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

Optimal. Leaf size=29 \[ \log (x) (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{2} b B x^2 \]

[Out]

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

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Rubi [A]  time = 0.0685896, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \log (x) (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{2} b B x^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0185427, size = 29, normalized size = 1. \[ \log (x) (a B+A b)-\frac{a A}{2 x^2}+\frac{1}{2} b B x^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 26, normalized size = 0.9 \[{\frac{bB{x}^{2}}{2}}+A\ln \left ( x \right ) b+B\ln \left ( x \right ) a-{\frac{Aa}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(B*x^2+A)/x^3,x)

[Out]

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

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Maxima [A]  time = 1.37265, size = 38, normalized size = 1.31 \[ \frac{1}{2} \, B b x^{2} + \frac{1}{2} \,{\left (B a + A b\right )} \log \left (x^{2}\right ) - \frac{A a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.24138, size = 41, normalized size = 1.41 \[ \frac{B b x^{4} + 2 \,{\left (B a + A b\right )} x^{2} \log \left (x\right ) - A a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.3499, size = 26, normalized size = 0.9 \[ - \frac{A a}{2 x^{2}} + \frac{B b x^{2}}{2} + \left (A b + B a\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.262869, size = 57, normalized size = 1.97 \[ \frac{1}{2} \, B b x^{2} + \frac{1}{2} \,{\left (B a + A b\right )}{\rm ln}\left (x^{2}\right ) - \frac{B a x^{2} + A b x^{2} + A a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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