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

Optimal. Leaf size=63 \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{17} b^2 B x^{17/2} \]

[Out]

(2*a^2*A*x^(5/2))/5 + (2*a*(2*A*b + a*B)*x^(9/2))/9 + (2*b*(A*b + 2*a*B)*x^(13/2
))/13 + (2*b^2*B*x^(17/2))/17

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Rubi [A]  time = 0.0898352, antiderivative size = 63, 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 \[ \frac{2}{5} a^2 A x^{5/2}+\frac{2}{13} b x^{13/2} (2 a B+A b)+\frac{2}{9} a x^{9/2} (a B+2 A b)+\frac{2}{17} b^2 B x^{17/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a^2*A*x^(5/2))/5 + (2*a*(2*A*b + a*B)*x^(9/2))/9 + (2*b*(A*b + 2*a*B)*x^(13/2
))/13 + (2*b^2*B*x^(17/2))/17

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Rubi in Sympy [A]  time = 12.9284, size = 63, normalized size = 1. \[ \frac{2 A a^{2} x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{17}{2}}}{17} + \frac{2 a x^{\frac{9}{2}} \left (2 A b + B a\right )}{9} + \frac{2 b x^{\frac{13}{2}} \left (A b + 2 B a\right )}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*x**(5/2)/5 + 2*B*b**2*x**(17/2)/17 + 2*a*x**(9/2)*(2*A*b + B*a)/9 + 2*b
*x**(13/2)*(A*b + 2*B*a)/13

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Mathematica [A]  time = 0.0327183, size = 53, normalized size = 0.84 \[ \frac{2 x^{5/2} \left (1989 a^2 A+765 b x^4 (2 a B+A b)+1105 a x^2 (a B+2 A b)+585 b^2 B x^6\right )}{9945} \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(5/2)*(1989*a^2*A + 1105*a*(2*A*b + a*B)*x^2 + 765*b*(A*b + 2*a*B)*x^4 + 58
5*b^2*B*x^6))/9945

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[{\frac{1170\,{b}^{2}B{x}^{6}+1530\,A{b}^{2}{x}^{4}+3060\,{x}^{4}abB+4420\,aAb{x}^{2}+2210\,{x}^{2}{a}^{2}B+3978\,{a}^{2}A}{9945}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9945*x^(5/2)*(585*B*b^2*x^6+765*A*b^2*x^4+1530*B*a*b*x^4+2210*A*a*b*x^2+1105*B
*a^2*x^2+1989*A*a^2)

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Maxima [A]  time = 1.38131, size = 69, normalized size = 1.1 \[ \frac{2}{17} \, B b^{2} x^{\frac{17}{2}} + \frac{2}{13} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{13}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} + \frac{2}{9} \,{\left (B a^{2} + 2 \, A a b\right )} x^{\frac{9}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*B*b^2*x^(17/2) + 2/13*(2*B*a*b + A*b^2)*x^(13/2) + 2/5*A*a^2*x^(5/2) + 2/9*
(B*a^2 + 2*A*a*b)*x^(9/2)

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Fricas [A]  time = 0.215985, size = 76, normalized size = 1.21 \[ \frac{2}{9945} \,{\left (585 \, B b^{2} x^{8} + 765 \,{\left (2 \, B a b + A b^{2}\right )} x^{6} + 1989 \, A a^{2} x^{2} + 1105 \,{\left (B a^{2} + 2 \, A a b\right )} x^{4}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9945*(585*B*b^2*x^8 + 765*(2*B*a*b + A*b^2)*x^6 + 1989*A*a^2*x^2 + 1105*(B*a^2
+ 2*A*a*b)*x^4)*sqrt(x)

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Sympy [A]  time = 20.4965, size = 80, normalized size = 1.27 \[ \frac{2 A a^{2} x^{\frac{5}{2}}}{5} + \frac{4 A a b x^{\frac{9}{2}}}{9} + \frac{2 A b^{2} x^{\frac{13}{2}}}{13} + \frac{2 B a^{2} x^{\frac{9}{2}}}{9} + \frac{4 B a b x^{\frac{13}{2}}}{13} + \frac{2 B b^{2} x^{\frac{17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*x**(5/2)/5 + 4*A*a*b*x**(9/2)/9 + 2*A*b**2*x**(13/2)/13 + 2*B*a**2*x**(
9/2)/9 + 4*B*a*b*x**(13/2)/13 + 2*B*b**2*x**(17/2)/17

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GIAC/XCAS [A]  time = 0.251165, size = 72, normalized size = 1.14 \[ \frac{2}{17} \, B b^{2} x^{\frac{17}{2}} + \frac{4}{13} \, B a b x^{\frac{13}{2}} + \frac{2}{13} \, A b^{2} x^{\frac{13}{2}} + \frac{2}{9} \, B a^{2} x^{\frac{9}{2}} + \frac{4}{9} \, A a b x^{\frac{9}{2}} + \frac{2}{5} \, A a^{2} x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*B*b^2*x^(17/2) + 4/13*B*a*b*x^(13/2) + 2/13*A*b^2*x^(13/2) + 2/9*B*a^2*x^(9
/2) + 4/9*A*a*b*x^(9/2) + 2/5*A*a^2*x^(5/2)