3.32.82 \(\int (a+b x)^2 (A+B x) (d+e x)^m \, dx\) [3182]

Optimal. Leaf size=138 \[ -\frac {(b d-a e)^2 (B d-A e) (d+e x)^{1+m}}{e^4 (1+m)}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{2+m}}{e^4 (2+m)}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^{3+m}}{e^4 (3+m)}+\frac {b^2 B (d+e x)^{4+m}}{e^4 (4+m)} \]

[Out]

-(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^(1+m)/e^4/(1+m)+(-a*e+b*d)*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^(2+m)/e^4/(2+m)-b
*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^(3+m)/e^4/(3+m)+b^2*B*(e*x+d)^(4+m)/e^4/(4+m)

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Rubi [A]
time = 0.06, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {78} \begin {gather*} -\frac {(b d-a e)^2 (B d-A e) (d+e x)^{m+1}}{e^4 (m+1)}+\frac {(b d-a e) (d+e x)^{m+2} (-a B e-2 A b e+3 b B d)}{e^4 (m+2)}-\frac {b (d+e x)^{m+3} (-2 a B e-A b e+3 b B d)}{e^4 (m+3)}+\frac {b^2 B (d+e x)^{m+4}}{e^4 (m+4)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(A + B*x)*(d + e*x)^m,x]

[Out]

-(((b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(1 + m))/(e^4*(1 + m))) + ((b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d +
e*x)^(2 + m))/(e^4*(2 + m)) - (b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(3 + m))/(e^4*(3 + m)) + (b^2*B*(d + e
*x)^(4 + m))/(e^4*(4 + m))

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

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

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Mathematica [A]
time = 0.14, size = 122, normalized size = 0.88 \begin {gather*} \frac {(d+e x)^{1+m} \left (-\frac {(b d-a e)^2 (B d-A e)}{1+m}+\frac {(b d-a e) (3 b B d-2 A b e-a B e) (d+e x)}{2+m}-\frac {b (3 b B d-A b e-2 a B e) (d+e x)^2}{3+m}+\frac {b^2 B (d+e x)^3}{4+m}\right )}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^m,x]

[Out]

((d + e*x)^(1 + m)*(-(((b*d - a*e)^2*(B*d - A*e))/(1 + m)) + ((b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x
))/(2 + m) - (b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^2)/(3 + m) + (b^2*B*(d + e*x)^3)/(4 + m)))/e^4

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(575\) vs. \(2(138)=276\).
time = 0.11, size = 576, normalized size = 4.17

method	result	size
gosper	\(\frac {\left (e x +d \right )^{1+m} \left (B \,b^{2} e^{3} m^{3} x^{3}+A \,b^{2} e^{3} m^{3} x^{2}+2 B a b \,e^{3} m^{3} x^{2}+6 B \,b^{2} e^{3} m^{2} x^{3}+2 A a b \,e^{3} m^{3} x +7 A \,b^{2} e^{3} m^{2} x^{2}+B \,a^{2} e^{3} m^{3} x +14 B a b \,e^{3} m^{2} x^{2}-3 B \,b^{2} d \,e^{2} m^{2} x^{2}+11 B \,b^{2} e^{3} m \,x^{3}+A \,a^{2} e^{3} m^{3}+16 A a b \,e^{3} m^{2} x -2 A \,b^{2} d \,e^{2} m^{2} x +14 A \,b^{2} e^{3} m \,x^{2}+8 B \,a^{2} e^{3} m^{2} x -4 B a b d \,e^{2} m^{2} x +28 B a b \,e^{3} m \,x^{2}-9 B \,b^{2} d \,e^{2} m \,x^{2}+6 b^{2} B \,x^{3} e^{3}+9 A \,a^{2} e^{3} m^{2}-2 A a b d \,e^{2} m^{2}+38 A a b \,e^{3} m x -10 A \,b^{2} d \,e^{2} m x +8 A \,b^{2} e^{3} x^{2}-B \,a^{2} d \,e^{2} m^{2}+19 B \,a^{2} e^{3} m x -20 B a b d \,e^{2} m x +16 B a b \,e^{3} x^{2}+6 B \,b^{2} d^{2} e m x -6 B \,b^{2} d \,e^{2} x^{2}+26 A \,a^{2} e^{3} m -14 A a b d \,e^{2} m +24 A a b \,e^{3} x +2 A \,b^{2} d^{2} e m -8 A \,b^{2} d \,e^{2} x -7 B \,a^{2} d \,e^{2} m +12 B \,a^{2} e^{3} x +4 B a b \,d^{2} e m -16 B a b d \,e^{2} x +6 B \,b^{2} d^{2} e x +24 a^{2} A \,e^{3}-24 A a b d \,e^{2}+8 A \,b^{2} d^{2} e -12 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -6 b^{2} B \,d^{3}\right )}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}\)	\(576\)
norman	\(\frac {b^{2} B \,x^{4} {\mathrm e}^{m \ln \left (e x +d \right )}}{4+m}+\frac {d \left (A \,a^{2} e^{3} m^{3}+9 A \,a^{2} e^{3} m^{2}-2 A a b d \,e^{2} m^{2}-B \,a^{2} d \,e^{2} m^{2}+26 A \,a^{2} e^{3} m -14 A a b d \,e^{2} m +2 A \,b^{2} d^{2} e m -7 B \,a^{2} d \,e^{2} m +4 B a b \,d^{2} e m +24 a^{2} A \,e^{3}-24 A a b d \,e^{2}+8 A \,b^{2} d^{2} e -12 B \,a^{2} d \,e^{2}+16 B a b \,d^{2} e -6 b^{2} B \,d^{3}\right ) {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{4} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {\left (2 A a b \,e^{2} m^{2}+A \,b^{2} d e \,m^{2}+B \,a^{2} e^{2} m^{2}+2 B a b d e \,m^{2}+14 A a b \,e^{2} m +4 A \,b^{2} d e m +7 B \,a^{2} e^{2} m +8 B a b d e m -3 B \,b^{2} d^{2} m +24 A a b \,e^{2}+12 B \,a^{2} e^{2}\right ) x^{2} {\mathrm e}^{m \ln \left (e x +d \right )}}{e^{2} \left (m^{3}+9 m^{2}+26 m +24\right )}+\frac {\left (A \,a^{2} e^{3} m^{3}+2 A a b d \,e^{2} m^{3}+B \,a^{2} d \,e^{2} m^{3}+9 A \,a^{2} e^{3} m^{2}+14 A a b d \,e^{2} m^{2}-2 A \,b^{2} d^{2} e \,m^{2}+7 B \,a^{2} d \,e^{2} m^{2}-4 B a b \,d^{2} e \,m^{2}+26 A \,a^{2} e^{3} m +24 A a b d \,e^{2} m -8 A \,b^{2} d^{2} e m +12 B \,a^{2} d \,e^{2} m -16 B a b \,d^{2} e m +6 B \,b^{2} d^{3} m +24 a^{2} A \,e^{3}\right ) x \,{\mathrm e}^{m \ln \left (e x +d \right )}}{e^{3} \left (m^{4}+10 m^{3}+35 m^{2}+50 m +24\right )}+\frac {b \left (A b e m +2 B a e m +B b d m +4 A b e +8 B a e \right ) x^{3} {\mathrm e}^{m \ln \left (e x +d \right )}}{e \left (m^{2}+7 m +12\right )}\)	\(609\)
risch	\(\frac {\left (B \,b^{2} e^{4} m^{3} x^{4}+A \,b^{2} e^{4} m^{3} x^{3}+2 B a b \,e^{4} m^{3} x^{3}+B \,b^{2} d \,e^{3} m^{3} x^{3}+6 B \,b^{2} e^{4} m^{2} x^{4}+2 A a b \,e^{4} m^{3} x^{2}+A \,b^{2} d \,e^{3} m^{3} x^{2}+7 A \,b^{2} e^{4} m^{2} x^{3}+B \,a^{2} e^{4} m^{3} x^{2}+2 B a b d \,e^{3} m^{3} x^{2}+14 B a b \,e^{4} m^{2} x^{3}+3 B \,b^{2} d \,e^{3} m^{2} x^{3}+11 B \,b^{2} e^{4} m \,x^{4}+A \,a^{2} e^{4} m^{3} x +2 A a b d \,e^{3} m^{3} x +16 A a b \,e^{4} m^{2} x^{2}+5 A \,b^{2} d \,e^{3} m^{2} x^{2}+14 A \,b^{2} e^{4} m \,x^{3}+B \,a^{2} d \,e^{3} m^{3} x +8 B \,a^{2} e^{4} m^{2} x^{2}+10 B a b d \,e^{3} m^{2} x^{2}+28 B a b \,e^{4} m \,x^{3}-3 B \,b^{2} d^{2} e^{2} m^{2} x^{2}+2 B \,b^{2} d \,e^{3} m \,x^{3}+6 b^{2} B \,x^{4} e^{4}+A \,a^{2} d \,e^{3} m^{3}+9 A \,a^{2} e^{4} m^{2} x +14 A a b d \,e^{3} m^{2} x +38 A a b \,e^{4} m \,x^{2}-2 A \,b^{2} d^{2} e^{2} m^{2} x +4 A \,b^{2} d \,e^{3} m \,x^{2}+8 A \,b^{2} e^{4} x^{3}+7 B \,a^{2} d \,e^{3} m^{2} x +19 B \,a^{2} e^{4} m \,x^{2}-4 B a b \,d^{2} e^{2} m^{2} x +8 B a b d \,e^{3} m \,x^{2}+16 B a b \,e^{4} x^{3}-3 B \,b^{2} d^{2} e^{2} m \,x^{2}+9 A \,a^{2} d \,e^{3} m^{2}+26 A \,a^{2} e^{4} m x -2 A a b \,d^{2} e^{2} m^{2}+24 A a b d \,e^{3} m x +24 A a b \,e^{4} x^{2}-8 A \,b^{2} d^{2} e^{2} m x -B \,a^{2} d^{2} e^{2} m^{2}+12 B \,a^{2} d \,e^{3} m x +12 B \,a^{2} e^{4} x^{2}-16 B a b \,d^{2} e^{2} m x +6 B \,b^{2} d^{3} e m x +26 A \,a^{2} d \,e^{3} m +24 A \,a^{2} e^{4} x -14 A a b \,d^{2} e^{2} m +2 A \,b^{2} d^{3} e m -7 B \,a^{2} d^{2} e^{2} m +4 B a b \,d^{3} e m +24 a^{2} A d \,e^{3}-24 A a b \,d^{2} e^{2}+8 A \,b^{2} d^{3} e -12 B \,a^{2} d^{2} e^{2}+16 B a b \,d^{3} e -6 b^{2} B \,d^{4}\right ) \left (e x +d \right )^{m}}{\left (3+m \right ) \left (4+m \right ) \left (2+m \right ) \left (1+m \right ) e^{4}}\)	\(828\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)*(e*x+d)^m,x,method=_RETURNVERBOSE)

[Out]

(e*x+d)^(1+m)*(B*b^2*e^3*m^3*x^3+A*b^2*e^3*m^3*x^2+2*B*a*b*e^3*m^3*x^2+6*B*b^2*e^3*m^2*x^3+2*A*a*b*e^3*m^3*x+7
*A*b^2*e^3*m^2*x^2+B*a^2*e^3*m^3*x+14*B*a*b*e^3*m^2*x^2-3*B*b^2*d*e^2*m^2*x^2+11*B*b^2*e^3*m*x^3+A*a^2*e^3*m^3
+16*A*a*b*e^3*m^2*x-2*A*b^2*d*e^2*m^2*x+14*A*b^2*e^3*m*x^2+8*B*a^2*e^3*m^2*x-4*B*a*b*d*e^2*m^2*x+28*B*a*b*e^3*
m*x^2-9*B*b^2*d*e^2*m*x^2+6*B*b^2*e^3*x^3+9*A*a^2*e^3*m^2-2*A*a*b*d*e^2*m^2+38*A*a*b*e^3*m*x-10*A*b^2*d*e^2*m*
x+8*A*b^2*e^3*x^2-B*a^2*d*e^2*m^2+19*B*a^2*e^3*m*x-20*B*a*b*d*e^2*m*x+16*B*a*b*e^3*x^2+6*B*b^2*d^2*e*m*x-6*B*b
^2*d*e^2*x^2+26*A*a^2*e^3*m-14*A*a*b*d*e^2*m+24*A*a*b*e^3*x+2*A*b^2*d^2*e*m-8*A*b^2*d*e^2*x-7*B*a^2*d*e^2*m+12
*B*a^2*e^3*x+4*B*a*b*d^2*e*m-16*B*a*b*d*e^2*x+6*B*b^2*d^2*e*x+24*A*a^2*e^3-24*A*a*b*d*e^2+8*A*b^2*d^2*e-12*B*a
^2*d*e^2+16*B*a*b*d^2*e-6*B*b^2*d^3)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 370 vs. \(2 (145) = 290\).
time = 0.36, size = 370, normalized size = 2.68 \begin {gather*} \frac {{\left (x e + d\right )}^{m + 1} A a^{2} e^{\left (-1\right )}}{m + 1} + \frac {{\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} B a^{2} e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {2 \, {\left ({\left (m + 1\right )} x^{2} e^{2} + d m x e - d^{2}\right )} A a b e^{\left (m \log \left (x e + d\right ) - 2\right )}}{m^{2} + 3 \, m + 2} + \frac {2 \, {\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 )} B a b e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\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 b^{2} e^{\left (m \log \left (x e + d\right ) - 3\right )}}{m^{3} + 6 \, m^{2} + 11 \, m + 6} + \frac {{\left ({\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} x^{4} e^{4} + {\left (m^{3} + 3 \, m^{2} + 2 \, m\right )} d x^{3} e^{3} - 3 \, {\left (m^{2} + m\right )} d^{2} x^{2} e^{2} + 6 \, d^{3} m x e - 6 \, d^{4}\right )} B b^{2} e^{\left (m \log \left (x e + d\right ) - 4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^m,x, algorithm="maxima")

[Out]

(x*e + d)^(m + 1)*A*a^2*e^(-1)/(m + 1) + ((m + 1)*x^2*e^2 + d*m*x*e - d^2)*B*a^2*e^(m*log(x*e + d) - 2)/(m^2 +
3*m + 2) + 2*((m + 1)*x^2*e^2 + d*m*x*e - d^2)*A*a*b*e^(m*log(x*e + d) - 2)/(m^2 + 3*m + 2) + 2*((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)*B*a*b*e^(m*log(x*e + d) - 3)/(m^3 + 6*m^2 + 11*m + 6)
+ ((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*b^2*e^(m*log(x*e + d) - 3)/(m^3 + 6
*m^2 + 11*m + 6) + ((m^3 + 6*m^2 + 11*m + 6)*x^4*e^4 + (m^3 + 3*m^2 + 2*m)*d*x^3*e^3 - 3*(m^2 + m)*d^2*x^2*e^2
+ 6*d^3*m*x*e - 6*d^4)*B*b^2*e^(m*log(x*e + d) - 4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 580 vs. \(2 (145) = 290\).
time = 1.16, size = 580, normalized size = 4.20 \begin {gather*} -\frac {{\left (6 \, B b^{2} d^{4} - {\left ({\left (B b^{2} m^{3} + 6 \, B b^{2} m^{2} + 11 \, B b^{2} m + 6 \, B b^{2}\right )} x^{4} + {\left ({\left (2 \, B a b + A b^{2}\right )} m^{3} + 16 \, B a b + 8 \, A b^{2} + 7 \, {\left (2 \, B a b + A b^{2}\right )} m^{2} + 14 \, {\left (2 \, B a b + A b^{2}\right )} m\right )} x^{3} + {\left ({\left (B a^{2} + 2 \, A a b\right )} m^{3} + 12 \, B a^{2} + 24 \, A a b + 8 \, {\left (B a^{2} + 2 \, A a b\right )} m^{2} + 19 \, {\left (B a^{2} + 2 \, A a b\right )} m\right )} x^{2} + {\left (A a^{2} m^{3} + 9 \, A a^{2} m^{2} + 26 \, A a^{2} m + 24 \, A a^{2}\right )} x\right )} e^{4} - {\left (A a^{2} d m^{3} + 9 \, A a^{2} d m^{2} + 26 \, A a^{2} d m + 24 \, A a^{2} d + {\left (B b^{2} d m^{3} + 3 \, B b^{2} d m^{2} + 2 \, B b^{2} d m\right )} x^{3} + {\left ({\left (2 \, B a b + A b^{2}\right )} d m^{3} + 5 \, {\left (2 \, B a b + A b^{2}\right )} d m^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d m\right )} x^{2} + {\left ({\left (B a^{2} + 2 \, A a b\right )} d m^{3} + 7 \, {\left (B a^{2} + 2 \, A a b\right )} d m^{2} + 12 \, {\left (B a^{2} + 2 \, A a b\right )} d m\right )} x\right )} e^{3} + {\left ({\left (B a^{2} + 2 \, A a b\right )} d^{2} m^{2} + 7 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} m + 12 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} + 3 \, {\left (B b^{2} d^{2} m^{2} + B b^{2} d^{2} m\right )} x^{2} + 2 \, {\left ({\left (2 \, B a b + A b^{2}\right )} d^{2} m^{2} + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} m\right )} x\right )} e^{2} - 2 \, {\left (3 \, B b^{2} d^{3} m x + {\left (2 \, B a b + A b^{2}\right )} d^{3} m + 4 \, {\left (2 \, B a b + A b^{2}\right )} d^{3}\right )} e\right )} {\left (x e + d\right )}^{m} e^{\left (-4\right )}}{m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^m,x, algorithm="fricas")

[Out]

-(6*B*b^2*d^4 - ((B*b^2*m^3 + 6*B*b^2*m^2 + 11*B*b^2*m + 6*B*b^2)*x^4 + ((2*B*a*b + A*b^2)*m^3 + 16*B*a*b + 8*
A*b^2 + 7*(2*B*a*b + A*b^2)*m^2 + 14*(2*B*a*b + A*b^2)*m)*x^3 + ((B*a^2 + 2*A*a*b)*m^3 + 12*B*a^2 + 24*A*a*b +
8*(B*a^2 + 2*A*a*b)*m^2 + 19*(B*a^2 + 2*A*a*b)*m)*x^2 + (A*a^2*m^3 + 9*A*a^2*m^2 + 26*A*a^2*m + 24*A*a^2)*x)*
e^4 - (A*a^2*d*m^3 + 9*A*a^2*d*m^2 + 26*A*a^2*d*m + 24*A*a^2*d + (B*b^2*d*m^3 + 3*B*b^2*d*m^2 + 2*B*b^2*d*m)*x
^3 + ((2*B*a*b + A*b^2)*d*m^3 + 5*(2*B*a*b + A*b^2)*d*m^2 + 4*(2*B*a*b + A*b^2)*d*m)*x^2 + ((B*a^2 + 2*A*a*b)*
d*m^3 + 7*(B*a^2 + 2*A*a*b)*d*m^2 + 12*(B*a^2 + 2*A*a*b)*d*m)*x)*e^3 + ((B*a^2 + 2*A*a*b)*d^2*m^2 + 7*(B*a^2 +
2*A*a*b)*d^2*m + 12*(B*a^2 + 2*A*a*b)*d^2 + 3*(B*b^2*d^2*m^2 + B*b^2*d^2*m)*x^2 + 2*((2*B*a*b + A*b^2)*d^2*m^
2 + 4*(2*B*a*b + A*b^2)*d^2*m)*x)*e^2 - 2*(3*B*b^2*d^3*m*x + (2*B*a*b + A*b^2)*d^3*m + 4*(2*B*a*b + A*b^2)*d^3
)*e)*(x*e + d)^m*e^(-4)/(m^4 + 10*m^3 + 35*m^2 + 50*m + 24)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 6186 vs. \(2 (126) = 252\).
time = 1.44, size = 6186, normalized size = 44.83 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(B*x+A)*(e*x+d)**m,x)

[Out]

Piecewise((d**m*(A*a**2*x + A*a*b*x**2 + A*b**2*x**3/3 + B*a**2*x**2/2 + 2*B*a*b*x**3/3 + B*b**2*x**4/4), Eq(e
, 0)), (-2*A*a**2*e**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 2*A*a*b*d*e**2/(6*d**3*
e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*A*a*b*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e*
*6*x**2 + 6*e**7*x**3) - 2*A*b**2*d**2*e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*A*b
**2*d*e**2*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 6*A*b**2*e**3*x**2/(6*d**3*e**4 +
18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - B*a**2*d*e**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2
+ 6*e**7*x**3) - 3*B*a**2*e**3*x/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 4*B*a*b*d**2
*e/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 12*B*a*b*d*e**2*x/(6*d**3*e**4 + 18*d**2*e*
*5*x + 18*d*e**6*x**2 + 6*e**7*x**3) - 12*B*a*b*e**3*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e
**7*x**3) + 6*B*b**2*d**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 11*B*b*
*2*d**3/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 18*B*b**2*d**2*e*x*log(d/e + x)/(6*d**
3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 27*B*b**2*d**2*e*x/(6*d**3*e**4 + 18*d**2*e**5*x + 1
8*d*e**6*x**2 + 6*e**7*x**3) + 18*B*b**2*d*e**2*x**2*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x*
*2 + 6*e**7*x**3) + 18*B*b**2*d*e**2*x**2/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3) + 6*B*
b**2*e**3*x**3*log(d/e + x)/(6*d**3*e**4 + 18*d**2*e**5*x + 18*d*e**6*x**2 + 6*e**7*x**3), Eq(m, -4)), (-A*a**
2*e**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 2*A*a*b*d*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 4*
A*a*b*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*A*b**2*d**2*e*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x
+ 2*e**6*x**2) + 3*A*b**2*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*A*b**2*d*e**2*x*log(d/e + x)/(2
*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*A*b**2*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*A*b**2
*e**3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - B*a**2*d*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2
*e**6*x**2) - 2*B*a**2*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*B*a*b*d**2*e*log(d/e + x)/(2*d**2*e
**4 + 4*d*e**5*x + 2*e**6*x**2) + 6*B*a*b*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 8*B*a*b*d*e**2*x*l
og(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 8*B*a*b*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**
2) + 4*B*a*b*e**3*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*B*b**2*d**3*log(d/e + x)/(2*d
**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 9*B*b**2*d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*B*b**2*d**2
*e*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*B*b**2*d**2*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*
e**6*x**2) - 6*B*b**2*d*e**2*x**2*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*B*b**2*e**3*x**3/(
2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2), Eq(m, -3)), (-2*A*a**2*e**3/(2*d*e**4 + 2*e**5*x) + 4*A*a*b*d*e**2*lo
g(d/e + x)/(2*d*e**4 + 2*e**5*x) + 4*A*a*b*d*e**2/(2*d*e**4 + 2*e**5*x) + 4*A*a*b*e**3*x*log(d/e + x)/(2*d*e**
4 + 2*e**5*x) - 4*A*b**2*d**2*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*A*b**2*d**2*e/(2*d*e**4 + 2*e**5*x) - 4
*A*b**2*d*e**2*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*A*b**2*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 2*B*a**2*d*e*
*2*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*B*a**2*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*B*a**2*e**3*x*log(d/e + x)/(
2*d*e**4 + 2*e**5*x) - 8*B*a*b*d**2*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 8*B*a*b*d**2*e/(2*d*e**4 + 2*e**5*x
) - 8*B*a*b*d*e**2*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 4*B*a*b*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 6*B*b**2*d
**3*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 6*B*b**2*d**3/(2*d*e**4 + 2*e**5*x) + 6*B*b**2*d**2*e*x*log(d/e + x)/
(2*d*e**4 + 2*e**5*x) - 3*B*b**2*d*e**2*x**2/(2*d*e**4 + 2*e**5*x) + B*b**2*e**3*x**3/(2*d*e**4 + 2*e**5*x), E
q(m, -2)), (A*a**2*log(d/e + x)/e - 2*A*a*b*d*log(d/e + x)/e**2 + 2*A*a*b*x/e + A*b**2*d**2*log(d/e + x)/e**3
- A*b**2*d*x/e**2 + A*b**2*x**2/(2*e) - B*a**2*d*log(d/e + x)/e**2 + B*a**2*x/e + 2*B*a*b*d**2*log(d/e + x)/e*
*3 - 2*B*a*b*d*x/e**2 + B*a*b*x**2/e - B*b**2*d**3*log(d/e + x)/e**4 + B*b**2*d**2*x/e**3 - B*b**2*d*x**2/(2*e
**2) + B*b**2*x**3/(3*e), Eq(m, -1)), (A*a**2*d*e**3*m**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**
2 + 50*e**4*m + 24*e**4) + 9*A*a**2*d*e**3*m**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**
4*m + 24*e**4) + 26*A*a**2*d*e**3*m*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**
4) + 24*A*a**2*d*e**3*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + A*a**2*e*
*4*m**3*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 9*A*a**2*e**4*m**2*x*
(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) + 26*A*a**2*e**4*m*x*(d + e*x)**m
/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*...

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1267 vs. \(2 (145) = 290\).
time = 1.27, size = 1267, normalized size = 9.18 \begin {gather*} \frac {{\left (x e + d\right )}^{m} B b^{2} m^{3} x^{4} e^{4} + {\left (x e + d\right )}^{m} B b^{2} d m^{3} x^{3} e^{3} + 2 \, {\left (x e + d\right )}^{m} B a b m^{3} x^{3} e^{4} + {\left (x e + d\right )}^{m} A b^{2} m^{3} x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} B b^{2} m^{2} x^{4} e^{4} + 2 \, {\left (x e + d\right )}^{m} B a b d m^{3} x^{2} e^{3} + {\left (x e + d\right )}^{m} A b^{2} d m^{3} x^{2} e^{3} + 3 \, {\left (x e + d\right )}^{m} B b^{2} d m^{2} x^{3} e^{3} - 3 \, {\left (x e + d\right )}^{m} B b^{2} d^{2} m^{2} x^{2} e^{2} + {\left (x e + d\right )}^{m} B a^{2} m^{3} x^{2} e^{4} + 2 \, {\left (x e + d\right )}^{m} A a b m^{3} x^{2} e^{4} + 14 \, {\left (x e + d\right )}^{m} B a b m^{2} x^{3} e^{4} + 7 \, {\left (x e + d\right )}^{m} A b^{2} m^{2} x^{3} e^{4} + 11 \, {\left (x e + d\right )}^{m} B b^{2} m x^{4} e^{4} + {\left (x e + d\right )}^{m} B a^{2} d m^{3} x e^{3} + 2 \, {\left (x e + d\right )}^{m} A a b d m^{3} x e^{3} + 10 \, {\left (x e + d\right )}^{m} B a b d m^{2} x^{2} e^{3} + 5 \, {\left (x e + d\right )}^{m} A b^{2} d m^{2} x^{2} e^{3} + 2 \, {\left (x e + d\right )}^{m} B b^{2} d m x^{3} e^{3} - 4 \, {\left (x e + d\right )}^{m} B a b d^{2} m^{2} x e^{2} - 2 \, {\left (x e + d\right )}^{m} A b^{2} d^{2} m^{2} x e^{2} - 3 \, {\left (x e + d\right )}^{m} B b^{2} d^{2} m x^{2} e^{2} + 6 \, {\left (x e + d\right )}^{m} B b^{2} d^{3} m x e + {\left (x e + d\right )}^{m} A a^{2} m^{3} x e^{4} + 8 \, {\left (x e + d\right )}^{m} B a^{2} m^{2} x^{2} e^{4} + 16 \, {\left (x e + d\right )}^{m} A a b m^{2} x^{2} e^{4} + 28 \, {\left (x e + d\right )}^{m} B a b m x^{3} e^{4} + 14 \, {\left (x e + d\right )}^{m} A b^{2} m x^{3} e^{4} + 6 \, {\left (x e + d\right )}^{m} B b^{2} x^{4} e^{4} + {\left (x e + d\right )}^{m} A a^{2} d m^{3} e^{3} + 7 \, {\left (x e + d\right )}^{m} B a^{2} d m^{2} x e^{3} + 14 \, {\left (x e + d\right )}^{m} A a b d m^{2} x e^{3} + 8 \, {\left (x e + d\right )}^{m} B a b d m x^{2} e^{3} + 4 \, {\left (x e + d\right )}^{m} A b^{2} d m x^{2} e^{3} - {\left (x e + d\right )}^{m} B a^{2} d^{2} m^{2} e^{2} - 2 \, {\left (x e + d\right )}^{m} A a b d^{2} m^{2} e^{2} - 16 \, {\left (x e + d\right )}^{m} B a b d^{2} m x e^{2} - 8 \, {\left (x e + d\right )}^{m} A b^{2} d^{2} m x e^{2} + 4 \, {\left (x e + d\right )}^{m} B a b d^{3} m e + 2 \, {\left (x e + d\right )}^{m} A b^{2} d^{3} m e - 6 \, {\left (x e + d\right )}^{m} B b^{2} d^{4} + 9 \, {\left (x e + d\right )}^{m} A a^{2} m^{2} x e^{4} + 19 \, {\left (x e + d\right )}^{m} B a^{2} m x^{2} e^{4} + 38 \, {\left (x e + d\right )}^{m} A a b m x^{2} e^{4} + 16 \, {\left (x e + d\right )}^{m} B a b x^{3} e^{4} + 8 \, {\left (x e + d\right )}^{m} A b^{2} x^{3} e^{4} + 9 \, {\left (x e + d\right )}^{m} A a^{2} d m^{2} e^{3} + 12 \, {\left (x e + d\right )}^{m} B a^{2} d m x e^{3} + 24 \, {\left (x e + d\right )}^{m} A a b d m x e^{3} - 7 \, {\left (x e + d\right )}^{m} B a^{2} d^{2} m e^{2} - 14 \, {\left (x e + d\right )}^{m} A a b d^{2} m e^{2} + 16 \, {\left (x e + d\right )}^{m} B a b d^{3} e + 8 \, {\left (x e + d\right )}^{m} A b^{2} d^{3} e + 26 \, {\left (x e + d\right )}^{m} A a^{2} m x e^{4} + 12 \, {\left (x e + d\right )}^{m} B a^{2} x^{2} e^{4} + 24 \, {\left (x e + d\right )}^{m} A a b x^{2} e^{4} + 26 \, {\left (x e + d\right )}^{m} A a^{2} d m e^{3} - 12 \, {\left (x e + d\right )}^{m} B a^{2} d^{2} e^{2} - 24 \, {\left (x e + d\right )}^{m} A a b d^{2} e^{2} + 24 \, {\left (x e + d\right )}^{m} A a^{2} x e^{4} + 24 \, {\left (x e + d\right )}^{m} A a^{2} d e^{3}}{m^{4} e^{4} + 10 \, m^{3} e^{4} + 35 \, m^{2} e^{4} + 50 \, m e^{4} + 24 \, e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^m,x, algorithm="giac")

[Out]

((x*e + d)^m*B*b^2*m^3*x^4*e^4 + (x*e + d)^m*B*b^2*d*m^3*x^3*e^3 + 2*(x*e + d)^m*B*a*b*m^3*x^3*e^4 + (x*e + d)
^m*A*b^2*m^3*x^3*e^4 + 6*(x*e + d)^m*B*b^2*m^2*x^4*e^4 + 2*(x*e + d)^m*B*a*b*d*m^3*x^2*e^3 + (x*e + d)^m*A*b^2
*d*m^3*x^2*e^3 + 3*(x*e + d)^m*B*b^2*d*m^2*x^3*e^3 - 3*(x*e + d)^m*B*b^2*d^2*m^2*x^2*e^2 + (x*e + d)^m*B*a^2*m
^3*x^2*e^4 + 2*(x*e + d)^m*A*a*b*m^3*x^2*e^4 + 14*(x*e + d)^m*B*a*b*m^2*x^3*e^4 + 7*(x*e + d)^m*A*b^2*m^2*x^3*
e^4 + 11*(x*e + d)^m*B*b^2*m*x^4*e^4 + (x*e + d)^m*B*a^2*d*m^3*x*e^3 + 2*(x*e + d)^m*A*a*b*d*m^3*x*e^3 + 10*(x
*e + d)^m*B*a*b*d*m^2*x^2*e^3 + 5*(x*e + d)^m*A*b^2*d*m^2*x^2*e^3 + 2*(x*e + d)^m*B*b^2*d*m*x^3*e^3 - 4*(x*e +
d)^m*B*a*b*d^2*m^2*x*e^2 - 2*(x*e + d)^m*A*b^2*d^2*m^2*x*e^2 - 3*(x*e + d)^m*B*b^2*d^2*m*x^2*e^2 + 6*(x*e + d
)^m*B*b^2*d^3*m*x*e + (x*e + d)^m*A*a^2*m^3*x*e^4 + 8*(x*e + d)^m*B*a^2*m^2*x^2*e^4 + 16*(x*e + d)^m*A*a*b*m^2
*x^2*e^4 + 28*(x*e + d)^m*B*a*b*m*x^3*e^4 + 14*(x*e + d)^m*A*b^2*m*x^3*e^4 + 6*(x*e + d)^m*B*b^2*x^4*e^4 + (x*
e + d)^m*A*a^2*d*m^3*e^3 + 7*(x*e + d)^m*B*a^2*d*m^2*x*e^3 + 14*(x*e + d)^m*A*a*b*d*m^2*x*e^3 + 8*(x*e + d)^m*
B*a*b*d*m*x^2*e^3 + 4*(x*e + d)^m*A*b^2*d*m*x^2*e^3 - (x*e + d)^m*B*a^2*d^2*m^2*e^2 - 2*(x*e + d)^m*A*a*b*d^2*
m^2*e^2 - 16*(x*e + d)^m*B*a*b*d^2*m*x*e^2 - 8*(x*e + d)^m*A*b^2*d^2*m*x*e^2 + 4*(x*e + d)^m*B*a*b*d^3*m*e + 2
*(x*e + d)^m*A*b^2*d^3*m*e - 6*(x*e + d)^m*B*b^2*d^4 + 9*(x*e + d)^m*A*a^2*m^2*x*e^4 + 19*(x*e + d)^m*B*a^2*m*
x^2*e^4 + 38*(x*e + d)^m*A*a*b*m*x^2*e^4 + 16*(x*e + d)^m*B*a*b*x^3*e^4 + 8*(x*e + d)^m*A*b^2*x^3*e^4 + 9*(x*e
+ d)^m*A*a^2*d*m^2*e^3 + 12*(x*e + d)^m*B*a^2*d*m*x*e^3 + 24*(x*e + d)^m*A*a*b*d*m*x*e^3 - 7*(x*e + d)^m*B*a^
2*d^2*m*e^2 - 14*(x*e + d)^m*A*a*b*d^2*m*e^2 + 16*(x*e + d)^m*B*a*b*d^3*e + 8*(x*e + d)^m*A*b^2*d^3*e + 26*(x*
e + d)^m*A*a^2*m*x*e^4 + 12*(x*e + d)^m*B*a^2*x^2*e^4 + 24*(x*e + d)^m*A*a*b*x^2*e^4 + 26*(x*e + d)^m*A*a^2*d*
m*e^3 - 12*(x*e + d)^m*B*a^2*d^2*e^2 - 24*(x*e + d)^m*A*a*b*d^2*e^2 + 24*(x*e + d)^m*A*a^2*x*e^4 + 24*(x*e + d
)^m*A*a^2*d*e^3)/(m^4*e^4 + 10*m^3*e^4 + 35*m^2*e^4 + 50*m*e^4 + 24*e^4)

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Mupad [B]
time = 2.77, size = 676, normalized size = 4.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^m\,\left (-B\,a^2\,d^2\,e^2\,m^2-7\,B\,a^2\,d^2\,e^2\,m-12\,B\,a^2\,d^2\,e^2+A\,a^2\,d\,e^3\,m^3+9\,A\,a^2\,d\,e^3\,m^2+26\,A\,a^2\,d\,e^3\,m+24\,A\,a^2\,d\,e^3+4\,B\,a\,b\,d^3\,e\,m+16\,B\,a\,b\,d^3\,e-2\,A\,a\,b\,d^2\,e^2\,m^2-14\,A\,a\,b\,d^2\,e^2\,m-24\,A\,a\,b\,d^2\,e^2-6\,B\,b^2\,d^4+2\,A\,b^2\,d^3\,e\,m+8\,A\,b^2\,d^3\,e\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x\,{\left (d+e\,x\right )}^m\,\left (B\,a^2\,d\,e^3\,m^3+7\,B\,a^2\,d\,e^3\,m^2+12\,B\,a^2\,d\,e^3\,m+A\,a^2\,e^4\,m^3+9\,A\,a^2\,e^4\,m^2+26\,A\,a^2\,e^4\,m+24\,A\,a^2\,e^4-4\,B\,a\,b\,d^2\,e^2\,m^2-16\,B\,a\,b\,d^2\,e^2\,m+2\,A\,a\,b\,d\,e^3\,m^3+14\,A\,a\,b\,d\,e^3\,m^2+24\,A\,a\,b\,d\,e^3\,m+6\,B\,b^2\,d^3\,e\,m-2\,A\,b^2\,d^2\,e^2\,m^2-8\,A\,b^2\,d^2\,e^2\,m\right )}{e^4\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {x^2\,\left (m+1\right )\,{\left (d+e\,x\right )}^m\,\left (B\,a^2\,e^2\,m^2+7\,B\,a^2\,e^2\,m+12\,B\,a^2\,e^2+2\,B\,a\,b\,d\,e\,m^2+8\,B\,a\,b\,d\,e\,m+2\,A\,a\,b\,e^2\,m^2+14\,A\,a\,b\,e^2\,m+24\,A\,a\,b\,e^2-3\,B\,b^2\,d^2\,m+A\,b^2\,d\,e\,m^2+4\,A\,b^2\,d\,e\,m\right )}{e^2\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}+\frac {B\,b^2\,x^4\,{\left (d+e\,x\right )}^m\,\left (m^3+6\,m^2+11\,m+6\right )}{m^4+10\,m^3+35\,m^2+50\,m+24}+\frac {b\,x^3\,{\left (d+e\,x\right )}^m\,\left (m^2+3\,m+2\right )\,\left (4\,A\,b\,e+8\,B\,a\,e+A\,b\,e\,m+2\,B\,a\,e\,m+B\,b\,d\,m\right )}{e\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^2*(d + e*x)^m,x)

[Out]

((d + e*x)^m*(24*A*a^2*d*e^3 - 6*B*b^2*d^4 + 8*A*b^2*d^3*e - 12*B*a^2*d^2*e^2 + 9*A*a^2*d*e^3*m^2 + A*a^2*d*e^
3*m^3 - 7*B*a^2*d^2*e^2*m + 16*B*a*b*d^3*e - B*a^2*d^2*e^2*m^2 - 24*A*a*b*d^2*e^2 + 26*A*a^2*d*e^3*m + 2*A*b^2
*d^3*e*m - 14*A*a*b*d^2*e^2*m - 2*A*a*b*d^2*e^2*m^2 + 4*B*a*b*d^3*e*m))/(e^4*(50*m + 35*m^2 + 10*m^3 + m^4 + 2
4)) + (x*(d + e*x)^m*(24*A*a^2*e^4 + 26*A*a^2*e^4*m + 9*A*a^2*e^4*m^2 + A*a^2*e^4*m^3 - 8*A*b^2*d^2*e^2*m + 7*
B*a^2*d*e^3*m^2 + B*a^2*d*e^3*m^3 - 2*A*b^2*d^2*e^2*m^2 + 12*B*a^2*d*e^3*m + 6*B*b^2*d^3*e*m + 14*A*a*b*d*e^3*
m^2 + 2*A*a*b*d*e^3*m^3 - 16*B*a*b*d^2*e^2*m - 4*B*a*b*d^2*e^2*m^2 + 24*A*a*b*d*e^3*m))/(e^4*(50*m + 35*m^2 +
10*m^3 + m^4 + 24)) + (x^2*(m + 1)*(d + e*x)^m*(12*B*a^2*e^2 + 24*A*a*b*e^2 + 7*B*a^2*e^2*m - 3*B*b^2*d^2*m +
B*a^2*e^2*m^2 + 14*A*a*b*e^2*m + 4*A*b^2*d*e*m + 2*A*a*b*e^2*m^2 + A*b^2*d*e*m^2 + 8*B*a*b*d*e*m + 2*B*a*b*d*e
*m^2))/(e^2*(50*m + 35*m^2 + 10*m^3 + m^4 + 24)) + (B*b^2*x^4*(d + e*x)^m*(11*m + 6*m^2 + m^3 + 6))/(50*m + 35
*m^2 + 10*m^3 + m^4 + 24) + (b*x^3*(d + e*x)^m*(3*m + m^2 + 2)*(4*A*b*e + 8*B*a*e + A*b*e*m + 2*B*a*e*m + B*b*
d*m))/(e*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))

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