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

Optimal. Leaf size=29 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{4 x^4}+b B \log (x) \]

[Out]

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

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Rubi [A]  time = 0.0634178, 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 \[ -\frac{a B+A b}{2 x^2}-\frac{a A}{4 x^4}+b B \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.45912, size = 31, normalized size = 1.07 \[ - \frac{A a}{4 x^{4}} + \frac{B b \log{\left (x^{2} \right )}}{2} - \frac{\frac{A b}{2} + \frac{B a}{2}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0307526, size = 31, normalized size = 1.07 \[ \frac{-a B-A b}{2 x^2}-\frac{a A}{4 x^4}+b B \log (x) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 28, normalized size = 1. \[ bB\ln \left ( x \right ) -{\frac{Aa}{4\,{x}^{4}}}-{\frac{Ab}{2\,{x}^{2}}}-{\frac{Ba}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)*(B*x^2+A)/x^5,x)

[Out]

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

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Maxima [A]  time = 1.34606, size = 41, normalized size = 1.41 \[ \frac{1}{2} \, B b \log \left (x^{2}\right ) - \frac{2 \,{\left (B a + A b\right )} x^{2} + A a}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.244359, size = 42, normalized size = 1.45 \[ \frac{4 \, B b x^{4} \log \left (x\right ) - 2 \,{\left (B a + A b\right )} x^{2} - A a}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.06332, size = 27, normalized size = 0.93 \[ B b \log{\left (x \right )} - \frac{A a + x^{2} \left (2 A b + 2 B a\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.246021, size = 53, normalized size = 1.83 \[ \frac{1}{2} \, B b{\rm ln}\left (x^{2}\right ) - \frac{3 \, B b x^{4} + 2 \, B a x^{2} + 2 \, A b x^{2} + A a}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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