3.8.21 \(\int (d+e x)^m (a+c x^2)^3 \, dx\) [721]

Optimal. Leaf size=223 \[ \frac {\left (c d^2+a e^2\right )^3 (d+e x)^{1+m}}{e^7 (1+m)}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^{2+m}}{e^7 (2+m)}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^{3+m}}{e^7 (3+m)}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{4+m}}{e^7 (4+m)}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{5+m}}{e^7 (5+m)}-\frac {6 c^3 d (d+e x)^{6+m}}{e^7 (6+m)}+\frac {c^3 (d+e x)^{7+m}}{e^7 (7+m)} \]

[Out]

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

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Rubi [A]
time = 0.09, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \begin {gather*} -\frac {4 c^2 d \left (3 a e^2+5 c d^2\right ) (d+e x)^{m+4}}{e^7 (m+4)}+\frac {3 c^2 \left (a e^2+5 c d^2\right ) (d+e x)^{m+5}}{e^7 (m+5)}+\frac {\left (a e^2+c d^2\right )^3 (d+e x)^{m+1}}{e^7 (m+1)}-\frac {6 c d \left (a e^2+c d^2\right )^2 (d+e x)^{m+2}}{e^7 (m+2)}+\frac {3 c \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right ) (d+e x)^{m+3}}{e^7 (m+3)}-\frac {6 c^3 d (d+e x)^{m+6}}{e^7 (m+6)}+\frac {c^3 (d+e x)^{m+7}}{e^7 (m+7)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^m*(a + c*x^2)^3,x]

[Out]

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

Rule 711

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

Rubi steps

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

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Mathematica [A]
time = 0.24, size = 398, normalized size = 1.78 \begin {gather*} \frac {(d+e x)^{1+m} \left (a^3 e^6 \left (5040+8028 m+5104 m^2+1665 m^3+295 m^4+27 m^5+m^6\right )+3 a^2 c e^4 \left (840+638 m+179 m^2+22 m^3+m^4\right ) \left (2 d^2-2 d e (1+m) x+e^2 \left (2+3 m+m^2\right ) x^2\right )+3 a c^2 e^2 \left (42+13 m+m^2\right ) \left (24 d^4-24 d^3 e (1+m) x+12 d^2 e^2 \left (2+3 m+m^2\right ) x^2-4 d e^3 \left (6+11 m+6 m^2+m^3\right ) x^3+e^4 \left (24+50 m+35 m^2+10 m^3+m^4\right ) x^4\right )+c^3 \left (720 d^6-720 d^5 e (1+m) x+360 d^4 e^2 \left (2+3 m+m^2\right ) x^2-120 d^3 e^3 \left (6+11 m+6 m^2+m^3\right ) x^3+30 d^2 e^4 \left (24+50 m+35 m^2+10 m^3+m^4\right ) x^4-6 d e^5 \left (120+274 m+225 m^2+85 m^3+15 m^4+m^5\right ) x^5+e^6 \left (720+1764 m+1624 m^2+735 m^3+175 m^4+21 m^5+m^6\right ) x^6\right )\right )}{e^7 (1+m) (2+m) (3+m) (4+m) (5+m) (6+m) (7+m)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(a^3*e^6*(5040 + 8028*m + 5104*m^2 + 1665*m^3 + 295*m^4 + 27*m^5 + m^6) + 3*a^2*c*e^4*(840
+ 638*m + 179*m^2 + 22*m^3 + m^4)*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2) + 3*a*c^2*e^2*(42 + 13*m
+ m^2)*(24*d^4 - 24*d^3*e*(1 + m)*x + 12*d^2*e^2*(2 + 3*m + m^2)*x^2 - 4*d*e^3*(6 + 11*m + 6*m^2 + m^3)*x^3 +
e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4) + c^3*(720*d^6 - 720*d^5*e*(1 + m)*x + 360*d^4*e^2*(2 + 3*m + m^
2)*x^2 - 120*d^3*e^3*(6 + 11*m + 6*m^2 + m^3)*x^3 + 30*d^2*e^4*(24 + 50*m + 35*m^2 + 10*m^3 + m^4)*x^4 - 6*d*e
^5*(120 + 274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5)*x^5 + e^6*(720 + 1764*m + 1624*m^2 + 735*m^3 + 175*m^4 + 21
*m^5 + m^6)*x^6)))/(e^7*(1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m)*(6 + m)*(7 + m))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(953\) vs. \(2(223)=446\).
time = 0.49, size = 954, normalized size = 4.28 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c^3/(7+m)*x^7*exp(m*ln(e*x+d))+d*(a^3*e^6*m^6+27*a^3*e^6*m^5+295*a^3*e^6*m^4+6*a^2*c*d^2*e^4*m^4+1665*a^3*e^6*
m^3+132*a^2*c*d^2*e^4*m^3+5104*a^3*e^6*m^2+1074*a^2*c*d^2*e^4*m^2+72*a*c^2*d^4*e^2*m^2+8028*a^3*e^6*m+3828*a^2
*c*d^2*e^4*m+936*a*c^2*d^4*e^2*m+5040*a^3*e^6+5040*a^2*c*d^2*e^4+3024*a*c^2*d^4*e^2+720*c^3*d^6)/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))+(a^3*e^6*m^6+27*a^3*e^6*m^5-6*a^2*c*d^2*
e^4*m^5+295*a^3*e^6*m^4-132*a^2*c*d^2*e^4*m^4+1665*a^3*e^6*m^3-1074*a^2*c*d^2*e^4*m^3-72*a*c^2*d^4*e^2*m^3+510
4*a^3*e^6*m^2-3828*a^2*c*d^2*e^4*m^2-936*a*c^2*d^4*e^2*m^2+8028*a^3*e^6*m-5040*a^2*c*d^2*e^4*m-3024*a*c^2*d^4*
e^2*m-720*c^3*d^6*m+5040*a^3*e^6)/e^6/(m^7+28*m^6+322*m^5+1960*m^4+6769*m^3+13132*m^2+13068*m+5040)*x*exp(m*ln
(e*x+d))+c^3*d*m/e/(m^2+13*m+42)*x^6*exp(m*ln(e*x+d))+3*(a*e^2*m^2+13*a*e^2*m-2*c*d^2*m+42*a*e^2)*c^2/e^2/(m^3
+18*m^2+107*m+210)*x^5*exp(m*ln(e*x+d))+3*(a^2*e^4*m^4+22*a^2*e^4*m^3-4*a*c*d^2*e^2*m^3+179*a^2*e^4*m^2-52*a*c
*d^2*e^2*m^2+638*a^2*e^4*m-168*a*c*d^2*e^2*m-40*c^2*d^4*m+840*a^2*e^4)*c/e^4/(m^5+25*m^4+245*m^3+1175*m^2+2754
*m+2520)*x^3*exp(m*ln(e*x+d))+3*(a*e^2*m^2+13*a*e^2*m+42*a*e^2+10*c*d^2)*c^2*d/e^3*m/(m^4+22*m^3+179*m^2+638*m
+840)*x^4*exp(m*ln(e*x+d))+3*(a^2*e^4*m^4+22*a^2*e^4*m^3+179*a^2*e^4*m^2+12*a*c*d^2*e^2*m^2+638*a^2*e^4*m+156*
a*c*d^2*e^2*m+840*a^2*e^4+504*a*c*d^2*e^2+120*c^2*d^4)*c*d/e^5*m/(m^6+27*m^5+295*m^4+1665*m^3+5104*m^2+8028*m+
5040)*x^2*exp(m*ln(e*x+d))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 469 vs. \(2 (217) = 434\).
time = 0.35, size = 469, normalized size = 2.10 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} a^{3} e^{\left (-1\right )}}{m + 1} + \frac {3 \, {\left ({\left (m^{2} + 3 \, m + 2\right )} x^{3} e^{3} + {\left (m^{2} + m\right )} d x^{2} e^{2} - 2 \, d^{2} m x e + 2 \, d^{3}\right )} a^{2} c e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {3 \, {\left ({\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} x^{5} e^{5} + {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d x^{4} e^{4} - 4 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{2} x^{3} e^{3} + 12 \, {\left (m^{2} + m\right )} d^{3} x^{2} e^{2} - 24 \, d^{4} m x e + 24 \, d^{5}\right )} a c^{2} e^{\left (m \log \left (x e + d\right ) - 5\right )}}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} + \frac {{\left ({\left (m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720\right )} x^{7} e^{7} + {\left (m^{6} + 15 \, m^{5} + 85 \, m^{4} + 225 \, m^{3} + 274 \, m^{2} + 120 \, m\right )} d x^{6} e^{6} - 6 \, {\left (m^{5} + 10 \, m^{4} + 35 \, m^{3} + 50 \, m^{2} + 24 \, m\right )} d^{2} x^{5} e^{5} + 30 \, {\left (m^{4} + 6 \, m^{3} + 11 \, m^{2} + 6 \, m\right )} d^{3} x^{4} e^{4} - 120 \, {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d^{4} x^{3} e^{3} + 360 \, {\left (m^{2} + m\right )} d^{5} x^{2} e^{2} - 720 \, d^{6} m x e + 720 \, d^{7}\right )} c^{3} e^{\left (m \log \left (x e + d\right ) - 7\right )}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*e + d)^(m + 1)*a^3*e^(-1)/(m + 1) + 3*((m^2 + 3*m + 2)*x^3*e^3 + (m^2 + m)*d*x^2*e^2 - 2*d^2*m*x*e + 2*d^3)
*a^2*c*e^(m*log(x*e + d) - 3)/(m^3 + 6*m^2 + 11*m + 6) + 3*((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*x^5*e^5 + (m^4
+ 6*m^3 + 11*m^2 + 6*m)*d*x^4*e^4 - 4*(m^3 + 3*m^2 + 2*m)*d^2*x^3*e^3 + 12*(m^2 + m)*d^3*x^2*e^2 - 24*d^4*m*x
*e + 24*d^5)*a*c^2*e^(m*log(x*e + d) - 5)/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120) + ((m^6 + 21*m^5 + 1
75*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)*x^7*e^7 + (m^6 + 15*m^5 + 85*m^4 + 225*m^3 + 274*m^2 + 120*m)*d*x^
6*e^6 - 6*(m^5 + 10*m^4 + 35*m^3 + 50*m^2 + 24*m)*d^2*x^5*e^5 + 30*(m^4 + 6*m^3 + 11*m^2 + 6*m)*d^3*x^4*e^4 -
120*(m^3 + 3*m^2 + 2*m)*d^4*x^3*e^3 + 360*(m^2 + m)*d^5*x^2*e^2 - 720*d^6*m*x*e + 720*d^7)*c^3*e^(m*log(x*e +
d) - 7)/(m^7 + 28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1030 vs. \(2 (217) = 434\).
time = 2.68, size = 1030, normalized size = 4.62 \begin {gather*} -\frac {{\left (720 \, c^{3} d^{6} m x e - 720 \, c^{3} d^{7} - {\left ({\left (c^{3} m^{6} + 21 \, c^{3} m^{5} + 175 \, c^{3} m^{4} + 735 \, c^{3} m^{3} + 1624 \, c^{3} m^{2} + 1764 \, c^{3} m + 720 \, c^{3}\right )} x^{7} + 3 \, {\left (a c^{2} m^{6} + 23 \, a c^{2} m^{5} + 207 \, a c^{2} m^{4} + 925 \, a c^{2} m^{3} + 2144 \, a c^{2} m^{2} + 2412 \, a c^{2} m + 1008 \, a c^{2}\right )} x^{5} + 3 \, {\left (a^{2} c m^{6} + 25 \, a^{2} c m^{5} + 247 \, a^{2} c m^{4} + 1219 \, a^{2} c m^{3} + 3112 \, a^{2} c m^{2} + 3796 \, a^{2} c m + 1680 \, a^{2} c\right )} x^{3} + {\left (a^{3} m^{6} + 27 \, a^{3} m^{5} + 295 \, a^{3} m^{4} + 1665 \, a^{3} m^{3} + 5104 \, a^{3} m^{2} + 8028 \, a^{3} m + 5040 \, a^{3}\right )} x\right )} e^{7} - {\left (a^{3} d m^{6} + 27 \, a^{3} d m^{5} + 295 \, a^{3} d m^{4} + 1665 \, a^{3} d m^{3} + {\left (c^{3} d m^{6} + 15 \, c^{3} d m^{5} + 85 \, c^{3} d m^{4} + 225 \, c^{3} d m^{3} + 274 \, c^{3} d m^{2} + 120 \, c^{3} d m\right )} x^{6} + 5104 \, a^{3} d m^{2} + 8028 \, a^{3} d m + 3 \, {\left (a c^{2} d m^{6} + 19 \, a c^{2} d m^{5} + 131 \, a c^{2} d m^{4} + 401 \, a c^{2} d m^{3} + 540 \, a c^{2} d m^{2} + 252 \, a c^{2} d m\right )} x^{4} + 5040 \, a^{3} d + 3 \, {\left (a^{2} c d m^{6} + 23 \, a^{2} c d m^{5} + 201 \, a^{2} c d m^{4} + 817 \, a^{2} c d m^{3} + 1478 \, a^{2} c d m^{2} + 840 \, a^{2} c d m\right )} x^{2}\right )} e^{6} + 6 \, {\left ({\left (c^{3} d^{2} m^{5} + 10 \, c^{3} d^{2} m^{4} + 35 \, c^{3} d^{2} m^{3} + 50 \, c^{3} d^{2} m^{2} + 24 \, c^{3} d^{2} m\right )} x^{5} + 2 \, {\left (a c^{2} d^{2} m^{5} + 16 \, a c^{2} d^{2} m^{4} + 83 \, a c^{2} d^{2} m^{3} + 152 \, a c^{2} d^{2} m^{2} + 84 \, a c^{2} d^{2} m\right )} x^{3} + {\left (a^{2} c d^{2} m^{5} + 22 \, a^{2} c d^{2} m^{4} + 179 \, a^{2} c d^{2} m^{3} + 638 \, a^{2} c d^{2} m^{2} + 840 \, a^{2} c d^{2} m\right )} x\right )} e^{5} - 6 \, {\left (a^{2} c d^{3} m^{4} + 22 \, a^{2} c d^{3} m^{3} + 179 \, a^{2} c d^{3} m^{2} + 638 \, a^{2} c d^{3} m + 840 \, a^{2} c d^{3} + 5 \, {\left (c^{3} d^{3} m^{4} + 6 \, c^{3} d^{3} m^{3} + 11 \, c^{3} d^{3} m^{2} + 6 \, c^{3} d^{3} m\right )} x^{4} + 6 \, {\left (a c^{2} d^{3} m^{4} + 14 \, a c^{2} d^{3} m^{3} + 55 \, a c^{2} d^{3} m^{2} + 42 \, a c^{2} d^{3} m\right )} x^{2}\right )} e^{4} + 24 \, {\left (5 \, {\left (c^{3} d^{4} m^{3} + 3 \, c^{3} d^{4} m^{2} + 2 \, c^{3} d^{4} m\right )} x^{3} + 3 \, {\left (a c^{2} d^{4} m^{3} + 13 \, a c^{2} d^{4} m^{2} + 42 \, a c^{2} d^{4} m\right )} x\right )} e^{3} - 72 \, {\left (a c^{2} d^{5} m^{2} + 13 \, a c^{2} d^{5} m + 42 \, a c^{2} d^{5} + 5 \, {\left (c^{3} d^{5} m^{2} + c^{3} d^{5} m\right )} x^{2}\right )} e^{2}\right )} {\left (x e + d\right )}^{m} e^{\left (-7\right )}}{m^{7} + 28 \, m^{6} + 322 \, m^{5} + 1960 \, m^{4} + 6769 \, m^{3} + 13132 \, m^{2} + 13068 \, m + 5040} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(720*c^3*d^6*m*x*e - 720*c^3*d^7 - ((c^3*m^6 + 21*c^3*m^5 + 175*c^3*m^4 + 735*c^3*m^3 + 1624*c^3*m^2 + 1764*c
^3*m + 720*c^3)*x^7 + 3*(a*c^2*m^6 + 23*a*c^2*m^5 + 207*a*c^2*m^4 + 925*a*c^2*m^3 + 2144*a*c^2*m^2 + 2412*a*c^
2*m + 1008*a*c^2)*x^5 + 3*(a^2*c*m^6 + 25*a^2*c*m^5 + 247*a^2*c*m^4 + 1219*a^2*c*m^3 + 3112*a^2*c*m^2 + 3796*a
^2*c*m + 1680*a^2*c)*x^3 + (a^3*m^6 + 27*a^3*m^5 + 295*a^3*m^4 + 1665*a^3*m^3 + 5104*a^3*m^2 + 8028*a^3*m + 50
40*a^3)*x)*e^7 - (a^3*d*m^6 + 27*a^3*d*m^5 + 295*a^3*d*m^4 + 1665*a^3*d*m^3 + (c^3*d*m^6 + 15*c^3*d*m^5 + 85*c
^3*d*m^4 + 225*c^3*d*m^3 + 274*c^3*d*m^2 + 120*c^3*d*m)*x^6 + 5104*a^3*d*m^2 + 8028*a^3*d*m + 3*(a*c^2*d*m^6 +
19*a*c^2*d*m^5 + 131*a*c^2*d*m^4 + 401*a*c^2*d*m^3 + 540*a*c^2*d*m^2 + 252*a*c^2*d*m)*x^4 + 5040*a^3*d + 3*(a
^2*c*d*m^6 + 23*a^2*c*d*m^5 + 201*a^2*c*d*m^4 + 817*a^2*c*d*m^3 + 1478*a^2*c*d*m^2 + 840*a^2*c*d*m)*x^2)*e^6 +
6*((c^3*d^2*m^5 + 10*c^3*d^2*m^4 + 35*c^3*d^2*m^3 + 50*c^3*d^2*m^2 + 24*c^3*d^2*m)*x^5 + 2*(a*c^2*d^2*m^5 + 1
6*a*c^2*d^2*m^4 + 83*a*c^2*d^2*m^3 + 152*a*c^2*d^2*m^2 + 84*a*c^2*d^2*m)*x^3 + (a^2*c*d^2*m^5 + 22*a^2*c*d^2*m
^4 + 179*a^2*c*d^2*m^3 + 638*a^2*c*d^2*m^2 + 840*a^2*c*d^2*m)*x)*e^5 - 6*(a^2*c*d^3*m^4 + 22*a^2*c*d^3*m^3 + 1
79*a^2*c*d^3*m^2 + 638*a^2*c*d^3*m + 840*a^2*c*d^3 + 5*(c^3*d^3*m^4 + 6*c^3*d^3*m^3 + 11*c^3*d^3*m^2 + 6*c^3*d
^3*m)*x^4 + 6*(a*c^2*d^3*m^4 + 14*a*c^2*d^3*m^3 + 55*a*c^2*d^3*m^2 + 42*a*c^2*d^3*m)*x^2)*e^4 + 24*(5*(c^3*d^4
*m^3 + 3*c^3*d^4*m^2 + 2*c^3*d^4*m)*x^3 + 3*(a*c^2*d^4*m^3 + 13*a*c^2*d^4*m^2 + 42*a*c^2*d^4*m)*x)*e^3 - 72*(a
*c^2*d^5*m^2 + 13*a*c^2*d^5*m + 42*a*c^2*d^5 + 5*(c^3*d^5*m^2 + c^3*d^5*m)*x^2)*e^2)*(x*e + d)^m*e^(-7)/(m^7 +
28*m^6 + 322*m^5 + 1960*m^4 + 6769*m^3 + 13132*m^2 + 13068*m + 5040)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 15990 vs. \(2 (207) = 414\).
time = 4.26, size = 15990, normalized size = 71.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(a**3*x + a**2*c*x**3 + 3*a*c**2*x**5/5 + c**3*x**7/7), Eq(e, 0)), (-10*a**3*e**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) - 3*a**2*c*d**2*e**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) - 18*a**2*c*d*e**5*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) - 45*a**
2*c*e**6*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) - 6*a*c**2*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*a*c**2*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*a*c**2*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*a*c**2*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*a*c**2*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**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 + 90
0*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**1
1*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)),
(-2*a**3*e**6/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10
*e**12*x**5) - a**2*c*d**2*e**4/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50
*d*e**11*x**4 + 10*e**12*x**5) - 5*a**2*c*d*e**5*x/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d
**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 10*a**2*c*e**6*x**2/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d
**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 6*a*c**2*d**4*e**2/(10*d**5*e**7 + 50
*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 30*a*c**2*d**3*e*
*3*x/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x*
*5) - 60*a*c**2*d**2*e**4*x**2/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*
d*e**11*x**4 + 10*e**12*x**5) - 60*a*c**2*d*e**5*x**3/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 10
0*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 30*a*c**2*e**6*x**4/(10*d**5*e**7 + 50*d**4*e**8*x + 10
0*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 60*c**3*d**6*log(d/e + x)/(10*d**5
*e**7 + 50*d**4*e**8*x + 100*d**3*e**9*x**2 + 100*d**2*e**10*x**3 + 50*d*e**11*x**4 + 10*e**12*x**5) - 137*c**
3*d**6/(10*d**5*e**7 + 50*d**4*e**8*x + 100*d**...

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2080 vs. \(2 (217) = 434\).
time = 0.78, size = 2080, normalized size = 9.33 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x*e + d)^m*c^3*m^6*x^7*e^7 + (x*e + d)^m*c^3*d*m^6*x^6*e^6 + 21*(x*e + d)^m*c^3*m^5*x^7*e^7 + 15*(x*e + d)^m
*c^3*d*m^5*x^6*e^6 - 6*(x*e + d)^m*c^3*d^2*m^5*x^5*e^5 + 3*(x*e + d)^m*a*c^2*m^6*x^5*e^7 + 175*(x*e + d)^m*c^3
*m^4*x^7*e^7 + 3*(x*e + d)^m*a*c^2*d*m^6*x^4*e^6 + 85*(x*e + d)^m*c^3*d*m^4*x^6*e^6 - 60*(x*e + d)^m*c^3*d^2*m
^4*x^5*e^5 + 30*(x*e + d)^m*c^3*d^3*m^4*x^4*e^4 + 69*(x*e + d)^m*a*c^2*m^5*x^5*e^7 + 735*(x*e + d)^m*c^3*m^3*x
^7*e^7 + 57*(x*e + d)^m*a*c^2*d*m^5*x^4*e^6 + 225*(x*e + d)^m*c^3*d*m^3*x^6*e^6 - 12*(x*e + d)^m*a*c^2*d^2*m^5
*x^3*e^5 - 210*(x*e + d)^m*c^3*d^2*m^3*x^5*e^5 + 180*(x*e + d)^m*c^3*d^3*m^3*x^4*e^4 - 120*(x*e + d)^m*c^3*d^4
*m^3*x^3*e^3 + 3*(x*e + d)^m*a^2*c*m^6*x^3*e^7 + 621*(x*e + d)^m*a*c^2*m^4*x^5*e^7 + 1624*(x*e + d)^m*c^3*m^2*
x^7*e^7 + 3*(x*e + d)^m*a^2*c*d*m^6*x^2*e^6 + 393*(x*e + d)^m*a*c^2*d*m^4*x^4*e^6 + 274*(x*e + d)^m*c^3*d*m^2*
x^6*e^6 - 192*(x*e + d)^m*a*c^2*d^2*m^4*x^3*e^5 - 300*(x*e + d)^m*c^3*d^2*m^2*x^5*e^5 + 36*(x*e + d)^m*a*c^2*d
^3*m^4*x^2*e^4 + 330*(x*e + d)^m*c^3*d^3*m^2*x^4*e^4 - 360*(x*e + d)^m*c^3*d^4*m^2*x^3*e^3 + 360*(x*e + d)^m*c
^3*d^5*m^2*x^2*e^2 + 75*(x*e + d)^m*a^2*c*m^5*x^3*e^7 + 2775*(x*e + d)^m*a*c^2*m^3*x^5*e^7 + 1764*(x*e + d)^m*
c^3*m*x^7*e^7 + 69*(x*e + d)^m*a^2*c*d*m^5*x^2*e^6 + 1203*(x*e + d)^m*a*c^2*d*m^3*x^4*e^6 + 120*(x*e + d)^m*c^
3*d*m*x^6*e^6 - 6*(x*e + d)^m*a^2*c*d^2*m^5*x*e^5 - 996*(x*e + d)^m*a*c^2*d^2*m^3*x^3*e^5 - 144*(x*e + d)^m*c^
3*d^2*m*x^5*e^5 + 504*(x*e + d)^m*a*c^2*d^3*m^3*x^2*e^4 + 180*(x*e + d)^m*c^3*d^3*m*x^4*e^4 - 72*(x*e + d)^m*a
*c^2*d^4*m^3*x*e^3 - 240*(x*e + d)^m*c^3*d^4*m*x^3*e^3 + 360*(x*e + d)^m*c^3*d^5*m*x^2*e^2 - 720*(x*e + d)^m*c
^3*d^6*m*x*e + (x*e + d)^m*a^3*m^6*x*e^7 + 741*(x*e + d)^m*a^2*c*m^4*x^3*e^7 + 6432*(x*e + d)^m*a*c^2*m^2*x^5*
e^7 + 720*(x*e + d)^m*c^3*x^7*e^7 + (x*e + d)^m*a^3*d*m^6*e^6 + 603*(x*e + d)^m*a^2*c*d*m^4*x^2*e^6 + 1620*(x*
e + d)^m*a*c^2*d*m^2*x^4*e^6 - 132*(x*e + d)^m*a^2*c*d^2*m^4*x*e^5 - 1824*(x*e + d)^m*a*c^2*d^2*m^2*x^3*e^5 +
6*(x*e + d)^m*a^2*c*d^3*m^4*e^4 + 1980*(x*e + d)^m*a*c^2*d^3*m^2*x^2*e^4 - 936*(x*e + d)^m*a*c^2*d^4*m^2*x*e^3
+ 72*(x*e + d)^m*a*c^2*d^5*m^2*e^2 + 720*(x*e + d)^m*c^3*d^7 + 27*(x*e + d)^m*a^3*m^5*x*e^7 + 3657*(x*e + d)^
m*a^2*c*m^3*x^3*e^7 + 7236*(x*e + d)^m*a*c^2*m*x^5*e^7 + 27*(x*e + d)^m*a^3*d*m^5*e^6 + 2451*(x*e + d)^m*a^2*c
*d*m^3*x^2*e^6 + 756*(x*e + d)^m*a*c^2*d*m*x^4*e^6 - 1074*(x*e + d)^m*a^2*c*d^2*m^3*x*e^5 - 1008*(x*e + d)^m*a
*c^2*d^2*m*x^3*e^5 + 132*(x*e + d)^m*a^2*c*d^3*m^3*e^4 + 1512*(x*e + d)^m*a*c^2*d^3*m*x^2*e^4 - 3024*(x*e + d)
^m*a*c^2*d^4*m*x*e^3 + 936*(x*e + d)^m*a*c^2*d^5*m*e^2 + 295*(x*e + d)^m*a^3*m^4*x*e^7 + 9336*(x*e + d)^m*a^2*
c*m^2*x^3*e^7 + 3024*(x*e + d)^m*a*c^2*x^5*e^7 + 295*(x*e + d)^m*a^3*d*m^4*e^6 + 4434*(x*e + d)^m*a^2*c*d*m^2*
x^2*e^6 - 3828*(x*e + d)^m*a^2*c*d^2*m^2*x*e^5 + 1074*(x*e + d)^m*a^2*c*d^3*m^2*e^4 + 3024*(x*e + d)^m*a*c^2*d
^5*e^2 + 1665*(x*e + d)^m*a^3*m^3*x*e^7 + 11388*(x*e + d)^m*a^2*c*m*x^3*e^7 + 1665*(x*e + d)^m*a^3*d*m^3*e^6 +
2520*(x*e + d)^m*a^2*c*d*m*x^2*e^6 - 5040*(x*e + d)^m*a^2*c*d^2*m*x*e^5 + 3828*(x*e + d)^m*a^2*c*d^3*m*e^4 +
5104*(x*e + d)^m*a^3*m^2*x*e^7 + 5040*(x*e + d)^m*a^2*c*x^3*e^7 + 5104*(x*e + d)^m*a^3*d*m^2*e^6 + 5040*(x*e +
d)^m*a^2*c*d^3*e^4 + 8028*(x*e + d)^m*a^3*m*x*e^7 + 8028*(x*e + d)^m*a^3*d*m*e^6 + 5040*(x*e + d)^m*a^3*x*e^7
+ 5040*(x*e + d)^m*a^3*d*e^6)/(m^7*e^7 + 28*m^6*e^7 + 322*m^5*e^7 + 1960*m^4*e^7 + 6769*m^3*e^7 + 13132*m^2*e
^7 + 13068*m*e^7 + 5040*e^7)

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Mupad [B]
time = 1.05, size = 1144, normalized size = 5.13 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (a^3\,d\,e^6\,m^6+27\,a^3\,d\,e^6\,m^5+295\,a^3\,d\,e^6\,m^4+1665\,a^3\,d\,e^6\,m^3+5104\,a^3\,d\,e^6\,m^2+8028\,a^3\,d\,e^6\,m+5040\,a^3\,d\,e^6+6\,a^2\,c\,d^3\,e^4\,m^4+132\,a^2\,c\,d^3\,e^4\,m^3+1074\,a^2\,c\,d^3\,e^4\,m^2+3828\,a^2\,c\,d^3\,e^4\,m+5040\,a^2\,c\,d^3\,e^4+72\,a\,c^2\,d^5\,e^2\,m^2+936\,a\,c^2\,d^5\,e^2\,m+3024\,a\,c^2\,d^5\,e^2+720\,c^3\,d^7\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\,{\left (d+e\,x\right )}^m\,\left (-a^3\,e^7\,m^6-27\,a^3\,e^7\,m^5-295\,a^3\,e^7\,m^4-1665\,a^3\,e^7\,m^3-5104\,a^3\,e^7\,m^2-8028\,a^3\,e^7\,m-5040\,a^3\,e^7+6\,a^2\,c\,d^2\,e^5\,m^5+132\,a^2\,c\,d^2\,e^5\,m^4+1074\,a^2\,c\,d^2\,e^5\,m^3+3828\,a^2\,c\,d^2\,e^5\,m^2+5040\,a^2\,c\,d^2\,e^5\,m+72\,a\,c^2\,d^4\,e^3\,m^3+936\,a\,c^2\,d^4\,e^3\,m^2+3024\,a\,c^2\,d^4\,e^3\,m+720\,c^3\,d^6\,e\,m\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 {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 {3\,c^2\,x^5\,{\left (d+e\,x\right )}^m\,\left (-2\,c\,d^2\,m+a\,e^2\,m^2+13\,a\,e^2\,m+42\,a\,e^2\right )\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\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 {3\,c\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (a^2\,e^4\,m^4+22\,a^2\,e^4\,m^3+179\,a^2\,e^4\,m^2+638\,a^2\,e^4\,m+840\,a^2\,e^4-4\,a\,c\,d^2\,e^2\,m^3-52\,a\,c\,d^2\,e^2\,m^2-168\,a\,c\,d^2\,e^2\,m-40\,c^2\,d^4\,m\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 {c^3\,d\,m\,x^6\,{\left (d+e\,x\right )}^m\,\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 {3\,c^2\,d\,m\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )\,\left (10\,c\,d^2+a\,e^2\,m^2+13\,a\,e^2\,m+42\,a\,e^2\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\,d\,m\,x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (a^2\,e^4\,m^4+22\,a^2\,e^4\,m^3+179\,a^2\,e^4\,m^2+638\,a^2\,e^4\,m+840\,a^2\,e^4+12\,a\,c\,d^2\,e^2\,m^2+156\,a\,c\,d^2\,e^2\,m+504\,a\,c\,d^2\,e^2+120\,c^2\,d^4\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c*x^2)^3*(d + e*x)^m,x)

[Out]

((d + e*x)^m*(720*c^3*d^7 + 5040*a^3*d*e^6 + 3024*a*c^2*d^5*e^2 + 5040*a^2*c*d^3*e^4 + 5104*a^3*d*e^6*m^2 + 16
65*a^3*d*e^6*m^3 + 295*a^3*d*e^6*m^4 + 27*a^3*d*e^6*m^5 + a^3*d*e^6*m^6 + 8028*a^3*d*e^6*m + 936*a*c^2*d^5*e^2
*m + 3828*a^2*c*d^3*e^4*m + 72*a*c^2*d^5*e^2*m^2 + 1074*a^2*c*d^3*e^4*m^2 + 132*a^2*c*d^3*e^4*m^3 + 6*a^2*c*d^
3*e^4*m^4))/(e^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) - (x*(d + e*x)^m
*(720*c^3*d^6*e*m - 8028*a^3*e^7*m - 5104*a^3*e^7*m^2 - 1665*a^3*e^7*m^3 - 295*a^3*e^7*m^4 - 27*a^3*e^7*m^5 -
a^3*e^7*m^6 - 5040*a^3*e^7 + 3024*a*c^2*d^4*e^3*m + 5040*a^2*c*d^2*e^5*m + 936*a*c^2*d^4*e^3*m^2 + 3828*a^2*c*
d^2*e^5*m^2 + 72*a*c^2*d^4*e^3*m^3 + 1074*a^2*c*d^2*e^5*m^3 + 132*a^2*c*d^2*e^5*m^4 + 6*a^2*c*d^2*e^5*m^5))/(e
^7*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (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) + (3*c^2*x^5*(d + e*x)^m*(42*a*e^2 + a*e^2*m^2 + 13*a*e^2*m - 2*c*d^2*m)*(50*m + 35*m^2 +
10*m^3 + m^4 + 24))/(e^2*(13068*m + 13132*m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*c*
x^3*(d + e*x)^m*(3*m + m^2 + 2)*(840*a^2*e^4 + 638*a^2*e^4*m - 40*c^2*d^4*m + 179*a^2*e^4*m^2 + 22*a^2*e^4*m^3
+ a^2*e^4*m^4 - 168*a*c*d^2*e^2*m - 52*a*c*d^2*e^2*m^2 - 4*a*c*d^2*e^2*m^3))/(e^4*(13068*m + 13132*m^2 + 6769
*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (c^3*d*m*x^6*(d + e*x)^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)) + (3*c^2*d*m*x
^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6)*(42*a*e^2 + 10*c*d^2 + a*e^2*m^2 + 13*a*e^2*m))/(e^3*(13068*m + 13132*
m^2 + 6769*m^3 + 1960*m^4 + 322*m^5 + 28*m^6 + m^7 + 5040)) + (3*c*d*m*x^2*(m + 1)*(d + e*x)^m*(840*a^2*e^4 +
120*c^2*d^4 + 638*a^2*e^4*m + 179*a^2*e^4*m^2 + 22*a^2*e^4*m^3 + a^2*e^4*m^4 + 504*a*c*d^2*e^2 + 156*a*c*d^2*e
^2*m + 12*a*c*d^2*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))

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