3.364 \(\int \frac{\left (a+b x^2\right )^3 \left (A+B x^2\right )}{x^{3/2}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{3} a^2 x^{3/2} (a B+3 A b)+\frac{2}{11} b^2 x^{11/2} (3 a B+A b)+\frac{6}{7} a b x^{7/2} (a B+A b)+\frac{2}{15} b^3 B x^{15/2} \]

[Out]

(-2*a^3*A)/Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(3/2))/3 + (6*a*b*(A*b + a*B)*x^(7/2
))/7 + (2*b^2*(A*b + 3*a*B)*x^(11/2))/11 + (2*b^3*B*x^(15/2))/15

_______________________________________________________________________________________

Rubi [A]  time = 0.114803, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 a^3 A}{\sqrt{x}}+\frac{2}{3} a^2 x^{3/2} (a B+3 A b)+\frac{2}{11} b^2 x^{11/2} (3 a B+A b)+\frac{6}{7} a b x^{7/2} (a B+A b)+\frac{2}{15} b^3 B x^{15/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^3*A)/Sqrt[x] + (2*a^2*(3*A*b + a*B)*x^(3/2))/3 + (6*a*b*(A*b + a*B)*x^(7/2
))/7 + (2*b^2*(A*b + 3*a*B)*x^(11/2))/11 + (2*b^3*B*x^(15/2))/15

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 16.7358, size = 82, normalized size = 0.99 \[ - \frac{2 A a^{3}}{\sqrt{x}} + \frac{2 B b^{3} x^{\frac{15}{2}}}{15} + 2 a^{2} x^{\frac{3}{2}} \left (A b + \frac{B a}{3}\right ) + \frac{6 a b x^{\frac{7}{2}} \left (A b + B a\right )}{7} + \frac{2 b^{2} x^{\frac{11}{2}} \left (A b + 3 B a\right )}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**3/sqrt(x) + 2*B*b**3*x**(15/2)/15 + 2*a**2*x**(3/2)*(A*b + B*a/3) + 6*a*
b*x**(7/2)*(A*b + B*a)/7 + 2*b**2*x**(11/2)*(A*b + 3*B*a)/11

_______________________________________________________________________________________

Mathematica [A]  time = 0.0448136, size = 71, normalized size = 0.86 \[ \frac{2 \left (-1155 a^3 A+385 a^2 x^2 (a B+3 A b)+105 b^2 x^6 (3 a B+A b)+495 a b x^4 (a B+A b)+77 b^3 B x^8\right )}{1155 \sqrt{x}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-1155*a^3*A + 385*a^2*(3*A*b + a*B)*x^2 + 495*a*b*(A*b + a*B)*x^4 + 105*b^2*
(A*b + 3*a*B)*x^6 + 77*b^3*B*x^8))/(1155*Sqrt[x])

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 80, normalized size = 1. \[ -{\frac{-154\,{b}^{3}B{x}^{8}-210\,{x}^{6}{b}^{3}A-630\,{x}^{6}a{b}^{2}B-990\,{x}^{4}a{b}^{2}A-990\,{x}^{4}{a}^{2}bB-2310\,{x}^{2}A{a}^{2}b-770\,{x}^{2}B{a}^{3}+2310\,{a}^{3}A}{1155}{\frac{1}{\sqrt{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/1155*(-77*B*b^3*x^8-105*A*b^3*x^6-315*B*a*b^2*x^6-495*A*a*b^2*x^4-495*B*a^2*b
*x^4-1155*A*a^2*b*x^2-385*B*a^3*x^2+1155*A*a^3)/x^(1/2)

_______________________________________________________________________________________

Maxima [A]  time = 1.34287, size = 99, normalized size = 1.19 \[ \frac{2}{15} \, B b^{3} x^{\frac{15}{2}} + \frac{2}{11} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{11}{2}} + \frac{6}{7} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{7}{2}} - \frac{2 \, A a^{3}}{\sqrt{x}} + \frac{2}{3} \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*B*b^3*x^(15/2) + 2/11*(3*B*a*b^2 + A*b^3)*x^(11/2) + 6/7*(B*a^2*b + A*a*b^2
)*x^(7/2) - 2*A*a^3/sqrt(x) + 2/3*(B*a^3 + 3*A*a^2*b)*x^(3/2)

_______________________________________________________________________________________

Fricas [A]  time = 0.224956, size = 101, normalized size = 1.22 \[ \frac{2 \,{\left (77 \, B b^{3} x^{8} + 105 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 495 \,{\left (B a^{2} b + A a b^{2}\right )} x^{4} - 1155 \, A a^{3} + 385 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{1155 \, \sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/1155*(77*B*b^3*x^8 + 105*(3*B*a*b^2 + A*b^3)*x^6 + 495*(B*a^2*b + A*a*b^2)*x^4
- 1155*A*a^3 + 385*(B*a^3 + 3*A*a^2*b)*x^2)/sqrt(x)

_______________________________________________________________________________________

Sympy [A]  time = 20.2476, size = 110, normalized size = 1.33 \[ - \frac{2 A a^{3}}{\sqrt{x}} + 2 A a^{2} b x^{\frac{3}{2}} + \frac{6 A a b^{2} x^{\frac{7}{2}}}{7} + \frac{2 A b^{3} x^{\frac{11}{2}}}{11} + \frac{2 B a^{3} x^{\frac{3}{2}}}{3} + \frac{6 B a^{2} b x^{\frac{7}{2}}}{7} + \frac{6 B a b^{2} x^{\frac{11}{2}}}{11} + \frac{2 B b^{3} x^{\frac{15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**3/sqrt(x) + 2*A*a**2*b*x**(3/2) + 6*A*a*b**2*x**(7/2)/7 + 2*A*b**3*x**(1
1/2)/11 + 2*B*a**3*x**(3/2)/3 + 6*B*a**2*b*x**(7/2)/7 + 6*B*a*b**2*x**(11/2)/11
+ 2*B*b**3*x**(15/2)/15

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.211181, size = 104, normalized size = 1.25 \[ \frac{2}{15} \, B b^{3} x^{\frac{15}{2}} + \frac{6}{11} \, B a b^{2} x^{\frac{11}{2}} + \frac{2}{11} \, A b^{3} x^{\frac{11}{2}} + \frac{6}{7} \, B a^{2} b x^{\frac{7}{2}} + \frac{6}{7} \, A a b^{2} x^{\frac{7}{2}} + \frac{2}{3} \, B a^{3} x^{\frac{3}{2}} + 2 \, A a^{2} b x^{\frac{3}{2}} - \frac{2 \, A a^{3}}{\sqrt{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/15*B*b^3*x^(15/2) + 6/11*B*a*b^2*x^(11/2) + 2/11*A*b^3*x^(11/2) + 6/7*B*a^2*b*
x^(7/2) + 6/7*A*a*b^2*x^(7/2) + 2/3*B*a^3*x^(3/2) + 2*A*a^2*b*x^(3/2) - 2*A*a^3/
sqrt(x)