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

Optimal. Leaf size=48 \[ -\frac{a^2 A}{x}+\frac{1}{3} b x^3 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]

[Out]

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

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Rubi [A]  time = 0.080669, antiderivative size = 48, 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 \[ -\frac{a^2 A}{x}+\frac{1}{3} b x^3 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{2}}{x} + \frac{B b^{2} x^{5}}{5} + \frac{b x^{3} \left (A b + 2 B a\right )}{3} + \frac{a \left (2 A b + B a\right ) \int B\, dx}{B} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0265244, size = 48, normalized size = 1. \[ -\frac{a^2 A}{x}+\frac{1}{3} b x^3 (2 a B+A b)+a x (a B+2 A b)+\frac{1}{5} b^2 B x^5 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.006, size = 49, normalized size = 1. \[{\frac{{b}^{2}B{x}^{5}}{5}}+{\frac{A{x}^{3}{b}^{2}}{3}}+{\frac{2\,B{x}^{3}ab}{3}}+2\,Axab+Bx{a}^{2}-{\frac{A{a}^{2}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b^2*B*x^5+1/3*A*x^3*b^2+2/3*B*x^3*a*b+2*A*x*a*b+B*x*a^2-a^2*A/x

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Maxima [A]  time = 1.34916, size = 65, normalized size = 1.35 \[ \frac{1}{5} \, B b^{2} x^{5} + \frac{1}{3} \,{\left (2 \, B a b + A b^{2}\right )} x^{3} - \frac{A a^{2}}{x} +{\left (B a^{2} + 2 \, A a b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*b^2*x^5 + 1/3*(2*B*a*b + A*b^2)*x^3 - A*a^2/x + (B*a^2 + 2*A*a*b)*x

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Fricas [A]  time = 0.240336, size = 72, normalized size = 1.5 \[ \frac{3 \, B b^{2} x^{6} + 5 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} - 15 \, A a^{2} + 15 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2}}{15 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/15*(3*B*b^2*x^6 + 5*(2*B*a*b + A*b^2)*x^4 - 15*A*a^2 + 15*(B*a^2 + 2*A*a*b)*x^
2)/x

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Sympy [A]  time = 1.21373, size = 48, normalized size = 1. \[ - \frac{A a^{2}}{x} + \frac{B b^{2} x^{5}}{5} + x^{3} \left (\frac{A b^{2}}{3} + \frac{2 B a b}{3}\right ) + x \left (2 A a b + B a^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.230963, size = 65, normalized size = 1.35 \[ \frac{1}{5} \, B b^{2} x^{5} + \frac{2}{3} \, B a b x^{3} + \frac{1}{3} \, A b^{2} x^{3} + B a^{2} x + 2 \, A a b x - \frac{A a^{2}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*B*b^2*x^5 + 2/3*B*a*b*x^3 + 1/3*A*b^2*x^3 + B*a^2*x + 2*A*a*b*x - A*a^2/x