3.2350 \(\int x (d+e x)^m (a+b x+c x^2) \, dx\)

Optimal. Leaf size=121 \[ -\frac{d (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}+\frac{(d+e x)^{m+2} \left (3 c d^2-e (2 b d-a e)\right )}{e^4 (m+2)}-\frac{(3 c d-b e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac{c (d+e x)^{m+4}}{e^4 (m+4)} \]

[Out]

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

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Rubi [A]  time = 0.0813408, antiderivative size = 121, normalized size of antiderivative = 1., 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 = {771} \[ -\frac{d (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}+\frac{(d+e x)^{m+2} \left (3 c d^2-e (2 b d-a e)\right )}{e^4 (m+2)}-\frac{(3 c d-b e) (d+e x)^{m+3}}{e^4 (m+3)}+\frac{c (d+e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

Mathematica [A]  time = 0.264268, size = 142, normalized size = 1.17 \[ \frac{(d+e x)^{m+1} \left (e (m+4) \left (a e (m+3) (e (m+1) x-d)+b \left (2 d^2-2 d e (m+1) x+e^2 \left (m^2+3 m+2\right ) x^2\right )\right )+c \left (6 d^2 e (m+1) x-6 d^3-3 d e^2 \left (m^2+3 m+2\right ) x^2+e^3 \left (m^3+6 m^2+11 m+6\right ) x^3\right )\right )}{e^4 (m+1) (m+2) (m+3) (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((d + e*x)^(1 + m)*(c*(-6*d^3 + 6*d^2*e*(1 + m)*x - 3*d*e^2*(2 + 3*m + m^2)*x^2 + e^3*(6 + 11*m + 6*m^2 + m^3)
*x^3) + e*(4 + m)*(a*e*(3 + m)*(-d + e*(1 + m)*x) + b*(2*d^2 - 2*d*e*(1 + m)*x + e^2*(2 + 3*m + m^2)*x^2))))/(
e^4*(1 + m)*(2 + m)*(3 + m)*(4 + m))

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Maple [B]  time = 0.006, size = 281, normalized size = 2.3 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{1+m} \left ( -c{e}^{3}{m}^{3}{x}^{3}-b{e}^{3}{m}^{3}{x}^{2}-6\,c{e}^{3}{m}^{2}{x}^{3}-a{e}^{3}{m}^{3}x-7\,b{e}^{3}{m}^{2}{x}^{2}+3\,cd{e}^{2}{m}^{2}{x}^{2}-11\,c{e}^{3}m{x}^{3}-8\,a{e}^{3}{m}^{2}x+2\,bd{e}^{2}{m}^{2}x-14\,b{e}^{3}m{x}^{2}+9\,cd{e}^{2}m{x}^{2}-6\,c{x}^{3}{e}^{3}+ad{e}^{2}{m}^{2}-19\,a{e}^{3}mx+10\,bd{e}^{2}mx-8\,b{e}^{3}{x}^{2}-6\,c{d}^{2}emx+6\,cd{e}^{2}{x}^{2}+7\,ad{e}^{2}m-12\,a{e}^{3}x-2\,b{d}^{2}em+8\,bd{e}^{2}x-6\,c{d}^{2}ex+12\,ad{e}^{2}-8\,b{d}^{2}e+6\,c{d}^{3} \right ) }{{e}^{4} \left ({m}^{4}+10\,{m}^{3}+35\,{m}^{2}+50\,m+24 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(e*x+d)^(1+m)*(-c*e^3*m^3*x^3-b*e^3*m^3*x^2-6*c*e^3*m^2*x^3-a*e^3*m^3*x-7*b*e^3*m^2*x^2+3*c*d*e^2*m^2*x^2-11*
c*e^3*m*x^3-8*a*e^3*m^2*x+2*b*d*e^2*m^2*x-14*b*e^3*m*x^2+9*c*d*e^2*m*x^2-6*c*e^3*x^3+a*d*e^2*m^2-19*a*e^3*m*x+
10*b*d*e^2*m*x-8*b*e^3*x^2-6*c*d^2*e*m*x+6*c*d*e^2*x^2+7*a*d*e^2*m-12*a*e^3*x-2*b*d^2*e*m+8*b*d*e^2*x-6*c*d^2*
e*x+12*a*d*e^2-8*b*d^2*e+6*c*d^3)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

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Maxima [A]  time = 1.03889, size = 290, normalized size = 2.4 \begin{align*} \frac{{\left (e^{2}{\left (m + 1\right )} x^{2} + d e m x - d^{2}\right )}{\left (e x + d\right )}^{m} a}{{\left (m^{2} + 3 \, m + 2\right )} e^{2}} + \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}{{\left (m^{3} + 6 \, m^{2} + 11 \, m + 6\right )} e^{3}} + \frac{{\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} c}{{\left (m^{4} + 10 \, m^{3} + 35 \, m^{2} + 50 \, m + 24\right )} e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B]  time = 1.38795, size = 702, normalized size = 5.8 \begin{align*} -\frac{{\left (a d^{2} e^{2} m^{2} + 6 \, c d^{4} - 8 \, b d^{3} e + 12 \, a d^{2} e^{2} -{\left (c e^{4} m^{3} + 6 \, c e^{4} m^{2} + 11 \, c e^{4} m + 6 \, c e^{4}\right )} x^{4} -{\left (8 \, b e^{4} +{\left (c d e^{3} + b e^{4}\right )} m^{3} +{\left (3 \, c d e^{3} + 7 \, b e^{4}\right )} m^{2} + 2 \,{\left (c d e^{3} + 7 \, b e^{4}\right )} m\right )} x^{3} -{\left (12 \, a e^{4} +{\left (b d e^{3} + a e^{4}\right )} m^{3} -{\left (3 \, c d^{2} e^{2} - 5 \, b d e^{3} - 8 \, a e^{4}\right )} m^{2} -{\left (3 \, c d^{2} e^{2} - 4 \, b d e^{3} - 19 \, a e^{4}\right )} m\right )} x^{2} -{\left (2 \, b d^{3} e - 7 \, a d^{2} e^{2}\right )} m -{\left (a d e^{3} m^{3} -{\left (2 \, b d^{2} e^{2} - 7 \, a d e^{3}\right )} m^{2} + 2 \,{\left (3 \, c d^{3} e - 4 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} m\right )} x\right )}{\left (e x + d\right )}^{m}}{e^{4} m^{4} + 10 \, e^{4} m^{3} + 35 \, e^{4} m^{2} + 50 \, e^{4} m + 24 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a*d^2*e^2*m^2 + 6*c*d^4 - 8*b*d^3*e + 12*a*d^2*e^2 - (c*e^4*m^3 + 6*c*e^4*m^2 + 11*c*e^4*m + 6*c*e^4)*x^4 -
(8*b*e^4 + (c*d*e^3 + b*e^4)*m^3 + (3*c*d*e^3 + 7*b*e^4)*m^2 + 2*(c*d*e^3 + 7*b*e^4)*m)*x^3 - (12*a*e^4 + (b*d
*e^3 + a*e^4)*m^3 - (3*c*d^2*e^2 - 5*b*d*e^3 - 8*a*e^4)*m^2 - (3*c*d^2*e^2 - 4*b*d*e^3 - 19*a*e^4)*m)*x^2 - (2
*b*d^3*e - 7*a*d^2*e^2)*m - (a*d*e^3*m^3 - (2*b*d^2*e^2 - 7*a*d*e^3)*m^2 + 2*(3*c*d^3*e - 4*b*d^2*e^2 + 6*a*d*
e^3)*m)*x)*(e*x + d)^m/(e^4*m^4 + 10*e^4*m^3 + 35*e^4*m^2 + 50*e^4*m + 24*e^4)

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Sympy [A]  time = 3.48243, size = 3267, normalized size = 27. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(a*x**2/2 + b*x**3/3 + c*x**4/4), Eq(e, 0)), (-a*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*a*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*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) - 6*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) - 6*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*c*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*c*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*c*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*c*d**2*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*c*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*c*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*c*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*d*e**2/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) -
2*a*e**3*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*b*d**2*e*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e
**6*x**2) + 3*b*d**2*e/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*b*d*e**2*x*log(d/e + x)/(2*d**2*e**4 + 4*d
*e**5*x + 2*e**6*x**2) + 4*b*d*e**2*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*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*c*d**3*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 9*c*
d**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*c*d**2*e*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*
x**2) - 12*c*d**2*e*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c*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*c*e**3*x**3/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2), Eq(m, -3)), (2*a*d*e**2*log
(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*a*d*e**2/(2*d*e**4 + 2*e**5*x) + 2*a*e**3*x*log(d/e + x)/(2*d*e**4 + 2*e**
5*x) - 4*b*d**2*e*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*b*d**2*e/(2*d*e**4 + 2*e**5*x) - 4*b*d*e**2*x*log(d/e
+ x)/(2*d*e**4 + 2*e**5*x) + 2*b*e**3*x**2/(2*d*e**4 + 2*e**5*x) + 6*c*d**3*log(d/e + x)/(2*d*e**4 + 2*e**5*x
) + 6*c*d**3/(2*d*e**4 + 2*e**5*x) + 6*c*d**2*e*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 3*c*d*e**2*x**2/(2*d*e*
*4 + 2*e**5*x) + c*e**3*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (-a*d*log(d/e + x)/e**2 + a*x/e + b*d**2*log(d
/e + x)/e**3 - b*d*x/e**2 + b*x**2/(2*e) - c*d**3*log(d/e + x)/e**4 + c*d**2*x/e**3 - c*d*x**2/(2*e**2) + c*x*
*3/(3*e), Eq(m, -1)), (-a*d**2*e**2*m**2*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 2
4*e**4) - 7*a*d**2*e**2*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) - 12*a*
d**2*e**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) + a*d*e**3*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) + 7*a*d*e**3*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) + 12*a*d*e**3*m*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) + a*e**4*m**3*x**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) + 8*a*e**4*m**2*x**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) + 19*a*e**4*m*x**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) + 12*a*e**4*x**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) + 2*b*d**3*e
*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) + 8*b*d**3*e*(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) - 2*b*d**2*e**2*m**2*x*(d + e*x)**m/(e**4*m**4 + 1
0*e**4*m**3 + 35*e**4*m**2 + 50*e**4*m + 24*e**4) - 8*b*d**2*e**2*m*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) + b*d*e**3*m**3*x**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) + 5*b*d*e**3*m**2*x**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) + 4*b*d*e**3*m*x**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
) + b*e**4*m**3*x**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) + 7*b*e**4*m
**2*x**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) + 14*b*e**4*m*x**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) + 8*b*e**4*x**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) - 6*c*d**4*(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) + 6*c*d**3*e*m*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) - 3*c*d**2*e**2*m**2*x**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) - 3*c*d**2*e**2*m*x**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)
+ c*d*e**3*m**3*x**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) + 3*c*d*e**3
*m**2*x**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) + 2*c*d*e**3*m*x**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) + c*e**4*m**3*x**4*(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) + 6*c*e**4*m**2*x**4*(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) + 11*c*e**4*m*x**4*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 3
5*e**4*m**2 + 50*e**4*m + 24*e**4) + 6*c*e**4*x**4*(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), True))

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Giac [B]  time = 1.16993, size = 818, normalized size = 6.76 \begin{align*} \frac{{\left (x e + d\right )}^{m} c m^{3} x^{4} e^{4} +{\left (x e + d\right )}^{m} c d m^{3} x^{3} e^{3} +{\left (x e + d\right )}^{m} b m^{3} x^{3} e^{4} + 6 \,{\left (x e + d\right )}^{m} c m^{2} x^{4} e^{4} +{\left (x e + d\right )}^{m} b d m^{3} x^{2} e^{3} + 3 \,{\left (x e + d\right )}^{m} c d m^{2} x^{3} e^{3} - 3 \,{\left (x e + d\right )}^{m} c d^{2} m^{2} x^{2} e^{2} +{\left (x e + d\right )}^{m} a m^{3} x^{2} e^{4} + 7 \,{\left (x e + d\right )}^{m} b m^{2} x^{3} e^{4} + 11 \,{\left (x e + d\right )}^{m} c m x^{4} e^{4} +{\left (x e + d\right )}^{m} a d m^{3} x e^{3} + 5 \,{\left (x e + d\right )}^{m} b d m^{2} x^{2} e^{3} + 2 \,{\left (x e + d\right )}^{m} c d m x^{3} e^{3} - 2 \,{\left (x e + d\right )}^{m} b d^{2} m^{2} x e^{2} - 3 \,{\left (x e + d\right )}^{m} c d^{2} m x^{2} e^{2} + 6 \,{\left (x e + d\right )}^{m} c d^{3} m x e + 8 \,{\left (x e + d\right )}^{m} a m^{2} x^{2} e^{4} + 14 \,{\left (x e + d\right )}^{m} b m x^{3} e^{4} + 6 \,{\left (x e + d\right )}^{m} c x^{4} e^{4} + 7 \,{\left (x e + d\right )}^{m} a d m^{2} x e^{3} + 4 \,{\left (x e + d\right )}^{m} b d m x^{2} e^{3} -{\left (x e + d\right )}^{m} a d^{2} m^{2} e^{2} - 8 \,{\left (x e + d\right )}^{m} b d^{2} m x e^{2} + 2 \,{\left (x e + d\right )}^{m} b d^{3} m e - 6 \,{\left (x e + d\right )}^{m} c d^{4} + 19 \,{\left (x e + d\right )}^{m} a m x^{2} e^{4} + 8 \,{\left (x e + d\right )}^{m} b x^{3} e^{4} + 12 \,{\left (x e + d\right )}^{m} a d m x e^{3} - 7 \,{\left (x e + d\right )}^{m} a d^{2} m e^{2} + 8 \,{\left (x e + d\right )}^{m} b d^{3} e + 12 \,{\left (x e + d\right )}^{m} a x^{2} e^{4} - 12 \,{\left (x e + d\right )}^{m} a d^{2} e^{2}}{m^{4} e^{4} + 10 \, m^{3} e^{4} + 35 \, m^{2} e^{4} + 50 \, m e^{4} + 24 \, e^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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