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

Optimal. Leaf size=39 \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{17} b B x^{17/2} \]

[Out]

(2*a*A*x^(5/2))/5 + (2*(A*b + a*B)*x^(11/2))/11 + (2*b*B*x^(17/2))/17

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Rubi [A]  time = 0.0570354, 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 \[ \frac{2}{11} x^{11/2} (a B+A b)+\frac{2}{5} a A x^{5/2}+\frac{2}{17} b B x^{17/2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*A*x^(5/2))/5 + (2*(A*b + a*B)*x^(11/2))/11 + (2*b*B*x^(17/2))/17

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Rubi in Sympy [A]  time = 6.07963, size = 41, normalized size = 1.05 \[ \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 B b x^{\frac{17}{2}}}{17} + x^{\frac{11}{2}} \left (\frac{2 A b}{11} + \frac{2 B a}{11}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a*x**(5/2)/5 + 2*B*b*x**(17/2)/17 + x**(11/2)*(2*A*b/11 + 2*B*a/11)

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Mathematica [A]  time = 0.0192323, size = 33, normalized size = 0.85 \[ \frac{2}{935} x^{5/2} \left (85 x^3 (a B+A b)+187 a A+55 b B x^6\right ) \]

Antiderivative was successfully verified.

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

[Out]

(2*x^(5/2)*(187*a*A + 85*(A*b + a*B)*x^3 + 55*b*B*x^6))/935

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Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \[{\frac{110\,bB{x}^{6}+170\,A{x}^{3}b+170\,B{x}^{3}a+374\,Aa}{935}{x}^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/935*x^(5/2)*(55*B*b*x^6+85*A*b*x^3+85*B*a*x^3+187*A*a)

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Maxima [A]  time = 1.36883, size = 36, normalized size = 0.92 \[ \frac{2}{17} \, B b x^{\frac{17}{2}} + \frac{2}{11} \,{\left (B a + A b\right )} x^{\frac{11}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*B*b*x^(17/2) + 2/11*(B*a + A*b)*x^(11/2) + 2/5*A*a*x^(5/2)

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Fricas [A]  time = 0.234927, size = 43, normalized size = 1.1 \[ \frac{2}{935} \,{\left (55 \, B b x^{8} + 85 \,{\left (B a + A b\right )} x^{5} + 187 \, A a x^{2}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/935*(55*B*b*x^8 + 85*(B*a + A*b)*x^5 + 187*A*a*x^2)*sqrt(x)

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Sympy [A]  time = 19.4076, size = 46, normalized size = 1.18 \[ \frac{2 A a x^{\frac{5}{2}}}{5} + \frac{2 A b x^{\frac{11}{2}}}{11} + \frac{2 B a x^{\frac{11}{2}}}{11} + \frac{2 B b x^{\frac{17}{2}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a*x**(5/2)/5 + 2*A*b*x**(11/2)/11 + 2*B*a*x**(11/2)/11 + 2*B*b*x**(17/2)/17

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GIAC/XCAS [A]  time = 0.210095, size = 39, normalized size = 1. \[ \frac{2}{17} \, B b x^{\frac{17}{2}} + \frac{2}{11} \, B a x^{\frac{11}{2}} + \frac{2}{11} \, A b x^{\frac{11}{2}} + \frac{2}{5} \, A a x^{\frac{5}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/17*B*b*x^(17/2) + 2/11*B*a*x^(11/2) + 2/11*A*b*x^(11/2) + 2/5*A*a*x^(5/2)