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

Optimal. Leaf size=144 \[ \frac{(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{(d+e x)^{m+2} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^4 (m+2)}+\frac{(d+e x)^{m+3} (b e g-3 c d g+c e f)}{e^4 (m+3)}+\frac{c g (d+e x)^{m+4}}{e^4 (m+4)} \]

[Out]

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

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Rubi [A]  time = 0.113269, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {771} \[ \frac{(e f-d g) (d+e x)^{m+1} \left (a e^2-b d e+c d^2\right )}{e^4 (m+1)}-\frac{(d+e x)^{m+2} (c d (2 e f-3 d g)-e (a e g-2 b d g+b e f))}{e^4 (m+2)}+\frac{(d+e x)^{m+3} (b e g-3 c d g+c e f)}{e^4 (m+3)}+\frac{c g (d+e x)^{m+4}}{e^4 (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.352825, size = 180, normalized size = 1.25 \[ \frac{(d+e x)^{m+1} \left (\frac{(d+e x) \left (c e (2 a e g (m+3)+b d g (m-2)+b e f (m+4))-b^2 e^2 g (m+2)+2 c^2 d (3 d g-e f (m+4))\right )}{e^2 (m+2)}-\frac{\left (e (a e-b d)+c d^2\right ) (b e g (m+1)+6 c d g-2 c e f (m+4))}{e^2 (m+1)}+(a+x (b+c x)) (b e g+c (-3 d g+e f (m+4)+e g (m+3) x))\right )}{c e^2 (m+3) (m+4)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.052, size = 503, normalized size = 3.5 \begin{align*} -{\frac{ \left ( ex+d \right ) ^{1+m} \left ( -c{e}^{3}g{m}^{3}{x}^{3}-b{e}^{3}g{m}^{3}{x}^{2}-c{e}^{3}f{m}^{3}{x}^{2}-6\,c{e}^{3}g{m}^{2}{x}^{3}-a{e}^{3}g{m}^{3}x-b{e}^{3}f{m}^{3}x-7\,b{e}^{3}g{m}^{2}{x}^{2}+3\,cd{e}^{2}g{m}^{2}{x}^{2}-7\,c{e}^{3}f{m}^{2}{x}^{2}-11\,c{e}^{3}gm{x}^{3}-a{e}^{3}f{m}^{3}-8\,a{e}^{3}g{m}^{2}x+2\,bd{e}^{2}g{m}^{2}x-8\,b{e}^{3}f{m}^{2}x-14\,b{e}^{3}gm{x}^{2}+2\,cd{e}^{2}f{m}^{2}x+9\,cd{e}^{2}gm{x}^{2}-14\,c{e}^{3}fm{x}^{2}-6\,cg{x}^{3}{e}^{3}+ad{e}^{2}g{m}^{2}-9\,a{e}^{3}f{m}^{2}-19\,a{e}^{3}gmx+bd{e}^{2}f{m}^{2}+10\,bd{e}^{2}gmx-19\,b{e}^{3}fmx-8\,b{e}^{3}g{x}^{2}-6\,c{d}^{2}egmx+10\,cd{e}^{2}fmx+6\,cd{e}^{2}g{x}^{2}-8\,c{e}^{3}f{x}^{2}+7\,ad{e}^{2}gm-26\,a{e}^{3}fm-12\,a{e}^{3}gx-2\,b{d}^{2}egm+7\,bd{e}^{2}fm+8\,bd{e}^{2}gx-12\,b{e}^{3}fx-2\,c{d}^{2}efm-6\,c{d}^{2}egx+8\,cd{e}^{2}fx+12\,ad{e}^{2}g-24\,a{e}^{3}f-8\,b{d}^{2}eg+12\,bd{e}^{2}f+6\,c{d}^{3}g-8\,c{d}^{2}ef \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((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a),x)

[Out]

-(e*x+d)^(1+m)*(-c*e^3*g*m^3*x^3-b*e^3*g*m^3*x^2-c*e^3*f*m^3*x^2-6*c*e^3*g*m^2*x^3-a*e^3*g*m^3*x-b*e^3*f*m^3*x
-7*b*e^3*g*m^2*x^2+3*c*d*e^2*g*m^2*x^2-7*c*e^3*f*m^2*x^2-11*c*e^3*g*m*x^3-a*e^3*f*m^3-8*a*e^3*g*m^2*x+2*b*d*e^
2*g*m^2*x-8*b*e^3*f*m^2*x-14*b*e^3*g*m*x^2+2*c*d*e^2*f*m^2*x+9*c*d*e^2*g*m*x^2-14*c*e^3*f*m*x^2-6*c*e^3*g*x^3+
a*d*e^2*g*m^2-9*a*e^3*f*m^2-19*a*e^3*g*m*x+b*d*e^2*f*m^2+10*b*d*e^2*g*m*x-19*b*e^3*f*m*x-8*b*e^3*g*x^2-6*c*d^2
*e*g*m*x+10*c*d*e^2*f*m*x+6*c*d*e^2*g*x^2-8*c*e^3*f*x^2+7*a*d*e^2*g*m-26*a*e^3*f*m-12*a*e^3*g*x-2*b*d^2*e*g*m+
7*b*d*e^2*f*m+8*b*d*e^2*g*x-12*b*e^3*f*x-2*c*d^2*e*f*m-6*c*d^2*e*g*x+8*c*d*e^2*f*x+12*a*d*e^2*g-24*a*e^3*f-8*b
*d^2*e*g+12*b*d*e^2*f+6*c*d^3*g-8*c*d^2*e*f)/e^4/(m^4+10*m^3+35*m^2+50*m+24)

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.01193, size = 1297, normalized size = 9.01 \begin{align*} \frac{{\left (a d e^{3} f m^{3} +{\left (c e^{4} g m^{3} + 6 \, c e^{4} g m^{2} + 11 \, c e^{4} g m + 6 \, c e^{4} g\right )} x^{4} +{\left (8 \, c e^{4} f + 8 \, b e^{4} g +{\left (c e^{4} f +{\left (c d e^{3} + b e^{4}\right )} g\right )} m^{3} +{\left (7 \, c e^{4} f +{\left (3 \, c d e^{3} + 7 \, b e^{4}\right )} g\right )} m^{2} + 2 \,{\left (7 \, c e^{4} f +{\left (c d e^{3} + 7 \, b e^{4}\right )} g\right )} m\right )} x^{3} -{\left (a d^{2} e^{2} g +{\left (b d^{2} e^{2} - 9 \, a d e^{3}\right )} f\right )} m^{2} +{\left (12 \, b e^{4} f + 12 \, a e^{4} g +{\left ({\left (c d e^{3} + b e^{4}\right )} f +{\left (b d e^{3} + a e^{4}\right )} g\right )} m^{3} +{\left ({\left (5 \, c d e^{3} + 8 \, b e^{4}\right )} f -{\left (3 \, c d^{2} e^{2} - 5 \, b d e^{3} - 8 \, a e^{4}\right )} g\right )} m^{2} +{\left ({\left (4 \, c d e^{3} + 19 \, b e^{4}\right )} f -{\left (3 \, c d^{2} e^{2} - 4 \, b d e^{3} - 19 \, a e^{4}\right )} g\right )} m\right )} x^{2} + 4 \,{\left (2 \, c d^{3} e - 3 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} f - 2 \,{\left (3 \, c d^{4} - 4 \, b d^{3} e + 6 \, a d^{2} e^{2}\right )} g +{\left ({\left (2 \, c d^{3} e - 7 \, b d^{2} e^{2} + 26 \, a d e^{3}\right )} f +{\left (2 \, b d^{3} e - 7 \, a d^{2} e^{2}\right )} g\right )} m +{\left (24 \, a e^{4} f +{\left (a d e^{3} g +{\left (b d e^{3} + a e^{4}\right )} f\right )} m^{3} -{\left ({\left (2 \, c d^{2} e^{2} - 7 \, b d e^{3} - 9 \, a e^{4}\right )} f +{\left (2 \, b d^{2} e^{2} - 7 \, a d e^{3}\right )} g\right )} m^{2} - 2 \,{\left ({\left (4 \, c d^{2} e^{2} - 6 \, b d e^{3} - 13 \, a e^{4}\right )} f -{\left (3 \, c d^{3} e - 4 \, b d^{2} e^{2} + 6 \, a d e^{3}\right )} g\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((e*x+d)^m*(g*x+f)*(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

(a*d*e^3*f*m^3 + (c*e^4*g*m^3 + 6*c*e^4*g*m^2 + 11*c*e^4*g*m + 6*c*e^4*g)*x^4 + (8*c*e^4*f + 8*b*e^4*g + (c*e^
4*f + (c*d*e^3 + b*e^4)*g)*m^3 + (7*c*e^4*f + (3*c*d*e^3 + 7*b*e^4)*g)*m^2 + 2*(7*c*e^4*f + (c*d*e^3 + 7*b*e^4
)*g)*m)*x^3 - (a*d^2*e^2*g + (b*d^2*e^2 - 9*a*d*e^3)*f)*m^2 + (12*b*e^4*f + 12*a*e^4*g + ((c*d*e^3 + b*e^4)*f
+ (b*d*e^3 + a*e^4)*g)*m^3 + ((5*c*d*e^3 + 8*b*e^4)*f - (3*c*d^2*e^2 - 5*b*d*e^3 - 8*a*e^4)*g)*m^2 + ((4*c*d*e
^3 + 19*b*e^4)*f - (3*c*d^2*e^2 - 4*b*d*e^3 - 19*a*e^4)*g)*m)*x^2 + 4*(2*c*d^3*e - 3*b*d^2*e^2 + 6*a*d*e^3)*f
- 2*(3*c*d^4 - 4*b*d^3*e + 6*a*d^2*e^2)*g + ((2*c*d^3*e - 7*b*d^2*e^2 + 26*a*d*e^3)*f + (2*b*d^3*e - 7*a*d^2*e
^2)*g)*m + (24*a*e^4*f + (a*d*e^3*g + (b*d*e^3 + a*e^4)*f)*m^3 - ((2*c*d^2*e^2 - 7*b*d*e^3 - 9*a*e^4)*f + (2*b
*d^2*e^2 - 7*a*d*e^3)*g)*m^2 - 2*((4*c*d^2*e^2 - 6*b*d*e^3 - 13*a*e^4)*f - (3*c*d^3*e - 4*b*d^2*e^2 + 6*a*d*e^
3)*g)*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 = 6.69103, size = 5795, normalized size = 40.24 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((d**m*(a*f*x + a*g*x**2/2 + b*f*x**2/2 + b*g*x**3/3 + c*f*x**3/3 + c*g*x**4/4), Eq(e, 0)), (-a*d**2*
e**2*g/(6*d**4*e**4 + 18*d**3*e**5*x + 18*d**2*e**6*x**2 + 6*d*e**7*x**3) - 2*a*d*e**3*f/(6*d**4*e**4 + 18*d**
3*e**5*x + 18*d**2*e**6*x**2 + 6*d*e**7*x**3) - 3*a*d*e**3*g*x/(6*d**4*e**4 + 18*d**3*e**5*x + 18*d**2*e**6*x*
*2 + 6*d*e**7*x**3) - b*d**2*e**2*f/(6*d**4*e**4 + 18*d**3*e**5*x + 18*d**2*e**6*x**2 + 6*d*e**7*x**3) - 3*b*d
*e**3*f*x/(6*d**4*e**4 + 18*d**3*e**5*x + 18*d**2*e**6*x**2 + 6*d*e**7*x**3) + 2*b*e**4*g*x**3/(6*d**4*e**4 +
18*d**3*e**5*x + 18*d**2*e**6*x**2 + 6*d*e**7*x**3) + 6*c*d**4*g*log(d/e + x)/(6*d**4*e**4 + 18*d**3*e**5*x +
18*d**2*e**6*x**2 + 6*d*e**7*x**3) + 5*c*d**4*g/(6*d**4*e**4 + 18*d**3*e**5*x + 18*d**2*e**6*x**2 + 6*d*e**7*x
**3) + 18*c*d**3*e*g*x*log(d/e + x)/(6*d**4*e**4 + 18*d**3*e**5*x + 18*d**2*e**6*x**2 + 6*d*e**7*x**3) + 9*c*d
**3*e*g*x/(6*d**4*e**4 + 18*d**3*e**5*x + 18*d**2*e**6*x**2 + 6*d*e**7*x**3) + 18*c*d**2*e**2*g*x**2*log(d/e +
x)/(6*d**4*e**4 + 18*d**3*e**5*x + 18*d**2*e**6*x**2 + 6*d*e**7*x**3) + 6*c*d*e**3*g*x**3*log(d/e + x)/(6*d**
4*e**4 + 18*d**3*e**5*x + 18*d**2*e**6*x**2 + 6*d*e**7*x**3) - 6*c*d*e**3*g*x**3/(6*d**4*e**4 + 18*d**3*e**5*x
+ 18*d**2*e**6*x**2 + 6*d*e**7*x**3) + 2*c*e**4*f*x**3/(6*d**4*e**4 + 18*d**3*e**5*x + 18*d**2*e**6*x**2 + 6*
d*e**7*x**3), Eq(m, -4)), (-a*d*e**2*g/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - a*e**3*f/(2*d**2*e**4 + 4*d*
e**5*x + 2*e**6*x**2) - 2*a*e**3*g*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*b*d**2*e*g*log(d/e + x)/(2*d
**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*b*d**2*e*g/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - b*d*e**2*f/(2*d
**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*b*d*e**2*g*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*g*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 2*b*e**3*f*x/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2
) + 2*b*e**3*g*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*g*log(d/e + x)/(2*d**2*e*
*4 + 4*d*e**5*x + 2*e**6*x**2) - 9*c*d**3*g/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 2*c*d**2*e*f*log(d/e +
x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 3*c*d**2*e*f/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) - 12*c*d**
2*e*g*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*g*x/(2*d**2*e**4 + 4*d*e**5*x + 2*
e**6*x**2) + 4*c*d*e**2*f*x*log(d/e + x)/(2*d**2*e**4 + 4*d*e**5*x + 2*e**6*x**2) + 4*c*d*e**2*f*x/(2*d**2*e**
4 + 4*d*e**5*x + 2*e**6*x**2) - 6*c*d*e**2*g*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*f*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*g*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*g*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*a*d*e**2*g/(2*d*e**4 + 2*e**5
*x) - 2*a*e**3*f/(2*d*e**4 + 2*e**5*x) + 2*a*e**3*g*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*b*d**2*e*g*log(d/
e + x)/(2*d*e**4 + 2*e**5*x) - 4*b*d**2*e*g/(2*d*e**4 + 2*e**5*x) + 2*b*d*e**2*f*log(d/e + x)/(2*d*e**4 + 2*e*
*5*x) + 2*b*d*e**2*f/(2*d*e**4 + 2*e**5*x) - 4*b*d*e**2*g*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*b*e**3*f*x*
log(d/e + x)/(2*d*e**4 + 2*e**5*x) + 2*b*e**3*g*x**2/(2*d*e**4 + 2*e**5*x) + 6*c*d**3*g*log(d/e + x)/(2*d*e**4
+ 2*e**5*x) + 6*c*d**3*g/(2*d*e**4 + 2*e**5*x) - 4*c*d**2*e*f*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*c*d**2*e
*f/(2*d*e**4 + 2*e**5*x) + 6*c*d**2*e*g*x*log(d/e + x)/(2*d*e**4 + 2*e**5*x) - 4*c*d*e**2*f*x*log(d/e + x)/(2*
d*e**4 + 2*e**5*x) - 3*c*d*e**2*g*x**2/(2*d*e**4 + 2*e**5*x) + 2*c*e**3*f*x**2/(2*d*e**4 + 2*e**5*x) + c*e**3*
g*x**3/(2*d*e**4 + 2*e**5*x), Eq(m, -2)), (-a*d*g*log(d/e + x)/e**2 + a*f*log(d/e + x)/e + a*g*x/e + b*d**2*g*
log(d/e + x)/e**3 - b*d*f*log(d/e + x)/e**2 - b*d*g*x/e**2 + b*f*x/e + b*g*x**2/(2*e) - c*d**3*g*log(d/e + x)/
e**4 + c*d**2*f*log(d/e + x)/e**3 + c*d**2*g*x/e**3 - c*d*f*x/e**2 - c*d*g*x**2/(2*e**2) + c*f*x**2/(2*e) + c*
g*x**3/(3*e), Eq(m, -1)), (-a*d**2*e**2*g*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) - 7*a*d**2*e**2*g*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*g*(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*f*
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*d*e**3*f*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*d*e**3*f*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*d*e**3*f*(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*g*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*g*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*g*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*f*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*e**4*f*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*e**4*f*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) + 24*a*e**4*f*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) + a*e**4*g*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*g*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*g*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*g*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*g*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*g
*(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**2*e**2*f*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) - 7*b*d**2*e**2*f*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*b*d**2*e**2*f*(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*g*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) - 8*b*d**2*e**2*g*m*x*(d + e*x)**m/(e**4*m**4 + 10*e**4*m**3 + 35*e**4*m**2 + 5
0*e**4*m + 24*e**4) + b*d*e**3*f*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*b*d*e**3*f*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
*b*d*e**3*f*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*g*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*g*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*g*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*f*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*b*e**4*f*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*b*e**4*f*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*b*e**4*f*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*g*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*g*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*g*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*g*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*g*(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**3*e*f*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*c*d**3*e*f*(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*g*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) - 2*c*d**2*e**2*f*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)
- 8*c*d**2*e**2*f*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*g*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
*g*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*f*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*c*d*e**3*f*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*c*d*e**3*f*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*g*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*g*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*g*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*f*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*c*e**4*f*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*c*e**4*f*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*c
*e**4*f*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*g*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*g*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*g*m*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*g*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))

________________________________________________________________________________________

Giac [B]  time = 1.16435, size = 1569, normalized size = 10.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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