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

Optimal. Leaf size=39 \[ \frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{9} b B x^{9/2} \]

[Out]

(-2*a*A)/(3*x^(3/2)) + (2*(A*b + a*B)*x^(3/2))/3 + (2*b*B*x^(9/2))/9

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Rubi [A]  time = 0.0574405, 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 \[ \frac{2}{3} x^{3/2} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{9} b B x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*A)/(3*x^(3/2)) + (2*(A*b + a*B)*x^(3/2))/3 + (2*b*B*x^(9/2))/9

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Rubi in Sympy [A]  time = 6.0919, size = 41, normalized size = 1.05 \[ - \frac{2 A a}{3 x^{\frac{3}{2}}} + \frac{2 B b x^{\frac{9}{2}}}{9} + x^{\frac{3}{2}} \left (\frac{2 A b}{3} + \frac{2 B a}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a/(3*x**(3/2)) + 2*B*b*x**(9/2)/9 + x**(3/2)*(2*A*b/3 + 2*B*a/3)

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Mathematica [A]  time = 0.0202908, size = 34, normalized size = 0.87 \[ \frac{2 \left (-3 a A+3 a B x^3+3 A b x^3+b B x^6\right )}{9 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-3*a*A + 3*A*b*x^3 + 3*a*B*x^3 + b*B*x^6))/(9*x^(3/2))

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Maple [A]  time = 0.006, size = 32, normalized size = 0.8 \[ -{\frac{-2\,bB{x}^{6}-6\,A{x}^{3}b-6\,B{x}^{3}a+6\,Aa}{9}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/9*(-B*b*x^6-3*A*b*x^3-3*B*a*x^3+3*A*a)/x^(3/2)

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Maxima [A]  time = 1.58397, size = 36, normalized size = 0.92 \[ \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{3} \,{\left (B a + A b\right )} x^{\frac{3}{2}} - \frac{2 \, A a}{3 \, x^{\frac{3}{2}}} \]

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]

2/9*B*b*x^(9/2) + 2/3*(B*a + A*b)*x^(3/2) - 2/3*A*a/x^(3/2)

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Fricas [A]  time = 0.227699, size = 38, normalized size = 0.97 \[ \frac{2 \,{\left (B b x^{6} + 3 \,{\left (B a + A b\right )} x^{3} - 3 \, A a\right )}}{9 \, x^{\frac{3}{2}}} \]

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="fricas")

[Out]

2/9*(B*b*x^6 + 3*(B*a + A*b)*x^3 - 3*A*a)/x^(3/2)

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Sympy [A]  time = 9.95857, size = 46, normalized size = 1.18 \[ - \frac{2 A a}{3 x^{\frac{3}{2}}} + \frac{2 A b x^{\frac{3}{2}}}{3} + \frac{2 B a x^{\frac{3}{2}}}{3} + \frac{2 B b x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a/(3*x**(3/2)) + 2*A*b*x**(3/2)/3 + 2*B*a*x**(3/2)/3 + 2*B*b*x**(9/2)/9

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GIAC/XCAS [A]  time = 0.212472, size = 39, normalized size = 1. \[ \frac{2}{9} \, B b x^{\frac{9}{2}} + \frac{2}{3} \, B a x^{\frac{3}{2}} + \frac{2}{3} \, A b x^{\frac{3}{2}} - \frac{2 \, A a}{3 \, x^{\frac{3}{2}}} \]

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]

2/9*B*b*x^(9/2) + 2/3*B*a*x^(3/2) + 2/3*A*b*x^(3/2) - 2/3*A*a/x^(3/2)