3.68 \(\int (e x)^m (a+b x) (a c-b c x)^4 \, dx\)
Optimal. Leaf size=145 \[ \frac{2 a^3 b^2 c^4 (e x)^{m+3}}{e^3 (m+3)}+\frac{2 a^2 b^3 c^4 (e x)^{m+4}}{e^4 (m+4)}-\frac{3 a^4 b c^4 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^5 c^4 (e x)^{m+1}}{e (m+1)}-\frac{3 a b^4 c^4 (e x)^{m+5}}{e^5 (m+5)}+\frac{b^5 c^4 (e x)^{m+6}}{e^6 (m+6)} \]
[Out]
(a^5*c^4*(e*x)^(1 + m))/(e*(1 + m)) - (3*a^4*b*c^4*(e*x)^(2 + m))/(e^2*(2 + m)) + (2*a^3*b^2*c^4*(e*x)^(3 + m)
)/(e^3*(3 + m)) + (2*a^2*b^3*c^4*(e*x)^(4 + m))/(e^4*(4 + m)) - (3*a*b^4*c^4*(e*x)^(5 + m))/(e^5*(5 + m)) + (b
^5*c^4*(e*x)^(6 + m))/(e^6*(6 + m))
________________________________________________________________________________________
Rubi [A] time = 0.084603, antiderivative size = 145, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used =
{75} \[ \frac{2 a^3 b^2 c^4 (e x)^{m+3}}{e^3 (m+3)}+\frac{2 a^2 b^3 c^4 (e x)^{m+4}}{e^4 (m+4)}-\frac{3 a^4 b c^4 (e x)^{m+2}}{e^2 (m+2)}+\frac{a^5 c^4 (e x)^{m+1}}{e (m+1)}-\frac{3 a b^4 c^4 (e x)^{m+5}}{e^5 (m+5)}+\frac{b^5 c^4 (e x)^{m+6}}{e^6 (m+6)} \]
Antiderivative was successfully verified.
[In]
Int[(e*x)^m*(a + b*x)*(a*c - b*c*x)^4,x]
[Out]
(a^5*c^4*(e*x)^(1 + m))/(e*(1 + m)) - (3*a^4*b*c^4*(e*x)^(2 + m))/(e^2*(2 + m)) + (2*a^3*b^2*c^4*(e*x)^(3 + m)
)/(e^3*(3 + m)) + (2*a^2*b^3*c^4*(e*x)^(4 + m))/(e^4*(4 + m)) - (3*a*b^4*c^4*(e*x)^(5 + m))/(e^5*(5 + m)) + (b
^5*c^4*(e*x)^(6 + m))/(e^6*(6 + m))
Rule 75
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && !(ILtQ[n
+ p + 2, 0] && GtQ[n + 2*p, 0])
Rubi steps
\begin{align*} \int (e x)^m (a+b x) (a c-b c x)^4 \, dx &=\int \left (a^5 c^4 (e x)^m-\frac{3 a^4 b c^4 (e x)^{1+m}}{e}+\frac{2 a^3 b^2 c^4 (e x)^{2+m}}{e^2}+\frac{2 a^2 b^3 c^4 (e x)^{3+m}}{e^3}-\frac{3 a b^4 c^4 (e x)^{4+m}}{e^4}+\frac{b^5 c^4 (e x)^{5+m}}{e^5}\right ) \, dx\\ &=\frac{a^5 c^4 (e x)^{1+m}}{e (1+m)}-\frac{3 a^4 b c^4 (e x)^{2+m}}{e^2 (2+m)}+\frac{2 a^3 b^2 c^4 (e x)^{3+m}}{e^3 (3+m)}+\frac{2 a^2 b^3 c^4 (e x)^{4+m}}{e^4 (4+m)}-\frac{3 a b^4 c^4 (e x)^{5+m}}{e^5 (5+m)}+\frac{b^5 c^4 (e x)^{6+m}}{e^6 (6+m)}\\ \end{align*}
Mathematica [A] time = 0.100314, size = 96, normalized size = 0.66 \[ \frac{c^4 x (e x)^m \left (a (2 m+7) \left (\frac{6 a^2 b^2 x^2}{m+3}-\frac{4 a^3 b x}{m+2}+\frac{a^4}{m+1}-\frac{4 a b^3 x^3}{m+4}+\frac{b^4 x^4}{m+5}\right )+(b x-a)^5\right )}{m+6} \]
Antiderivative was successfully verified.
[In]
Integrate[(e*x)^m*(a + b*x)*(a*c - b*c*x)^4,x]
[Out]
(c^4*x*(e*x)^m*((-a + b*x)^5 + a*(7 + 2*m)*(a^4/(1 + m) - (4*a^3*b*x)/(2 + m) + (6*a^2*b^2*x^2)/(3 + m) - (4*a
*b^3*x^3)/(4 + m) + (b^4*x^4)/(5 + m))))/(6 + m)
________________________________________________________________________________________
Maple [B] time = 0.006, size = 424, normalized size = 2.9 \begin{align*}{\frac{{c}^{4} \left ( ex \right ) ^{m} \left ({b}^{5}{m}^{5}{x}^{5}-3\,a{b}^{4}{m}^{5}{x}^{4}+15\,{b}^{5}{m}^{4}{x}^{5}+2\,{a}^{2}{b}^{3}{m}^{5}{x}^{3}-48\,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}-285\,a{b}^{4}{m}^{3}{x}^{4}+225\,{b}^{5}{m}^{2}{x}^{5}-3\,{a}^{4}b{m}^{5}x+36\,{a}^{3}{b}^{2}{m}^{4}{x}^{2}+214\,{a}^{2}{b}^{3}{m}^{3}{x}^{3}-780\,a{b}^{4}{m}^{2}{x}^{4}+274\,{b}^{5}m{x}^{5}+{a}^{5}{m}^{5}-57\,{a}^{4}b{m}^{4}x+242\,{a}^{3}{b}^{2}{m}^{3}{x}^{2}+614\,{a}^{2}{b}^{3}{m}^{2}{x}^{3}-972\,a{b}^{4}m{x}^{4}+120\,{b}^{5}{x}^{5}+20\,{a}^{5}{m}^{4}-411\,{a}^{4}b{m}^{3}x+744\,{a}^{3}{b}^{2}{m}^{2}{x}^{2}+792\,{a}^{2}{b}^{3}m{x}^{3}-432\,a{b}^{4}{x}^{4}+155\,{a}^{5}{m}^{3}-1383\,{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}-2106\,{a}^{4}bmx+480\,{a}^{3}{b}^{2}{x}^{2}+1044\,{a}^{5}m-1080\,{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)*(-b*c*x+a*c)^4,x)
[Out]
c^4*(e*x)^m*(b^5*m^5*x^5-3*a*b^4*m^5*x^4+15*b^5*m^4*x^5+2*a^2*b^3*m^5*x^3-48*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^3-285*a*b^4*m^3*x^4+225*b^5*m^2*x^5-3*a^4*b*m^5*x+36*a^3*b^2*m^4*x^2+214*a^2*b^
3*m^3*x^3-780*a*b^4*m^2*x^4+274*b^5*m*x^5+a^5*m^5-57*a^4*b*m^4*x+242*a^3*b^2*m^3*x^2+614*a^2*b^3*m^2*x^3-972*a
*b^4*m*x^4+120*b^5*x^5+20*a^5*m^4-411*a^4*b*m^3*x+744*a^3*b^2*m^2*x^2+792*a^2*b^3*m*x^3-432*a*b^4*x^4+155*a^5*
m^3-1383*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^2-2106*a^4*b*m*x+480*a^3*b^2*x^2+1044*a^5*m-
1080*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)*(-b*c*x+a*c)^4,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.01059, size = 1033, normalized size = 7.12 \begin{align*} \frac{{\left ({\left (b^{5} c^{4} m^{5} + 15 \, b^{5} c^{4} m^{4} + 85 \, b^{5} c^{4} m^{3} + 225 \, b^{5} c^{4} m^{2} + 274 \, b^{5} c^{4} m + 120 \, b^{5} c^{4}\right )} x^{6} - 3 \,{\left (a b^{4} c^{4} m^{5} + 16 \, a b^{4} c^{4} m^{4} + 95 \, a b^{4} c^{4} m^{3} + 260 \, a b^{4} c^{4} m^{2} + 324 \, a b^{4} c^{4} m + 144 \, a b^{4} c^{4}\right )} x^{5} + 2 \,{\left (a^{2} b^{3} c^{4} m^{5} + 17 \, a^{2} b^{3} c^{4} m^{4} + 107 \, a^{2} b^{3} c^{4} m^{3} + 307 \, a^{2} b^{3} c^{4} m^{2} + 396 \, a^{2} b^{3} c^{4} m + 180 \, a^{2} b^{3} c^{4}\right )} x^{4} + 2 \,{\left (a^{3} b^{2} c^{4} m^{5} + 18 \, a^{3} b^{2} c^{4} m^{4} + 121 \, a^{3} b^{2} c^{4} m^{3} + 372 \, a^{3} b^{2} c^{4} m^{2} + 508 \, a^{3} b^{2} c^{4} m + 240 \, a^{3} b^{2} c^{4}\right )} x^{3} - 3 \,{\left (a^{4} b c^{4} m^{5} + 19 \, a^{4} b c^{4} m^{4} + 137 \, a^{4} b c^{4} m^{3} + 461 \, a^{4} b c^{4} m^{2} + 702 \, a^{4} b c^{4} m + 360 \, a^{4} b c^{4}\right )} x^{2} +{\left (a^{5} c^{4} m^{5} + 20 \, a^{5} c^{4} m^{4} + 155 \, a^{5} c^{4} m^{3} + 580 \, a^{5} c^{4} m^{2} + 1044 \, a^{5} c^{4} m + 720 \, a^{5} c^{4}\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)*(-b*c*x+a*c)^4,x, algorithm="fricas")
[Out]
((b^5*c^4*m^5 + 15*b^5*c^4*m^4 + 85*b^5*c^4*m^3 + 225*b^5*c^4*m^2 + 274*b^5*c^4*m + 120*b^5*c^4)*x^6 - 3*(a*b^
4*c^4*m^5 + 16*a*b^4*c^4*m^4 + 95*a*b^4*c^4*m^3 + 260*a*b^4*c^4*m^2 + 324*a*b^4*c^4*m + 144*a*b^4*c^4)*x^5 + 2
*(a^2*b^3*c^4*m^5 + 17*a^2*b^3*c^4*m^4 + 107*a^2*b^3*c^4*m^3 + 307*a^2*b^3*c^4*m^2 + 396*a^2*b^3*c^4*m + 180*a
^2*b^3*c^4)*x^4 + 2*(a^3*b^2*c^4*m^5 + 18*a^3*b^2*c^4*m^4 + 121*a^3*b^2*c^4*m^3 + 372*a^3*b^2*c^4*m^2 + 508*a^
3*b^2*c^4*m + 240*a^3*b^2*c^4)*x^3 - 3*(a^4*b*c^4*m^5 + 19*a^4*b*c^4*m^4 + 137*a^4*b*c^4*m^3 + 461*a^4*b*c^4*m
^2 + 702*a^4*b*c^4*m + 360*a^4*b*c^4)*x^2 + (a^5*c^4*m^5 + 20*a^5*c^4*m^4 + 155*a^5*c^4*m^3 + 580*a^5*c^4*m^2
+ 1044*a^5*c^4*m + 720*a^5*c^4)*x)*(e*x)^m/(m^6 + 21*m^5 + 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)
________________________________________________________________________________________
Sympy [A] time = 2.19047, size = 2338, normalized size = 16.12 \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)*(-b*c*x+a*c)**4,x)
[Out]
Piecewise(((-a**5*c**4/(5*x**5) + 3*a**4*b*c**4/(4*x**4) - 2*a**3*b**2*c**4/(3*x**3) - a**2*b**3*c**4/x**2 + 3
*a*b**4*c**4/x + b**5*c**4*log(x))/e**6, Eq(m, -6)), ((-a**5*c**4/(4*x**4) + a**4*b*c**4/x**3 - a**3*b**2*c**4
/x**2 - 2*a**2*b**3*c**4/x - 3*a*b**4*c**4*log(x) + b**5*c**4*x)/e**5, Eq(m, -5)), ((-a**5*c**4/(3*x**3) + 3*a
**4*b*c**4/(2*x**2) - 2*a**3*b**2*c**4/x + 2*a**2*b**3*c**4*log(x) - 3*a*b**4*c**4*x + b**5*c**4*x**2/2)/e**4,
Eq(m, -4)), ((-a**5*c**4/(2*x**2) + 3*a**4*b*c**4/x + 2*a**3*b**2*c**4*log(x) + 2*a**2*b**3*c**4*x - 3*a*b**4
*c**4*x**2/2 + b**5*c**4*x**3/3)/e**3, Eq(m, -3)), ((-a**5*c**4/x - 3*a**4*b*c**4*log(x) + 2*a**3*b**2*c**4*x
+ a**2*b**3*c**4*x**2 - a*b**4*c**4*x**3 + b**5*c**4*x**4/4)/e**2, Eq(m, -2)), ((a**5*c**4*log(x) - 3*a**4*b*c
**4*x + a**3*b**2*c**4*x**2 + 2*a**2*b**3*c**4*x**3/3 - 3*a*b**4*c**4*x**4/4 + b**5*c**4*x**5/5)/e, Eq(m, -1))
, (a**5*c**4*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*c**4
*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*c**4*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*c**4*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*c**4*e**m*m*x*x**m/(m**6 + 21*m**5 +
175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 720*a**5*c**4*e**m*x*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*
m**3 + 1624*m**2 + 1764*m + 720) - 3*a**4*b*c**4*e**m*m**5*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1
624*m**2 + 1764*m + 720) - 57*a**4*b*c**4*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) - 411*a**4*b*c**4*e**m*m**3*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 17
64*m + 720) - 1383*a**4*b*c**4*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) - 2106*a**4*b*c**4*e**m*m*x**2*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) -
1080*a**4*b*c**4*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*c**4*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*c
**4*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*c**4
*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*c**4*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*c**4*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*c**4*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*c**4*e**m*m**5*x**4*x**m/
(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m + 720) + 34*a**2*b**3*c**4*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*c**4*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*c**4*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*c**4*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*c**4*e**m*x**4*x**m/(m**6 + 21*m**5 + 175*m*
*4 + 735*m**3 + 1624*m**2 + 1764*m + 720) - 3*a*b**4*c**4*e**m*m**5*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735
*m**3 + 1624*m**2 + 1764*m + 720) - 48*a*b**4*c**4*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) - 285*a*b**4*c**4*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) - 780*a*b**4*c**4*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) - 972*a*b**4*c**4*e**m*m*x**5*x**m/(m**6 + 21*m**5 + 175*m**4 + 735*m**3 + 1624*m**2 + 1764*m +
720) - 432*a*b**4*c**4*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*c**4*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*c**4*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*c**4*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*c**4*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*c**4*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*c**4*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 = 1.29863, size = 972, normalized size = 6.7 \begin{align*} \frac{b^{5} c^{4} m^{5} x^{6} x^{m} e^{m} - 3 \, a b^{4} c^{4} m^{5} x^{5} x^{m} e^{m} + 15 \, b^{5} c^{4} m^{4} x^{6} x^{m} e^{m} + 2 \, a^{2} b^{3} c^{4} m^{5} x^{4} x^{m} e^{m} - 48 \, a b^{4} c^{4} m^{4} x^{5} x^{m} e^{m} + 85 \, b^{5} c^{4} m^{3} x^{6} x^{m} e^{m} + 2 \, a^{3} b^{2} c^{4} m^{5} x^{3} x^{m} e^{m} + 34 \, a^{2} b^{3} c^{4} m^{4} x^{4} x^{m} e^{m} - 285 \, a b^{4} c^{4} m^{3} x^{5} x^{m} e^{m} + 225 \, b^{5} c^{4} m^{2} x^{6} x^{m} e^{m} - 3 \, a^{4} b c^{4} m^{5} x^{2} x^{m} e^{m} + 36 \, a^{3} b^{2} c^{4} m^{4} x^{3} x^{m} e^{m} + 214 \, a^{2} b^{3} c^{4} m^{3} x^{4} x^{m} e^{m} - 780 \, a b^{4} c^{4} m^{2} x^{5} x^{m} e^{m} + 274 \, b^{5} c^{4} m x^{6} x^{m} e^{m} + a^{5} c^{4} m^{5} x x^{m} e^{m} - 57 \, a^{4} b c^{4} m^{4} x^{2} x^{m} e^{m} + 242 \, a^{3} b^{2} c^{4} m^{3} x^{3} x^{m} e^{m} + 614 \, a^{2} b^{3} c^{4} m^{2} x^{4} x^{m} e^{m} - 972 \, a b^{4} c^{4} m x^{5} x^{m} e^{m} + 120 \, b^{5} c^{4} x^{6} x^{m} e^{m} + 20 \, a^{5} c^{4} m^{4} x x^{m} e^{m} - 411 \, a^{4} b c^{4} m^{3} x^{2} x^{m} e^{m} + 744 \, a^{3} b^{2} c^{4} m^{2} x^{3} x^{m} e^{m} + 792 \, a^{2} b^{3} c^{4} m x^{4} x^{m} e^{m} - 432 \, a b^{4} c^{4} x^{5} x^{m} e^{m} + 155 \, a^{5} c^{4} m^{3} x x^{m} e^{m} - 1383 \, a^{4} b c^{4} m^{2} x^{2} x^{m} e^{m} + 1016 \, a^{3} b^{2} c^{4} m x^{3} x^{m} e^{m} + 360 \, a^{2} b^{3} c^{4} x^{4} x^{m} e^{m} + 580 \, a^{5} c^{4} m^{2} x x^{m} e^{m} - 2106 \, a^{4} b c^{4} m x^{2} x^{m} e^{m} + 480 \, a^{3} b^{2} c^{4} x^{3} x^{m} e^{m} + 1044 \, a^{5} c^{4} m x x^{m} e^{m} - 1080 \, a^{4} b c^{4} x^{2} x^{m} e^{m} + 720 \, a^{5} c^{4} 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)*(-b*c*x+a*c)^4,x, algorithm="giac")
[Out]
(b^5*c^4*m^5*x^6*x^m*e^m - 3*a*b^4*c^4*m^5*x^5*x^m*e^m + 15*b^5*c^4*m^4*x^6*x^m*e^m + 2*a^2*b^3*c^4*m^5*x^4*x^
m*e^m - 48*a*b^4*c^4*m^4*x^5*x^m*e^m + 85*b^5*c^4*m^3*x^6*x^m*e^m + 2*a^3*b^2*c^4*m^5*x^3*x^m*e^m + 34*a^2*b^3
*c^4*m^4*x^4*x^m*e^m - 285*a*b^4*c^4*m^3*x^5*x^m*e^m + 225*b^5*c^4*m^2*x^6*x^m*e^m - 3*a^4*b*c^4*m^5*x^2*x^m*e
^m + 36*a^3*b^2*c^4*m^4*x^3*x^m*e^m + 214*a^2*b^3*c^4*m^3*x^4*x^m*e^m - 780*a*b^4*c^4*m^2*x^5*x^m*e^m + 274*b^
5*c^4*m*x^6*x^m*e^m + a^5*c^4*m^5*x*x^m*e^m - 57*a^4*b*c^4*m^4*x^2*x^m*e^m + 242*a^3*b^2*c^4*m^3*x^3*x^m*e^m +
614*a^2*b^3*c^4*m^2*x^4*x^m*e^m - 972*a*b^4*c^4*m*x^5*x^m*e^m + 120*b^5*c^4*x^6*x^m*e^m + 20*a^5*c^4*m^4*x*x^
m*e^m - 411*a^4*b*c^4*m^3*x^2*x^m*e^m + 744*a^3*b^2*c^4*m^2*x^3*x^m*e^m + 792*a^2*b^3*c^4*m*x^4*x^m*e^m - 432*
a*b^4*c^4*x^5*x^m*e^m + 155*a^5*c^4*m^3*x*x^m*e^m - 1383*a^4*b*c^4*m^2*x^2*x^m*e^m + 1016*a^3*b^2*c^4*m*x^3*x^
m*e^m + 360*a^2*b^3*c^4*x^4*x^m*e^m + 580*a^5*c^4*m^2*x*x^m*e^m - 2106*a^4*b*c^4*m*x^2*x^m*e^m + 480*a^3*b^2*c
^4*x^3*x^m*e^m + 1044*a^5*c^4*m*x*x^m*e^m - 1080*a^4*b*c^4*x^2*x^m*e^m + 720*a^5*c^4*x*x^m*e^m)/(m^6 + 21*m^5
+ 175*m^4 + 735*m^3 + 1624*m^2 + 1764*m + 720)