3.4.69 \(\int (d+e x)^m (b x+c x^2)^2 \, dx\)
Optimal. Leaf size=159 \[ \frac {\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}+\frac {d^2 (c d-b e)^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {2 d (c d-b e) (2 c d-b e) (d+e x)^{m+2}}{e^5 (m+2)}-\frac {2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)} \]
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Rubi [A] time = 0.08, antiderivative size = 159, normalized size of antiderivative = 1.00,
number of steps used = 2, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used =
{698} \begin {gather*} \frac {\left (b^2 e^2-6 b c d e+6 c^2 d^2\right ) (d+e x)^{m+3}}{e^5 (m+3)}+\frac {d^2 (c d-b e)^2 (d+e x)^{m+1}}{e^5 (m+1)}-\frac {2 d (c d-b e) (2 c d-b e) (d+e x)^{m+2}}{e^5 (m+2)}-\frac {2 c (2 c d-b e) (d+e x)^{m+4}}{e^5 (m+4)}+\frac {c^2 (d+e x)^{m+5}}{e^5 (m+5)} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(d + e*x)^m*(b*x + c*x^2)^2,x]
[Out]
(d^2*(c*d - b*e)^2*(d + e*x)^(1 + m))/(e^5*(1 + m)) - (2*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x)^(2 + m))/(e^5*(
2 + m)) + ((6*c^2*d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^(3 + m))/(e^5*(3 + m)) - (2*c*(2*c*d - b*e)*(d + e*x)^(
4 + m))/(e^5*(4 + m)) + (c^2*(d + e*x)^(5 + m))/(e^5*(5 + m))
Rule 698
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*} \int (d+e x)^m \left (b x+c x^2\right )^2 \, dx &=\int \left (\frac {d^2 (c d-b e)^2 (d+e x)^m}{e^4}+\frac {2 d (c d-b e) (-2 c d+b e) (d+e x)^{1+m}}{e^4}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{2+m}}{e^4}-\frac {2 c (2 c d-b e) (d+e x)^{3+m}}{e^4}+\frac {c^2 (d+e x)^{4+m}}{e^4}\right ) \, dx\\ &=\frac {d^2 (c d-b e)^2 (d+e x)^{1+m}}{e^5 (1+m)}-\frac {2 d (c d-b e) (2 c d-b e) (d+e x)^{2+m}}{e^5 (2+m)}+\frac {\left (6 c^2 d^2-6 b c d e+b^2 e^2\right ) (d+e x)^{3+m}}{e^5 (3+m)}-\frac {2 c (2 c d-b e) (d+e x)^{4+m}}{e^5 (4+m)}+\frac {c^2 (d+e x)^{5+m}}{e^5 (5+m)}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 138, normalized size = 0.87 \begin {gather*} \frac {(d+e x)^{m+1} \left (\frac {(d+e x)^2 \left (b^2 e^2-6 b c d e+6 c^2 d^2\right )}{m+3}+\frac {d^2 (c d-b e)^2}{m+1}-\frac {2 c (d+e x)^3 (2 c d-b e)}{m+4}-\frac {2 d (d+e x) (c d-b e) (2 c d-b e)}{m+2}+\frac {c^2 (d+e x)^4}{m+5}\right )}{e^5} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(d + e*x)^m*(b*x + c*x^2)^2,x]
[Out]
((d + e*x)^(1 + m)*((d^2*(c*d - b*e)^2)/(1 + m) - (2*d*(c*d - b*e)*(2*c*d - b*e)*(d + e*x))/(2 + m) + ((6*c^2*
d^2 - 6*b*c*d*e + b^2*e^2)*(d + e*x)^2)/(3 + m) - (2*c*(2*c*d - b*e)*(d + e*x)^3)/(4 + m) + (c^2*(d + e*x)^4)/
(5 + m)))/e^5
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IntegrateAlgebraic [F] time = 0.08, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (b x+c x^2\right )^2 \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[(d + e*x)^m*(b*x + c*x^2)^2,x]
[Out]
Defer[IntegrateAlgebraic][(d + e*x)^m*(b*x + c*x^2)^2, x]
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fricas [B] time = 0.43, size = 584, normalized size = 3.67 \begin {gather*} \frac {{\left (2 \, b^{2} d^{3} e^{2} m^{2} + 24 \, c^{2} d^{5} - 60 \, b c d^{4} e + 40 \, b^{2} d^{3} e^{2} + {\left (c^{2} e^{5} m^{4} + 10 \, c^{2} e^{5} m^{3} + 35 \, c^{2} e^{5} m^{2} + 50 \, c^{2} e^{5} m + 24 \, c^{2} e^{5}\right )} x^{5} + {\left (60 \, b c e^{5} + {\left (c^{2} d e^{4} + 2 \, b c e^{5}\right )} m^{4} + 2 \, {\left (3 \, c^{2} d e^{4} + 11 \, b c e^{5}\right )} m^{3} + {\left (11 \, c^{2} d e^{4} + 82 \, b c e^{5}\right )} m^{2} + 2 \, {\left (3 \, c^{2} d e^{4} + 61 \, b c e^{5}\right )} m\right )} x^{4} + {\left (40 \, b^{2} e^{5} + {\left (2 \, b c d e^{4} + b^{2} e^{5}\right )} m^{4} - 4 \, {\left (c^{2} d^{2} e^{3} - 4 \, b c d e^{4} - 3 \, b^{2} e^{5}\right )} m^{3} - {\left (12 \, c^{2} d^{2} e^{3} - 34 \, b c d e^{4} - 49 \, b^{2} e^{5}\right )} m^{2} - 2 \, {\left (4 \, c^{2} d^{2} e^{3} - 10 \, b c d e^{4} - 39 \, b^{2} e^{5}\right )} m\right )} x^{3} + {\left (b^{2} d e^{4} m^{4} - 2 \, {\left (3 \, b c d^{2} e^{3} - 5 \, b^{2} d e^{4}\right )} m^{3} + {\left (12 \, c^{2} d^{3} e^{2} - 36 \, b c d^{2} e^{3} + 29 \, b^{2} d e^{4}\right )} m^{2} + 2 \, {\left (6 \, c^{2} d^{3} e^{2} - 15 \, b c d^{2} e^{3} + 10 \, b^{2} d e^{4}\right )} m\right )} x^{2} - 6 \, {\left (2 \, b c d^{4} e - 3 \, b^{2} d^{3} e^{2}\right )} m - 2 \, {\left (b^{2} d^{2} e^{3} m^{3} - 3 \, {\left (2 \, b c d^{3} e^{2} - 3 \, b^{2} d^{2} e^{3}\right )} m^{2} + 2 \, {\left (6 \, c^{2} d^{4} e - 15 \, b c d^{3} e^{2} + 10 \, b^{2} d^{2} e^{3}\right )} m\right )} x\right )} {\left (e x + d\right )}^{m}}{e^{5} m^{5} + 15 \, e^{5} m^{4} + 85 \, e^{5} m^{3} + 225 \, e^{5} m^{2} + 274 \, e^{5} m + 120 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(c*x^2+b*x)^2,x, algorithm="fricas")
[Out]
(2*b^2*d^3*e^2*m^2 + 24*c^2*d^5 - 60*b*c*d^4*e + 40*b^2*d^3*e^2 + (c^2*e^5*m^4 + 10*c^2*e^5*m^3 + 35*c^2*e^5*m
^2 + 50*c^2*e^5*m + 24*c^2*e^5)*x^5 + (60*b*c*e^5 + (c^2*d*e^4 + 2*b*c*e^5)*m^4 + 2*(3*c^2*d*e^4 + 11*b*c*e^5)
*m^3 + (11*c^2*d*e^4 + 82*b*c*e^5)*m^2 + 2*(3*c^2*d*e^4 + 61*b*c*e^5)*m)*x^4 + (40*b^2*e^5 + (2*b*c*d*e^4 + b^
2*e^5)*m^4 - 4*(c^2*d^2*e^3 - 4*b*c*d*e^4 - 3*b^2*e^5)*m^3 - (12*c^2*d^2*e^3 - 34*b*c*d*e^4 - 49*b^2*e^5)*m^2
- 2*(4*c^2*d^2*e^3 - 10*b*c*d*e^4 - 39*b^2*e^5)*m)*x^3 + (b^2*d*e^4*m^4 - 2*(3*b*c*d^2*e^3 - 5*b^2*d*e^4)*m^3
+ (12*c^2*d^3*e^2 - 36*b*c*d^2*e^3 + 29*b^2*d*e^4)*m^2 + 2*(6*c^2*d^3*e^2 - 15*b*c*d^2*e^3 + 10*b^2*d*e^4)*m)*
x^2 - 6*(2*b*c*d^4*e - 3*b^2*d^3*e^2)*m - 2*(b^2*d^2*e^3*m^3 - 3*(2*b*c*d^3*e^2 - 3*b^2*d^2*e^3)*m^2 + 2*(6*c^
2*d^4*e - 15*b*c*d^3*e^2 + 10*b^2*d^2*e^3)*m)*x)*(e*x + d)^m/(e^5*m^5 + 15*e^5*m^4 + 85*e^5*m^3 + 225*e^5*m^2
+ 274*e^5*m + 120*e^5)
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giac [B] time = 0.19, size = 1002, normalized size = 6.30 \begin {gather*} \frac {{\left (x e + d\right )}^{m} c^{2} m^{4} x^{5} e^{5} + {\left (x e + d\right )}^{m} c^{2} d m^{4} x^{4} e^{4} + 2 \, {\left (x e + d\right )}^{m} b c m^{4} x^{4} e^{5} + 10 \, {\left (x e + d\right )}^{m} c^{2} m^{3} x^{5} e^{5} + 2 \, {\left (x e + d\right )}^{m} b c d m^{4} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m^{3} x^{4} e^{4} - 4 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{3} x^{3} e^{3} + {\left (x e + d\right )}^{m} b^{2} m^{4} x^{3} e^{5} + 22 \, {\left (x e + d\right )}^{m} b c m^{3} x^{4} e^{5} + 35 \, {\left (x e + d\right )}^{m} c^{2} m^{2} x^{5} e^{5} + {\left (x e + d\right )}^{m} b^{2} d m^{4} x^{2} e^{4} + 16 \, {\left (x e + d\right )}^{m} b c d m^{3} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} c^{2} d m^{2} x^{4} e^{4} - 6 \, {\left (x e + d\right )}^{m} b c d^{2} m^{3} x^{2} e^{3} - 12 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m^{2} x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m^{2} x^{2} e^{2} + 12 \, {\left (x e + d\right )}^{m} b^{2} m^{3} x^{3} e^{5} + 82 \, {\left (x e + d\right )}^{m} b c m^{2} x^{4} e^{5} + 50 \, {\left (x e + d\right )}^{m} c^{2} m x^{5} e^{5} + 10 \, {\left (x e + d\right )}^{m} b^{2} d m^{3} x^{2} e^{4} + 34 \, {\left (x e + d\right )}^{m} b c d m^{2} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} c^{2} d m x^{4} e^{4} - 2 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m^{3} x e^{3} - 36 \, {\left (x e + d\right )}^{m} b c d^{2} m^{2} x^{2} e^{3} - 8 \, {\left (x e + d\right )}^{m} c^{2} d^{2} m x^{3} e^{3} + 12 \, {\left (x e + d\right )}^{m} b c d^{3} m^{2} x e^{2} + 12 \, {\left (x e + d\right )}^{m} c^{2} d^{3} m x^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} c^{2} d^{4} m x e + 49 \, {\left (x e + d\right )}^{m} b^{2} m^{2} x^{3} e^{5} + 122 \, {\left (x e + d\right )}^{m} b c m x^{4} e^{5} + 24 \, {\left (x e + d\right )}^{m} c^{2} x^{5} e^{5} + 29 \, {\left (x e + d\right )}^{m} b^{2} d m^{2} x^{2} e^{4} + 20 \, {\left (x e + d\right )}^{m} b c d m x^{3} e^{4} - 18 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m^{2} x e^{3} - 30 \, {\left (x e + d\right )}^{m} b c d^{2} m x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} b^{2} d^{3} m^{2} e^{2} + 60 \, {\left (x e + d\right )}^{m} b c d^{3} m x e^{2} - 12 \, {\left (x e + d\right )}^{m} b c d^{4} m e + 24 \, {\left (x e + d\right )}^{m} c^{2} d^{5} + 78 \, {\left (x e + d\right )}^{m} b^{2} m x^{3} e^{5} + 60 \, {\left (x e + d\right )}^{m} b c x^{4} e^{5} + 20 \, {\left (x e + d\right )}^{m} b^{2} d m x^{2} e^{4} - 40 \, {\left (x e + d\right )}^{m} b^{2} d^{2} m x e^{3} + 18 \, {\left (x e + d\right )}^{m} b^{2} d^{3} m e^{2} - 60 \, {\left (x e + d\right )}^{m} b c d^{4} e + 40 \, {\left (x e + d\right )}^{m} b^{2} x^{3} e^{5} + 40 \, {\left (x e + d\right )}^{m} b^{2} d^{3} e^{2}}{m^{5} e^{5} + 15 \, m^{4} e^{5} + 85 \, m^{3} e^{5} + 225 \, m^{2} e^{5} + 274 \, m e^{5} + 120 \, e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(c*x^2+b*x)^2,x, algorithm="giac")
[Out]
((x*e + d)^m*c^2*m^4*x^5*e^5 + (x*e + d)^m*c^2*d*m^4*x^4*e^4 + 2*(x*e + d)^m*b*c*m^4*x^4*e^5 + 10*(x*e + d)^m*
c^2*m^3*x^5*e^5 + 2*(x*e + d)^m*b*c*d*m^4*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m^3*x^4*e^4 - 4*(x*e + d)^m*c^2*d^2*m^
3*x^3*e^3 + (x*e + d)^m*b^2*m^4*x^3*e^5 + 22*(x*e + d)^m*b*c*m^3*x^4*e^5 + 35*(x*e + d)^m*c^2*m^2*x^5*e^5 + (x
*e + d)^m*b^2*d*m^4*x^2*e^4 + 16*(x*e + d)^m*b*c*d*m^3*x^3*e^4 + 11*(x*e + d)^m*c^2*d*m^2*x^4*e^4 - 6*(x*e + d
)^m*b*c*d^2*m^3*x^2*e^3 - 12*(x*e + d)^m*c^2*d^2*m^2*x^3*e^3 + 12*(x*e + d)^m*c^2*d^3*m^2*x^2*e^2 + 12*(x*e +
d)^m*b^2*m^3*x^3*e^5 + 82*(x*e + d)^m*b*c*m^2*x^4*e^5 + 50*(x*e + d)^m*c^2*m*x^5*e^5 + 10*(x*e + d)^m*b^2*d*m^
3*x^2*e^4 + 34*(x*e + d)^m*b*c*d*m^2*x^3*e^4 + 6*(x*e + d)^m*c^2*d*m*x^4*e^4 - 2*(x*e + d)^m*b^2*d^2*m^3*x*e^3
- 36*(x*e + d)^m*b*c*d^2*m^2*x^2*e^3 - 8*(x*e + d)^m*c^2*d^2*m*x^3*e^3 + 12*(x*e + d)^m*b*c*d^3*m^2*x*e^2 + 1
2*(x*e + d)^m*c^2*d^3*m*x^2*e^2 - 24*(x*e + d)^m*c^2*d^4*m*x*e + 49*(x*e + d)^m*b^2*m^2*x^3*e^5 + 122*(x*e + d
)^m*b*c*m*x^4*e^5 + 24*(x*e + d)^m*c^2*x^5*e^5 + 29*(x*e + d)^m*b^2*d*m^2*x^2*e^4 + 20*(x*e + d)^m*b*c*d*m*x^3
*e^4 - 18*(x*e + d)^m*b^2*d^2*m^2*x*e^3 - 30*(x*e + d)^m*b*c*d^2*m*x^2*e^3 + 2*(x*e + d)^m*b^2*d^3*m^2*e^2 + 6
0*(x*e + d)^m*b*c*d^3*m*x*e^2 - 12*(x*e + d)^m*b*c*d^4*m*e + 24*(x*e + d)^m*c^2*d^5 + 78*(x*e + d)^m*b^2*m*x^3
*e^5 + 60*(x*e + d)^m*b*c*x^4*e^5 + 20*(x*e + d)^m*b^2*d*m*x^2*e^4 - 40*(x*e + d)^m*b^2*d^2*m*x*e^3 + 18*(x*e
+ d)^m*b^2*d^3*m*e^2 - 60*(x*e + d)^m*b*c*d^4*e + 40*(x*e + d)^m*b^2*x^3*e^5 + 40*(x*e + d)^m*b^2*d^3*e^2)/(m^
5*e^5 + 15*m^4*e^5 + 85*m^3*e^5 + 225*m^2*e^5 + 274*m*e^5 + 120*e^5)
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maple [B] time = 0.05, size = 547, normalized size = 3.44 \begin {gather*} \frac {\left (c^{2} e^{4} m^{4} x^{4}+2 b c \,e^{4} m^{4} x^{3}+10 c^{2} e^{4} m^{3} x^{4}+b^{2} e^{4} m^{4} x^{2}+22 b c \,e^{4} m^{3} x^{3}-4 c^{2} d \,e^{3} m^{3} x^{3}+35 c^{2} e^{4} m^{2} x^{4}+12 b^{2} e^{4} m^{3} x^{2}-6 b c d \,e^{3} m^{3} x^{2}+82 b c \,e^{4} m^{2} x^{3}-24 c^{2} d \,e^{3} m^{2} x^{3}+50 c^{2} e^{4} m \,x^{4}-2 b^{2} d \,e^{3} m^{3} x +49 b^{2} e^{4} m^{2} x^{2}-48 b c d \,e^{3} m^{2} x^{2}+122 b c \,e^{4} m \,x^{3}+12 c^{2} d^{2} e^{2} m^{2} x^{2}-44 c^{2} d \,e^{3} m \,x^{3}+24 c^{2} x^{4} e^{4}-20 b^{2} d \,e^{3} m^{2} x +78 b^{2} e^{4} m \,x^{2}+12 b c \,d^{2} e^{2} m^{2} x -102 b c d \,e^{3} m \,x^{2}+60 b c \,e^{4} x^{3}+36 c^{2} d^{2} e^{2} m \,x^{2}-24 c^{2} d \,e^{3} x^{3}+2 b^{2} d^{2} e^{2} m^{2}-58 b^{2} d \,e^{3} m x +40 b^{2} e^{4} x^{2}+72 b c \,d^{2} e^{2} m x -60 b c d \,e^{3} x^{2}-24 c^{2} d^{3} e m x +24 c^{2} d^{2} e^{2} x^{2}+18 b^{2} d^{2} e^{2} m -40 b^{2} d \,e^{3} x -12 b c \,d^{3} e m +60 b c \,d^{2} e^{2} x -24 c^{2} d^{3} e x +40 b^{2} d^{2} e^{2}-60 b c \,d^{3} e +24 c^{2} d^{4}\right ) \left (e x +d \right )^{m +1}}{\left (m^{5}+15 m^{4}+85 m^{3}+225 m^{2}+274 m +120\right ) e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((e*x+d)^m*(c*x^2+b*x)^2,x)
[Out]
(e*x+d)^(m+1)*(c^2*e^4*m^4*x^4+2*b*c*e^4*m^4*x^3+10*c^2*e^4*m^3*x^4+b^2*e^4*m^4*x^2+22*b*c*e^4*m^3*x^3-4*c^2*d
*e^3*m^3*x^3+35*c^2*e^4*m^2*x^4+12*b^2*e^4*m^3*x^2-6*b*c*d*e^3*m^3*x^2+82*b*c*e^4*m^2*x^3-24*c^2*d*e^3*m^2*x^3
+50*c^2*e^4*m*x^4-2*b^2*d*e^3*m^3*x+49*b^2*e^4*m^2*x^2-48*b*c*d*e^3*m^2*x^2+122*b*c*e^4*m*x^3+12*c^2*d^2*e^2*m
^2*x^2-44*c^2*d*e^3*m*x^3+24*c^2*e^4*x^4-20*b^2*d*e^3*m^2*x+78*b^2*e^4*m*x^2+12*b*c*d^2*e^2*m^2*x-102*b*c*d*e^
3*m*x^2+60*b*c*e^4*x^3+36*c^2*d^2*e^2*m*x^2-24*c^2*d*e^3*x^3+2*b^2*d^2*e^2*m^2-58*b^2*d*e^3*m*x+40*b^2*e^4*x^2
+72*b*c*d^2*e^2*m*x-60*b*c*d*e^3*x^2-24*c^2*d^3*e*m*x+24*c^2*d^2*e^2*x^2+18*b^2*d^2*e^2*m-40*b^2*d*e^3*x-12*b*
c*d^3*e*m+60*b*c*d^2*e^2*x-24*c^2*d^3*e*x+40*b^2*d^2*e^2-60*b*c*d^3*e+24*c^2*d^4)/e^5/(m^5+15*m^4+85*m^3+225*m
^2+274*m+120)
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maxima [A] time = 1.50, size = 318, normalized size = 2.00 \begin {gather*} \frac {{\left ({\left (m^{2} + 3 \, m + 2\right )} e^{3} x^{3} + {\left (m^{2} + m\right )} d e^{2} x^{2} - 2 \, d^{2} e m x + 2 \, d^{3}\right )} {\left (e x + d\right )}^{m} b^{2}}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac {2 \, {\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 c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} + \frac {{\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} c^{2}}{{\left (m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120\right )} e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)^m*(c*x^2+b*x)^2,x, algorithm="maxima")
[Out]
((m^2 + 3*m + 2)*e^3*x^3 + (m^2 + m)*d*e^2*x^2 - 2*d^2*e*m*x + 2*d^3)*(e*x + d)^m*b^2/((m^3 + 6*m^2 + 11*m + 6
)*e^3) + 2*((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*c/((m^4 + 10*m^3 + 35*m^2 + 50*m + 24)*e^4) + ((m^4 + 10*m^3 + 35*m^2 + 50*m + 2
4)*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*c^2/((m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)*e^5)
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mupad [B] time = 0.61, size = 464, normalized size = 2.92 \begin {gather*} {\left (d+e\,x\right )}^m\,\left (\frac {c^2\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120}+\frac {2\,d^3\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2-6\,b\,c\,d\,e\,m-30\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^5\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {x^3\,\left (m^2+3\,m+2\right )\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2+2\,b\,c\,d\,e\,m^2+10\,b\,c\,d\,e\,m-4\,c^2\,d^2\,m\right )}{e^2\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {c\,x^4\,\left (10\,b\,e+2\,b\,e\,m+c\,d\,m\right )\,\left (m^3+6\,m^2+11\,m+6\right )}{e\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}-\frac {2\,d^2\,m\,x\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2-6\,b\,c\,d\,e\,m-30\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^4\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}+\frac {d\,m\,x^2\,\left (m+1\right )\,\left (b^2\,e^2\,m^2+9\,b^2\,e^2\,m+20\,b^2\,e^2-6\,b\,c\,d\,e\,m-30\,b\,c\,d\,e+12\,c^2\,d^2\right )}{e^3\,\left (m^5+15\,m^4+85\,m^3+225\,m^2+274\,m+120\right )}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x + c*x^2)^2*(d + e*x)^m,x)
[Out]
(d + e*x)^m*((c^2*x^5*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120) + (
2*d^3*(20*b^2*e^2 + 12*c^2*d^2 + 9*b^2*e^2*m + b^2*e^2*m^2 - 30*b*c*d*e - 6*b*c*d*e*m))/(e^5*(274*m + 225*m^2
+ 85*m^3 + 15*m^4 + m^5 + 120)) + (x^3*(3*m + m^2 + 2)*(20*b^2*e^2 + 9*b^2*e^2*m - 4*c^2*d^2*m + b^2*e^2*m^2 +
10*b*c*d*e*m + 2*b*c*d*e*m^2))/(e^2*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) + (c*x^4*(10*b*e + 2*b*e
*m + c*d*m)*(11*m + 6*m^2 + m^3 + 6))/(e*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)) - (2*d^2*m*x*(20*b^2
*e^2 + 12*c^2*d^2 + 9*b^2*e^2*m + b^2*e^2*m^2 - 30*b*c*d*e - 6*b*c*d*e*m))/(e^4*(274*m + 225*m^2 + 85*m^3 + 15
*m^4 + m^5 + 120)) + (d*m*x^2*(m + 1)*(20*b^2*e^2 + 12*c^2*d^2 + 9*b^2*e^2*m + b^2*e^2*m^2 - 30*b*c*d*e - 6*b*
c*d*e*m))/(e^3*(274*m + 225*m^2 + 85*m^3 + 15*m^4 + m^5 + 120)))
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sympy [A] time = 6.67, size = 6418, normalized size = 40.36
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((e*x+d)**m*(c*x**2+b*x)**2,x)
[Out]
Piecewise((d**m*(b**2*x**3/3 + b*c*x**4/2 + c**2*x**5/5), Eq(e, 0)), (-b**2*d**2*e**2/(12*d**4*e**5 + 48*d**3*
e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 4*b**2*d*e**3*x/(12*d**4*e**5 + 48*d**3*e**6*x +
72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 6*b**2*e**4*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**
2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9*x**4) - 6*b*c*d**3*e/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2
+ 48*d*e**8*x**3 + 12*e**9*x**4) - 24*b*c*d**2*e**2*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48
*d*e**8*x**3 + 12*e**9*x**4) - 36*b*c*d*e**3*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e*
*8*x**3 + 12*e**9*x**4) - 24*b*c*e**4*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3
+ 12*e**9*x**4) + 12*c**2*d**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x*
*3 + 12*e**9*x**4) + 25*c**2*d**4/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**
9*x**4) + 48*c**2*d**3*e*x*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 +
12*e**9*x**4) + 88*c**2*d**3*e*x/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x**3 + 12*e**9
*x**4) + 72*c**2*d**2*e**2*x**2*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x*
*3 + 12*e**9*x**4) + 108*c**2*d**2*e**2*x**2/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e**8*x*
*3 + 12*e**9*x**4) + 48*c**2*d*e**3*x**3*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*
d*e**8*x**3 + 12*e**9*x**4) + 48*c**2*d*e**3*x**3/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 + 48*d*e*
*8*x**3 + 12*e**9*x**4) + 12*c**2*e**4*x**4*log(d/e + x)/(12*d**4*e**5 + 48*d**3*e**6*x + 72*d**2*e**7*x**2 +
48*d*e**8*x**3 + 12*e**9*x**4), Eq(m, -5)), (-b**2*d**2*e**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*
e**8*x**3) - 3*b**2*d*e**3*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 3*b**2*e**4*x**2/(3
*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 6*b*c*d**3*e*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e*
*6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 11*b*c*d**3*e/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3
) + 18*b*c*d**2*e**2*x*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 27*b*c*d**2*
e**2*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 18*b*c*d*e**3*x**2*log(d/e + x)/(3*d**3*e
**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 18*b*c*d*e**3*x**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**
7*x**2 + 3*e**8*x**3) + 6*b*c*e**4*x**3*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**
3) - 12*c**2*d**4*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 22*c**2*d**4/(3*d
**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 36*c**2*d**3*e*x*log(d/e + x)/(3*d**3*e**5 + 9*d**2*
e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 54*c**2*d**3*e*x/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8
*x**3) - 36*c**2*d**2*e**2*x**2*log(d/e + x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 36*
c**2*d**2*e**2*x**2/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) - 12*c**2*d*e**3*x**3*log(d/e
+ x)/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x**3) + 3*c**2*e**4*x**4/(3*d**3*e**5 + 9*d**2*e**6
*x + 9*d*e**7*x**2 + 3*e**8*x**3), Eq(m, -4)), (2*b**2*d**2*e**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e*
*7*x**2) + 3*b**2*d**2*e**2/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b**2*d*e**3*x*log(d/e + x)/(2*d**2*e*
*5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b**2*d*e**3*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 2*b**2*e**4*x**2*l
og(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*c*d**3*e*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x +
2*e**7*x**2) - 18*b*c*d**3*e/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*b*c*d**2*e**2*x*log(d/e + x)/(2*d**
2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 24*b*c*d**2*e**2*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) - 12*b*c*d*e*
*3*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 4*b*c*e**4*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*
e**7*x**2) + 12*c**2*d**4*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 18*c**2*d**4/(2*d**2*e**5 +
4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d**3*e*x*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 24*c**2*d
**3*e*x/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + 12*c**2*d**2*e**2*x**2*log(d/e + x)/(2*d**2*e**5 + 4*d*e**6
*x + 2*e**7*x**2) - 4*c**2*d*e**3*x**3/(2*d**2*e**5 + 4*d*e**6*x + 2*e**7*x**2) + c**2*e**4*x**4/(2*d**2*e**5
+ 4*d*e**6*x + 2*e**7*x**2), Eq(m, -3)), (-6*b**2*d**2*e**2*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 6*b**2*d**2*e
**2/(3*d*e**5 + 3*e**6*x) - 6*b**2*d*e**3*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 3*b**2*e**4*x**2/(3*d*e**5 +
3*e**6*x) + 18*b*c*d**3*e*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 18*b*c*d**3*e/(3*d*e**5 + 3*e**6*x) + 18*b*c*d*
*2*e**2*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 9*b*c*d*e**3*x**2/(3*d*e**5 + 3*e**6*x) + 3*b*c*e**4*x**3/(3*d*
e**5 + 3*e**6*x) - 12*c**2*d**4*log(d/e + x)/(3*d*e**5 + 3*e**6*x) - 12*c**2*d**4/(3*d*e**5 + 3*e**6*x) - 12*c
**2*d**3*e*x*log(d/e + x)/(3*d*e**5 + 3*e**6*x) + 6*c**2*d**2*e**2*x**2/(3*d*e**5 + 3*e**6*x) - 2*c**2*d*e**3*
x**3/(3*d*e**5 + 3*e**6*x) + c**2*e**4*x**4/(3*d*e**5 + 3*e**6*x), Eq(m, -2)), (b**2*d**2*log(d/e + x)/e**3 -
b**2*d*x/e**2 + b**2*x**2/(2*e) - 2*b*c*d**3*log(d/e + x)/e**4 + 2*b*c*d**2*x/e**3 - b*c*d*x**2/e**2 + 2*b*c*x
**3/(3*e) + c**2*d**4*log(d/e + x)/e**5 - c**2*d**3*x/e**4 + c**2*d**2*x**2/(2*e**3) - c**2*d*x**3/(3*e**2) +
c**2*x**4/(4*e), Eq(m, -1)), (2*b**2*d**3*e**2*m**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 22
5*e**5*m**2 + 274*e**5*m + 120*e**5) + 18*b**2*d**3*e**2*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m*
*3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 40*b**2*d**3*e**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e*
*5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 2*b**2*d**2*e**3*m**3*x*(d + e*x)**m/(e**5*m**5 + 15*e**5*m
**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 18*b**2*d**2*e**3*m**2*x*(d + e*x)**m/(e**5*m**5
+ 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 40*b**2*d**2*e**3*m*x*(d + e*x)**m/(
e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + b**2*d*e**4*m**4*x**2*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 10*b**2*d*e**4*m**
3*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 29*b**
2*d*e**4*m**2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e*
*5) + 20*b**2*d*e**4*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m
+ 120*e**5) + b**2*e**5*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274
*e**5*m + 120*e**5) + 12*b**2*e**5*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*
m**2 + 274*e**5*m + 120*e**5) + 49*b**2*e**5*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 +
225*e**5*m**2 + 274*e**5*m + 120*e**5) + 78*b**2*e**5*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5
*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 40*b**2*e**5*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85
*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 12*b*c*d**4*e*m*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 +
85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 60*b*c*d**4*e*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4
+ 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*b*c*d**3*e**2*m**2*x*(d + e*x)**m/(e**5*m**5 + 15
*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 60*b*c*d**3*e**2*m*x*(d + e*x)**m/(e**5*m
**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 6*b*c*d**2*e**3*m**3*x**2*(d + e*
x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 36*b*c*d**2*e**3*m**
2*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 30*b*c
*d**2*e**3*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**
5) + 2*b*c*d*e**4*m**4*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m
+ 120*e**5) + 16*b*c*d*e**4*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 +
274*e**5*m + 120*e**5) + 34*b*c*d*e**4*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*
e**5*m**2 + 274*e**5*m + 120*e**5) + 20*b*c*d*e**4*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**
3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 2*b*c*e**5*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*
e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 22*b*c*e**5*m**3*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m
**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 82*b*c*e**5*m**2*x**4*(d + e*x)**m/(e**5*m**5 +
15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 122*b*c*e**5*m*x**4*(d + e*x)**m/(e**5*
m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 60*b*c*e**5*x**4*(d + e*x)**m/(e
**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 24*c**2*d**5*(d + e*x)**m/(e
**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) - 24*c**2*d**4*e*m*x*(d + e*x)
**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*c**2*d**3*e**2*m**2
*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 12*c**2
*d**3*e**2*m*x**2*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**
5) - 4*c**2*d**2*e**3*m**3*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e*
*5*m + 120*e**5) - 12*c**2*d**2*e**3*m**2*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**
5*m**2 + 274*e**5*m + 120*e**5) - 8*c**2*d**2*e**3*m*x**3*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**
3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + c**2*d*e**4*m**4*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85
*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*c**2*d*e**4*m**3*x**4*(d + e*x)**m/(e**5*m**5 + 15*e**
5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 11*c**2*d*e**4*m**2*x**4*(d + e*x)**m/(e**5*m
**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 6*c**2*d*e**4*m*x**4*(d + e*x)**m
/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + c**2*e**5*m**4*x**5*(d +
e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 10*c**2*e**5*m**3*
x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5) + 35*c**2*
e**5*m**2*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120*e**5)
+ 50*c**2*e**5*m*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m + 120
*e**5) + 24*c**2*e**5*x**5*(d + e*x)**m/(e**5*m**5 + 15*e**5*m**4 + 85*e**5*m**3 + 225*e**5*m**2 + 274*e**5*m
+ 120*e**5), True))
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