3.26 \(\int (a+b x)^2 (c+d x)^n \left (A+B x+C x^2+D x^3\right ) \, dx\)

Optimal. Leaf size=338 \[ \frac{(c+d x)^{n+3} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^6 (n+3)}+\frac{(c+d x)^{n+4} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^6 (n+4)}+\frac{(b c-a d)^2 (c+d x)^{n+1} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6 (n+1)}+\frac{(b c-a d) (c+d x)^{n+2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^6 (n+2)}+\frac{b (c+d x)^{n+5} (2 a d D-5 b c D+b C d)}{d^6 (n+5)}+\frac{b^2 D (c+d x)^{n+6}}{d^6 (n+6)} \]

[Out]

((b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^6*(1 +
n)) + ((b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 +
2*A*d^3 - 5*c^3*D))*(c + d*x)^(2 + n))/(d^6*(2 + n)) + ((a^2*d^2*(C*d - 3*c*D)
- 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*
c^3*D))*(c + d*x)^(3 + n))/(d^6*(3 + n)) + ((a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) -
b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^(4 + n))/(d^6*(4 + n)) + (b*(b*C*d
- 5*b*c*D + 2*a*d*D)*(c + d*x)^(5 + n))/(d^6*(5 + n)) + (b^2*D*(c + d*x)^(6 + n)
)/(d^6*(6 + n))

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Rubi [A]  time = 0.580939, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.033 \[ \frac{(c+d x)^{n+3} \left (a^2 d^2 (C d-3 c D)-2 a b d \left (-B d^2-6 c^2 D+3 c C d\right )+b^2 \left (A d^3-3 B c d^2-10 c^3 D+6 c^2 C d\right )\right )}{d^6 (n+3)}+\frac{(c+d x)^{n+4} \left (a^2 d^2 D+2 a b d (C d-4 c D)+b^2 \left (-\left (-B d^2-10 c^2 D+4 c C d\right )\right )\right )}{d^6 (n+4)}+\frac{(b c-a d)^2 (c+d x)^{n+1} \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^6 (n+1)}+\frac{(b c-a d) (c+d x)^{n+2} \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (2 A d^3-3 B c d^2-5 c^3 D+4 c^2 C d\right )\right )}{d^6 (n+2)}+\frac{b (c+d x)^{n+5} (2 a d D-5 b c D+b C d)}{d^6 (n+5)}+\frac{b^2 D (c+d x)^{n+6}}{d^6 (n+6)} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]

[Out]

((b*c - a*d)^2*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*(c + d*x)^(1 + n))/(d^6*(1 +
n)) + ((b*c - a*d)*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(4*c^2*C*d - 3*B*c*d^2 +
2*A*d^3 - 5*c^3*D))*(c + d*x)^(2 + n))/(d^6*(2 + n)) + ((a^2*d^2*(C*d - 3*c*D)
- 2*a*b*d*(3*c*C*d - B*d^2 - 6*c^2*D) + b^2*(6*c^2*C*d - 3*B*c*d^2 + A*d^3 - 10*
c^3*D))*(c + d*x)^(3 + n))/(d^6*(3 + n)) + ((a^2*d^2*D + 2*a*b*d*(C*d - 4*c*D) -
b^2*(4*c*C*d - B*d^2 - 10*c^2*D))*(c + d*x)^(4 + n))/(d^6*(4 + n)) + (b*(b*C*d
- 5*b*c*D + 2*a*d*D)*(c + d*x)^(5 + n))/(d^6*(5 + n)) + (b^2*D*(c + d*x)^(6 + n)
)/(d^6*(6 + n))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**2*(d*x+c)**n*(D*x**3+C*x**2+B*x+A),x)

[Out]

Timed out

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Mathematica [A]  time = 2.20084, size = 619, normalized size = 1.83 \[ \frac{(c+d x)^{n+1} \left (a^2 d^2 \left (n^2+11 n+30\right ) \left (d^3 \left (A \left (n^3+9 n^2+26 n+24\right )+(n+1) x \left (B \left (n^2+7 n+12\right )+(n+2) x (C (n+4)+D (n+3) x)\right )\right )-c d^2 \left (B \left (n^2+7 n+12\right )+(n+1) x (2 C (n+4)+3 D (n+2) x)\right )-6 c^3 D+2 c^2 d (C (n+4)+3 D (n+1) x)\right )+2 a b d (n+6) \left (-c d^3 \left (A \left (n^3+12 n^2+47 n+60\right )+(n+1) x \left (2 B \left (n^2+9 n+20\right )+(n+2) x (3 C (n+5)+4 D (n+3) x)\right )\right )+d^4 (n+1) x \left (A \left (n^3+12 n^2+47 n+60\right )+(n+2) x \left (B \left (n^2+9 n+20\right )+(n+3) x (C (n+5)+D (n+4) x)\right )\right )+2 c^2 d^2 \left (B \left (n^2+9 n+20\right )+3 (n+1) x (C (n+5)+2 D (n+2) x)\right )+24 c^4 D-6 c^3 d (C (n+5)+4 D (n+1) x)\right )+b^2 \left (-\left (-2 c^2 d^3 \left (A \left (n^3+15 n^2+74 n+120\right )+(n+1) x \left (3 B \left (n^2+11 n+30\right )+2 (n+2) x (3 C (n+6)+5 D (n+3) x)\right )\right )+c d^4 (n+1) x \left (2 A \left (n^3+15 n^2+74 n+120\right )+(n+2) x \left (3 B \left (n^2+11 n+30\right )+(n+3) x (4 C (n+6)+5 D (n+4) x)\right )\right )-d^5 \left (n^2+3 n+2\right ) x^2 \left (A \left (n^3+15 n^2+74 n+120\right )+(n+3) x \left (B \left (n^2+11 n+30\right )+(n+4) x (C (n+6)+D (n+5) x)\right )\right )+6 c^3 d^2 \left (B \left (n^2+11 n+30\right )+2 (n+1) x (2 C (n+6)+5 D (n+2) x)\right )+120 c^5 D-24 c^4 d (C (n+6)+5 D (n+1) x)\right )\right )\right )}{d^6 (n+1) (n+2) (n+3) (n+4) (n+5) (n+6)} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^2*(c + d*x)^n*(A + B*x + C*x^2 + D*x^3),x]

[Out]

((c + d*x)^(1 + n)*(a^2*d^2*(30 + 11*n + n^2)*(-6*c^3*D + 2*c^2*d*(C*(4 + n) + 3
*D*(1 + n)*x) - c*d^2*(B*(12 + 7*n + n^2) + (1 + n)*x*(2*C*(4 + n) + 3*D*(2 + n)
*x)) + d^3*(A*(24 + 26*n + 9*n^2 + n^3) + (1 + n)*x*(B*(12 + 7*n + n^2) + (2 + n
)*x*(C*(4 + n) + D*(3 + n)*x)))) + 2*a*b*d*(6 + n)*(24*c^4*D - 6*c^3*d*(C*(5 + n
) + 4*D*(1 + n)*x) + 2*c^2*d^2*(B*(20 + 9*n + n^2) + 3*(1 + n)*x*(C*(5 + n) + 2*
D*(2 + n)*x)) - c*d^3*(A*(60 + 47*n + 12*n^2 + n^3) + (1 + n)*x*(2*B*(20 + 9*n +
n^2) + (2 + n)*x*(3*C*(5 + n) + 4*D*(3 + n)*x))) + d^4*(1 + n)*x*(A*(60 + 47*n
+ 12*n^2 + n^3) + (2 + n)*x*(B*(20 + 9*n + n^2) + (3 + n)*x*(C*(5 + n) + D*(4 +
n)*x)))) - b^2*(120*c^5*D - 24*c^4*d*(C*(6 + n) + 5*D*(1 + n)*x) + 6*c^3*d^2*(B*
(30 + 11*n + n^2) + 2*(1 + n)*x*(2*C*(6 + n) + 5*D*(2 + n)*x)) - 2*c^2*d^3*(A*(1
20 + 74*n + 15*n^2 + n^3) + (1 + n)*x*(3*B*(30 + 11*n + n^2) + 2*(2 + n)*x*(3*C*
(6 + n) + 5*D*(3 + n)*x))) + c*d^4*(1 + n)*x*(2*A*(120 + 74*n + 15*n^2 + n^3) +
(2 + n)*x*(3*B*(30 + 11*n + n^2) + (3 + n)*x*(4*C*(6 + n) + 5*D*(4 + n)*x))) - d
^5*(2 + 3*n + n^2)*x^2*(A*(120 + 74*n + 15*n^2 + n^3) + (3 + n)*x*(B*(30 + 11*n
+ n^2) + (4 + n)*x*(C*(6 + n) + D*(5 + n)*x))))))/(d^6*(1 + n)*(2 + n)*(3 + n)*(
4 + n)*(5 + n)*(6 + n))

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Maple [B]  time = 0.02, size = 2588, normalized size = 7.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2*(d*x+c)^n*(D*x^3+C*x^2+B*x+A),x)

[Out]

(d*x+c)^(1+n)*(D*b^2*d^5*n^5*x^5+C*b^2*d^5*n^5*x^4+2*D*a*b*d^5*n^5*x^4+15*D*b^2*
d^5*n^4*x^5+B*b^2*d^5*n^5*x^3+2*C*a*b*d^5*n^5*x^3+16*C*b^2*d^5*n^4*x^4+D*a^2*d^5
*n^5*x^3+32*D*a*b*d^5*n^4*x^4-5*D*b^2*c*d^4*n^4*x^4+85*D*b^2*d^5*n^3*x^5+A*b^2*d
^5*n^5*x^2+2*B*a*b*d^5*n^5*x^2+17*B*b^2*d^5*n^4*x^3+C*a^2*d^5*n^5*x^2+34*C*a*b*d
^5*n^4*x^3-4*C*b^2*c*d^4*n^4*x^3+95*C*b^2*d^5*n^3*x^4+17*D*a^2*d^5*n^4*x^3-8*D*a
*b*c*d^4*n^4*x^3+190*D*a*b*d^5*n^3*x^4-50*D*b^2*c*d^4*n^3*x^4+225*D*b^2*d^5*n^2*
x^5+2*A*a*b*d^5*n^5*x+18*A*b^2*d^5*n^4*x^2+B*a^2*d^5*n^5*x+36*B*a*b*d^5*n^4*x^2-
3*B*b^2*c*d^4*n^4*x^2+107*B*b^2*d^5*n^3*x^3+18*C*a^2*d^5*n^4*x^2-6*C*a*b*c*d^4*n
^4*x^2+214*C*a*b*d^5*n^3*x^3-48*C*b^2*c*d^4*n^3*x^3+260*C*b^2*d^5*n^2*x^4-3*D*a^
2*c*d^4*n^4*x^2+107*D*a^2*d^5*n^3*x^3-96*D*a*b*c*d^4*n^3*x^3+520*D*a*b*d^5*n^2*x
^4+20*D*b^2*c^2*d^3*n^3*x^3-175*D*b^2*c*d^4*n^2*x^4+274*D*b^2*d^5*n*x^5+A*a^2*d^
5*n^5+38*A*a*b*d^5*n^4*x-2*A*b^2*c*d^4*n^4*x+121*A*b^2*d^5*n^3*x^2+19*B*a^2*d^5*
n^4*x-4*B*a*b*c*d^4*n^4*x+242*B*a*b*d^5*n^3*x^2-42*B*b^2*c*d^4*n^3*x^2+307*B*b^2
*d^5*n^2*x^3-2*C*a^2*c*d^4*n^4*x+121*C*a^2*d^5*n^3*x^2-84*C*a*b*c*d^4*n^3*x^2+61
4*C*a*b*d^5*n^2*x^3+12*C*b^2*c^2*d^3*n^3*x^2-188*C*b^2*c*d^4*n^2*x^3+324*C*b^2*d
^5*n*x^4-42*D*a^2*c*d^4*n^3*x^2+307*D*a^2*d^5*n^2*x^3+24*D*a*b*c^2*d^3*n^3*x^2-3
76*D*a*b*c*d^4*n^2*x^3+648*D*a*b*d^5*n*x^4+120*D*b^2*c^2*d^3*n^2*x^3-250*D*b^2*c
*d^4*n*x^4+120*D*b^2*d^5*x^5+20*A*a^2*d^5*n^4-2*A*a*b*c*d^4*n^4+274*A*a*b*d^5*n^
3*x-32*A*b^2*c*d^4*n^3*x+372*A*b^2*d^5*n^2*x^2-B*a^2*c*d^4*n^4+137*B*a^2*d^5*n^3
*x-64*B*a*b*c*d^4*n^3*x+744*B*a*b*d^5*n^2*x^2+6*B*b^2*c^2*d^3*n^3*x-195*B*b^2*c*
d^4*n^2*x^2+396*B*b^2*d^5*n*x^3-32*C*a^2*c*d^4*n^3*x+372*C*a^2*d^5*n^2*x^2+12*C*
a*b*c^2*d^3*n^3*x-390*C*a*b*c*d^4*n^2*x^2+792*C*a*b*d^5*n*x^3+108*C*b^2*c^2*d^3*
n^2*x^2-288*C*b^2*c*d^4*n*x^3+144*C*b^2*d^5*x^4+6*D*a^2*c^2*d^3*n^3*x-195*D*a^2*
c*d^4*n^2*x^2+396*D*a^2*d^5*n*x^3+216*D*a*b*c^2*d^3*n^2*x^2-576*D*a*b*c*d^4*n*x^
3+288*D*a*b*d^5*x^4-60*D*b^2*c^3*d^2*n^2*x^2+220*D*b^2*c^2*d^3*n*x^3-120*D*b^2*c
*d^4*x^4+155*A*a^2*d^5*n^3-36*A*a*b*c*d^4*n^3+922*A*a*b*d^5*n^2*x+2*A*b^2*c^2*d^
3*n^3-178*A*b^2*c*d^4*n^2*x+508*A*b^2*d^5*n*x^2-18*B*a^2*c*d^4*n^3+461*B*a^2*d^5
*n^2*x+4*B*a*b*c^2*d^3*n^3-356*B*a*b*c*d^4*n^2*x+1016*B*a*b*d^5*n*x^2+72*B*b^2*c
^2*d^3*n^2*x-336*B*b^2*c*d^4*n*x^2+180*B*b^2*d^5*x^3+2*C*a^2*c^2*d^3*n^3-178*C*a
^2*c*d^4*n^2*x+508*C*a^2*d^5*n*x^2+144*C*a*b*c^2*d^3*n^2*x-672*C*a*b*c*d^4*n*x^2
+360*C*a*b*d^5*x^3-24*C*b^2*c^3*d^2*n^2*x+240*C*b^2*c^2*d^3*n*x^2-144*C*b^2*c*d^
4*x^3+72*D*a^2*c^2*d^3*n^2*x-336*D*a^2*c*d^4*n*x^2+180*D*a^2*d^5*x^3-48*D*a*b*c^
3*d^2*n^2*x+480*D*a*b*c^2*d^3*n*x^2-288*D*a*b*c*d^4*x^3-180*D*b^2*c^3*d^2*n*x^2+
120*D*b^2*c^2*d^3*x^3+580*A*a^2*d^5*n^2-238*A*a*b*c*d^4*n^2+1404*A*a*b*d^5*n*x+3
0*A*b^2*c^2*d^3*n^2-388*A*b^2*c*d^4*n*x+240*A*b^2*d^5*x^2-119*B*a^2*c*d^4*n^2+70
2*B*a^2*d^5*n*x+60*B*a*b*c^2*d^3*n^2-776*B*a*b*c*d^4*n*x+480*B*a*b*d^5*x^2-6*B*b
^2*c^3*d^2*n^2+246*B*b^2*c^2*d^3*n*x-180*B*b^2*c*d^4*x^2+30*C*a^2*c^2*d^3*n^2-38
8*C*a^2*c*d^4*n*x+240*C*a^2*d^5*x^2-12*C*a*b*c^3*d^2*n^2+492*C*a*b*c^2*d^3*n*x-3
60*C*a*b*c*d^4*x^2-168*C*b^2*c^3*d^2*n*x+144*C*b^2*c^2*d^3*x^2-6*D*a^2*c^3*d^2*n
^2+246*D*a^2*c^2*d^3*n*x-180*D*a^2*c*d^4*x^2-336*D*a*b*c^3*d^2*n*x+288*D*a*b*c^2
*d^3*x^2+120*D*b^2*c^4*d*n*x-120*D*b^2*c^3*d^2*x^2+1044*A*a^2*d^5*n-684*A*a*b*c*
d^4*n+720*A*a*b*d^5*x+148*A*b^2*c^2*d^3*n-240*A*b^2*c*d^4*x-342*B*a^2*c*d^4*n+36
0*B*a^2*d^5*x+296*B*a*b*c^2*d^3*n-480*B*a*b*c*d^4*x-66*B*b^2*c^3*d^2*n+180*B*b^2
*c^2*d^3*x+148*C*a^2*c^2*d^3*n-240*C*a^2*c*d^4*x-132*C*a*b*c^3*d^2*n+360*C*a*b*c
^2*d^3*x+24*C*b^2*c^4*d*n-144*C*b^2*c^3*d^2*x-66*D*a^2*c^3*d^2*n+180*D*a^2*c^2*d
^3*x+48*D*a*b*c^4*d*n-288*D*a*b*c^3*d^2*x+120*D*b^2*c^4*d*x+720*A*a^2*d^5-720*A*
a*b*c*d^4+240*A*b^2*c^2*d^3-360*B*a^2*c*d^4+480*B*a*b*c^2*d^3-180*B*b^2*c^3*d^2+
240*C*a^2*c^2*d^3-360*C*a*b*c^3*d^2+144*C*b^2*c^4*d-180*D*a^2*c^3*d^2+288*D*a*b*
c^4*d-120*D*b^2*c^5)/d^6/(n^6+21*n^5+175*n^4+735*n^3+1624*n^2+1764*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((D*x^3 + C*x^2 + B*x + A)*(b*x + a)^2*(d*x + c)^n,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246987, size = 2911, normalized size = 8.61 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)^2*(d*x + c)^n,x, algorithm="fricas")

[Out]

(A*a^2*c*d^5*n^5 - 120*D*b^2*c^6 + 720*A*a^2*c*d^5 + 144*(2*D*a*b + C*b^2)*c^5*d
- 180*(D*a^2 + 2*C*a*b + B*b^2)*c^4*d^2 + 240*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^3
- 360*(B*a^2 + 2*A*a*b)*c^2*d^4 + (D*b^2*d^6*n^5 + 15*D*b^2*d^6*n^4 + 85*D*b^2*
d^6*n^3 + 225*D*b^2*d^6*n^2 + 274*D*b^2*d^6*n + 120*D*b^2*d^6)*x^6 + (144*(2*D*a
*b + C*b^2)*d^6 + (D*b^2*c*d^5 + (2*D*a*b + C*b^2)*d^6)*n^5 + 2*(5*D*b^2*c*d^5 +
8*(2*D*a*b + C*b^2)*d^6)*n^4 + 5*(7*D*b^2*c*d^5 + 19*(2*D*a*b + C*b^2)*d^6)*n^3
+ 10*(5*D*b^2*c*d^5 + 26*(2*D*a*b + C*b^2)*d^6)*n^2 + 12*(2*D*b^2*c*d^5 + 27*(2
*D*a*b + C*b^2)*d^6)*n)*x^5 + (20*A*a^2*c*d^5 - (B*a^2 + 2*A*a*b)*c^2*d^4)*n^4 +
(180*(D*a^2 + 2*C*a*b + B*b^2)*d^6 + ((2*D*a*b + C*b^2)*c*d^5 + (D*a^2 + 2*C*a*
b + B*b^2)*d^6)*n^5 - (5*D*b^2*c^2*d^4 - 12*(2*D*a*b + C*b^2)*c*d^5 - 17*(D*a^2
+ 2*C*a*b + B*b^2)*d^6)*n^4 - (30*D*b^2*c^2*d^4 - 47*(2*D*a*b + C*b^2)*c*d^5 - 1
07*(D*a^2 + 2*C*a*b + B*b^2)*d^6)*n^3 - (55*D*b^2*c^2*d^4 - 72*(2*D*a*b + C*b^2)
*c*d^5 - 307*(D*a^2 + 2*C*a*b + B*b^2)*d^6)*n^2 - 6*(5*D*b^2*c^2*d^4 - 6*(2*D*a*
b + C*b^2)*c*d^5 - 66*(D*a^2 + 2*C*a*b + B*b^2)*d^6)*n)*x^4 + (155*A*a^2*c*d^5 +
2*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^3 - 18*(B*a^2 + 2*A*a*b)*c^2*d^4)*n^3 + (240*
(C*a^2 + 2*B*a*b + A*b^2)*d^6 + ((D*a^2 + 2*C*a*b + B*b^2)*c*d^5 + (C*a^2 + 2*B*
a*b + A*b^2)*d^6)*n^5 - 2*(2*(2*D*a*b + C*b^2)*c^2*d^4 - 7*(D*a^2 + 2*C*a*b + B*
b^2)*c*d^5 - 9*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*n^4 + (20*D*b^2*c^3*d^3 - 36*(2*D*
a*b + C*b^2)*c^2*d^4 + 65*(D*a^2 + 2*C*a*b + B*b^2)*c*d^5 + 121*(C*a^2 + 2*B*a*b
+ A*b^2)*d^6)*n^3 + 4*(15*D*b^2*c^3*d^3 - 20*(2*D*a*b + C*b^2)*c^2*d^4 + 28*(D*
a^2 + 2*C*a*b + B*b^2)*c*d^5 + 93*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*n^2 + 4*(10*D*b
^2*c^3*d^3 - 12*(2*D*a*b + C*b^2)*c^2*d^4 + 15*(D*a^2 + 2*C*a*b + B*b^2)*c*d^5 +
127*(C*a^2 + 2*B*a*b + A*b^2)*d^6)*n)*x^3 + (580*A*a^2*c*d^5 - 6*(D*a^2 + 2*C*a
*b + B*b^2)*c^4*d^2 + 30*(C*a^2 + 2*B*a*b + A*b^2)*c^3*d^3 - 119*(B*a^2 + 2*A*a*
b)*c^2*d^4)*n^2 + (360*(B*a^2 + 2*A*a*b)*d^6 + ((C*a^2 + 2*B*a*b + A*b^2)*c*d^5
+ (B*a^2 + 2*A*a*b)*d^6)*n^5 - (3*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^4 - 16*(C*a^2
+ 2*B*a*b + A*b^2)*c*d^5 - 19*(B*a^2 + 2*A*a*b)*d^6)*n^4 + (12*(2*D*a*b + C*b^2)
*c^3*d^3 - 36*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^4 + 89*(C*a^2 + 2*B*a*b + A*b^2)*c
*d^5 + 137*(B*a^2 + 2*A*a*b)*d^6)*n^3 - (60*D*b^2*c^4*d^2 - 84*(2*D*a*b + C*b^2)
*c^3*d^3 + 123*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^4 - 194*(C*a^2 + 2*B*a*b + A*b^2)
*c*d^5 - 461*(B*a^2 + 2*A*a*b)*d^6)*n^2 - 6*(10*D*b^2*c^4*d^2 - 12*(2*D*a*b + C*
b^2)*c^3*d^3 + 15*(D*a^2 + 2*C*a*b + B*b^2)*c^2*d^4 - 20*(C*a^2 + 2*B*a*b + A*b^
2)*c*d^5 - 117*(B*a^2 + 2*A*a*b)*d^6)*n)*x^2 + 2*(522*A*a^2*c*d^5 + 12*(2*D*a*b
+ C*b^2)*c^5*d - 33*(D*a^2 + 2*C*a*b + B*b^2)*c^4*d^2 + 74*(C*a^2 + 2*B*a*b + A*
b^2)*c^3*d^3 - 171*(B*a^2 + 2*A*a*b)*c^2*d^4)*n + (720*A*a^2*d^6 + (A*a^2*d^6 +
(B*a^2 + 2*A*a*b)*c*d^5)*n^5 + 2*(10*A*a^2*d^6 - (C*a^2 + 2*B*a*b + A*b^2)*c^2*d
^4 + 9*(B*a^2 + 2*A*a*b)*c*d^5)*n^4 + (155*A*a^2*d^6 + 6*(D*a^2 + 2*C*a*b + B*b^
2)*c^3*d^3 - 30*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 + 119*(B*a^2 + 2*A*a*b)*c*d^5)
*n^3 + 2*(290*A*a^2*d^6 - 12*(2*D*a*b + C*b^2)*c^4*d^2 + 33*(D*a^2 + 2*C*a*b + B
*b^2)*c^3*d^3 - 74*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 + 171*(B*a^2 + 2*A*a*b)*c*d
^5)*n^2 + 12*(10*D*b^2*c^5*d + 87*A*a^2*d^6 - 12*(2*D*a*b + C*b^2)*c^4*d^2 + 15*
(D*a^2 + 2*C*a*b + B*b^2)*c^3*d^3 - 20*(C*a^2 + 2*B*a*b + A*b^2)*c^2*d^4 + 30*(B
*a^2 + 2*A*a*b)*c*d^5)*n)*x)*(d*x + c)^n/(d^6*n^6 + 21*d^6*n^5 + 175*d^6*n^4 + 7
35*d^6*n^3 + 1624*d^6*n^2 + 1764*d^6*n + 720*d^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((b*x+a)**2*(d*x+c)**n*(D*x**3+C*x**2+B*x+A),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.221591, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((D*x^3 + C*x^2 + B*x + A)*(b*x + a)^2*(d*x + c)^n,x, algorithm="giac")

[Out]

Done