3.344 \(\int x^{5/2} \left (a+b x^2\right ) \left (A+B x^2\right ) \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{15} b B x^{15/2} \]

[Out]

(2*a*A*x^(7/2))/7 + (2*(A*b + a*B)*x^(11/2))/11 + (2*b*B*x^(15/2))/15

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Rubi [A]  time = 0.0500869, 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 \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{15} b B x^{15/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a + b*x^2)*(A + B*x^2),x]

[Out]

(2*a*A*x^(7/2))/7 + (2*(A*b + a*B)*x^(11/2))/11 + (2*b*B*x^(15/2))/15

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Rubi in Sympy [A]  time = 7.10942, size = 41, normalized size = 1.05 \[ \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 B b x^{\frac{15}{2}}}{15} + x^{\frac{11}{2}} \left (\frac{2 A b}{11} + \frac{2 B a}{11}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(b*x**2+a)*(B*x**2+A),x)

[Out]

2*A*a*x**(7/2)/7 + 2*B*b*x**(15/2)/15 + x**(11/2)*(2*A*b/11 + 2*B*a/11)

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Mathematica [A]  time = 0.0175703, size = 33, normalized size = 0.85 \[ \frac{2 x^{7/2} \left (105 x^2 (a B+A b)+165 a A+77 b B x^4\right )}{1155} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(a + b*x^2)*(A + B*x^2),x]

[Out]

(2*x^(7/2)*(165*a*A + 105*(A*b + a*B)*x^2 + 77*b*B*x^4))/1155

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \[{\frac{154\,bB{x}^{4}+210\,A{x}^{2}b+210\,B{x}^{2}a+330\,Aa}{1155}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x^2+a)*(B*x^2+A),x)

[Out]

2/1155*x^(7/2)*(77*B*b*x^4+105*A*b*x^2+105*B*a*x^2+165*A*a)

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Maxima [A]  time = 1.34799, size = 36, normalized size = 0.92 \[ \frac{2}{15} \, B b x^{\frac{15}{2}} + \frac{2}{11} \,{\left (B a + A b\right )} x^{\frac{11}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)*x^(5/2),x, algorithm="maxima")

[Out]

2/15*B*b*x^(15/2) + 2/11*(B*a + A*b)*x^(11/2) + 2/7*A*a*x^(7/2)

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Fricas [A]  time = 0.225491, size = 43, normalized size = 1.1 \[ \frac{2}{1155} \,{\left (77 \, B b x^{7} + 105 \,{\left (B a + A b\right )} x^{5} + 165 \, A a x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)*x^(5/2),x, algorithm="fricas")

[Out]

2/1155*(77*B*b*x^7 + 105*(B*a + A*b)*x^5 + 165*A*a*x^3)*sqrt(x)

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Sympy [A]  time = 20.0121, size = 46, normalized size = 1.18 \[ \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 A b x^{\frac{11}{2}}}{11} + \frac{2 B a x^{\frac{11}{2}}}{11} + \frac{2 B b x^{\frac{15}{2}}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(b*x**2+a)*(B*x**2+A),x)

[Out]

2*A*a*x**(7/2)/7 + 2*A*b*x**(11/2)/11 + 2*B*a*x**(11/2)/11 + 2*B*b*x**(15/2)/15

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GIAC/XCAS [A]  time = 0.216875, size = 39, normalized size = 1. \[ \frac{2}{15} \, B b x^{\frac{15}{2}} + \frac{2}{11} \, B a x^{\frac{11}{2}} + \frac{2}{11} \, A b x^{\frac{11}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^2 + A)*(b*x^2 + a)*x^(5/2),x, algorithm="giac")

[Out]

2/15*B*b*x^(15/2) + 2/11*B*a*x^(11/2) + 2/11*A*b*x^(11/2) + 2/7*A*a*x^(7/2)