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

Optimal. Leaf size=39 \[ \frac{2}{9} x^{9/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{13} b B x^{13/2} \]

[Out]

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

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Rubi [A]  time = 0.0497932, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{2}{9} x^{9/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{13} b B x^{13/2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.06465, size = 41, normalized size = 1.05 \[ \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 B b x^{\frac{13}{2}}}{13} + x^{\frac{9}{2}} \left (\frac{2 A b}{9} + \frac{2 B a}{9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0169249, size = 33, normalized size = 0.85 \[ \frac{2}{585} x^{5/2} \left (65 x^2 (a B+A b)+117 a A+45 b B x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(5/2)*(117*a*A + 65*(A*b + a*B)*x^2 + 45*b*B*x^4))/585

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \[{\frac{90\,bB{x}^{4}+130\,A{x}^{2}b+130\,B{x}^{2}a+234\,Aa}{585}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/585*x^(5/2)*(45*B*b*x^4+65*A*b*x^2+65*B*a*x^2+117*A*a)

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Maxima [A]  time = 1.35172, size = 36, normalized size = 0.92 \[ \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{9} \,{\left (B a + A b\right )} x^{\frac{9}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.21123, size = 43, normalized size = 1.1 \[ \frac{2}{585} \,{\left (45 \, B b x^{6} + 65 \,{\left (B a + A b\right )} x^{4} + 117 \, A a x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/585*(45*B*b*x^6 + 65*(B*a + A*b)*x^4 + 117*A*a*x^2)*sqrt(x)

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Sympy [A]  time = 7.92613, size = 46, normalized size = 1.18 \[ \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 A b x^{\frac{9}{2}}}{9} + \frac{2 B a x^{\frac{9}{2}}}{9} + \frac{2 B b x^{\frac{13}{2}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216553, size = 39, normalized size = 1. \[ \frac{2}{13} \, B b x^{\frac{13}{2}} + \frac{2}{9} \, B a x^{\frac{9}{2}} + \frac{2}{9} \, A b x^{\frac{9}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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