3.18 \(\int \frac{1}{x^2 \left (a x+b x^3\right )} \, dx\)

Optimal. Leaf size=35 \[ \frac{b \log \left (a+b x^2\right )}{2 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2} \]

[Out]

-1/(2*a*x^2) - (b*Log[x])/a^2 + (b*Log[a + b*x^2])/(2*a^2)

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Rubi [A]  time = 0.057307, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{b \log \left (a+b x^2\right )}{2 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a*x + b*x^3)),x]

[Out]

-1/(2*a*x^2) - (b*Log[x])/a^2 + (b*Log[a + b*x^2])/(2*a^2)

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Rubi in Sympy [A]  time = 9.65802, size = 34, normalized size = 0.97 \[ - \frac{1}{2 a x^{2}} - \frac{b \log{\left (x^{2} \right )}}{2 a^{2}} + \frac{b \log{\left (a + b x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x**3+a*x),x)

[Out]

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

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Mathematica [A]  time = 0.0102577, size = 35, normalized size = 1. \[ \frac{b \log \left (a+b x^2\right )}{2 a^2}-\frac{b \log (x)}{a^2}-\frac{1}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a*x + b*x^3)),x]

[Out]

-1/(2*a*x^2) - (b*Log[x])/a^2 + (b*Log[a + b*x^2])/(2*a^2)

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Maple [A]  time = 0.009, size = 32, normalized size = 0.9 \[ -{\frac{1}{2\,a{x}^{2}}}-{\frac{b\ln \left ( x \right ) }{{a}^{2}}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^3+a*x),x)

[Out]

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

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Maxima [A]  time = 1.37685, size = 42, normalized size = 1.2 \[ \frac{b \log \left (b x^{2} + a\right )}{2 \, a^{2}} - \frac{b \log \left (x\right )}{a^{2}} - \frac{1}{2 \, a x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x)*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.206515, size = 45, normalized size = 1.29 \[ \frac{b x^{2} \log \left (b x^{2} + a\right ) - 2 \, b x^{2} \log \left (x\right ) - a}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.64168, size = 31, normalized size = 0.89 \[ - \frac{1}{2 a x^{2}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x**3+a*x),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.218645, size = 58, normalized size = 1.66 \[ -\frac{b{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} + \frac{b{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac{b x^{2} - a}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a*x)*x^2),x, algorithm="giac")

[Out]

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