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

Optimal. Leaf size=37 \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{5} b B x^{5/2} \]

[Out]

(-2*a*A)/(3*x^(3/2)) + 2*(A*b + a*B)*Sqrt[x] + (2*b*B*x^(5/2))/5

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Rubi [A]  time = 0.0499817, antiderivative size = 37, 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 \[ 2 \sqrt{x} (a B+A b)-\frac{2 a A}{3 x^{3/2}}+\frac{2}{5} b B x^{5/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a*A)/(3*x^(3/2)) + 2*(A*b + a*B)*Sqrt[x] + (2*b*B*x^(5/2))/5

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Rubi in Sympy [A]  time = 6.97056, size = 37, normalized size = 1. \[ - \frac{2 A a}{3 x^{\frac{3}{2}}} + \frac{2 B b x^{\frac{5}{2}}}{5} + \sqrt{x} \left (2 A b + 2 B a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a/(3*x**(3/2)) + 2*B*b*x**(5/2)/5 + sqrt(x)*(2*A*b + 2*B*a)

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Mathematica [A]  time = 0.0198245, size = 33, normalized size = 0.89 \[ \frac{2 \left (15 x^2 (a B+A b)-5 a A+3 b B x^4\right )}{15 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-5*a*A + 15*(A*b + a*B)*x^2 + 3*b*B*x^4))/(15*x^(3/2))

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Maple [A]  time = 0.006, size = 32, normalized size = 0.9 \[ -{\frac{-6\,bB{x}^{4}-30\,A{x}^{2}b-30\,B{x}^{2}a+10\,Aa}{15}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/15*(-3*B*b*x^4-15*A*b*x^2-15*B*a*x^2+5*A*a)/x^(3/2)

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Maxima [A]  time = 1.32995, size = 36, normalized size = 0.97 \[ \frac{2}{5} \, B b x^{\frac{5}{2}} + 2 \,{\left (B a + A b\right )} \sqrt{x} - \frac{2 \, A a}{3 \, x^{\frac{3}{2}}} \]

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]

2/5*B*b*x^(5/2) + 2*(B*a + A*b)*sqrt(x) - 2/3*A*a/x^(3/2)

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Fricas [A]  time = 0.224072, size = 39, normalized size = 1.05 \[ \frac{2 \,{\left (3 \, B b x^{4} + 15 \,{\left (B a + A b\right )} x^{2} - 5 \, A a\right )}}{15 \, x^{\frac{3}{2}}} \]

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="fricas")

[Out]

2/15*(3*B*b*x^4 + 15*(B*a + A*b)*x^2 - 5*A*a)/x^(3/2)

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Sympy [A]  time = 4.07331, size = 42, normalized size = 1.14 \[ - \frac{2 A a}{3 x^{\frac{3}{2}}} + 2 A b \sqrt{x} + 2 B a \sqrt{x} + \frac{2 B b x^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a/(3*x**(3/2)) + 2*A*b*sqrt(x) + 2*B*a*sqrt(x) + 2*B*b*x**(5/2)/5

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GIAC/XCAS [A]  time = 0.213255, size = 39, normalized size = 1.05 \[ \frac{2}{5} \, B b x^{\frac{5}{2}} + 2 \, B a \sqrt{x} + 2 \, A b \sqrt{x} - \frac{2 \, A a}{3 \, x^{\frac{3}{2}}} \]

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]

2/5*B*b*x^(5/2) + 2*B*a*sqrt(x) + 2*A*b*sqrt(x) - 2/3*A*a/x^(3/2)