3.638 \(\int \frac{1}{x \left (a^2+2 a b x^2+b^2 x^4\right )^{3/2}} \, dx\)

Optimal. Leaf size=147 \[ \frac{1}{4 a \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\log (x) \left (a+b x^2\right )}{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

1/(2*a^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(4*a*(a + b*x^2)*Sqrt[a^2 + 2*a*b*
x^2 + b^2*x^4]) + ((a + b*x^2)*Log[x])/(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (
(a + b*x^2)*Log[a + b*x^2])/(2*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi [A]  time = 0.205017, antiderivative size = 147, 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 \[ \frac{1}{4 a \left (a+b x^2\right ) \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{1}{2 a^2 \sqrt{a^2+2 a b x^2+b^2 x^4}}+\frac{\log (x) \left (a+b x^2\right )}{a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}-\frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 a^3 \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

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

[Out]

1/(2*a^2*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) + 1/(4*a*(a + b*x^2)*Sqrt[a^2 + 2*a*b*
x^2 + b^2*x^4]) + ((a + b*x^2)*Log[x])/(a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4]) - (
(a + b*x^2)*Log[a + b*x^2])/(2*a^3*Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4])

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Rubi in Sympy [A]  time = 27.6618, size = 144, normalized size = 0.98 \[ \frac{2 a + 2 b x^{2}}{8 a \left (a^{2} + 2 a b x^{2} + b^{2} x^{4}\right )^{\frac{3}{2}}} + \frac{1}{2 a^{2} \sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}}} + \frac{\sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (x^{2} \right )}}{2 a^{3} \left (a + b x^{2}\right )} - \frac{\sqrt{a^{2} + 2 a b x^{2} + b^{2} x^{4}} \log{\left (a + b x^{2} \right )}}{2 a^{3} \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a + 2*b*x**2)/(8*a*(a**2 + 2*a*b*x**2 + b**2*x**4)**(3/2)) + 1/(2*a**2*sqrt(a
**2 + 2*a*b*x**2 + b**2*x**4)) + sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)*log(x**2)/(
2*a**3*(a + b*x**2)) - sqrt(a**2 + 2*a*b*x**2 + b**2*x**4)*log(a + b*x**2)/(2*a*
*3*(a + b*x**2))

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Mathematica [A]  time = 0.0441868, size = 74, normalized size = 0.5 \[ \frac{a \left (3 a+2 b x^2\right )+4 \log (x) \left (a+b x^2\right )^2-2 \left (a+b x^2\right )^2 \log \left (a+b x^2\right )}{4 a^3 \left (a+b x^2\right ) \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

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

[Out]

(a*(3*a + 2*b*x^2) + 4*(a + b*x^2)^2*Log[x] - 2*(a + b*x^2)^2*Log[a + b*x^2])/(4
*a^3*(a + b*x^2)*Sqrt[(a + b*x^2)^2])

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Maple [A]  time = 0.024, size = 107, normalized size = 0.7 \[ -{\frac{ \left ( 2\,\ln \left ( b{x}^{2}+a \right ){x}^{4}{b}^{2}-4\,\ln \left ( x \right ){x}^{4}{b}^{2}+4\,\ln \left ( b{x}^{2}+a \right ){x}^{2}ab-8\,\ln \left ( x \right ){x}^{2}ab-2\,ab{x}^{2}+2\,{a}^{2}\ln \left ( b{x}^{2}+a \right ) -4\,{a}^{2}\ln \left ( x \right ) -3\,{a}^{2} \right ) \left ( b{x}^{2}+a \right ) }{4\,{a}^{3}} \left ( \left ( b{x}^{2}+a \right ) ^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(2*ln(b*x^2+a)*x^4*b^2-4*ln(x)*x^4*b^2+4*ln(b*x^2+a)*x^2*a*b-8*ln(x)*x^2*a*
b-2*a*b*x^2+2*a^2*ln(b*x^2+a)-4*a^2*ln(x)-3*a^2)*(b*x^2+a)/a^3/((b*x^2+a)^2)^(3/
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/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.268827, size = 122, normalized size = 0.83 \[ \frac{2 \, a b x^{2} + 3 \, a^{2} - 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{3} b^{2} x^{4} + 2 \, a^{4} b x^{2} + a^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x),x, algorithm="fricas")

[Out]

1/4*(2*a*b*x^2 + 3*a^2 - 2*(b^2*x^4 + 2*a*b*x^2 + a^2)*log(b*x^2 + a) + 4*(b^2*x
^4 + 2*a*b*x^2 + a^2)*log(x))/(a^3*b^2*x^4 + 2*a^4*b*x^2 + a^5)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \left (\left (a + b x^{2}\right )^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.639614, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b^2*x^4 + 2*a*b*x^2 + a^2)^(3/2)*x),x, algorithm="giac")

[Out]

sage0*x