3.63 \(\int (e x)^m (a+b x)^3 (a d-b d x)^2 \, dx\)

Optimal. Leaf size=143 \[ -\frac{2 a^3 b^2 d^2 (e x)^{m+3}}{e^3 (m+3)}-\frac{2 a^2 b^3 d^2 (e x)^{m+4}}{e^4 (m+4)}+\frac{a^4 b d^2 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^5 d^2 (e x)^{m+1}}{e (m+1)}+\frac{a b^4 d^2 (e x)^{m+5}}{e^5 (m+5)}+\frac{b^5 d^2 (e x)^{m+6}}{e^6 (m+6)} \]

[Out]

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

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Rubi [A]  time = 0.0809989, antiderivative size = 143, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {88} \[ -\frac{2 a^3 b^2 d^2 (e x)^{m+3}}{e^3 (m+3)}-\frac{2 a^2 b^3 d^2 (e x)^{m+4}}{e^4 (m+4)}+\frac{a^4 b d^2 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^5 d^2 (e x)^{m+1}}{e (m+1)}+\frac{a b^4 d^2 (e x)^{m+5}}{e^5 (m+5)}+\frac{b^5 d^2 (e x)^{m+6}}{e^6 (m+6)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x)^3*(a*d - b*d*x)^2,x]

[Out]

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

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

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

Mathematica [A]  time = 0.0480851, size = 88, normalized size = 0.62 \[ d^2 x (e x)^m \left (-\frac{2 a^3 b^2 x^2}{m+3}-\frac{2 a^2 b^3 x^3}{m+4}+\frac{a^4 b x}{m+2}+\frac{a^5}{m+1}+\frac{a b^4 x^4}{m+5}+\frac{b^5 x^5}{m+6}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.007, size = 422, normalized size = 3. \begin{align*}{\frac{{d}^{2} \left ( ex \right ) ^{m} \left ({b}^{5}{m}^{5}{x}^{5}+a{b}^{4}{m}^{5}{x}^{4}+15\,{b}^{5}{m}^{4}{x}^{5}-2\,{a}^{2}{b}^{3}{m}^{5}{x}^{3}+16\,a{b}^{4}{m}^{4}{x}^{4}+85\,{b}^{5}{m}^{3}{x}^{5}-2\,{a}^{3}{b}^{2}{m}^{5}{x}^{2}-34\,{a}^{2}{b}^{3}{m}^{4}{x}^{3}+95\,a{b}^{4}{m}^{3}{x}^{4}+225\,{b}^{5}{m}^{2}{x}^{5}+{a}^{4}b{m}^{5}x-36\,{a}^{3}{b}^{2}{m}^{4}{x}^{2}-214\,{a}^{2}{b}^{3}{m}^{3}{x}^{3}+260\,a{b}^{4}{m}^{2}{x}^{4}+274\,{b}^{5}m{x}^{5}+{a}^{5}{m}^{5}+19\,{a}^{4}b{m}^{4}x-242\,{a}^{3}{b}^{2}{m}^{3}{x}^{2}-614\,{a}^{2}{b}^{3}{m}^{2}{x}^{3}+324\,a{b}^{4}m{x}^{4}+120\,{b}^{5}{x}^{5}+20\,{a}^{5}{m}^{4}+137\,{a}^{4}b{m}^{3}x-744\,{a}^{3}{b}^{2}{m}^{2}{x}^{2}-792\,{a}^{2}{b}^{3}m{x}^{3}+144\,a{b}^{4}{x}^{4}+155\,{a}^{5}{m}^{3}+461\,{a}^{4}b{m}^{2}x-1016\,{a}^{3}{b}^{2}m{x}^{2}-360\,{a}^{2}{b}^{3}{x}^{3}+580\,{a}^{5}{m}^{2}+702\,{a}^{4}bmx-480\,{a}^{3}{b}^{2}{x}^{2}+1044\,{a}^{5}m+360\,{a}^{4}bx+720\,{a}^{5} \right ) x}{ \left ( 6+m \right ) \left ( 5+m \right ) \left ( 4+m \right ) \left ( 3+m \right ) \left ( 2+m \right ) \left ( 1+m \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x+a)^3*(-b*d*x+a*d)^2,x)

[Out]

d^2*(e*x)^m*(b^5*m^5*x^5+a*b^4*m^5*x^4+15*b^5*m^4*x^5-2*a^2*b^3*m^5*x^3+16*a*b^4*m^4*x^4+85*b^5*m^3*x^5-2*a^3*
b^2*m^5*x^2-34*a^2*b^3*m^4*x^3+95*a*b^4*m^3*x^4+225*b^5*m^2*x^5+a^4*b*m^5*x-36*a^3*b^2*m^4*x^2-214*a^2*b^3*m^3
*x^3+260*a*b^4*m^2*x^4+274*b^5*m*x^5+a^5*m^5+19*a^4*b*m^4*x-242*a^3*b^2*m^3*x^2-614*a^2*b^3*m^2*x^3+324*a*b^4*
m*x^4+120*b^5*x^5+20*a^5*m^4+137*a^4*b*m^3*x-744*a^3*b^2*m^2*x^2-792*a^2*b^3*m*x^3+144*a*b^4*x^4+155*a^5*m^3+4
61*a^4*b*m^2*x-1016*a^3*b^2*m*x^2-360*a^2*b^3*x^3+580*a^5*m^2+702*a^4*b*m*x-480*a^3*b^2*x^2+1044*a^5*m+360*a^4
*b*x+720*a^5)*x/(6+m)/(5+m)/(4+m)/(3+m)/(2+m)/(1+m)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.95209, size = 1027, normalized size = 7.18 \begin{align*} \frac{{\left ({\left (b^{5} d^{2} m^{5} + 15 \, b^{5} d^{2} m^{4} + 85 \, b^{5} d^{2} m^{3} + 225 \, b^{5} d^{2} m^{2} + 274 \, b^{5} d^{2} m + 120 \, b^{5} d^{2}\right )} x^{6} +{\left (a b^{4} d^{2} m^{5} + 16 \, a b^{4} d^{2} m^{4} + 95 \, a b^{4} d^{2} m^{3} + 260 \, a b^{4} d^{2} m^{2} + 324 \, a b^{4} d^{2} m + 144 \, a b^{4} d^{2}\right )} x^{5} - 2 \,{\left (a^{2} b^{3} d^{2} m^{5} + 17 \, a^{2} b^{3} d^{2} m^{4} + 107 \, a^{2} b^{3} d^{2} m^{3} + 307 \, a^{2} b^{3} d^{2} m^{2} + 396 \, a^{2} b^{3} d^{2} m + 180 \, a^{2} b^{3} d^{2}\right )} x^{4} - 2 \,{\left (a^{3} b^{2} d^{2} m^{5} + 18 \, a^{3} b^{2} d^{2} m^{4} + 121 \, a^{3} b^{2} d^{2} m^{3} + 372 \, a^{3} b^{2} d^{2} m^{2} + 508 \, a^{3} b^{2} d^{2} m + 240 \, a^{3} b^{2} d^{2}\right )} x^{3} +{\left (a^{4} b d^{2} m^{5} + 19 \, a^{4} b d^{2} m^{4} + 137 \, a^{4} b d^{2} m^{3} + 461 \, a^{4} b d^{2} m^{2} + 702 \, a^{4} b d^{2} m + 360 \, a^{4} b d^{2}\right )} x^{2} +{\left (a^{5} d^{2} m^{5} + 20 \, a^{5} d^{2} m^{4} + 155 \, a^{5} d^{2} m^{3} + 580 \, a^{5} d^{2} m^{2} + 1044 \, a^{5} d^{2} m + 720 \, a^{5} d^{2}\right )} x\right )} \left (e x\right )^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((b^5*d^2*m^5 + 15*b^5*d^2*m^4 + 85*b^5*d^2*m^3 + 225*b^5*d^2*m^2 + 274*b^5*d^2*m + 120*b^5*d^2)*x^6 + (a*b^4*
d^2*m^5 + 16*a*b^4*d^2*m^4 + 95*a*b^4*d^2*m^3 + 260*a*b^4*d^2*m^2 + 324*a*b^4*d^2*m + 144*a*b^4*d^2)*x^5 - 2*(
a^2*b^3*d^2*m^5 + 17*a^2*b^3*d^2*m^4 + 107*a^2*b^3*d^2*m^3 + 307*a^2*b^3*d^2*m^2 + 396*a^2*b^3*d^2*m + 180*a^2
*b^3*d^2)*x^4 - 2*(a^3*b^2*d^2*m^5 + 18*a^3*b^2*d^2*m^4 + 121*a^3*b^2*d^2*m^3 + 372*a^3*b^2*d^2*m^2 + 508*a^3*
b^2*d^2*m + 240*a^3*b^2*d^2)*x^3 + (a^4*b*d^2*m^5 + 19*a^4*b*d^2*m^4 + 137*a^4*b*d^2*m^3 + 461*a^4*b*d^2*m^2 +
702*a^4*b*d^2*m + 360*a^4*b*d^2)*x^2 + (a^5*d^2*m^5 + 20*a^5*d^2*m^4 + 155*a^5*d^2*m^3 + 580*a^5*d^2*m^2 + 10
44*a^5*d^2*m + 720*a^5*d^2)*x)*(e*x)^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)

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Sympy [A]  time = 2.16412, size = 2320, normalized size = 16.22 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise(((-a**5*d**2/(5*x**5) - a**4*b*d**2/(4*x**4) + 2*a**3*b**2*d**2/(3*x**3) + a**2*b**3*d**2/x**2 - a*b
**4*d**2/x + b**5*d**2*log(x))/e**6, Eq(m, -6)), ((-a**5*d**2/(4*x**4) - a**4*b*d**2/(3*x**3) + a**3*b**2*d**2
/x**2 + 2*a**2*b**3*d**2/x + a*b**4*d**2*log(x) + b**5*d**2*x)/e**5, Eq(m, -5)), ((-a**5*d**2/(3*x**3) - a**4*
b*d**2/(2*x**2) + 2*a**3*b**2*d**2/x - 2*a**2*b**3*d**2*log(x) + a*b**4*d**2*x + b**5*d**2*x**2/2)/e**4, Eq(m,
-4)), ((-a**5*d**2/(2*x**2) - a**4*b*d**2/x - 2*a**3*b**2*d**2*log(x) - 2*a**2*b**3*d**2*x + a*b**4*d**2*x**2
/2 + b**5*d**2*x**3/3)/e**3, Eq(m, -3)), ((-a**5*d**2/x + a**4*b*d**2*log(x) - 2*a**3*b**2*d**2*x - a**2*b**3*
d**2*x**2 + a*b**4*d**2*x**3/3 + b**5*d**2*x**4/4)/e**2, Eq(m, -2)), ((a**5*d**2*log(x) + a**4*b*d**2*x - a**3
*b**2*d**2*x**2 - 2*a**2*b**3*d**2*x**3/3 + a*b**4*d**2*x**4/4 + b**5*d**2*x**5/5)/e, Eq(m, -1)), (a**5*d**2*e
**m*m**5*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 20*a**5*d**2*e**m*m**4*x*x
**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 155*a**5*d**2*e**m*m**3*x*x**m/(m**6 +
21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 580*a**5*d**2*e**m*m**2*x*x**m/(m**6 + 21*m**5 +
175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 1044*a**5*d**2*e**m*m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 73
5*m**3 + 1624*m**2 + 1764*m + 720) + 720*a**5*d**2*e**m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m*
*2 + 1764*m + 720) + a**4*b*d**2*e**m*m**5*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*
m + 720) + 19*a**4*b*d**2*e**m*m**4*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720
) + 137*a**4*b*d**2*e**m*m**3*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 46
1*a**4*b*d**2*e**m*m**2*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 702*a**4
*b*d**2*e**m*m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 360*a**4*b*d**2*e
**m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 2*a**3*b**2*d**2*e**m*m**5*x
**3*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 36*a**3*b**2*d**2*e**m*m**4*x**3*
x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 242*a**3*b**2*d**2*e**m*m**3*x**3*x**
m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 744*a**3*b**2*d**2*e**m*m**2*x**3*x**m/(
m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 1016*a**3*b**2*d**2*e**m*m*x**3*x**m/(m**6
+ 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 480*a**3*b**2*d**2*e**m*x**3*x**m/(m**6 + 21*m**
5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 2*a**2*b**3*d**2*e**m*m**5*x**4*x**m/(m**6 + 21*m**5 + 1
75*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 34*a**2*b**3*d**2*e**m*m**4*x**4*x**m/(m**6 + 21*m**5 + 175*m
**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 214*a**2*b**3*d**2*e**m*m**3*x**4*x**m/(m**6 + 21*m**5 + 175*m**4
+ 735*m**3 + 1624*m**2 + 1764*m + 720) - 614*a**2*b**3*d**2*e**m*m**2*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 +
735*m**3 + 1624*m**2 + 1764*m + 720) - 792*a**2*b**3*d**2*e**m*m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m*
*3 + 1624*m**2 + 1764*m + 720) - 360*a**2*b**3*d**2*e**m*x**4*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 162
4*m**2 + 1764*m + 720) + a*b**4*d**2*e**m*m**5*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1
764*m + 720) + 16*a*b**4*d**2*e**m*m**4*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m +
720) + 95*a*b**4*d**2*e**m*m**3*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) +
260*a*b**4*d**2*e**m*m**2*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 324*a
*b**4*d**2*e**m*m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 144*a*b**4*d**
2*e**m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + b**5*d**2*e**m*m**5*x**6*
x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 15*b**5*d**2*e**m*m**4*x**6*x**m/(m**
6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 85*b**5*d**2*e**m*m**3*x**6*x**m/(m**6 + 21*m*
*5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 225*b**5*d**2*e**m*m**2*x**6*x**m/(m**6 + 21*m**5 + 175
*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 274*b**5*d**2*e**m*m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735
*m**3 + 1624*m**2 + 1764*m + 720) + 120*b**5*d**2*e**m*x**6*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*
m**2 + 1764*m + 720), True))

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Giac [B]  time = 1.21642, size = 969, normalized size = 6.78 \begin{align*} \frac{b^{5} d^{2} m^{5} x^{6} x^{m} e^{m} + a b^{4} d^{2} m^{5} x^{5} x^{m} e^{m} + 15 \, b^{5} d^{2} m^{4} x^{6} x^{m} e^{m} - 2 \, a^{2} b^{3} d^{2} m^{5} x^{4} x^{m} e^{m} + 16 \, a b^{4} d^{2} m^{4} x^{5} x^{m} e^{m} + 85 \, b^{5} d^{2} m^{3} x^{6} x^{m} e^{m} - 2 \, a^{3} b^{2} d^{2} m^{5} x^{3} x^{m} e^{m} - 34 \, a^{2} b^{3} d^{2} m^{4} x^{4} x^{m} e^{m} + 95 \, a b^{4} d^{2} m^{3} x^{5} x^{m} e^{m} + 225 \, b^{5} d^{2} m^{2} x^{6} x^{m} e^{m} + a^{4} b d^{2} m^{5} x^{2} x^{m} e^{m} - 36 \, a^{3} b^{2} d^{2} m^{4} x^{3} x^{m} e^{m} - 214 \, a^{2} b^{3} d^{2} m^{3} x^{4} x^{m} e^{m} + 260 \, a b^{4} d^{2} m^{2} x^{5} x^{m} e^{m} + 274 \, b^{5} d^{2} m x^{6} x^{m} e^{m} + a^{5} d^{2} m^{5} x x^{m} e^{m} + 19 \, a^{4} b d^{2} m^{4} x^{2} x^{m} e^{m} - 242 \, a^{3} b^{2} d^{2} m^{3} x^{3} x^{m} e^{m} - 614 \, a^{2} b^{3} d^{2} m^{2} x^{4} x^{m} e^{m} + 324 \, a b^{4} d^{2} m x^{5} x^{m} e^{m} + 120 \, b^{5} d^{2} x^{6} x^{m} e^{m} + 20 \, a^{5} d^{2} m^{4} x x^{m} e^{m} + 137 \, a^{4} b d^{2} m^{3} x^{2} x^{m} e^{m} - 744 \, a^{3} b^{2} d^{2} m^{2} x^{3} x^{m} e^{m} - 792 \, a^{2} b^{3} d^{2} m x^{4} x^{m} e^{m} + 144 \, a b^{4} d^{2} x^{5} x^{m} e^{m} + 155 \, a^{5} d^{2} m^{3} x x^{m} e^{m} + 461 \, a^{4} b d^{2} m^{2} x^{2} x^{m} e^{m} - 1016 \, a^{3} b^{2} d^{2} m x^{3} x^{m} e^{m} - 360 \, a^{2} b^{3} d^{2} x^{4} x^{m} e^{m} + 580 \, a^{5} d^{2} m^{2} x x^{m} e^{m} + 702 \, a^{4} b d^{2} m x^{2} x^{m} e^{m} - 480 \, a^{3} b^{2} d^{2} x^{3} x^{m} e^{m} + 1044 \, a^{5} d^{2} m x x^{m} e^{m} + 360 \, a^{4} b d^{2} x^{2} x^{m} e^{m} + 720 \, a^{5} d^{2} x x^{m} e^{m}}{m^{6} + 21 \, m^{5} + 175 \, m^{4} + 735 \, m^{3} + 1624 \, m^{2} + 1764 \, m + 720} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b^5*d^2*m^5*x^6*x^m*e^m + a*b^4*d^2*m^5*x^5*x^m*e^m + 15*b^5*d^2*m^4*x^6*x^m*e^m - 2*a^2*b^3*d^2*m^5*x^4*x^m*
e^m + 16*a*b^4*d^2*m^4*x^5*x^m*e^m + 85*b^5*d^2*m^3*x^6*x^m*e^m - 2*a^3*b^2*d^2*m^5*x^3*x^m*e^m - 34*a^2*b^3*d
^2*m^4*x^4*x^m*e^m + 95*a*b^4*d^2*m^3*x^5*x^m*e^m + 225*b^5*d^2*m^2*x^6*x^m*e^m + a^4*b*d^2*m^5*x^2*x^m*e^m -
36*a^3*b^2*d^2*m^4*x^3*x^m*e^m - 214*a^2*b^3*d^2*m^3*x^4*x^m*e^m + 260*a*b^4*d^2*m^2*x^5*x^m*e^m + 274*b^5*d^2
*m*x^6*x^m*e^m + a^5*d^2*m^5*x*x^m*e^m + 19*a^4*b*d^2*m^4*x^2*x^m*e^m - 242*a^3*b^2*d^2*m^3*x^3*x^m*e^m - 614*
a^2*b^3*d^2*m^2*x^4*x^m*e^m + 324*a*b^4*d^2*m*x^5*x^m*e^m + 120*b^5*d^2*x^6*x^m*e^m + 20*a^5*d^2*m^4*x*x^m*e^m
+ 137*a^4*b*d^2*m^3*x^2*x^m*e^m - 744*a^3*b^2*d^2*m^2*x^3*x^m*e^m - 792*a^2*b^3*d^2*m*x^4*x^m*e^m + 144*a*b^4
*d^2*x^5*x^m*e^m + 155*a^5*d^2*m^3*x*x^m*e^m + 461*a^4*b*d^2*m^2*x^2*x^m*e^m - 1016*a^3*b^2*d^2*m*x^3*x^m*e^m
- 360*a^2*b^3*d^2*x^4*x^m*e^m + 580*a^5*d^2*m^2*x*x^m*e^m + 702*a^4*b*d^2*m*x^2*x^m*e^m - 480*a^3*b^2*d^2*x^3*
x^m*e^m + 1044*a^5*d^2*m*x*x^m*e^m + 360*a^4*b*d^2*x^2*x^m*e^m + 720*a^5*d^2*x*x^m*e^m)/(m^6 + 21*m^5 + 175*m^
4 + 735*m^3 + 1624*m^2 + 1764*m + 720)