3.805 \(\int (d+e x)^3 (f+g x)^n \left (a+2 c d x+c e x^2\right ) \, dx\)

Optimal. Leaf size=275 \[ \frac{(e f-d g)^2 (f+g x)^{n+2} \left (3 a e g^2+c \left (2 d^2 g^2-10 d e f g+5 e^2 f^2\right )\right )}{g^6 (n+2)}-\frac{e (e f-d g) (f+g x)^{n+3} \left (3 a e g^2+c \left (7 d^2 g^2-20 d e f g+10 e^2 f^2\right )\right )}{g^6 (n+3)}+\frac{e^2 (f+g x)^{n+4} \left (a e g^2+c \left (9 d^2 g^2-20 d e f g+10 e^2 f^2\right )\right )}{g^6 (n+4)}-\frac{(e f-d g)^3 (f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^6 (n+1)}-\frac{5 c e^3 (e f-d g) (f+g x)^{n+5}}{g^6 (n+5)}+\frac{c e^4 (f+g x)^{n+6}}{g^6 (n+6)} \]

[Out]

-(((e*f - d*g)^3*(a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^6*(1 + n))) +
((e*f - d*g)^2*(3*a*e*g^2 + c*(5*e^2*f^2 - 10*d*e*f*g + 2*d^2*g^2))*(f + g*x)^(
2 + n))/(g^6*(2 + n)) - (e*(e*f - d*g)*(3*a*e*g^2 + c*(10*e^2*f^2 - 20*d*e*f*g +
7*d^2*g^2))*(f + g*x)^(3 + n))/(g^6*(3 + n)) + (e^2*(a*e*g^2 + c*(10*e^2*f^2 -
20*d*e*f*g + 9*d^2*g^2))*(f + g*x)^(4 + n))/(g^6*(4 + n)) - (5*c*e^3*(e*f - d*g)
*(f + g*x)^(5 + n))/(g^6*(5 + n)) + (c*e^4*(f + g*x)^(6 + n))/(g^6*(6 + n))

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Rubi [A]  time = 0.568992, antiderivative size = 275, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.036 \[ \frac{(e f-d g)^2 (f+g x)^{n+2} \left (3 a e g^2+c \left (2 d^2 g^2-10 d e f g+5 e^2 f^2\right )\right )}{g^6 (n+2)}-\frac{e (e f-d g) (f+g x)^{n+3} \left (3 a e g^2+c \left (7 d^2 g^2-20 d e f g+10 e^2 f^2\right )\right )}{g^6 (n+3)}+\frac{e^2 (f+g x)^{n+4} \left (a e g^2+c \left (9 d^2 g^2-20 d e f g+10 e^2 f^2\right )\right )}{g^6 (n+4)}-\frac{(e f-d g)^3 (f+g x)^{n+1} \left (a g^2+c f (e f-2 d g)\right )}{g^6 (n+1)}-\frac{5 c e^3 (e f-d g) (f+g x)^{n+5}}{g^6 (n+5)}+\frac{c e^4 (f+g x)^{n+6}}{g^6 (n+6)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^3*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]

[Out]

-(((e*f - d*g)^3*(a*g^2 + c*f*(e*f - 2*d*g))*(f + g*x)^(1 + n))/(g^6*(1 + n))) +
((e*f - d*g)^2*(3*a*e*g^2 + c*(5*e^2*f^2 - 10*d*e*f*g + 2*d^2*g^2))*(f + g*x)^(
2 + n))/(g^6*(2 + n)) - (e*(e*f - d*g)*(3*a*e*g^2 + c*(10*e^2*f^2 - 20*d*e*f*g +
7*d^2*g^2))*(f + g*x)^(3 + n))/(g^6*(3 + n)) + (e^2*(a*e*g^2 + c*(10*e^2*f^2 -
20*d*e*f*g + 9*d^2*g^2))*(f + g*x)^(4 + n))/(g^6*(4 + n)) - (5*c*e^3*(e*f - d*g)
*(f + g*x)^(5 + n))/(g^6*(5 + n)) + (c*e^4*(f + g*x)^(6 + n))/(g^6*(6 + n))

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Rubi in Sympy [A]  time = 118.36, size = 274, normalized size = 1. \[ \frac{c e^{4} \left (f + g x\right )^{n + 6}}{g^{6} \left (n + 6\right )} + \frac{5 c e^{3} \left (f + g x\right )^{n + 5} \left (d g - e f\right )}{g^{6} \left (n + 5\right )} + \frac{e^{2} \left (f + g x\right )^{n + 4} \left (a e g^{2} + 9 c d^{2} g^{2} - 20 c d e f g + 10 c e^{2} f^{2}\right )}{g^{6} \left (n + 4\right )} + \frac{e \left (f + g x\right )^{n + 3} \left (d g - e f\right ) \left (3 a e g^{2} + 7 c d^{2} g^{2} - 20 c d e f g + 10 c e^{2} f^{2}\right )}{g^{6} \left (n + 3\right )} + \frac{\left (f + g x\right )^{n + 1} \left (d g - e f\right )^{3} \left (a g^{2} - 2 c d f g + c e f^{2}\right )}{g^{6} \left (n + 1\right )} + \frac{\left (f + g x\right )^{n + 2} \left (d g - e f\right )^{2} \left (3 a e g^{2} + 2 c d^{2} g^{2} - 10 c d e f g + 5 c e^{2} f^{2}\right )}{g^{6} \left (n + 2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**3*(g*x+f)**n*(c*e*x**2+2*c*d*x+a),x)

[Out]

c*e**4*(f + g*x)**(n + 6)/(g**6*(n + 6)) + 5*c*e**3*(f + g*x)**(n + 5)*(d*g - e*
f)/(g**6*(n + 5)) + e**2*(f + g*x)**(n + 4)*(a*e*g**2 + 9*c*d**2*g**2 - 20*c*d*e
*f*g + 10*c*e**2*f**2)/(g**6*(n + 4)) + e*(f + g*x)**(n + 3)*(d*g - e*f)*(3*a*e*
g**2 + 7*c*d**2*g**2 - 20*c*d*e*f*g + 10*c*e**2*f**2)/(g**6*(n + 3)) + (f + g*x)
**(n + 1)*(d*g - e*f)**3*(a*g**2 - 2*c*d*f*g + c*e*f**2)/(g**6*(n + 1)) + (f + g
*x)**(n + 2)*(d*g - e*f)**2*(3*a*e*g**2 + 2*c*d**2*g**2 - 10*c*d*e*f*g + 5*c*e**
2*f**2)/(g**6*(n + 2))

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Mathematica [B]  time = 1.36352, size = 577, normalized size = 2.1 \[ \frac{(f+g x)^{n+1} \left (a g^2 \left (n^2+11 n+30\right ) \left (d^3 g^3 \left (n^3+9 n^2+26 n+24\right )+3 d^2 e g^2 \left (n^2+7 n+12\right ) (g (n+1) x-f)+3 d e^2 g (n+4) \left (2 f^2-2 f g (n+1) x+g^2 \left (n^2+3 n+2\right ) x^2\right )+e^3 \left (-6 f^3+6 f^2 g (n+1) x-3 f g^2 \left (n^2+3 n+2\right ) x^2+g^3 \left (n^3+6 n^2+11 n+6\right ) x^3\right )\right )+c \left (2 d^4 g^4 \left (n^4+18 n^3+119 n^2+342 n+360\right ) (g (n+1) x-f)+7 d^3 e g^3 \left (n^3+15 n^2+74 n+120\right ) \left (2 f^2-2 f g (n+1) x+g^2 \left (n^2+3 n+2\right ) x^2\right )+9 d^2 e^2 g^2 \left (n^2+11 n+30\right ) \left (-6 f^3+6 f^2 g (n+1) x-3 f g^2 \left (n^2+3 n+2\right ) x^2+g^3 \left (n^3+6 n^2+11 n+6\right ) x^3\right )+5 d e^3 g (n+6) \left (24 f^4-24 f^3 g (n+1) x+12 f^2 g^2 \left (n^2+3 n+2\right ) x^2-4 f g^3 \left (n^3+6 n^2+11 n+6\right ) x^3+g^4 \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^4\right )+e^4 \left (-\left (120 f^5-120 f^4 g (n+1) x+60 f^3 g^2 \left (n^2+3 n+2\right ) x^2-20 f^2 g^3 \left (n^3+6 n^2+11 n+6\right ) x^3+5 f g^4 \left (n^4+10 n^3+35 n^2+50 n+24\right ) x^4-g^5 \left (n^5+15 n^4+85 n^3+225 n^2+274 n+120\right ) x^5\right )\right )\right )\right )}{g^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^3*(f + g*x)^n*(a + 2*c*d*x + c*e*x^2),x]

[Out]

((f + g*x)^(1 + n)*(a*g^2*(30 + 11*n + n^2)*(d^3*g^3*(24 + 26*n + 9*n^2 + n^3) +
3*d^2*e*g^2*(12 + 7*n + n^2)*(-f + g*(1 + n)*x) + 3*d*e^2*g*(4 + n)*(2*f^2 - 2*
f*g*(1 + n)*x + g^2*(2 + 3*n + n^2)*x^2) + e^3*(-6*f^3 + 6*f^2*g*(1 + n)*x - 3*f
*g^2*(2 + 3*n + n^2)*x^2 + g^3*(6 + 11*n + 6*n^2 + n^3)*x^3)) + c*(2*d^4*g^4*(36
0 + 342*n + 119*n^2 + 18*n^3 + n^4)*(-f + g*(1 + n)*x) + 7*d^3*e*g^3*(120 + 74*n
+ 15*n^2 + n^3)*(2*f^2 - 2*f*g*(1 + n)*x + g^2*(2 + 3*n + n^2)*x^2) + 9*d^2*e^2
*g^2*(30 + 11*n + n^2)*(-6*f^3 + 6*f^2*g*(1 + n)*x - 3*f*g^2*(2 + 3*n + n^2)*x^2
+ g^3*(6 + 11*n + 6*n^2 + n^3)*x^3) + 5*d*e^3*g*(6 + n)*(24*f^4 - 24*f^3*g*(1 +
n)*x + 12*f^2*g^2*(2 + 3*n + n^2)*x^2 - 4*f*g^3*(6 + 11*n + 6*n^2 + n^3)*x^3 +
g^4*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)*x^4) - e^4*(120*f^5 - 120*f^4*g*(1 + n)*
x + 60*f^3*g^2*(2 + 3*n + n^2)*x^2 - 20*f^2*g^3*(6 + 11*n + 6*n^2 + n^3)*x^3 + 5
*f*g^4*(24 + 50*n + 35*n^2 + 10*n^3 + n^4)*x^4 - g^5*(120 + 274*n + 225*n^2 + 85
*n^3 + 15*n^4 + n^5)*x^5))))/(g^6*(1 + n)*(2 + n)*(3 + n)*(4 + n)*(5 + n)*(6 + n
))

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Maple [B]  time = 0.019, size = 2017, normalized size = 7.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^3*(g*x+f)^n*(c*e*x^2+2*c*d*x+a),x)

[Out]

(g*x+f)^(1+n)*(c*e^4*g^5*n^5*x^5+5*c*d*e^3*g^5*n^5*x^4+15*c*e^4*g^5*n^4*x^5+9*c*
d^2*e^2*g^5*n^5*x^3+80*c*d*e^3*g^5*n^4*x^4-5*c*e^4*f*g^4*n^4*x^4+85*c*e^4*g^5*n^
3*x^5+a*e^3*g^5*n^5*x^3+7*c*d^3*e*g^5*n^5*x^2+153*c*d^2*e^2*g^5*n^4*x^3-20*c*d*e
^3*f*g^4*n^4*x^3+475*c*d*e^3*g^5*n^3*x^4-50*c*e^4*f*g^4*n^3*x^4+225*c*e^4*g^5*n^
2*x^5+3*a*d*e^2*g^5*n^5*x^2+17*a*e^3*g^5*n^4*x^3+2*c*d^4*g^5*n^5*x+126*c*d^3*e*g
^5*n^4*x^2-27*c*d^2*e^2*f*g^4*n^4*x^2+963*c*d^2*e^2*g^5*n^3*x^3-240*c*d*e^3*f*g^
4*n^3*x^3+1300*c*d*e^3*g^5*n^2*x^4+20*c*e^4*f^2*g^3*n^3*x^3-175*c*e^4*f*g^4*n^2*
x^4+274*c*e^4*g^5*n*x^5+3*a*d^2*e*g^5*n^5*x+54*a*d*e^2*g^5*n^4*x^2-3*a*e^3*f*g^4
*n^4*x^2+107*a*e^3*g^5*n^3*x^3+38*c*d^4*g^5*n^4*x-14*c*d^3*e*f*g^4*n^4*x+847*c*d
^3*e*g^5*n^3*x^2-378*c*d^2*e^2*f*g^4*n^3*x^2+2763*c*d^2*e^2*g^5*n^2*x^3+60*c*d*e
^3*f^2*g^3*n^3*x^2-940*c*d*e^3*f*g^4*n^2*x^3+1620*c*d*e^3*g^5*n*x^4+120*c*e^4*f^
2*g^3*n^2*x^3-250*c*e^4*f*g^4*n*x^4+120*c*e^4*g^5*x^5+a*d^3*g^5*n^5+57*a*d^2*e*g
^5*n^4*x-6*a*d*e^2*f*g^4*n^4*x+363*a*d*e^2*g^5*n^3*x^2-42*a*e^3*f*g^4*n^3*x^2+30
7*a*e^3*g^5*n^2*x^3-2*c*d^4*f*g^4*n^4+274*c*d^4*g^5*n^3*x-224*c*d^3*e*f*g^4*n^3*
x+2604*c*d^3*e*g^5*n^2*x^2+54*c*d^2*e^2*f^2*g^3*n^3*x-1755*c*d^2*e^2*f*g^4*n^2*x
^2+3564*c*d^2*e^2*g^5*n*x^3+540*c*d*e^3*f^2*g^3*n^2*x^2-1440*c*d*e^3*f*g^4*n*x^3
+720*c*d*e^3*g^5*x^4-60*c*e^4*f^3*g^2*n^2*x^2+220*c*e^4*f^2*g^3*n*x^3-120*c*e^4*
f*g^4*x^4+20*a*d^3*g^5*n^4-3*a*d^2*e*f*g^4*n^4+411*a*d^2*e*g^5*n^3*x-96*a*d*e^2*
f*g^4*n^3*x+1116*a*d*e^2*g^5*n^2*x^2+6*a*e^3*f^2*g^3*n^3*x-195*a*e^3*f*g^4*n^2*x
^2+396*a*e^3*g^5*n*x^3-36*c*d^4*f*g^4*n^3+922*c*d^4*g^5*n^2*x+14*c*d^3*e*f^2*g^3
*n^3-1246*c*d^3*e*f*g^4*n^2*x+3556*c*d^3*e*g^5*n*x^2+648*c*d^2*e^2*f^2*g^3*n^2*x
-3024*c*d^2*e^2*f*g^4*n*x^2+1620*c*d^2*e^2*g^5*x^3-120*c*d*e^3*f^3*g^2*n^2*x+120
0*c*d*e^3*f^2*g^3*n*x^2-720*c*d*e^3*f*g^4*x^3-180*c*e^4*f^3*g^2*n*x^2+120*c*e^4*
f^2*g^3*x^3+155*a*d^3*g^5*n^3-54*a*d^2*e*f*g^4*n^3+1383*a*d^2*e*g^5*n^2*x+6*a*d*
e^2*f^2*g^3*n^3-534*a*d*e^2*f*g^4*n^2*x+1524*a*d*e^2*g^5*n*x^2+72*a*e^3*f^2*g^3*
n^2*x-336*a*e^3*f*g^4*n*x^2+180*a*e^3*g^5*x^3-238*c*d^4*f*g^4*n^2+1404*c*d^4*g^5
*n*x+210*c*d^3*e*f^2*g^3*n^2-2716*c*d^3*e*f*g^4*n*x+1680*c*d^3*e*g^5*x^2-54*c*d^
2*e^2*f^3*g^2*n^2+2214*c*d^2*e^2*f^2*g^3*n*x-1620*c*d^2*e^2*f*g^4*x^2-840*c*d*e^
3*f^3*g^2*n*x+720*c*d*e^3*f^2*g^3*x^2+120*c*e^4*f^4*g*n*x-120*c*e^4*f^3*g^2*x^2+
580*a*d^3*g^5*n^2-357*a*d^2*e*f*g^4*n^2+2106*a*d^2*e*g^5*n*x+90*a*d*e^2*f^2*g^3*
n^2-1164*a*d*e^2*f*g^4*n*x+720*a*d*e^2*g^5*x^2-6*a*e^3*f^3*g^2*n^2+246*a*e^3*f^2
*g^3*n*x-180*a*e^3*f*g^4*x^2-684*c*d^4*f*g^4*n+720*c*d^4*g^5*x+1036*c*d^3*e*f^2*
g^3*n-1680*c*d^3*e*f*g^4*x-594*c*d^2*e^2*f^3*g^2*n+1620*c*d^2*e^2*f^2*g^3*x+120*
c*d*e^3*f^4*g*n-720*c*d*e^3*f^3*g^2*x+120*c*e^4*f^4*g*x+1044*a*d^3*g^5*n-1026*a*
d^2*e*f*g^4*n+1080*a*d^2*e*g^5*x+444*a*d*e^2*f^2*g^3*n-720*a*d*e^2*f*g^4*x-66*a*
e^3*f^3*g^2*n+180*a*e^3*f^2*g^3*x-720*c*d^4*f*g^4+1680*c*d^3*e*f^2*g^3-1620*c*d^
2*e^2*f^3*g^2+720*c*d*e^3*f^4*g-120*c*e^4*f^5+720*a*d^3*g^5-1080*a*d^2*e*f*g^4+7
20*a*d*e^2*f^2*g^3-180*a*e^3*f^3*g^2)/g^6/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1
764*n+720)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e*x^2 + 2*c*d*x + a)*(e*x + d)^3*(g*x + f)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.313283, size = 2743, normalized size = 9.97 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e*x^2 + 2*c*d*x + a)*(e*x + d)^3*(g*x + f)^n,x, algorithm="fricas")

[Out]

(a*d^3*f*g^5*n^5 - 120*c*e^4*f^6 + 720*c*d*e^3*f^5*g + 720*a*d^3*f*g^5 - 180*(9*
c*d^2*e^2 + a*e^3)*f^4*g^2 + 240*(7*c*d^3*e + 3*a*d*e^2)*f^3*g^3 - 360*(2*c*d^4
+ 3*a*d^2*e)*f^2*g^4 + (c*e^4*g^6*n^5 + 15*c*e^4*g^6*n^4 + 85*c*e^4*g^6*n^3 + 22
5*c*e^4*g^6*n^2 + 274*c*e^4*g^6*n + 120*c*e^4*g^6)*x^6 + (720*c*d*e^3*g^6 + (c*e
^4*f*g^5 + 5*c*d*e^3*g^6)*n^5 + 10*(c*e^4*f*g^5 + 8*c*d*e^3*g^6)*n^4 + 5*(7*c*e^
4*f*g^5 + 95*c*d*e^3*g^6)*n^3 + 50*(c*e^4*f*g^5 + 26*c*d*e^3*g^6)*n^2 + 12*(2*c*
e^4*f*g^5 + 135*c*d*e^3*g^6)*n)*x^5 + (20*a*d^3*f*g^5 - (2*c*d^4 + 3*a*d^2*e)*f^
2*g^4)*n^4 + (180*(9*c*d^2*e^2 + a*e^3)*g^6 + (5*c*d*e^3*f*g^5 + (9*c*d^2*e^2 +
a*e^3)*g^6)*n^5 - (5*c*e^4*f^2*g^4 - 60*c*d*e^3*f*g^5 - 17*(9*c*d^2*e^2 + a*e^3)
*g^6)*n^4 - (30*c*e^4*f^2*g^4 - 235*c*d*e^3*f*g^5 - 107*(9*c*d^2*e^2 + a*e^3)*g^
6)*n^3 - (55*c*e^4*f^2*g^4 - 360*c*d*e^3*f*g^5 - 307*(9*c*d^2*e^2 + a*e^3)*g^6)*
n^2 - 6*(5*c*e^4*f^2*g^4 - 30*c*d*e^3*f*g^5 - 66*(9*c*d^2*e^2 + a*e^3)*g^6)*n)*x
^4 + (155*a*d^3*f*g^5 + 2*(7*c*d^3*e + 3*a*d*e^2)*f^3*g^3 - 18*(2*c*d^4 + 3*a*d^
2*e)*f^2*g^4)*n^3 + (240*(7*c*d^3*e + 3*a*d*e^2)*g^6 + ((9*c*d^2*e^2 + a*e^3)*f*
g^5 + (7*c*d^3*e + 3*a*d*e^2)*g^6)*n^5 - 2*(10*c*d*e^3*f^2*g^4 - 7*(9*c*d^2*e^2
+ a*e^3)*f*g^5 - 9*(7*c*d^3*e + 3*a*d*e^2)*g^6)*n^4 + (20*c*e^4*f^3*g^3 - 180*c*
d*e^3*f^2*g^4 + 65*(9*c*d^2*e^2 + a*e^3)*f*g^5 + 121*(7*c*d^3*e + 3*a*d*e^2)*g^6
)*n^3 + 4*(15*c*e^4*f^3*g^3 - 100*c*d*e^3*f^2*g^4 + 28*(9*c*d^2*e^2 + a*e^3)*f*g
^5 + 93*(7*c*d^3*e + 3*a*d*e^2)*g^6)*n^2 + 4*(10*c*e^4*f^3*g^3 - 60*c*d*e^3*f^2*
g^4 + 15*(9*c*d^2*e^2 + a*e^3)*f*g^5 + 127*(7*c*d^3*e + 3*a*d*e^2)*g^6)*n)*x^3 +
(580*a*d^3*f*g^5 - 6*(9*c*d^2*e^2 + a*e^3)*f^4*g^2 + 30*(7*c*d^3*e + 3*a*d*e^2)
*f^3*g^3 - 119*(2*c*d^4 + 3*a*d^2*e)*f^2*g^4)*n^2 + (360*(2*c*d^4 + 3*a*d^2*e)*g
^6 + ((7*c*d^3*e + 3*a*d*e^2)*f*g^5 + (2*c*d^4 + 3*a*d^2*e)*g^6)*n^5 - (3*(9*c*d
^2*e^2 + a*e^3)*f^2*g^4 - 16*(7*c*d^3*e + 3*a*d*e^2)*f*g^5 - 19*(2*c*d^4 + 3*a*d
^2*e)*g^6)*n^4 + (60*c*d*e^3*f^3*g^3 - 36*(9*c*d^2*e^2 + a*e^3)*f^2*g^4 + 89*(7*
c*d^3*e + 3*a*d*e^2)*f*g^5 + 137*(2*c*d^4 + 3*a*d^2*e)*g^6)*n^3 - (60*c*e^4*f^4*
g^2 - 420*c*d*e^3*f^3*g^3 + 123*(9*c*d^2*e^2 + a*e^3)*f^2*g^4 - 194*(7*c*d^3*e +
3*a*d*e^2)*f*g^5 - 461*(2*c*d^4 + 3*a*d^2*e)*g^6)*n^2 - 6*(10*c*e^4*f^4*g^2 - 6
0*c*d*e^3*f^3*g^3 + 15*(9*c*d^2*e^2 + a*e^3)*f^2*g^4 - 20*(7*c*d^3*e + 3*a*d*e^2
)*f*g^5 - 117*(2*c*d^4 + 3*a*d^2*e)*g^6)*n)*x^2 + 2*(60*c*d*e^3*f^5*g + 522*a*d^
3*f*g^5 - 33*(9*c*d^2*e^2 + a*e^3)*f^4*g^2 + 74*(7*c*d^3*e + 3*a*d*e^2)*f^3*g^3
- 171*(2*c*d^4 + 3*a*d^2*e)*f^2*g^4)*n + (720*a*d^3*g^6 + (a*d^3*g^6 + (2*c*d^4
+ 3*a*d^2*e)*f*g^5)*n^5 + 2*(10*a*d^3*g^6 - (7*c*d^3*e + 3*a*d*e^2)*f^2*g^4 + 9*
(2*c*d^4 + 3*a*d^2*e)*f*g^5)*n^4 + (155*a*d^3*g^6 + 6*(9*c*d^2*e^2 + a*e^3)*f^3*
g^3 - 30*(7*c*d^3*e + 3*a*d*e^2)*f^2*g^4 + 119*(2*c*d^4 + 3*a*d^2*e)*f*g^5)*n^3
- 2*(60*c*d*e^3*f^4*g^2 - 290*a*d^3*g^6 - 33*(9*c*d^2*e^2 + a*e^3)*f^3*g^3 + 74*
(7*c*d^3*e + 3*a*d*e^2)*f^2*g^4 - 171*(2*c*d^4 + 3*a*d^2*e)*f*g^5)*n^2 + 12*(10*
c*e^4*f^5*g - 60*c*d*e^3*f^4*g^2 + 87*a*d^3*g^6 + 15*(9*c*d^2*e^2 + a*e^3)*f^3*g
^3 - 20*(7*c*d^3*e + 3*a*d*e^2)*f^2*g^4 + 30*(2*c*d^4 + 3*a*d^2*e)*f*g^5)*n)*x)*
(g*x + f)^n/(g^6*n^6 + 21*g^6*n^5 + 175*g^6*n^4 + 735*g^6*n^3 + 1624*g^6*n^2 + 1
764*g^6*n + 720*g^6)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**3*(g*x+f)**n*(c*e*x**2+2*c*d*x+a),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.277802, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*e*x^2 + 2*c*d*x + a)*(e*x + d)^3*(g*x + f)^n,x, algorithm="giac")

[Out]

Done