3.627 \(\int \frac{1}{x \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=80 \[ \frac{\log (x) \left (a+b x^2\right )}{a \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.103734, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\log (x) \left (a+b x^2\right )}{a \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{\left (a + b x^{2}\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt((a + b*x**2)**2)), x)

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Mathematica [A]  time = 0.0195548, size = 42, normalized size = 0.52 \[ \frac{\left (a+b x^2\right ) \left (2 \log (x)-\log \left (a+b x^2\right )\right )}{2 a \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]

[Out]

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

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Maple [A]  time = 0.012, size = 37, normalized size = 0.5 \[ -{\frac{ \left ( b{x}^{2}+a \right ) \left ( \ln \left ( b{x}^{2}+a \right ) -2\,\ln \left ( x \right ) \right ) }{2\,a}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/((b*x^2+a)^2)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.261634, size = 24, normalized size = 0.3 \[ -\frac{\log \left (b x^{2} + a\right ) - 2 \, \log \left (x\right )}{2 \, a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(log(b*x^2 + a) - 2*log(x))/a

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Sympy [A]  time = 0.569871, size = 15, normalized size = 0.19 \[ \frac{\log{\left (x \right )}}{a} - \frac{\log{\left (\frac{a}{b} + x^{2} \right )}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(x)/a - log(a/b + x**2)/(2*a)

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GIAC/XCAS [A]  time = 0.271064, size = 45, normalized size = 0.56 \[ \frac{1}{2} \,{\left (\frac{{\rm ln}\left (x^{2}\right )}{a} - \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a}\right )}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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