3.628 \(\int \frac{1}{x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \, dx\)

Optimal. Leaf size=122 \[ -\frac{a+b x^2}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b \log (x) \left (a+b x^2\right )}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.135872, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ -\frac{a+b x^2}{2 a x^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{b \log (x) \left (a+b x^2\right )}{a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{b \left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]

[Out]

-(a + b*x^2)/(2*a*x^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (b*(a + b*x^2)*Log[x])/
(a^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + (b*(a + b*x^2)*Log[a + b*x^2])/(2*a^2*Sq
rt[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. \[ \int \frac{1}{x^{3} \sqrt{\left (a + b x^{2}\right )^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**3*sqrt((a + b*x**2)**2)), x)

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Mathematica [A]  time = 0.0300493, size = 54, normalized size = 0.44 \[ -\frac{\left (a+b x^2\right ) \left (-b x^2 \log \left (a+b x^2\right )+a+2 b x^2 \log (x)\right )}{2 a^2 x^2 \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]),x]

[Out]

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

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Maple [A]  time = 0.016, size = 52, normalized size = 0.4 \[{\frac{ \left ( b{x}^{2}+a \right ) \left ( b\ln \left ( b{x}^{2}+a \right ){x}^{2}-2\,b\ln \left ( x \right ){x}^{2}-a \right ) }{2\,{a}^{2}{x}^{2}}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.262705, size = 45, normalized size = 0.37 \[ \frac{b x^{2} \log \left (b x^{2} + a\right ) - 2 \, b x^{2} \log \left (x\right ) - a}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt((b*x^2 + a)^2)*x^3),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 = 1.59665, size = 31, normalized size = 0.25 \[ - \frac{1}{2 a x^{2}} - \frac{b \log{\left (x \right )}}{a^{2}} + \frac{b \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/((b*x**2+a)**2)**(1/2),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 = 0.271726, size = 70, normalized size = 0.57 \[ -\frac{1}{2} \,{\left (\frac{b{\rm ln}\left (x^{2}\right )}{a^{2}} - \frac{b{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{2}} - \frac{b x^{2} - a}{a^{2} x^{2}}\right )}{\rm sign}\left (b x^{2} + a\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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