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

Optimal. Leaf size=51 \[ -\frac{a^2 A}{2 x^2}+\frac{1}{2} b x^2 (2 a B+A b)+a \log (x) (a B+2 A b)+\frac{1}{4} b^2 B x^4 \]

[Out]

-(a^2*A)/(2*x^2) + (b*(A*b + 2*a*B)*x^2)/2 + (b^2*B*x^4)/4 + a*(2*A*b + a*B)*Log
[x]

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Rubi [A]  time = 0.133702, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^2 A}{2 x^2}+\frac{1}{2} b x^2 (2 a B+A b)+a \log (x) (a B+2 A b)+\frac{1}{4} b^2 B x^4 \]

Antiderivative was successfully verified.

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

[Out]

-(a^2*A)/(2*x^2) + (b*(A*b + 2*a*B)*x^2)/2 + (b^2*B*x^4)/4 + a*(2*A*b + a*B)*Log
[x]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0398264, size = 49, normalized size = 0.96 \[ \frac{1}{4} \left (-\frac{2 a^2 A}{x^2}+2 b x^2 (2 a B+A b)+4 a \log (x) (a B+2 A b)+b^2 B x^4\right ) \]

Antiderivative was successfully verified.

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

[Out]

((-2*a^2*A)/x^2 + 2*b*(A*b + 2*a*B)*x^2 + b^2*B*x^4 + 4*a*(2*A*b + a*B)*Log[x])/
4

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Maple [A]  time = 0.01, size = 50, normalized size = 1. \[{\frac{{b}^{2}B{x}^{4}}{4}}+{\frac{A{x}^{2}{b}^{2}}{2}}+B{x}^{2}ab+2\,A\ln \left ( x \right ) ab+B\ln \left ( x \right ){a}^{2}-{\frac{A{a}^{2}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*b^2*B*x^4+1/2*A*x^2*b^2+B*x^2*a*b+2*A*ln(x)*a*b+B*ln(x)*a^2-1/2*a^2*A/x^2

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Maxima [A]  time = 1.35219, size = 70, normalized size = 1.37 \[ \frac{1}{4} \, B b^{2} x^{4} + \frac{1}{2} \,{\left (2 \, B a b + A b^{2}\right )} x^{2} + \frac{1}{2} \,{\left (B a^{2} + 2 \, A a b\right )} \log \left (x^{2}\right ) - \frac{A a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*B*b^2*x^4 + 1/2*(2*B*a*b + A*b^2)*x^2 + 1/2*(B*a^2 + 2*A*a*b)*log(x^2) - 1/2
*A*a^2/x^2

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Fricas [A]  time = 0.233536, size = 73, normalized size = 1.43 \[ \frac{B b^{2} x^{6} + 2 \,{\left (2 \, B a b + A b^{2}\right )} x^{4} + 4 \,{\left (B a^{2} + 2 \, A a b\right )} x^{2} \log \left (x\right ) - 2 \, A a^{2}}{4 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.54329, size = 48, normalized size = 0.94 \[ - \frac{A a^{2}}{2 x^{2}} + \frac{B b^{2} x^{4}}{4} + a \left (2 A b + B a\right ) \log{\left (x \right )} + x^{2} \left (\frac{A b^{2}}{2} + B a b\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*a**2/(2*x**2) + B*b**2*x**4/4 + a*(2*A*b + B*a)*log(x) + x**2*(A*b**2/2 + B*a
*b)

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GIAC/XCAS [A]  time = 0.232184, size = 95, normalized size = 1.86 \[ \frac{1}{4} \, B b^{2} x^{4} + B a b x^{2} + \frac{1}{2} \, A b^{2} x^{2} + \frac{1}{2} \,{\left (B a^{2} + 2 \, A a b\right )}{\rm ln}\left (x^{2}\right ) - \frac{B a^{2} x^{2} + 2 \, A a b x^{2} + A a^{2}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*B*b^2*x^4 + B*a*b*x^2 + 1/2*A*b^2*x^2 + 1/2*(B*a^2 + 2*A*a*b)*ln(x^2) - 1/2*
(B*a^2*x^2 + 2*A*a*b*x^2 + A*a^2)/x^2