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

Optimal. Leaf size=434 \[ -\frac{2 (c+d x)^{3/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{3 d^7}+\frac{2 b (c+d x)^{7/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{7 d^7}+\frac{2 (c+d x)^{5/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{5 d^7}-\frac{2 \sqrt{c+d x} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7}+\frac{2 (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7 \sqrt{c+d x}}+\frac{2 b^2 (c+d x)^{9/2} (3 a d D-6 b c D+b C d)}{9 d^7}+\frac{2 b^3 D (c+d x)^{11/2}}{11 d^7} \]

[Out]

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

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Rubi [A]  time = 0.846959, 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 \[ -\frac{2 (c+d x)^{3/2} (b c-a d) \left (a^2 d^2 (C d-3 c D)-a b d \left (-3 B d^2-15 c^2 D+8 c C d\right )+b^2 \left (3 A d^3-6 B c d^2-15 c^3 D+10 c^2 C d\right )\right )}{3 d^7}+\frac{2 b (c+d x)^{7/2} \left (3 a^2 d^2 D+3 a b d (C d-5 c D)+b^2 \left (-\left (-B d^2-15 c^2 D+5 c C d\right )\right )\right )}{7 d^7}+\frac{2 (c+d x)^{5/2} \left (a^3 d^3 D+3 a^2 b d^2 (C d-4 c D)-3 a b^2 d \left (-B d^2-10 c^2 D+4 c C d\right )+b^3 \left (A d^3-4 B c d^2-20 c^3 D+10 c^2 C d\right )\right )}{5 d^7}-\frac{2 \sqrt{c+d x} (b c-a d)^2 \left (a d \left (-B d^2-3 c^2 D+2 c C d\right )-b \left (3 A d^3-4 B c d^2-6 c^3 D+5 c^2 C d\right )\right )}{d^7}+\frac{2 (b c-a d)^3 \left (A d^3-B c d^2+c^3 (-D)+c^2 C d\right )}{d^7 \sqrt{c+d x}}+\frac{2 b^2 (c+d x)^{9/2} (3 a d D-6 b c D+b C d)}{9 d^7}+\frac{2 b^3 D (c+d x)^{11/2}}{11 d^7} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*c - a*d)^3*(c^2*C*d - B*c*d^2 + A*d^3 - c^3*D))/(d^7*Sqrt[c + d*x]) - (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))*Sqrt[c + d*x])/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)^(3/2))/(3*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)^(5/2))/(5*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)^(7/2))/(7*d^7) + (2*b^2*(b*C*d - 6*b*c
*D + 3*a*d*D)*(c + d*x)^(9/2))/(9*d^7) + (2*b^3*D*(c + d*x)^(11/2))/(11*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)**(3/2),x)

[Out]

Timed out

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Mathematica [A]  time = 1.55113, size = 500, normalized size = 1.15 \[ \frac{2 \left (231 a^3 d^3 \left (d^3 \left (x \left (15 B+5 C x+3 D x^2\right )-15 A\right )+2 c d^2 (15 B-x (10 C+3 D x))+48 c^3 D-8 c^2 d (5 C-3 D x)\right )+99 a^2 b d^2 \left (2 c d^3 (105 A-x (70 B+3 x (7 C+4 D x)))+d^4 x (105 A+x (35 B+3 x (7 C+5 D x)))-8 c^2 d^2 (35 B-3 x (7 C+2 D x))-384 c^4 D+48 c^3 d (7 C-4 D x)\right )+33 a b^2 d \left (8 c^2 d^3 (x (63 B+2 x (9 C+5 D x))-105 A)-2 c d^4 x (210 A+x (63 B+x (36 C+25 D x)))+d^5 x^2 (105 A+x (63 B+5 x (9 C+7 D x)))+16 c^3 d^2 (63 B-2 x (18 C+5 D x))+1280 c^5 D-128 c^4 d (9 C-5 D x)\right )+b^3 \left (16 c^3 d^3 (693 A-2 x (198 B+5 x (11 C+6 D x)))+8 c^2 d^4 x (693 A+x (198 B+5 x (22 C+15 D x)))-2 c d^5 x^2 (693 A+x (396 B+5 x (55 C+42 D x)))+d^6 x^3 (693 A+5 x (99 B+7 x (11 C+9 D x)))-128 c^4 d^2 (99 B-5 x (11 C+3 D x))-15360 c^6 D+1280 c^5 d (11 C-6 D x)\right )\right )}{3465 d^7 \sqrt{c+d x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(231*a^3*d^3*(48*c^3*D - 8*c^2*d*(5*C - 3*D*x) + 2*c*d^2*(15*B - x*(10*C + 3*
D*x)) + d^3*(-15*A + x*(15*B + 5*C*x + 3*D*x^2))) + 99*a^2*b*d^2*(-384*c^4*D + 4
8*c^3*d*(7*C - 4*D*x) - 8*c^2*d^2*(35*B - 3*x*(7*C + 2*D*x)) + 2*c*d^3*(105*A -
x*(70*B + 3*x*(7*C + 4*D*x))) + d^4*x*(105*A + x*(35*B + 3*x*(7*C + 5*D*x)))) +
33*a*b^2*d*(1280*c^5*D - 128*c^4*d*(9*C - 5*D*x) + 16*c^3*d^2*(63*B - 2*x*(18*C
+ 5*D*x)) + 8*c^2*d^3*(-105*A + x*(63*B + 2*x*(9*C + 5*D*x))) + d^5*x^2*(105*A +
x*(63*B + 5*x*(9*C + 7*D*x))) - 2*c*d^4*x*(210*A + x*(63*B + x*(36*C + 25*D*x))
)) + b^3*(-15360*c^6*D + 1280*c^5*d*(11*C - 6*D*x) - 128*c^4*d^2*(99*B - 5*x*(11
*C + 3*D*x)) + 16*c^3*d^3*(693*A - 2*x*(198*B + 5*x*(11*C + 6*D*x))) + d^6*x^3*(
693*A + 5*x*(99*B + 7*x*(11*C + 9*D*x))) + 8*c^2*d^4*x*(693*A + x*(198*B + 5*x*(
22*C + 15*D*x))) - 2*c*d^5*x^2*(693*A + x*(396*B + 5*x*(55*C + 42*D*x))))))/(346
5*d^7*Sqrt[c + d*x])

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Maple [B]  time = 0.013, size = 841, normalized size = 1.9 \[ -{\frac{-630\,{b}^{3}D{x}^{6}{d}^{6}-770\,C{b}^{3}{d}^{6}{x}^{5}-2310\,Da{b}^{2}{d}^{6}{x}^{5}+840\,D{b}^{3}c{d}^{5}{x}^{5}-990\,B{b}^{3}{d}^{6}{x}^{4}-2970\,Ca{b}^{2}{d}^{6}{x}^{4}+1100\,C{b}^{3}c{d}^{5}{x}^{4}-2970\,D{a}^{2}b{d}^{6}{x}^{4}+3300\,Da{b}^{2}c{d}^{5}{x}^{4}-1200\,D{b}^{3}{c}^{2}{d}^{4}{x}^{4}-1386\,A{b}^{3}{d}^{6}{x}^{3}-4158\,Ba{b}^{2}{d}^{6}{x}^{3}+1584\,B{b}^{3}c{d}^{5}{x}^{3}-4158\,C{a}^{2}b{d}^{6}{x}^{3}+4752\,Ca{b}^{2}c{d}^{5}{x}^{3}-1760\,C{b}^{3}{c}^{2}{d}^{4}{x}^{3}-1386\,D{a}^{3}{d}^{6}{x}^{3}+4752\,D{a}^{2}bc{d}^{5}{x}^{3}-5280\,Da{b}^{2}{c}^{2}{d}^{4}{x}^{3}+1920\,D{b}^{3}{c}^{3}{d}^{3}{x}^{3}-6930\,Aa{b}^{2}{d}^{6}{x}^{2}+2772\,A{b}^{3}c{d}^{5}{x}^{2}-6930\,B{a}^{2}b{d}^{6}{x}^{2}+8316\,Ba{b}^{2}c{d}^{5}{x}^{2}-3168\,B{b}^{3}{c}^{2}{d}^{4}{x}^{2}-2310\,C{a}^{3}{d}^{6}{x}^{2}+8316\,C{a}^{2}bc{d}^{5}{x}^{2}-9504\,Ca{b}^{2}{c}^{2}{d}^{4}{x}^{2}+3520\,C{b}^{3}{c}^{3}{d}^{3}{x}^{2}+2772\,D{a}^{3}c{d}^{5}{x}^{2}-9504\,D{a}^{2}b{c}^{2}{d}^{4}{x}^{2}+10560\,Da{b}^{2}{c}^{3}{d}^{3}{x}^{2}-3840\,D{b}^{3}{c}^{4}{d}^{2}{x}^{2}-20790\,A{a}^{2}b{d}^{6}x+27720\,Aa{b}^{2}c{d}^{5}x-11088\,A{b}^{3}{c}^{2}{d}^{4}x-6930\,B{a}^{3}{d}^{6}x+27720\,B{a}^{2}bc{d}^{5}x-33264\,Ba{b}^{2}{c}^{2}{d}^{4}x+12672\,B{b}^{3}{c}^{3}{d}^{3}x+9240\,C{a}^{3}c{d}^{5}x-33264\,C{a}^{2}b{c}^{2}{d}^{4}x+38016\,Ca{b}^{2}{c}^{3}{d}^{3}x-14080\,C{b}^{3}{c}^{4}{d}^{2}x-11088\,D{a}^{3}{c}^{2}{d}^{4}x+38016\,D{a}^{2}b{c}^{3}{d}^{3}x-42240\,Da{b}^{2}{c}^{4}{d}^{2}x+15360\,D{b}^{3}{c}^{5}dx+6930\,{a}^{3}A{d}^{6}-41580\,A{a}^{2}bc{d}^{5}+55440\,Aa{b}^{2}{c}^{2}{d}^{4}-22176\,A{b}^{3}{c}^{3}{d}^{3}-13860\,B{a}^{3}c{d}^{5}+55440\,B{a}^{2}b{c}^{2}{d}^{4}-66528\,Ba{b}^{2}{c}^{3}{d}^{3}+25344\,B{b}^{3}{c}^{4}{d}^{2}+18480\,C{a}^{3}{c}^{2}{d}^{4}-66528\,C{a}^{2}b{c}^{3}{d}^{3}+76032\,Ca{b}^{2}{c}^{4}{d}^{2}-28160\,C{b}^{3}{c}^{5}d-22176\,D{a}^{3}{c}^{3}{d}^{3}+76032\,D{a}^{2}b{c}^{4}{d}^{2}-84480\,Da{b}^{2}{c}^{5}d+30720\,D{b}^{3}{c}^{6}}{3465\,{d}^{7}}{\frac{1}{\sqrt{dx+c}}}} \]

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)^(3/2),x)

[Out]

-2/3465/(d*x+c)^(1/2)*(-315*D*b^3*d^6*x^6-385*C*b^3*d^6*x^5-1155*D*a*b^2*d^6*x^5
+420*D*b^3*c*d^5*x^5-495*B*b^3*d^6*x^4-1485*C*a*b^2*d^6*x^4+550*C*b^3*c*d^5*x^4-
1485*D*a^2*b*d^6*x^4+1650*D*a*b^2*c*d^5*x^4-600*D*b^3*c^2*d^4*x^4-693*A*b^3*d^6*
x^3-2079*B*a*b^2*d^6*x^3+792*B*b^3*c*d^5*x^3-2079*C*a^2*b*d^6*x^3+2376*C*a*b^2*c
*d^5*x^3-880*C*b^3*c^2*d^4*x^3-693*D*a^3*d^6*x^3+2376*D*a^2*b*c*d^5*x^3-2640*D*a
*b^2*c^2*d^4*x^3+960*D*b^3*c^3*d^3*x^3-3465*A*a*b^2*d^6*x^2+1386*A*b^3*c*d^5*x^2
-3465*B*a^2*b*d^6*x^2+4158*B*a*b^2*c*d^5*x^2-1584*B*b^3*c^2*d^4*x^2-1155*C*a^3*d
^6*x^2+4158*C*a^2*b*c*d^5*x^2-4752*C*a*b^2*c^2*d^4*x^2+1760*C*b^3*c^3*d^3*x^2+13
86*D*a^3*c*d^5*x^2-4752*D*a^2*b*c^2*d^4*x^2+5280*D*a*b^2*c^3*d^3*x^2-1920*D*b^3*
c^4*d^2*x^2-10395*A*a^2*b*d^6*x+13860*A*a*b^2*c*d^5*x-5544*A*b^3*c^2*d^4*x-3465*
B*a^3*d^6*x+13860*B*a^2*b*c*d^5*x-16632*B*a*b^2*c^2*d^4*x+6336*B*b^3*c^3*d^3*x+4
620*C*a^3*c*d^5*x-16632*C*a^2*b*c^2*d^4*x+19008*C*a*b^2*c^3*d^3*x-7040*C*b^3*c^4
*d^2*x-5544*D*a^3*c^2*d^4*x+19008*D*a^2*b*c^3*d^3*x-21120*D*a*b^2*c^4*d^2*x+7680
*D*b^3*c^5*d*x+3465*A*a^3*d^6-20790*A*a^2*b*c*d^5+27720*A*a*b^2*c^2*d^4-11088*A*
b^3*c^3*d^3-6930*B*a^3*c*d^5+27720*B*a^2*b*c^2*d^4-33264*B*a*b^2*c^3*d^3+12672*B
*b^3*c^4*d^2+9240*C*a^3*c^2*d^4-33264*C*a^2*b*c^3*d^3+38016*C*a*b^2*c^4*d^2-1408
0*C*b^3*c^5*d-11088*D*a^3*c^3*d^3+38016*D*a^2*b*c^4*d^2-42240*D*a*b^2*c^5*d+1536
0*D*b^3*c^6)/d^7

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Maxima [A]  time = 1.36816, size = 849, normalized size = 1.96 \[ \frac{2 \,{\left (\frac{315 \,{\left (d x + c\right )}^{\frac{11}{2}} D b^{3} - 385 \,{\left (6 \, D b^{3} c -{\left (3 \, D a b^{2} + C b^{3}\right )} d\right )}{\left (d x + c\right )}^{\frac{9}{2}} + 495 \,{\left (15 \, D b^{3} c^{2} - 5 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c d +{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{2}\right )}{\left (d x + c\right )}^{\frac{7}{2}} - 693 \,{\left (20 \, D b^{3} c^{3} - 10 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d + 4 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{2} -{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{3}\right )}{\left (d x + c\right )}^{\frac{5}{2}} + 1155 \,{\left (15 \, D b^{3} c^{4} - 10 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d + 6 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{2} - 3 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{4}\right )}{\left (d x + c\right )}^{\frac{3}{2}} - 3465 \,{\left (6 \, D b^{3} c^{5} - 5 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d + 4 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{2} - 3 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{3} + 2 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{4} -{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} \sqrt{d x + c}}{d^{6}} - \frac{3465 \,{\left (D b^{3} c^{6} + A a^{3} d^{6} -{\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d +{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} -{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} +{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} -{\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5}\right )}}{\sqrt{d x + c} d^{6}}\right )}}{3465 \, d} \]

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)^(3/2),x, algorithm="maxima")

[Out]

2/3465*((315*(d*x + c)^(11/2)*D*b^3 - 385*(6*D*b^3*c - (3*D*a*b^2 + C*b^3)*d)*(d
*x + c)^(9/2) + 495*(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)^(7/2) - 693*(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)^(5/2) + 1155*(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)^(3/2) - 3465
*(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)*sqrt(d*x + c))/d^6 - 3465*(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^6))/d

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Fricas [A]  time = 0.220227, size = 841, normalized size = 1.94 \[ \frac{2 \,{\left (315 \, D b^{3} d^{6} x^{6} - 15360 \, D b^{3} c^{6} - 3465 \, A a^{3} d^{6} + 14080 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{5} d - 12672 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{4} d^{2} + 11088 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{3} d^{3} - 9240 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c^{2} d^{4} + 6930 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} c d^{5} - 35 \,{\left (12 \, D b^{3} c d^{5} - 11 \,{\left (3 \, D a b^{2} + C b^{3}\right )} d^{6}\right )} x^{5} + 5 \,{\left (120 \, D b^{3} c^{2} d^{4} - 110 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c d^{5} + 99 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} d^{6}\right )} x^{4} -{\left (960 \, D b^{3} c^{3} d^{3} - 880 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{2} d^{4} + 792 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c d^{5} - 693 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} d^{6}\right )} x^{3} +{\left (1920 \, D b^{3} c^{4} d^{2} - 1760 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{3} d^{3} + 1584 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{2} d^{4} - 1386 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c d^{5} + 1155 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} d^{6}\right )} x^{2} -{\left (7680 \, D b^{3} c^{5} d - 7040 \,{\left (3 \, D a b^{2} + C b^{3}\right )} c^{4} d^{2} + 6336 \,{\left (3 \, D a^{2} b + 3 \, C a b^{2} + B b^{3}\right )} c^{3} d^{3} - 5544 \,{\left (D a^{3} + 3 \, C a^{2} b + 3 \, B a b^{2} + A b^{3}\right )} c^{2} d^{4} + 4620 \,{\left (C a^{3} + 3 \, B a^{2} b + 3 \, A a b^{2}\right )} c d^{5} - 3465 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d^{6}\right )} x\right )}}{3465 \, \sqrt{d x + c} d^{7}} \]

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/(d*x + c)^(3/2),x, algorithm="fricas")

[Out]

2/3465*(315*D*b^3*d^6*x^6 - 15360*D*b^3*c^6 - 3465*A*a^3*d^6 + 14080*(3*D*a*b^2
+ C*b^3)*c^5*d - 12672*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^4*d^2 + 11088*(D*a^3 +
3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^3*d^3 - 9240*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c^
2*d^4 + 6930*(B*a^3 + 3*A*a^2*b)*c*d^5 - 35*(12*D*b^3*c*d^5 - 11*(3*D*a*b^2 + C*
b^3)*d^6)*x^5 + 5*(120*D*b^3*c^2*d^4 - 110*(3*D*a*b^2 + C*b^3)*c*d^5 + 99*(3*D*a
^2*b + 3*C*a*b^2 + B*b^3)*d^6)*x^4 - (960*D*b^3*c^3*d^3 - 880*(3*D*a*b^2 + C*b^3
)*c^2*d^4 + 792*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c*d^5 - 693*(D*a^3 + 3*C*a^2*b +
3*B*a*b^2 + A*b^3)*d^6)*x^3 + (1920*D*b^3*c^4*d^2 - 1760*(3*D*a*b^2 + C*b^3)*c^
3*d^3 + 1584*(3*D*a^2*b + 3*C*a*b^2 + B*b^3)*c^2*d^4 - 1386*(D*a^3 + 3*C*a^2*b +
3*B*a*b^2 + A*b^3)*c*d^5 + 1155*(C*a^3 + 3*B*a^2*b + 3*A*a*b^2)*d^6)*x^2 - (768
0*D*b^3*c^5*d - 7040*(3*D*a*b^2 + C*b^3)*c^4*d^2 + 6336*(3*D*a^2*b + 3*C*a*b^2 +
B*b^3)*c^3*d^3 - 5544*(D*a^3 + 3*C*a^2*b + 3*B*a*b^2 + A*b^3)*c^2*d^4 + 4620*(C
*a^3 + 3*B*a^2*b + 3*A*a*b^2)*c*d^5 - 3465*(B*a^3 + 3*A*a^2*b)*d^6)*x)/(sqrt(d*x
+ c)*d^7)

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

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)**(3/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226548, 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)^(3/2),x, algorithm="giac")

[Out]

Done