3.18 \(\int \frac{(a+b x)^3 \left (A+B x+C x^2+D x^3\right )}{(c+d x)^{5/2}} \, dx\)

Optimal. Leaf size=434 \[ -\frac{2 \sqrt{c+d x} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{d^7}+\frac{2 b (c+d x)^{5/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{5 d^7}+\frac{2 (c+d x)^{3/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{3 d^7}+\frac{2 (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7 \sqrt{c+d x}}+\frac{2 (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^7 (c+d x)^{3/2}}+\frac{2 b^2 (c+d x)^{7/2} (3 a d D-6 b c D+b C d)}{7 d^7}+\frac{2 b^3 D (c+d x)^{9/2}}{9 d^7} \]

[Out]

(2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^7*(c + d*x)^(3/2)) +
(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)))/(d^7*Sqrt[c + d*x]) - (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))*Sqrt[c + d*x])/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)^(3/2))/(3*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)^(5/2))/(5*d^7) + (2*b^2*(b*C*d - 6*b*c
*D + 3*a*d*D)*(c + d*x)^(7/2))/(7*d^7) + (2*b^3*D*(c + d*x)^(9/2))/(9*d^7)

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Rubi [A]  time = 0.862334, antiderivative size = 434, 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 \[ -\frac{2 \sqrt{c+d x} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{d^7}+\frac{2 b (c+d x)^{5/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{5 d^7}+\frac{2 (c+d x)^{3/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{3 d^7}+\frac{2 (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7 \sqrt{c+d x}}+\frac{2 (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{3 d^7 (c+d x)^{3/2}}+\frac{2 b^2 (c+d x)^{7/2} (3 a d D-6 b c D+b C d)}{7 d^7}+\frac{2 b^3 D (c+d x)^{9/2}}{9 d^7} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]

[Out]

(2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(3*d^7*(c + d*x)^(3/2)) +
(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)))/(d^7*Sqrt[c + d*x]) - (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))*Sqrt[c + d*x])/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)^(3/2))/(3*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)^(5/2))/(5*d^7) + (2*b^2*(b*C*d - 6*b*c
*D + 3*a*d*D)*(c + d*x)^(7/2))/(7*d^7) + (2*b^3*D*(c + d*x)^(9/2))/(9*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)**(5/2),x)

[Out]

Timed out

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Mathematica [A]  time = 2.38416, size = 485, normalized size = 1.12 \[ \frac{2 \left (-105 a^3 d^3 \left (d^3 \left (A+3 B x+x^2 (-(3 C+D x))\right )+2 c d^2 (B+3 x (D x-2 C))+16 c^3 D-8 c^2 d (C-3 D x)\right )+63 a^2 b d^2 \left (-2 c d^3 \left (5 A+x \left (-30 B+15 C x+4 D x^2\right )\right )+d^4 x \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+8 c^2 d^2 (5 B+3 x (2 D x-5 C))+128 c^4 D+c^3 (192 d D x-80 C d)\right )+9 a b^2 d \left (8 c^2 d^3 (35 A+x (2 x (21 C+5 D x)-105 B))-2 c d^4 x (x (105 B+x (28 C+15 D x))-210 A)+d^5 x^2 (105 A+x (35 B+3 x (7 C+5 D x)))-16 c^3 d^2 (35 B+6 x (5 D x-14 C))-1280 c^5 D+128 c^4 d (7 C-15 D x)\right )+b^3 \left (-16 c^3 d^3 (105 A+2 x (5 x (9 C+2 D x)-126 B))+24 c^2 d^4 x (x (42 B+5 x (2 C+D x))-105 A)-6 c d^5 x^2 (105 A+x (28 B+5 x (3 C+2 D x)))+d^6 x^3 (105 A+x (63 B+5 x (9 C+7 D x)))+384 c^4 d^2 (7 B+5 x (D x-3 C))+5120 c^6 D-3840 c^5 d (C-2 D x)\right )\right )}{315 d^7 (c+d x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^3*(A + B*x + C*x^2 + D*x^3))/(c + d*x)^(5/2),x]

[Out]

(2*(-105*a^3*d^3*(16*c^3*D - 8*c^2*d*(C - 3*D*x) + 2*c*d^2*(B + 3*x*(-2*C + D*x)
) + d^3*(A + 3*B*x - x^2*(3*C + D*x))) + 63*a^2*b*d^2*(128*c^4*D + c^3*(-80*C*d
+ 192*d*D*x) + 8*c^2*d^2*(5*B + 3*x*(-5*C + 2*D*x)) + d^4*x*(-15*A + x*(15*B + 5
*C*x + 3*D*x^2)) - 2*c*d^3*(5*A + x*(-30*B + 15*C*x + 4*D*x^2))) + b^3*(5120*c^6
*D - 3840*c^5*d*(C - 2*D*x) + 384*c^4*d^2*(7*B + 5*x*(-3*C + D*x)) + 24*c^2*d^4*
x*(-105*A + x*(42*B + 5*x*(2*C + D*x))) - 6*c*d^5*x^2*(105*A + x*(28*B + 5*x*(3*
C + 2*D*x))) - 16*c^3*d^3*(105*A + 2*x*(-126*B + 5*x*(9*C + 2*D*x))) + d^6*x^3*(
105*A + x*(63*B + 5*x*(9*C + 7*D*x)))) + 9*a*b^2*d*(-1280*c^5*D + 128*c^4*d*(7*C
- 15*D*x) - 16*c^3*d^2*(35*B + 6*x*(-14*C + 5*D*x)) + d^5*x^2*(105*A + x*(35*B
+ 3*x*(7*C + 5*D*x))) + 8*c^2*d^3*(35*A + x*(-105*B + 2*x*(21*C + 5*D*x))) - 2*c
*d^4*x*(-210*A + x*(105*B + x*(28*C + 15*D*x))))))/(315*d^7*(c + d*x)^(3/2))

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Maple [B]  time = 0.013, size = 841, normalized size = 1.9 \[ -{\frac{-70\,{b}^{3}D{x}^{6}{d}^{6}-90\,C{b}^{3}{d}^{6}{x}^{5}-270\,Da{b}^{2}{d}^{6}{x}^{5}+120\,D{b}^{3}c{d}^{5}{x}^{5}-126\,B{b}^{3}{d}^{6}{x}^{4}-378\,Ca{b}^{2}{d}^{6}{x}^{4}+180\,C{b}^{3}c{d}^{5}{x}^{4}-378\,D{a}^{2}b{d}^{6}{x}^{4}+540\,Da{b}^{2}c{d}^{5}{x}^{4}-240\,D{b}^{3}{c}^{2}{d}^{4}{x}^{4}-210\,A{b}^{3}{d}^{6}{x}^{3}-630\,Ba{b}^{2}{d}^{6}{x}^{3}+336\,B{b}^{3}c{d}^{5}{x}^{3}-630\,C{a}^{2}b{d}^{6}{x}^{3}+1008\,Ca{b}^{2}c{d}^{5}{x}^{3}-480\,C{b}^{3}{c}^{2}{d}^{4}{x}^{3}-210\,D{a}^{3}{d}^{6}{x}^{3}+1008\,D{a}^{2}bc{d}^{5}{x}^{3}-1440\,Da{b}^{2}{c}^{2}{d}^{4}{x}^{3}+640\,D{b}^{3}{c}^{3}{d}^{3}{x}^{3}-1890\,Aa{b}^{2}{d}^{6}{x}^{2}+1260\,A{b}^{3}c{d}^{5}{x}^{2}-1890\,B{a}^{2}b{d}^{6}{x}^{2}+3780\,Ba{b}^{2}c{d}^{5}{x}^{2}-2016\,B{b}^{3}{c}^{2}{d}^{4}{x}^{2}-630\,C{a}^{3}{d}^{6}{x}^{2}+3780\,C{a}^{2}bc{d}^{5}{x}^{2}-6048\,Ca{b}^{2}{c}^{2}{d}^{4}{x}^{2}+2880\,C{b}^{3}{c}^{3}{d}^{3}{x}^{2}+1260\,D{a}^{3}c{d}^{5}{x}^{2}-6048\,D{a}^{2}b{c}^{2}{d}^{4}{x}^{2}+8640\,Da{b}^{2}{c}^{3}{d}^{3}{x}^{2}-3840\,D{b}^{3}{c}^{4}{d}^{2}{x}^{2}+1890\,A{a}^{2}b{d}^{6}x-7560\,Aa{b}^{2}c{d}^{5}x+5040\,A{b}^{3}{c}^{2}{d}^{4}x+630\,B{a}^{3}{d}^{6}x-7560\,B{a}^{2}bc{d}^{5}x+15120\,Ba{b}^{2}{c}^{2}{d}^{4}x-8064\,B{b}^{3}{c}^{3}{d}^{3}x-2520\,C{a}^{3}c{d}^{5}x+15120\,C{a}^{2}b{c}^{2}{d}^{4}x-24192\,Ca{b}^{2}{c}^{3}{d}^{3}x+11520\,C{b}^{3}{c}^{4}{d}^{2}x+5040\,D{a}^{3}{c}^{2}{d}^{4}x-24192\,D{a}^{2}b{c}^{3}{d}^{3}x+34560\,Da{b}^{2}{c}^{4}{d}^{2}x-15360\,D{b}^{3}{c}^{5}dx+210\,{a}^{3}A{d}^{6}+1260\,A{a}^{2}bc{d}^{5}-5040\,Aa{b}^{2}{c}^{2}{d}^{4}+3360\,A{b}^{3}{c}^{3}{d}^{3}+420\,B{a}^{3}c{d}^{5}-5040\,B{a}^{2}b{c}^{2}{d}^{4}+10080\,Ba{b}^{2}{c}^{3}{d}^{3}-5376\,B{b}^{3}{c}^{4}{d}^{2}-1680\,C{a}^{3}{c}^{2}{d}^{4}+10080\,C{a}^{2}b{c}^{3}{d}^{3}-16128\,Ca{b}^{2}{c}^{4}{d}^{2}+7680\,C{b}^{3}{c}^{5}d+3360\,D{a}^{3}{c}^{3}{d}^{3}-16128\,D{a}^{2}b{c}^{4}{d}^{2}+23040\,Da{b}^{2}{c}^{5}d-10240\,D{b}^{3}{c}^{6}}{315\,{d}^{7}} \left ( dx+c \right ) ^{-{\frac{3}{2}}}} \]

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)^(5/2),x)

[Out]

-2/315/(d*x+c)^(3/2)*(-35*D*b^3*d^6*x^6-45*C*b^3*d^6*x^5-135*D*a*b^2*d^6*x^5+60*
D*b^3*c*d^5*x^5-63*B*b^3*d^6*x^4-189*C*a*b^2*d^6*x^4+90*C*b^3*c*d^5*x^4-189*D*a^
2*b*d^6*x^4+270*D*a*b^2*c*d^5*x^4-120*D*b^3*c^2*d^4*x^4-105*A*b^3*d^6*x^3-315*B*
a*b^2*d^6*x^3+168*B*b^3*c*d^5*x^3-315*C*a^2*b*d^6*x^3+504*C*a*b^2*c*d^5*x^3-240*
C*b^3*c^2*d^4*x^3-105*D*a^3*d^6*x^3+504*D*a^2*b*c*d^5*x^3-720*D*a*b^2*c^2*d^4*x^
3+320*D*b^3*c^3*d^3*x^3-945*A*a*b^2*d^6*x^2+630*A*b^3*c*d^5*x^2-945*B*a^2*b*d^6*
x^2+1890*B*a*b^2*c*d^5*x^2-1008*B*b^3*c^2*d^4*x^2-315*C*a^3*d^6*x^2+1890*C*a^2*b
*c*d^5*x^2-3024*C*a*b^2*c^2*d^4*x^2+1440*C*b^3*c^3*d^3*x^2+630*D*a^3*c*d^5*x^2-3
024*D*a^2*b*c^2*d^4*x^2+4320*D*a*b^2*c^3*d^3*x^2-1920*D*b^3*c^4*d^2*x^2+945*A*a^
2*b*d^6*x-3780*A*a*b^2*c*d^5*x+2520*A*b^3*c^2*d^4*x+315*B*a^3*d^6*x-3780*B*a^2*b
*c*d^5*x+7560*B*a*b^2*c^2*d^4*x-4032*B*b^3*c^3*d^3*x-1260*C*a^3*c*d^5*x+7560*C*a
^2*b*c^2*d^4*x-12096*C*a*b^2*c^3*d^3*x+5760*C*b^3*c^4*d^2*x+2520*D*a^3*c^2*d^4*x
-12096*D*a^2*b*c^3*d^3*x+17280*D*a*b^2*c^4*d^2*x-7680*D*b^3*c^5*d*x+105*A*a^3*d^
6+630*A*a^2*b*c*d^5-2520*A*a*b^2*c^2*d^4+1680*A*b^3*c^3*d^3+210*B*a^3*c*d^5-2520
*B*a^2*b*c^2*d^4+5040*B*a*b^2*c^3*d^3-2688*B*b^3*c^4*d^2-840*C*a^3*c^2*d^4+5040*
C*a^2*b*c^3*d^3-8064*C*a*b^2*c^4*d^2+3840*C*b^3*c^5*d+1680*D*a^3*c^3*d^3-8064*D*
a^2*b*c^4*d^2+11520*D*a*b^2*c^5*d-5120*D*b^3*c^6)/d^7

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Maxima [A]  time = 1.37251, size = 846, normalized size = 1.95 \[ \frac{2 \,{\left (\frac{35 \,{\left (d x + c\right )}^{\frac{9}{2}} D b^{3} - 45 \,{\left (6 \, D b^{3} c -{\left (3 \, D a b^{2} + C b^{3}\right )} d\right )}{\left (d x + c\right )}^{\frac{7}{2}} + 63 \,{\left (15 \, D b^{3} c^{2} - 5 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c d +{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )}{\left (d x + c\right )}^{\frac{5}{2}} - 105 \,{\left (20 \, D b^{3} c^{3} - 10 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d + 4 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{2} -{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )}{\left (d x + c\right )}^{\frac{3}{2}} + 315 \,{\left (15 \, D b^{3} c^{4} - 10 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d + 6 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{2} - 3 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )} \sqrt{d x + c}}{d^{6}} - \frac{105 \,{\left (D b^{3} c^{6} + A a^{3} d^{6} -{\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d +{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} -{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} -{\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} - 3 \,{\left (6 \, D b^{3} c^{5} - 5 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d + 4 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} - 3 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} + 2 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{4} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )}{\left (d x + c\right )}\right )}}{{\left (d x + c\right )}^{\frac{3}{2}} d^{6}}\right )}}{315 \, d} \]

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/(d*x + c)^(5/2),x, algorithm="maxima")

[Out]

2/315*((35*(d*x + c)^(9/2)*D*b^3 - 45*(6*D*b^3*c - (3*D*a*b^2 + C*b^3)*d)*(d*x +
c)^(7/2) + 63*(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)^(5/2) - 105*(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)^(3/2) + 315*(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)*sqrt(d*x + c))/d^6 - 105*(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 - 3*(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))/((d*x + c)^(3/2)*d^6))/d

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Fricas [A]  time = 0.231058, size = 856, normalized size = 1.97 \[ \frac{2 \,{\left (35 \, D b^{3} d^{6} x^{6} + 5120 \, D b^{3} c^{6} - 105 \, A a^{3} d^{6} - 3840 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d + 2688 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} - 1680 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} + 840 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} - 210 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} - 15 \,{\left (4 \, D b^{3} c d^{5} - 3 \,{\left (3 \, D a b^{2} + C b^{3}\right )} d^{6}\right )} x^{5} + 3 \,{\left (40 \, D b^{3} c^{2} d^{4} - 30 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c d^{5} + 21 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{6}\right )} x^{4} -{\left (320 \, D b^{3} c^{3} d^{3} - 240 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d^{4} + 168 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{5} - 105 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{6}\right )} x^{3} + 3 \,{\left (640 \, D b^{3} c^{4} d^{2} - 480 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d^{3} + 336 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{4} - 210 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{5} + 105 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6}\right )} x^{2} + 3 \,{\left (2560 \, D b^{3} c^{5} d - 1920 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d^{2} + 1344 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{3} - 840 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{4} + 420 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} - 105 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6}\right )} x\right )}}{315 \,{\left (d^{8} x + c d^{7}\right )} \sqrt{d x + c}} \]

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/(d*x + c)^(5/2),x, algorithm="fricas")

[Out]

2/315*(35*D*b^3*d^6*x^6 + 5120*D*b^3*c^6 - 105*A*a^3*d^6 - 3840*(3*D*a*b^2 + C*b
^3)*c^5*d + 2688*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 - 1680*(D*a^3 + 3*C*a^2
*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 + 840*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^2*d^4 -
210*(B*a^3 + 3*A*a^2*b)*c*d^5 - 15*(4*D*b^3*c*d^5 - 3*(3*D*a*b^2 + C*b^3)*d^6)*x
^5 + 3*(40*D*b^3*c^2*d^4 - 30*(3*D*a*b^2 + C*b^3)*c*d^5 + 21*(3*D*a^2*b + 3*C*a*
b^2 + B*b^3)*d^6)*x^4 - (320*D*b^3*c^3*d^3 - 240*(3*D*a*b^2 + C*b^3)*c^2*d^4 + 1
68*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^5 - 105*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 +
A*b^3)*d^6)*x^3 + 3*(640*D*b^3*c^4*d^2 - 480*(3*D*a*b^2 + C*b^3)*c^3*d^3 + 336*(
3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^2*d^4 - 210*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*
b^3)*c*d^5 + 105*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^6)*x^2 + 3*(2560*D*b^3*c^5*d
- 1920*(3*D*a*b^2 + C*b^3)*c^4*d^2 + 1344*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^3*d^
3 - 840*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^4 + 420*(C*a^3 + 3*B*a^2*b
+ 3*A*a*b^2)*c*d^5 - 105*(B*a^3 + 3*A*a^2*b)*d^6)*x)/((d^8*x + c*d^7)*sqrt(d*x
+ c))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (a + b x\right )^{3} \left (A + B x + C x^{2} + D x^{3}\right )}{\left (c + d x\right )^{\frac{5}{2}}}\, dx \]

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)**(5/2),x)

[Out]

Integral((a + b*x)**3*(A + B*x + C*x**2 + D*x**3)/(c + d*x)**(5/2), x)

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GIAC/XCAS [A]  time = 0.222604, 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/(d*x + c)^(5/2),x, algorithm="giac")

[Out]

Done