3.132 \(\int x^{5/2} \left (a+b x^3\right ) \left (A+B x^3\right ) \, dx\)

Optimal. Leaf size=39 \[ \frac{2}{13} x^{13/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{19} b B x^{19/2} \]

[Out]

(2*a*A*x^(7/2))/7 + (2*(A*b + a*B)*x^(13/2))/13 + (2*b*B*x^(19/2))/19

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Rubi [A]  time = 0.0576091, 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 \[ \frac{2}{13} x^{13/2} (a B+A b)+\frac{2}{7} a A x^{7/2}+\frac{2}{19} b B x^{19/2} \]

Antiderivative was successfully verified.

[In]  Int[x^(5/2)*(a + b*x^3)*(A + B*x^3),x]

[Out]

(2*a*A*x^(7/2))/7 + (2*(A*b + a*B)*x^(13/2))/13 + (2*b*B*x^(19/2))/19

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Rubi in Sympy [A]  time = 6.0477, size = 41, normalized size = 1.05 \[ \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 B b x^{\frac{19}{2}}}{19} + x^{\frac{13}{2}} \left (\frac{2 A b}{13} + \frac{2 B a}{13}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(5/2)*(b*x**3+a)*(B*x**3+A),x)

[Out]

2*A*a*x**(7/2)/7 + 2*B*b*x**(19/2)/19 + x**(13/2)*(2*A*b/13 + 2*B*a/13)

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Mathematica [A]  time = 0.0200636, size = 33, normalized size = 0.85 \[ \frac{2 x^{7/2} \left (133 x^3 (a B+A b)+247 a A+91 b B x^6\right )}{1729} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(5/2)*(a + b*x^3)*(A + B*x^3),x]

[Out]

(2*x^(7/2)*(247*a*A + 133*(A*b + a*B)*x^3 + 91*b*B*x^6))/1729

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Maple [A]  time = 0.005, size = 32, normalized size = 0.8 \[{\frac{182\,bB{x}^{6}+266\,A{x}^{3}b+266\,B{x}^{3}a+494\,Aa}{1729}{x}^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(5/2)*(b*x^3+a)*(B*x^3+A),x)

[Out]

2/1729*x^(7/2)*(91*B*b*x^6+133*A*b*x^3+133*B*a*x^3+247*A*a)

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Maxima [A]  time = 1.41554, size = 36, normalized size = 0.92 \[ \frac{2}{19} \, B b x^{\frac{19}{2}} + \frac{2}{13} \,{\left (B a + A b\right )} x^{\frac{13}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)*x^(5/2),x, algorithm="maxima")

[Out]

2/19*B*b*x^(19/2) + 2/13*(B*a + A*b)*x^(13/2) + 2/7*A*a*x^(7/2)

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Fricas [A]  time = 0.229348, size = 43, normalized size = 1.1 \[ \frac{2}{1729} \,{\left (91 \, B b x^{9} + 133 \,{\left (B a + A b\right )} x^{6} + 247 \, A a x^{3}\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)*x^(5/2),x, algorithm="fricas")

[Out]

2/1729*(91*B*b*x^9 + 133*(B*a + A*b)*x^6 + 247*A*a*x^3)*sqrt(x)

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Sympy [A]  time = 38.1018, size = 46, normalized size = 1.18 \[ \frac{2 A a x^{\frac{7}{2}}}{7} + \frac{2 A b x^{\frac{13}{2}}}{13} + \frac{2 B a x^{\frac{13}{2}}}{13} + \frac{2 B b x^{\frac{19}{2}}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(5/2)*(b*x**3+a)*(B*x**3+A),x)

[Out]

2*A*a*x**(7/2)/7 + 2*A*b*x**(13/2)/13 + 2*B*a*x**(13/2)/13 + 2*B*b*x**(19/2)/19

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GIAC/XCAS [A]  time = 0.213856, size = 39, normalized size = 1. \[ \frac{2}{19} \, B b x^{\frac{19}{2}} + \frac{2}{13} \, B a x^{\frac{13}{2}} + \frac{2}{13} \, A b x^{\frac{13}{2}} + \frac{2}{7} \, A a x^{\frac{7}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)*x^(5/2),x, algorithm="giac")

[Out]

2/19*B*b*x^(19/2) + 2/13*B*a*x^(13/2) + 2/13*A*b*x^(13/2) + 2/7*A*a*x^(7/2)