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

Optimal. Leaf size=61 \[ -\frac{2 a^2 A}{3 x^{3/2}}+\frac{2}{5} b x^{5/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{9} b^2 B x^{9/2} \]

[Out]

(-2*a^2*A)/(3*x^(3/2)) + 2*a*(2*A*b + a*B)*Sqrt[x] + (2*b*(A*b + 2*a*B)*x^(5/2))
/5 + (2*b^2*B*x^(9/2))/9

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Rubi [A]  time = 0.0899875, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{2 a^2 A}{3 x^{3/2}}+\frac{2}{5} b x^{5/2} (2 a B+A b)+2 a \sqrt{x} (a B+2 A b)+\frac{2}{9} b^2 B x^{9/2} \]

Antiderivative was successfully verified.

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

[Out]

(-2*a^2*A)/(3*x^(3/2)) + 2*a*(2*A*b + a*B)*Sqrt[x] + (2*b*(A*b + 2*a*B)*x^(5/2))
/5 + (2*b^2*B*x^(9/2))/9

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Rubi in Sympy [A]  time = 12.6919, size = 61, normalized size = 1. \[ - \frac{2 A a^{2}}{3 x^{\frac{3}{2}}} + \frac{2 B b^{2} x^{\frac{9}{2}}}{9} + 2 a \sqrt{x} \left (2 A b + B a\right ) + \frac{2 b x^{\frac{5}{2}} \left (A b + 2 B a\right )}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**2/(3*x**(3/2)) + 2*B*b**2*x**(9/2)/9 + 2*a*sqrt(x)*(2*A*b + B*a) + 2*b*x
**(5/2)*(A*b + 2*B*a)/5

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Mathematica [A]  time = 0.0343367, size = 53, normalized size = 0.87 \[ \frac{2 \left (-15 a^2 A+9 b x^4 (2 a B+A b)+45 a x^2 (a B+2 A b)+5 b^2 B x^6\right )}{45 x^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*(-15*a^2*A + 45*a*(2*A*b + a*B)*x^2 + 9*b*(A*b + 2*a*B)*x^4 + 5*b^2*B*x^6))/(
45*x^(3/2))

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Maple [A]  time = 0.008, size = 56, normalized size = 0.9 \[ -{\frac{-10\,{b}^{2}B{x}^{6}-18\,A{b}^{2}{x}^{4}-36\,{x}^{4}abB-180\,aAb{x}^{2}-90\,B{a}^{2}{x}^{2}+30\,{a}^{2}A}{45}{x}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/45*(-5*B*b^2*x^6-9*A*b^2*x^4-18*B*a*b*x^4-90*A*a*b*x^2-45*B*a^2*x^2+15*A*a^2)
/x^(3/2)

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Maxima [A]  time = 1.32307, size = 69, normalized size = 1.13 \[ \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{2}{5} \,{\left (2 \, B a b + A b^{2}\right )} x^{\frac{5}{2}} - \frac{2 \, A a^{2}}{3 \, x^{\frac{3}{2}}} + 2 \,{\left (B a^{2} + 2 \, A a b\right )} \sqrt{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*B*b^2*x^(9/2) + 2/5*(2*B*a*b + A*b^2)*x^(5/2) - 2/3*A*a^2/x^(3/2) + 2*(B*a^2
+ 2*A*a*b)*sqrt(x)

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Fricas [A]  time = 0.224474, size = 72, normalized size = 1.18 \[ \frac{2 \,{\left (5 \, B b^{2} x^{6} + 9 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 15 \, A a^{2} + 45 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}\right )}}{45 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/45*(5*B*b^2*x^6 + 9*(2*B*a*b + A*b^2)*x^4 - 15*A*a^2 + 45*(B*a^2 + 2*A*a*b)*x^
2)/x^(3/2)

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Sympy [A]  time = 10.2123, size = 76, normalized size = 1.25 \[ - \frac{2 A a^{2}}{3 x^{\frac{3}{2}}} + 4 A a b \sqrt{x} + \frac{2 A b^{2} x^{\frac{5}{2}}}{5} + 2 B a^{2} \sqrt{x} + \frac{4 B a b x^{\frac{5}{2}}}{5} + \frac{2 B b^{2} x^{\frac{9}{2}}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*A*a**2/(3*x**(3/2)) + 4*A*a*b*sqrt(x) + 2*A*b**2*x**(5/2)/5 + 2*B*a**2*sqrt(x
) + 4*B*a*b*x**(5/2)/5 + 2*B*b**2*x**(9/2)/9

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GIAC/XCAS [A]  time = 0.212402, size = 72, normalized size = 1.18 \[ \frac{2}{9} \, B b^{2} x^{\frac{9}{2}} + \frac{4}{5} \, B a b x^{\frac{5}{2}} + \frac{2}{5} \, A b^{2} x^{\frac{5}{2}} + 2 \, B a^{2} \sqrt{x} + 4 \, A a b \sqrt{x} - \frac{2 \, A a^{2}}{3 \, x^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/9*B*b^2*x^(9/2) + 4/5*B*a*b*x^(5/2) + 2/5*A*b^2*x^(5/2) + 2*B*a^2*sqrt(x) + 4*
A*a*b*sqrt(x) - 2/3*A*a^2/x^(3/2)