3.1 \(\int \frac{(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{\sqrt{c+d x}} \, dx\)

Optimal. Leaf size=436 \[ -\frac{2 (c+d x)^{5/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{5 d^7}+\frac{2 b (c+d x)^{9/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{9 d^7}+\frac{2 (c+d x)^{7/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{7 d^7}-\frac{2 (c+d x)^{3/2} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{3 d^7}-\frac{2 \sqrt{c+d x} (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7}+\frac{2 b^2 (c+d x)^{11/2} (3 a d D-6 b c D+b C d)}{11 d^7}+\frac{2 b^3 D (c+d x)^{13/2}}{13 d^7} \]

[Out]

(-2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^7 - (2*(b
*c - a*d)^2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^
3 - 6*c^3*D))*(c + d*x)^(3/2))/(3*d^7) - (2*(b*c - a*d)*(a^2*d^2*(C*d - 3*c*D) -
a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 -
15*c^3*D))*(c + d*x)^(5/2))/(5*d^7) + (2*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D)
- 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 -
20*c^3*D))*(c + d*x)^(7/2))/(7*d^7) + (2*b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D)
- b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(9/2))/(9*d^7) + (2*b^2*(b*C*d -
6*b*c*D + 3*a*d*D)*(c + d*x)^(11/2))/(11*d^7) + (2*b^3*D*(c + d*x)^(13/2))/(13*d
^7)

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Rubi [A]  time = 0.859434, antiderivative size = 436, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031 \[ -\frac{2 (c+d x)^{5/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{5 d^7}+\frac{2 b (c+d x)^{9/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{9 d^7}+\frac{2 (c+d x)^{7/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{7 d^7}-\frac{2 (c+d x)^{3/2} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{3 d^7}-\frac{2 \sqrt{c+d x} (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7}+\frac{2 b^2 (c+d x)^{11/2} (3 a d D-6 b c D+b C d)}{11 d^7}+\frac{2 b^3 D (c+d x)^{13/2}}{13 d^7} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D)*Sqrt[c + d*x])/d^7 - (2*(b
*c - a*d)^2*(a*d*(2*c*C*d - B*d^2 - 3*c^2*D) - b*(5*c^2*C*d - 4*B*c*d^2 + 3*A*d^
3 - 6*c^3*D))*(c + d*x)^(3/2))/(3*d^7) - (2*(b*c - a*d)*(a^2*d^2*(C*d - 3*c*D) -
a*b*d*(8*c*C*d - 3*B*d^2 - 15*c^2*D) + b^2*(10*c^2*C*d - 6*B*c*d^2 + 3*A*d^3 -
15*c^3*D))*(c + d*x)^(5/2))/(5*d^7) + (2*(a^3*d^3*D + 3*a^2*b*d^2*(C*d - 4*c*D)
- 3*a*b^2*d*(4*c*C*d - B*d^2 - 10*c^2*D) + b^3*(10*c^2*C*d - 4*B*c*d^2 + A*d^3 -
20*c^3*D))*(c + d*x)^(7/2))/(7*d^7) + (2*b*(3*a^2*d^2*D + 3*a*b*d*(C*d - 5*c*D)
- b^2*(5*c*C*d - B*d^2 - 15*c^2*D))*(c + d*x)^(9/2))/(9*d^7) + (2*b^2*(b*C*d -
6*b*c*D + 3*a*d*D)*(c + d*x)^(11/2))/(11*d^7) + (2*b^3*D*(c + d*x)^(13/2))/(13*d
^7)

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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)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2),x)

[Out]

Timed out

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Mathematica [A]  time = 1.42002, size = 495, normalized size = 1.14 \[ \frac{2 \sqrt{c+d x} \left (429 a^3 d^3 \left (d^3 (105 A+x (35 B+3 x (7 C+5 D x)))-2 c d^2 (35 B+x (14 C+9 D x))-48 c^3 D+8 c^2 d (7 C+3 D x)\right )+429 a^2 b d^2 \left (-2 c d^3 (105 A+x (42 B+x (27 C+20 D x)))+d^4 x (105 A+x (63 B+5 x (9 C+7 D x)))+24 c^2 d^2 (7 B+x (3 C+2 D x))+128 c^4 D-16 c^3 d (9 C+4 D x)\right )+39 a b^2 d \left (8 c^2 d^3 \left (231 A+x \left (99 B+66 C x+50 D x^2\right )\right )-2 c d^4 x (462 A+x (297 B+5 x (44 C+35 D x)))+d^5 x^2 (693 A+5 x (99 B+7 x (11 C+9 D x)))-16 c^3 d^2 \left (99 B+44 C x+30 D x^2\right )-1280 c^5 D+128 c^4 d (11 C+5 D x)\right )+b^3 \left (-16 c^3 d^3 \left (1287 A+572 B x+390 C x^2+300 D x^3\right )+8 c^2 d^4 x \left (1287 A+x \left (858 B+650 C x+525 D x^2\right )\right )-2 c d^5 x^2 (3861 A+5 x (572 B+7 x (65 C+54 D x)))+5 d^6 x^3 \left (1287 A+7 x \left (143 B+117 C x+99 D x^2\right )\right )+128 c^4 d^2 (143 B+5 x (13 C+9 D x))+15360 c^6 D-1280 c^5 d (13 C+6 D x)\right )\right )}{45045 d^7} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c + d*x]*(429*a^3*d^3*(-48*c^3*D + 8*c^2*d*(7*C + 3*D*x) - 2*c*d^2*(35*B
+ x*(14*C + 9*D*x)) + d^3*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)))) + 429*a^2*b*d
^2*(128*c^4*D - 16*c^3*d*(9*C + 4*D*x) + 24*c^2*d^2*(7*B + x*(3*C + 2*D*x)) + d^
4*x*(105*A + x*(63*B + 5*x*(9*C + 7*D*x))) - 2*c*d^3*(105*A + x*(42*B + x*(27*C
+ 20*D*x)))) + 39*a*b^2*d*(-1280*c^5*D + 128*c^4*d*(11*C + 5*D*x) - 16*c^3*d^2*(
99*B + 44*C*x + 30*D*x^2) + 8*c^2*d^3*(231*A + x*(99*B + 66*C*x + 50*D*x^2)) + d
^5*x^2*(693*A + 5*x*(99*B + 7*x*(11*C + 9*D*x))) - 2*c*d^4*x*(462*A + x*(297*B +
5*x*(44*C + 35*D*x)))) + b^3*(15360*c^6*D - 1280*c^5*d*(13*C + 6*D*x) - 16*c^3*
d^3*(1287*A + 572*B*x + 390*C*x^2 + 300*D*x^3) + 128*c^4*d^2*(143*B + 5*x*(13*C
+ 9*D*x)) + 5*d^6*x^3*(1287*A + 7*x*(143*B + 117*C*x + 99*D*x^2)) + 8*c^2*d^4*x*
(1287*A + x*(858*B + 650*C*x + 525*D*x^2)) - 2*c*d^5*x^2*(3861*A + 5*x*(572*B +
7*x*(65*C + 54*D*x))))))/(45045*d^7)

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Maple [B]  time = 0.018, size = 841, normalized size = 1.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(D*x^3+C*x^2+B*x+A)/(d*x+c)^(1/2),x)

[Out]

2/45045*(d*x+c)^(1/2)*(3465*D*b^3*d^6*x^6+4095*C*b^3*d^6*x^5+12285*D*a*b^2*d^6*x
^5-3780*D*b^3*c*d^5*x^5+5005*B*b^3*d^6*x^4+15015*C*a*b^2*d^6*x^4-4550*C*b^3*c*d^
5*x^4+15015*D*a^2*b*d^6*x^4-13650*D*a*b^2*c*d^5*x^4+4200*D*b^3*c^2*d^4*x^4+6435*
A*b^3*d^6*x^3+19305*B*a*b^2*d^6*x^3-5720*B*b^3*c*d^5*x^3+19305*C*a^2*b*d^6*x^3-1
7160*C*a*b^2*c*d^5*x^3+5200*C*b^3*c^2*d^4*x^3+6435*D*a^3*d^6*x^3-17160*D*a^2*b*c
*d^5*x^3+15600*D*a*b^2*c^2*d^4*x^3-4800*D*b^3*c^3*d^3*x^3+27027*A*a*b^2*d^6*x^2-
7722*A*b^3*c*d^5*x^2+27027*B*a^2*b*d^6*x^2-23166*B*a*b^2*c*d^5*x^2+6864*B*b^3*c^
2*d^4*x^2+9009*C*a^3*d^6*x^2-23166*C*a^2*b*c*d^5*x^2+20592*C*a*b^2*c^2*d^4*x^2-6
240*C*b^3*c^3*d^3*x^2-7722*D*a^3*c*d^5*x^2+20592*D*a^2*b*c^2*d^4*x^2-18720*D*a*b
^2*c^3*d^3*x^2+5760*D*b^3*c^4*d^2*x^2+45045*A*a^2*b*d^6*x-36036*A*a*b^2*c*d^5*x+
10296*A*b^3*c^2*d^4*x+15015*B*a^3*d^6*x-36036*B*a^2*b*c*d^5*x+30888*B*a*b^2*c^2*
d^4*x-9152*B*b^3*c^3*d^3*x-12012*C*a^3*c*d^5*x+30888*C*a^2*b*c^2*d^4*x-27456*C*a
*b^2*c^3*d^3*x+8320*C*b^3*c^4*d^2*x+10296*D*a^3*c^2*d^4*x-27456*D*a^2*b*c^3*d^3*
x+24960*D*a*b^2*c^4*d^2*x-7680*D*b^3*c^5*d*x+45045*A*a^3*d^6-90090*A*a^2*b*c*d^5
+72072*A*a*b^2*c^2*d^4-20592*A*b^3*c^3*d^3-30030*B*a^3*c*d^5+72072*B*a^2*b*c^2*d
^4-61776*B*a*b^2*c^3*d^3+18304*B*b^3*c^4*d^2+24024*C*a^3*c^2*d^4-61776*C*a^2*b*c
^3*d^3+54912*C*a*b^2*c^4*d^2-16640*C*b^3*c^5*d-20592*D*a^3*c^3*d^3+54912*D*a^2*b
*c^4*d^2-49920*D*a*b^2*c^5*d+15360*D*b^3*c^6)/d^7

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Maxima [A]  time = 1.36648, size = 838, normalized size = 1.92 \[ \frac{2 \,{\left (3465 \,{\left (d x + c\right )}^{\frac{13}{2}} D b^{3} - 4095 \,{\left (6 \, D b^{3} c -{\left (3 \, D a b^{2} + C b^{3}\right )} d\right )}{\left (d x + c\right )}^{\frac{11}{2}} + 5005 \,{\left (15 \, D b^{3} c^{2} - 5 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c d +{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )}{\left (d x + c\right )}^{\frac{9}{2}} - 6435 \,{\left (20 \, D b^{3} c^{3} - 10 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d + 4 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{2} -{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 9009 \,{\left (15 \, D b^{3} c^{4} - 10 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d + 6 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{2} - 3 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )}{\left (d x + c\right )}^{\frac{5}{2}} - 15015 \,{\left (6 \, D b^{3} c^{5} - 5 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d + 4 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} - 3 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} + 2 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{4} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 45045 \,{\left (D b^{3} c^{6} + A a^{3} d^{6} -{\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d +{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} -{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} -{\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5}\right )} \sqrt{d x + c}\right )}}{45045 \, d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3465*(d*x + c)^(13/2)*D*b^3 - 4095*(6*D*b^3*c - (3*D*a*b^2 + C*b^3)*d)*
(d*x + c)^(11/2) + 5005*(15*D*b^3*c^2 - 5*(3*D*a*b^2 + C*b^3)*c*d + (3*D*a^2*b +
3*C*a*b^2 + B*b^3)*d^2)*(d*x + c)^(9/2) - 6435*(20*D*b^3*c^3 - 10*(3*D*a*b^2 +
C*b^3)*c^2*d + 4*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^2 - (D*a^3 + 3*C*a^2*b + 3*
B*a*b^2 + A*b^3)*d^3)*(d*x + c)^(7/2) + 9009*(15*D*b^3*c^4 - 10*(3*D*a*b^2 + C*b
^3)*c^3*d + 6*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^2*d^2 - 3*(D*a^3 + 3*C*a^2*b + 3
*B*a*b^2 + A*b^3)*c*d^3 + (C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^4)*(d*x + c)^(5/2) -
15015*(6*D*b^3*c^5 - 5*(3*D*a*b^2 + C*b^3)*c^4*d + 4*(3*D*a^2*b + 3*C*a*b^2 + B
*b^3)*c^3*d^2 - 3*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^3 + 2*(C*a^3 + 3
*B*a^2*b + 3*A*a*b^2)*c*d^4 - (B*a^3 + 3*A*a^2*b)*d^5)*(d*x + c)^(3/2) + 45045*(
D*b^3*c^6 + A*a^3*d^6 - (3*D*a*b^2 + C*b^3)*c^5*d + (3*D*a^2*b + 3*C*a*b^2 + B*b
^3)*c^4*d^2 - (D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 + (C*a^3 + 3*B*a^2
*b + 3*A*a*b^2)*c^2*d^4 - (B*a^3 + 3*A*a^2*b)*c*d^5)*sqrt(d*x + c))/d^7

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Fricas [A]  time = 0.217346, size = 842, normalized size = 1.93 \[ \frac{2 \,{\left (3465 \, D b^{3} d^{6} x^{6} + 15360 \, D b^{3} c^{6} + 45045 \, A a^{3} d^{6} - 16640 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d + 18304 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} - 20592 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} + 24024 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - 30030 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} - 315 \,{\left (12 \, D b^{3} c d^{5} - 13 \,{\left (3 \, D a b^{2} + C b^{3}\right )} d^{6}\right )} x^{5} + 35 \,{\left (120 \, D b^{3} c^{2} d^{4} - 130 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c d^{5} + 143 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{6}\right )} x^{4} - 5 \,{\left (960 \, D b^{3} c^{3} d^{3} - 1040 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d^{4} + 1144 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{5} - 1287 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{6}\right )} x^{3} + 3 \,{\left (1920 \, D b^{3} c^{4} d^{2} - 2080 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d^{3} + 2288 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{4} - 2574 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{5} + 3003 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6}\right )} x^{2} -{\left (7680 \, D b^{3} c^{5} d - 8320 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d^{2} + 9152 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{3} - 10296 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{4} + 12012 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} - 15015 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6}\right )} x\right )} \sqrt{d x + c}}{45045 \, d^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45045*(3465*D*b^3*d^6*x^6 + 15360*D*b^3*c^6 + 45045*A*a^3*d^6 - 16640*(3*D*a*b
^2 + C*b^3)*c^5*d + 18304*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 - 20592*(D*a^3
+ 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 + 24024*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2
)*c^2*d^4 - 30030*(B*a^3 + 3*A*a^2*b)*c*d^5 - 315*(12*D*b^3*c*d^5 - 13*(3*D*a*b^
2 + C*b^3)*d^6)*x^5 + 35*(120*D*b^3*c^2*d^4 - 130*(3*D*a*b^2 + C*b^3)*c*d^5 + 14
3*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*d^6)*x^4 - 5*(960*D*b^3*c^3*d^3 - 1040*(3*D*a*
b^2 + C*b^3)*c^2*d^4 + 1144*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^5 - 1287*(D*a^3
+ 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*d^6)*x^3 + 3*(1920*D*b^3*c^4*d^2 - 2080*(3*D*a*
b^2 + C*b^3)*c^3*d^3 + 2288*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^2*d^4 - 2574*(D*a^
3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c*d^5 + 3003*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*
d^6)*x^2 - (7680*D*b^3*c^5*d - 8320*(3*D*a*b^2 + C*b^3)*c^4*d^2 + 9152*(3*D*a^2*
b + 3*C*a*b^2 + B*b^3)*c^3*d^3 - 10296*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c
^2*d^4 + 12012*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^5 - 15015*(B*a^3 + 3*A*a^2*b)
*d^6)*x)*sqrt(d*x + c)/d^7

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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)**3*(D*x**3+C*x**2+B*x+A)/(d*x+c)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.23415, 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)^3/sqrt(d*x + c),x, algorithm="giac")

[Out]

Done