(d/e + x)/e**4 + b**3*
d**2*x/e**3 - b**3*d*x**2/(2*e**2) + b**3*x**3/(3*e) + 3*b**2*c*d**4*log(d/e + x)/e**5 - 3*b**2*c*d**3*x/e**4
+ 3*b**2*c*d**2*x**2/(2*e**3) - b**2*c*d*x**3/e**2 + 3*b**2*c*x**4/(4*e) - 3*b*c**2*d**5*log(d/e + x)/e**6 + 3
*b*c**2*d**4*x/e**5 - 3*b*c**2*d**3*x**2/(2*e**4) + b*c**2*d**2*x**3/e**3 - 3*b*c**2*d*x**4/(4*e**2) + 3*b*c**
2*x**5/(5*e) + c**3*d**6*log(d/e + x)/e**7 - c**3*d**5*x/e**6 + c**3*d**4*x**2/(2*e**5) - c**3*d**3*x**3/(3*e*
*4) + c**3*d**2*x**4/(4*e**3) - c**3*d*x**5/(5*e**2) + c**3*x**6/(6*e), Eq(m, -1)), (-6*b**3*d**4*e**3*m**3*(d
+ e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 130
68*e**7*m + 5040*e**7) - 108*b**3*d**4*e**3*m**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960
*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 642*b**3*d**4*e**3*m*(d + e*x)**m/
(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m +
5040*e**7) - 1260*b**3*d**4*e**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 67
69*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 6*b**3*d**3*e**4*m**4*x*(d + e*x)**m/(e**7*m**7 +
28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7)
+ 108*b**3*d**3*e**4*m**3*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**
7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 642*b**3*d**3*e**4*m**2*x*(d + e*x)**m/(e**7*m**7 + 28*
e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 12
60*b**3*d**3*e**4*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3
+ 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 3*b**3*d**2*e**5*m**5*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*
m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 57*b**3
*d**2*e**5*m**4*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3
+ 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 375*b**3*d**2*e**5*m**3*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7
*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 951*b*
*3*d**2*e**5*m**2*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**
3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 630*b**3*d**2*e**5*m*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*
m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + b**3*d*
e**6*m**6*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 1313
2*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 21*b**3*d*e**6*m**5*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 32
2*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 163*b**3*d*e**6*
m**4*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**
7*m**2 + 13068*e**7*m + 5040*e**7) + 567*b**3*d*e**6*m**3*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e*
*7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 844*b**3*d*e**6*m**2
*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m*
*2 + 13068*e**7*m + 5040*e**7) + 420*b**3*d*e**6*m*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5
+ 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + b**3*e**7*m**6*x**4*(d + e*
x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e*
*7*m + 5040*e**7) + 24*b**3*e**7*m**5*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*
m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 226*b**3*e**7*m**4*x**4*(d + e*x)**m/(e*
*7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 50
40*e**7) + 1056*b**3*e**7*m**3*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 +
6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 2545*b**3*e**7*m**2*x**4*(d + e*x)**m/(e**7*m**
7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**
7) + 2952*b**3*e**7*m*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7
*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1260*b**3*e**7*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m*
*6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 72*b**2*c
*d**5*e**2*m**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 131
32*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 936*b**2*c*d**5*e**2*m*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322
*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3024*b**2*c*d**5*
e**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**
2 + 13068*e**7*m + 5040*e**7) - 72*b**2*c*d**4*e**3*m**3*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m
**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 936*b**2*c*d**4*e**3*m**
2*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2
+ 13068*e**7*m + 5040*e**7) - 3024*b**2*c*d**4*e**3*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**
5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 36*b**2*c*d**3*e**4*m**4*x
**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2
+ 13068*e**7*m + 5040*e**7) + 504*b**2*c*d**3*e**4*m**3*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**
7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1980*b**2*c*d**3*e**4
*m**2*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e*
*7*m**2 + 13068*e**7*m + 5040*e**7) + 1512*b**2*c*d**3*e**4*m*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 32
2*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 12*b**2*c*d**2*e
**5*m**5*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132
*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 192*b**2*c*d**2*e**5*m**4*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6
+ 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 996*b**2*c*
d**2*e**5*m**3*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 +
13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 1824*b**2*c*d**2*e**5*m**2*x**3*(d + e*x)**m/(e**7*m**7 + 28*e*
*7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 1008
*b**2*c*d**2*e**5*m*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m
**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*b**2*c*d*e**6*m**6*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**
7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 57*b*
*2*c*d*e**6*m**5*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3
+ 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 393*b**2*c*d*e**6*m**4*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7
*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1203*b
**2*c*d*e**6*m**3*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**
3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1620*b**2*c*d*e**6*m**2*x**4*(d + e*x)**m/(e**7*m**7 + 28*e*
*7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 756*
b**2*c*d*e**6*m*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3
+ 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*b**2*c*e**7*m**6*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6
+ 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 69*b**2*c*e
**7*m**5*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132
*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 621*b**2*c*e**7*m**4*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 32
2*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 2775*b**2*c*e**7
*m**3*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e*
*7*m**2 + 13068*e**7*m + 5040*e**7) + 6432*b**2*c*e**7*m**2*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*
e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 7236*b**2*c*e**7*m
*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m*
*2 + 13068*e**7*m + 5040*e**7) + 3024*b**2*c*e**7*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5
+ 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 360*b*c**2*d**6*e*m*(d + e*x
)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**
7*m + 5040*e**7) - 2520*b*c**2*d**6*e*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4
+ 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 360*b*c**2*d**5*e**2*m**2*x*(d + e*x)**m/(e**
7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 504
0*e**7) + 2520*b*c**2*d**5*e**2*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 +
6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 180*b*c**2*d**4*e**3*m**3*x**2*(d + e*x)**m/(e*
*7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 50
40*e**7) - 1440*b*c**2*d**4*e**3*m**2*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*
m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 1260*b*c**2*d**4*e**3*m*x**2*(d + e*x)**
m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m
+ 5040*e**7) + 60*b*c**2*d**3*e**4*m**4*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e*
*7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 600*b*c**2*d**3*e**4*m**3*x**3*(d + e
*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e
**7*m + 5040*e**7) + 1380*b*c**2*d**3*e**4*m**2*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 +
1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 840*b*c**2*d**3*e**4*m*x**3*(d
+ e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 130
68*e**7*m + 5040*e**7) - 15*b*c**2*d**2*e**5*m**5*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5
+ 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 195*b*c**2*d**2*e**5*m**4*x*
*4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2
+ 13068*e**7*m + 5040*e**7) - 795*b*c**2*d**2*e**5*m**3*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7
*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 1245*b*c**2*d**2*e**5*
m**2*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**
7*m**2 + 13068*e**7*m + 5040*e**7) - 630*b*c**2*d**2*e**5*m*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*
e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*b*c**2*d*e**6*m*
*6*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*
m**2 + 13068*e**7*m + 5040*e**7) + 51*b*c**2*d*e**6*m**5*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**
7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 315*b*c**2*d*e**6*m**
4*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m
**2 + 13068*e**7*m + 5040*e**7) + 885*b*c**2*d*e**6*m**3*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**
7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1122*b*c**2*d*e**6*m*
*2*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*
m**2 + 13068*e**7*m + 5040*e**7) + 504*b*c**2*d*e**6*m*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*
m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 3*b*c**2*e**7*m**6*x**6
*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 +
13068*e**7*m + 5040*e**7) + 66*b*c**2*e**7*m**5*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 +
1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 570*b*c**2*e**7*m**4*x**6*(d +
e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068
*e**7*m + 5040*e**7) + 2460*b*c**2*e**7*m**3*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 196
0*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 5547*b*c**2*e**7*m**2*x**6*(d + e
*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e
**7*m + 5040*e**7) + 6114*b*c**2*e**7*m*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**
7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 2520*b*c**2*e**7*x**6*(d + e*x)**m/(e*
*7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 50
40*e**7) + 720*c**3*d**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m
**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 720*c**3*d**6*e*m*x*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6
+ 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 360*c**3*d*
*5*e**2*m**2*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 1
3132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 360*c**3*d**5*e**2*m*x**2*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6
+ 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 120*c**3*d**
4*e**3*m**3*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13
132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 360*c**3*d**4*e**3*m**2*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**
6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 240*c**3*d
**4*e**3*m*x**3*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 131
32*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 30*c**3*d**3*e**4*m**4*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6
+ 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 180*c**3*d**
3*e**4*m**3*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13
132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 330*c**3*d**3*e**4*m**2*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**
6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 180*c**3*d
**3*e**4*m*x**4*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 131
32*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 6*c**3*d**2*e**5*m**5*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 +
322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 60*c**3*d**2*
e**5*m**4*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 1313
2*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 210*c**3*d**2*e**5*m**3*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6
+ 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 300*c**3*d**
2*e**5*m**2*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13
132*e**7*m**2 + 13068*e**7*m + 5040*e**7) - 144*c**3*d**2*e**5*m*x**5*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 +
322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + c**3*d*e**6*m
**6*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7
*m**2 + 13068*e**7*m + 5040*e**7) + 15*c**3*d*e**6*m**5*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7
*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 85*c**3*d*e**6*m**4*x*
*6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2
+ 13068*e**7*m + 5040*e**7) + 225*c**3*d*e**6*m**3*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5
+ 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 274*c**3*d*e**6*m**2*x**6*(
d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13
068*e**7*m + 5040*e**7) + 120*c**3*d*e**6*m*x**6*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960
*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + c**3*e**7*m**6*x**7*(d + e*x)**m/(
e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m +
5040*e**7) + 21*c**3*e**7*m**5*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 +
6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 175*c**3*e**7*m**4*x**7*(d + e*x)**m/(e**7*m**7
+ 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7
) + 735*c**3*e**7*m**3*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**
7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 1624*c**3*e**7*m**2*x**7*(d + e*x)**m/(e**7*m**7 + 28*e
**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 176
4*c**3*e**7*m*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 +
13132*e**7*m**2 + 13068*e**7*m + 5040*e**7) + 720*c**3*e**7*x**7*(d + e*x)**m/(e**7*m**7 + 28*e**7*m**6 + 322*
e**7*m**5 + 1960*e**7*m**4 + 6769*e**7*m**3 + 13132*e**7*m**2 + 13068*e**7*m + 5040*e**7), True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 669 vs. \(2 (267) = 534\).

Time = 0.22 (sec) , antiderivative size = 669, normalized size of antiderivative = 2.51 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{4} x^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d e^{3} x^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} e^{2} x^{2} + 6 \, d^{3} e m x - 6 \, d^{4}\right )} {\left (e x + d\right )}^{m} b^{3}}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{5} x^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d e^{4} x^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} e^{3} x^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} e^{2} x^{2} - 24 \, d^{4} e m x + 24 \, d^{5}\right )} {\left (e x + d\right )}^{m} b^{2} c}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} + \frac {3 \, {\left ({\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{6} x^{6} + {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d e^{5} x^{5} - 5 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{2} e^{4} x^{4} + 20 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{3} e^{3} x^{3} - 60 \, {\left (m^{2} + m\right )} d^{4} e^{2} x^{2} + 120 \, d^{5} e m x - 120 \, d^{6}\right )} {\left (e x + d\right )}^{m} b c^{2}}{{\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{6}} + \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} e^{7} x^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d e^{6} x^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} e^{5} x^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} e^{4} x^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} e^{3} x^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} e^{2} x^{2} - 720 \, d^{6} e m x + 720 \, d^{7}\right )} {\left (e x + d\right )}^{m} c^{3}}{{\left (m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040\right )} e^{7}} \]

[In]

integrate((e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

((m^3 + 6*m^2 + 11*m + 6)*e^4*x^4 + (m^3 + 3*m^2 + 2*m)*d*e^3*x^3 - 3*(m^2 + m)*d^2*e^2*x^2 + 6*d^3*e*m*x - 6*
d^4)*(e*x + d)^m*b^3/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^5*x^
5 + (m^4 + 6*m^3 + 11*m^2 + 6*m)*d*e^4*x^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*e^3*x^3 + 12*(m^2 + m)*d^3*e^2*x^2 - 24
*d^4*e*m*x + 24*d^5)*(e*x + d)^m*b^2*c/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5) + 3*((m^5 + 15*m^
4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^6*x^6 + (m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d*e^5*x^5 - 5*(m^4 + 6*m
^3 + 11*m^2 + 6*m)*d^2*e^4*x^4 + 20*(m^3 + 3*m^2 + 2*m)*d^3*e^3*x^3 - 60*(m^2 + m)*d^4*e^2*x^2 + 120*d^5*e*m*x
- 120*d^6)*(e*x + d)^m*b*c^2/((m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^6) + ((m^6 + 21*
m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*e^7*x^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*
m)*d*e^6*x^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*e^5*x^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*e^4
*x^4 - 120*(m^3 + 3*m^2 + 2*m)*d^4*e^3*x^3 + 360*(m^2 + m)*d^5*e^2*x^2 - 720*d^6*e*m*x + 720*d^7)*(e*x + d)^m*
c^3/((m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)*e^7)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2539 vs. \(2 (267) = 534\).

Time = 0.30 (sec) , antiderivative size = 2539, normalized size of antiderivative = 9.51 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\text {Too large to display} \]

[In]

integrate((e*x+d)^m*(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

((e*x + d)^m*c^3*e^7*m^6*x^7 + (e*x + d)^m*c^3*d*e^6*m^6*x^6 + 3*(e*x + d)^m*b*c^2*e^7*m^6*x^6 + 21*(e*x + d)^
m*c^3*e^7*m^5*x^7 + 3*(e*x + d)^m*b*c^2*d*e^6*m^6*x^5 + 3*(e*x + d)^m*b^2*c*e^7*m^6*x^5 + 15*(e*x + d)^m*c^3*d
*e^6*m^5*x^6 + 66*(e*x + d)^m*b*c^2*e^7*m^5*x^6 + 175*(e*x + d)^m*c^3*e^7*m^4*x^7 + 3*(e*x + d)^m*b^2*c*d*e^6*
m^6*x^4 + (e*x + d)^m*b^3*e^7*m^6*x^4 - 6*(e*x + d)^m*c^3*d^2*e^5*m^5*x^5 + 51*(e*x + d)^m*b*c^2*d*e^6*m^5*x^5
+ 69*(e*x + d)^m*b^2*c*e^7*m^5*x^5 + 85*(e*x + d)^m*c^3*d*e^6*m^4*x^6 + 570*(e*x + d)^m*b*c^2*e^7*m^4*x^6 + 7
35*(e*x + d)^m*c^3*e^7*m^3*x^7 + (e*x + d)^m*b^3*d*e^6*m^6*x^3 - 15*(e*x + d)^m*b*c^2*d^2*e^5*m^5*x^4 + 57*(e*
x + d)^m*b^2*c*d*e^6*m^5*x^4 + 24*(e*x + d)^m*b^3*e^7*m^5*x^4 - 60*(e*x + d)^m*c^3*d^2*e^5*m^4*x^5 + 315*(e*x
+ d)^m*b*c^2*d*e^6*m^4*x^5 + 621*(e*x + d)^m*b^2*c*e^7*m^4*x^5 + 225*(e*x + d)^m*c^3*d*e^6*m^3*x^6 + 2460*(e*x
+ d)^m*b*c^2*e^7*m^3*x^6 + 1624*(e*x + d)^m*c^3*e^7*m^2*x^7 - 12*(e*x + d)^m*b^2*c*d^2*e^5*m^5*x^3 + 21*(e*x
+ d)^m*b^3*d*e^6*m^5*x^3 + 30*(e*x + d)^m*c^3*d^3*e^4*m^4*x^4 - 195*(e*x + d)^m*b*c^2*d^2*e^5*m^4*x^4 + 393*(e
*x + d)^m*b^2*c*d*e^6*m^4*x^4 + 226*(e*x + d)^m*b^3*e^7*m^4*x^4 - 210*(e*x + d)^m*c^3*d^2*e^5*m^3*x^5 + 885*(e
*x + d)^m*b*c^2*d*e^6*m^3*x^5 + 2775*(e*x + d)^m*b^2*c*e^7*m^3*x^5 + 274*(e*x + d)^m*c^3*d*e^6*m^2*x^6 + 5547*
(e*x + d)^m*b*c^2*e^7*m^2*x^6 + 1764*(e*x + d)^m*c^3*e^7*m*x^7 - 3*(e*x + d)^m*b^3*d^2*e^5*m^5*x^2 + 60*(e*x +
d)^m*b*c^2*d^3*e^4*m^4*x^3 - 192*(e*x + d)^m*b^2*c*d^2*e^5*m^4*x^3 + 163*(e*x + d)^m*b^3*d*e^6*m^4*x^3 + 180*
(e*x + d)^m*c^3*d^3*e^4*m^3*x^4 - 795*(e*x + d)^m*b*c^2*d^2*e^5*m^3*x^4 + 1203*(e*x + d)^m*b^2*c*d*e^6*m^3*x^4
+ 1056*(e*x + d)^m*b^3*e^7*m^3*x^4 - 300*(e*x + d)^m*c^3*d^2*e^5*m^2*x^5 + 1122*(e*x + d)^m*b*c^2*d*e^6*m^2*x
^5 + 6432*(e*x + d)^m*b^2*c*e^7*m^2*x^5 + 120*(e*x + d)^m*c^3*d*e^6*m*x^6 + 6114*(e*x + d)^m*b*c^2*e^7*m*x^6 +
720*(e*x + d)^m*c^3*e^7*x^7 + 36*(e*x + d)^m*b^2*c*d^3*e^4*m^4*x^2 - 57*(e*x + d)^m*b^3*d^2*e^5*m^4*x^2 - 120
*(e*x + d)^m*c^3*d^4*e^3*m^3*x^3 + 600*(e*x + d)^m*b*c^2*d^3*e^4*m^3*x^3 - 996*(e*x + d)^m*b^2*c*d^2*e^5*m^3*x
^3 + 567*(e*x + d)^m*b^3*d*e^6*m^3*x^3 + 330*(e*x + d)^m*c^3*d^3*e^4*m^2*x^4 - 1245*(e*x + d)^m*b*c^2*d^2*e^5*
m^2*x^4 + 1620*(e*x + d)^m*b^2*c*d*e^6*m^2*x^4 + 2545*(e*x + d)^m*b^3*e^7*m^2*x^4 - 144*(e*x + d)^m*c^3*d^2*e^
5*m*x^5 + 504*(e*x + d)^m*b*c^2*d*e^6*m*x^5 + 7236*(e*x + d)^m*b^2*c*e^7*m*x^5 + 2520*(e*x + d)^m*b*c^2*e^7*x^
6 + 6*(e*x + d)^m*b^3*d^3*e^4*m^4*x - 180*(e*x + d)^m*b*c^2*d^4*e^3*m^3*x^2 + 504*(e*x + d)^m*b^2*c*d^3*e^4*m^
3*x^2 - 375*(e*x + d)^m*b^3*d^2*e^5*m^3*x^2 - 360*(e*x + d)^m*c^3*d^4*e^3*m^2*x^3 + 1380*(e*x + d)^m*b*c^2*d^3
*e^4*m^2*x^3 - 1824*(e*x + d)^m*b^2*c*d^2*e^5*m^2*x^3 + 844*(e*x + d)^m*b^3*d*e^6*m^2*x^3 + 180*(e*x + d)^m*c^
3*d^3*e^4*m*x^4 - 630*(e*x + d)^m*b*c^2*d^2*e^5*m*x^4 + 756*(e*x + d)^m*b^2*c*d*e^6*m*x^4 + 2952*(e*x + d)^m*b
^3*e^7*m*x^4 + 3024*(e*x + d)^m*b^2*c*e^7*x^5 - 72*(e*x + d)^m*b^2*c*d^4*e^3*m^3*x + 108*(e*x + d)^m*b^3*d^3*e
^4*m^3*x + 360*(e*x + d)^m*c^3*d^5*e^2*m^2*x^2 - 1440*(e*x + d)^m*b*c^2*d^4*e^3*m^2*x^2 + 1980*(e*x + d)^m*b^2
*c*d^3*e^4*m^2*x^2 - 951*(e*x + d)^m*b^3*d^2*e^5*m^2*x^2 - 240*(e*x + d)^m*c^3*d^4*e^3*m*x^3 + 840*(e*x + d)^m
*b*c^2*d^3*e^4*m*x^3 - 1008*(e*x + d)^m*b^2*c*d^2*e^5*m*x^3 + 420*(e*x + d)^m*b^3*d*e^6*m*x^3 + 1260*(e*x + d)
^m*b^3*e^7*x^4 - 6*(e*x + d)^m*b^3*d^4*e^3*m^3 + 360*(e*x + d)^m*b*c^2*d^5*e^2*m^2*x - 936*(e*x + d)^m*b^2*c*d
^4*e^3*m^2*x + 642*(e*x + d)^m*b^3*d^3*e^4*m^2*x + 360*(e*x + d)^m*c^3*d^5*e^2*m*x^2 - 1260*(e*x + d)^m*b*c^2*
d^4*e^3*m*x^2 + 1512*(e*x + d)^m*b^2*c*d^3*e^4*m*x^2 - 630*(e*x + d)^m*b^3*d^2*e^5*m*x^2 + 72*(e*x + d)^m*b^2*
c*d^5*e^2*m^2 - 108*(e*x + d)^m*b^3*d^4*e^3*m^2 - 720*(e*x + d)^m*c^3*d^6*e*m*x + 2520*(e*x + d)^m*b*c^2*d^5*e
^2*m*x - 3024*(e*x + d)^m*b^2*c*d^4*e^3*m*x + 1260*(e*x + d)^m*b^3*d^3*e^4*m*x - 360*(e*x + d)^m*b*c^2*d^6*e*m
+ 936*(e*x + d)^m*b^2*c*d^5*e^2*m - 642*(e*x + d)^m*b^3*d^4*e^3*m + 720*(e*x + d)^m*c^3*d^7 - 2520*(e*x + d)^
m*b*c^2*d^6*e + 3024*(e*x + d)^m*b^2*c*d^5*e^2 - 1260*(e*x + d)^m*b^3*d^4*e^3)/(e^7*m^7 + 28*e^7*m^6 + 322*e^7
*m^5 + 1960*e^7*m^4 + 6769*e^7*m^3 + 13132*e^7*m^2 + 13068*e^7*m + 5040*e^7)

Mupad [B] (verification not implemented)

Time = 10.36 (sec) , antiderivative size = 1085, normalized size of antiderivative = 4.06 \[ \int (d+e x)^m \left (b x+c x^2\right )^3 \, dx=\frac {c^3\,x^7\,{\left (d+e\,x\right )}^m\,\left (m^6+21\,m^5+175\,m^4+735\,m^3+1624\,m^2+1764\,m+720\right )}{m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040}-\frac {6\,d^4\,{\left (d+e\,x\right )}^m\,\left (b^3\,e^3\,m^3+18\,b^3\,e^3\,m^2+107\,b^3\,e^3\,m+210\,b^3\,e^3-12\,b^2\,c\,d\,e^2\,m^2-156\,b^2\,c\,d\,e^2\,m-504\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e\,m+420\,b\,c^2\,d^2\,e-120\,c^3\,d^3\right )}{e^7\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (b^3\,e^3\,m^3+18\,b^3\,e^3\,m^2+107\,b^3\,e^3\,m+210\,b^3\,e^3+3\,b^2\,c\,d\,e^2\,m^3+39\,b^2\,c\,d\,e^2\,m^2+126\,b^2\,c\,d\,e^2\,m-15\,b\,c^2\,d^2\,e\,m^2-105\,b\,c^2\,d^2\,e\,m+30\,c^3\,d^3\,m\right )}{e^3\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {3\,c\,x^5\,{\left (d+e\,x\right )}^m\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )\,\left (b^2\,e^2\,m^2+13\,b^2\,e^2\,m+42\,b^2\,e^2+b\,c\,d\,e\,m^2+7\,b\,c\,d\,e\,m-2\,c^2\,d^2\,m\right )}{e^2\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {6\,d^3\,m\,x\,{\left (d+e\,x\right )}^m\,\left (b^3\,e^3\,m^3+18\,b^3\,e^3\,m^2+107\,b^3\,e^3\,m+210\,b^3\,e^3-12\,b^2\,c\,d\,e^2\,m^2-156\,b^2\,c\,d\,e^2\,m-504\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e\,m+420\,b\,c^2\,d^2\,e-120\,c^3\,d^3\right )}{e^6\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {c^2\,x^6\,{\left (d+e\,x\right )}^m\,\left (21\,b\,e+3\,b\,e\,m+c\,d\,m\right )\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}{e\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}+\frac {d\,m\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (b^3\,e^3\,m^3+18\,b^3\,e^3\,m^2+107\,b^3\,e^3\,m+210\,b^3\,e^3-12\,b^2\,c\,d\,e^2\,m^2-156\,b^2\,c\,d\,e^2\,m-504\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e\,m+420\,b\,c^2\,d^2\,e-120\,c^3\,d^3\right )}{e^4\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )}-\frac {3\,d^2\,m\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (b^3\,e^3\,m^3+18\,b^3\,e^3\,m^2+107\,b^3\,e^3\,m+210\,b^3\,e^3-12\,b^2\,c\,d\,e^2\,m^2-156\,b^2\,c\,d\,e^2\,m-504\,b^2\,c\,d\,e^2+60\,b\,c^2\,d^2\,e\,m+420\,b\,c^2\,d^2\,e-120\,c^3\,d^3\right )}{e^5\,\left (m^7+28\,m^6+322\,m^5+1960\,m^4+6769\,m^3+13132\,m^2+13068\,m+5040\right )} \]

[In]

int((b*x + c*x^2)^3*(d + e*x)^m,x)

[Out]

(c^3*x^7*(d + e*x)^m*(1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21*m^5 + m^6 + 720))/(13068*m + 13132*m^2 + 6769
*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040) - (6*d^4*(d + e*x)^m*(210*b^3*e^3 - 120*c^3*d^3 + 107*b^3*e^3
*m + 18*b^3*e^3*m^2 + b^3*e^3*m^3 + 420*b*c^2*d^2*e - 504*b^2*c*d*e^2 + 60*b*c^2*d^2*e*m - 156*b^2*c*d*e^2*m -
12*b^2*c*d*e^2*m^2))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (x^4
*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(210*b^3*e^3 + 107*b^3*e^3*m + 30*c^3*d^3*m + 18*b^3*e^3*m^2 + b^3*e^3*m
^3 - 105*b*c^2*d^2*e*m + 126*b^2*c*d*e^2*m - 15*b*c^2*d^2*e*m^2 + 39*b^2*c*d*e^2*m^2 + 3*b^2*c*d*e^2*m^3))/(e^
3*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*c*x^5*(d + e*x)^m*(50*m +
35*m^2 + 10*m^3 + m^4 + 24)*(42*b^2*e^2 + 13*b^2*e^2*m - 2*c^2*d^2*m + b^2*e^2*m^2 + 7*b*c*d*e*m + b*c*d*e*m^2
))/(e^2*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (6*d^3*m*x*(d + e*x)^m*
(210*b^3*e^3 - 120*c^3*d^3 + 107*b^3*e^3*m + 18*b^3*e^3*m^2 + b^3*e^3*m^3 + 420*b*c^2*d^2*e - 504*b^2*c*d*e^2
+ 60*b*c^2*d^2*e*m - 156*b^2*c*d*e^2*m - 12*b^2*c*d*e^2*m^2))/(e^6*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4
+ 322*m^5 + 28*m^6 + m^7 + 5040)) + (c^2*x^6*(d + e*x)^m*(21*b*e + 3*b*e*m + c*d*m)*(274*m + 225*m^2 + 85*m^3
+ 15*m^4 + m^5 + 120))/(e*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (d*m*
x^3*(d + e*x)^m*(3*m + m^2 + 2)*(210*b^3*e^3 - 120*c^3*d^3 + 107*b^3*e^3*m + 18*b^3*e^3*m^2 + b^3*e^3*m^3 + 42
0*b*c^2*d^2*e - 504*b^2*c*d*e^2 + 60*b*c^2*d^2*e*m - 156*b^2*c*d*e^2*m - 12*b^2*c*d*e^2*m^2))/(e^4*(13068*m +
13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) - (3*d^2*m*x^2*(m + 1)*(d + e*x)^m*(210*b^3*
e^3 - 120*c^3*d^3 + 107*b^3*e^3*m + 18*b^3*e^3*m^2 + b^3*e^3*m^3 + 420*b*c^2*d^2*e - 504*b^2*c*d*e^2 + 60*b*c^
2*d^2*e*m - 156*b^2*c*d*e^2*m - 12*b^2*c*d*e^2*m^2))/(e^5*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5
+ 28*m^6 + m^7 + 5040))