3.387 \(\int x^3 (a+b x)^{3/2} (A+B x) \, dx\)

Optimal. Leaf size=122 \[ -\frac{2 a^3 (a+b x)^{5/2} (A b-a B)}{5 b^5}+\frac{2 a^2 (a+b x)^{7/2} (3 A b-4 a B)}{7 b^5}+\frac{2 (a+b x)^{11/2} (A b-4 a B)}{11 b^5}-\frac{2 a (a+b x)^{9/2} (A b-2 a B)}{3 b^5}+\frac{2 B (a+b x)^{13/2}}{13 b^5} \]

[Out]

(-2*a^3*(A*b - a*B)*(a + b*x)^(5/2))/(5*b^5) + (2*a^2*(3*A*b - 4*a*B)*(a + b*x)^
(7/2))/(7*b^5) - (2*a*(A*b - 2*a*B)*(a + b*x)^(9/2))/(3*b^5) + (2*(A*b - 4*a*B)*
(a + b*x)^(11/2))/(11*b^5) + (2*B*(a + b*x)^(13/2))/(13*b^5)

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Rubi [A]  time = 0.15479, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ -\frac{2 a^3 (a+b x)^{5/2} (A b-a B)}{5 b^5}+\frac{2 a^2 (a+b x)^{7/2} (3 A b-4 a B)}{7 b^5}+\frac{2 (a+b x)^{11/2} (A b-4 a B)}{11 b^5}-\frac{2 a (a+b x)^{9/2} (A b-2 a B)}{3 b^5}+\frac{2 B (a+b x)^{13/2}}{13 b^5} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^3*(A*b - a*B)*(a + b*x)^(5/2))/(5*b^5) + (2*a^2*(3*A*b - 4*a*B)*(a + b*x)^
(7/2))/(7*b^5) - (2*a*(A*b - 2*a*B)*(a + b*x)^(9/2))/(3*b^5) + (2*(A*b - 4*a*B)*
(a + b*x)^(11/2))/(11*b^5) + (2*B*(a + b*x)^(13/2))/(13*b^5)

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Rubi in Sympy [A]  time = 21.9568, size = 119, normalized size = 0.98 \[ \frac{2 B \left (a + b x\right )^{\frac{13}{2}}}{13 b^{5}} - \frac{2 a^{3} \left (a + b x\right )^{\frac{5}{2}} \left (A b - B a\right )}{5 b^{5}} + \frac{2 a^{2} \left (a + b x\right )^{\frac{7}{2}} \left (3 A b - 4 B a\right )}{7 b^{5}} - \frac{2 a \left (a + b x\right )^{\frac{9}{2}} \left (A b - 2 B a\right )}{3 b^{5}} + \frac{2 \left (a + b x\right )^{\frac{11}{2}} \left (A b - 4 B a\right )}{11 b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x+a)**(3/2)*(B*x+A),x)

[Out]

2*B*(a + b*x)**(13/2)/(13*b**5) - 2*a**3*(a + b*x)**(5/2)*(A*b - B*a)/(5*b**5) +
2*a**2*(a + b*x)**(7/2)*(3*A*b - 4*B*a)/(7*b**5) - 2*a*(a + b*x)**(9/2)*(A*b -
2*B*a)/(3*b**5) + 2*(a + b*x)**(11/2)*(A*b - 4*B*a)/(11*b**5)

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Mathematica [A]  time = 0.084017, size = 87, normalized size = 0.71 \[ \frac{2 (a+b x)^{5/2} \left (128 a^4 B-16 a^3 b (13 A+20 B x)+40 a^2 b^2 x (13 A+14 B x)-70 a b^3 x^2 (13 A+12 B x)+105 b^4 x^3 (13 A+11 B x)\right )}{15015 b^5} \]

Antiderivative was successfully verified.

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

[Out]

(2*(a + b*x)^(5/2)*(128*a^4*B + 105*b^4*x^3*(13*A + 11*B*x) - 70*a*b^3*x^2*(13*A
+ 12*B*x) + 40*a^2*b^2*x*(13*A + 14*B*x) - 16*a^3*b*(13*A + 20*B*x)))/(15015*b^
5)

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Maple [A]  time = 0.008, size = 95, normalized size = 0.8 \[ -{\frac{-2310\,B{x}^{4}{b}^{4}-2730\,A{b}^{4}{x}^{3}+1680\,Ba{b}^{3}{x}^{3}+1820\,Aa{b}^{3}{x}^{2}-1120\,B{a}^{2}{b}^{2}{x}^{2}-1040\,A{a}^{2}{b}^{2}x+640\,B{a}^{3}bx+416\,A{a}^{3}b-256\,B{a}^{4}}{15015\,{b}^{5}} \left ( bx+a \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^(3/2)*(B*x+A),x)

[Out]

-2/15015*(b*x+a)^(5/2)*(-1155*B*b^4*x^4-1365*A*b^4*x^3+840*B*a*b^3*x^3+910*A*a*b
^3*x^2-560*B*a^2*b^2*x^2-520*A*a^2*b^2*x+320*B*a^3*b*x+208*A*a^3*b-128*B*a^4)/b^
5

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Maxima [A]  time = 1.35179, size = 135, normalized size = 1.11 \[ \frac{2 \,{\left (1155 \,{\left (b x + a\right )}^{\frac{13}{2}} B - 1365 \,{\left (4 \, B a - A b\right )}{\left (b x + a\right )}^{\frac{11}{2}} + 5005 \,{\left (2 \, B a^{2} - A a b\right )}{\left (b x + a\right )}^{\frac{9}{2}} - 2145 \,{\left (4 \, B a^{3} - 3 \, A a^{2} b\right )}{\left (b x + a\right )}^{\frac{7}{2}} + 3003 \,{\left (B a^{4} - A a^{3} b\right )}{\left (b x + a\right )}^{\frac{5}{2}}\right )}}{15015 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(1155*(b*x + a)^(13/2)*B - 1365*(4*B*a - A*b)*(b*x + a)^(11/2) + 5005*(2
*B*a^2 - A*a*b)*(b*x + a)^(9/2) - 2145*(4*B*a^3 - 3*A*a^2*b)*(b*x + a)^(7/2) + 3
003*(B*a^4 - A*a^3*b)*(b*x + a)^(5/2))/b^5

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Fricas [A]  time = 0.207632, size = 193, normalized size = 1.58 \[ \frac{2 \,{\left (1155 \, B b^{6} x^{6} + 128 \, B a^{6} - 208 \, A a^{5} b + 105 \,{\left (14 \, B a b^{5} + 13 \, A b^{6}\right )} x^{5} + 35 \,{\left (B a^{2} b^{4} + 52 \, A a b^{5}\right )} x^{4} - 5 \,{\left (8 \, B a^{3} b^{3} - 13 \, A a^{2} b^{4}\right )} x^{3} + 6 \,{\left (8 \, B a^{4} b^{2} - 13 \, A a^{3} b^{3}\right )} x^{2} - 8 \,{\left (8 \, B a^{5} b - 13 \, A a^{4} b^{2}\right )} x\right )} \sqrt{b x + a}}{15015 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15015*(1155*B*b^6*x^6 + 128*B*a^6 - 208*A*a^5*b + 105*(14*B*a*b^5 + 13*A*b^6)*
x^5 + 35*(B*a^2*b^4 + 52*A*a*b^5)*x^4 - 5*(8*B*a^3*b^3 - 13*A*a^2*b^4)*x^3 + 6*(
8*B*a^4*b^2 - 13*A*a^3*b^3)*x^2 - 8*(8*B*a^5*b - 13*A*a^4*b^2)*x)*sqrt(b*x + a)/
b^5

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Sympy [A]  time = 5.39257, size = 298, normalized size = 2.44 \[ \frac{2 A a \left (- \frac{a^{3} \left (a + b x\right )^{\frac{3}{2}}}{3} + \frac{3 a^{2} \left (a + b x\right )^{\frac{5}{2}}}{5} - \frac{3 a \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{\left (a + b x\right )^{\frac{9}{2}}}{9}\right )}{b^{4}} + \frac{2 A \left (\frac{a^{4} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{4 a^{3} \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{6 a^{2} \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{4 a \left (a + b x\right )^{\frac{9}{2}}}{9} + \frac{\left (a + b x\right )^{\frac{11}{2}}}{11}\right )}{b^{4}} + \frac{2 B a \left (\frac{a^{4} \left (a + b x\right )^{\frac{3}{2}}}{3} - \frac{4 a^{3} \left (a + b x\right )^{\frac{5}{2}}}{5} + \frac{6 a^{2} \left (a + b x\right )^{\frac{7}{2}}}{7} - \frac{4 a \left (a + b x\right )^{\frac{9}{2}}}{9} + \frac{\left (a + b x\right )^{\frac{11}{2}}}{11}\right )}{b^{5}} + \frac{2 B \left (- \frac{a^{5} \left (a + b x\right )^{\frac{3}{2}}}{3} + a^{4} \left (a + b x\right )^{\frac{5}{2}} - \frac{10 a^{3} \left (a + b x\right )^{\frac{7}{2}}}{7} + \frac{10 a^{2} \left (a + b x\right )^{\frac{9}{2}}}{9} - \frac{5 a \left (a + b x\right )^{\frac{11}{2}}}{11} + \frac{\left (a + b x\right )^{\frac{13}{2}}}{13}\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x+a)**(3/2)*(B*x+A),x)

[Out]

2*A*a*(-a**3*(a + b*x)**(3/2)/3 + 3*a**2*(a + b*x)**(5/2)/5 - 3*a*(a + b*x)**(7/
2)/7 + (a + b*x)**(9/2)/9)/b**4 + 2*A*(a**4*(a + b*x)**(3/2)/3 - 4*a**3*(a + b*x
)**(5/2)/5 + 6*a**2*(a + b*x)**(7/2)/7 - 4*a*(a + b*x)**(9/2)/9 + (a + b*x)**(11
/2)/11)/b**4 + 2*B*a*(a**4*(a + b*x)**(3/2)/3 - 4*a**3*(a + b*x)**(5/2)/5 + 6*a*
*2*(a + b*x)**(7/2)/7 - 4*a*(a + b*x)**(9/2)/9 + (a + b*x)**(11/2)/11)/b**5 + 2*
B*(-a**5*(a + b*x)**(3/2)/3 + a**4*(a + b*x)**(5/2) - 10*a**3*(a + b*x)**(7/2)/7
+ 10*a**2*(a + b*x)**(9/2)/9 - 5*a*(a + b*x)**(11/2)/11 + (a + b*x)**(13/2)/13)
/b**5

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GIAC/XCAS [A]  time = 0.216745, size = 427, normalized size = 3.5 \[ \frac{2 \,{\left (\frac{143 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} b^{24} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a b^{24} + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} b^{24} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3} b^{24}\right )} A a}{b^{27}} + \frac{13 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{40} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{40} + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{40} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{40} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{40}\right )} B a}{b^{44}} + \frac{13 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} b^{40} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a b^{40} + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} b^{40} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} b^{40} + 1155 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} b^{40}\right )} A}{b^{43}} + \frac{5 \,{\left (693 \,{\left (b x + a\right )}^{\frac{13}{2}} b^{60} - 4095 \,{\left (b x + a\right )}^{\frac{11}{2}} a b^{60} + 10010 \,{\left (b x + a\right )}^{\frac{9}{2}} a^{2} b^{60} - 12870 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{3} b^{60} + 9009 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{4} b^{60} - 3003 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{5} b^{60}\right )} B}{b^{64}}\right )}}{45045 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^(3/2)*x^3,x, algorithm="giac")

[Out]

2/45045*(143*(35*(b*x + a)^(9/2)*b^24 - 135*(b*x + a)^(7/2)*a*b^24 + 189*(b*x +
a)^(5/2)*a^2*b^24 - 105*(b*x + a)^(3/2)*a^3*b^24)*A*a/b^27 + 13*(315*(b*x + a)^(
11/2)*b^40 - 1540*(b*x + a)^(9/2)*a*b^40 + 2970*(b*x + a)^(7/2)*a^2*b^40 - 2772*
(b*x + a)^(5/2)*a^3*b^40 + 1155*(b*x + a)^(3/2)*a^4*b^40)*B*a/b^44 + 13*(315*(b*
x + a)^(11/2)*b^40 - 1540*(b*x + a)^(9/2)*a*b^40 + 2970*(b*x + a)^(7/2)*a^2*b^40
- 2772*(b*x + a)^(5/2)*a^3*b^40 + 1155*(b*x + a)^(3/2)*a^4*b^40)*A/b^43 + 5*(69
3*(b*x + a)^(13/2)*b^60 - 4095*(b*x + a)^(11/2)*a*b^60 + 10010*(b*x + a)^(9/2)*a
^2*b^60 - 12870*(b*x + a)^(7/2)*a^3*b^60 + 9009*(b*x + a)^(5/2)*a^4*b^60 - 3003*
(b*x + a)^(3/2)*a^5*b^60)*B/b^64)/b