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

Optimal. Leaf size=49 \[ \frac{b \log \left (a+b x^2\right )}{a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{2 a^2 \left (a+b x^2\right )}-\frac{1}{2 a^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0816251, antiderivative size = 49, 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 \[ \frac{b \log \left (a+b x^2\right )}{a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{2 a^2 \left (a+b x^2\right )}-\frac{1}{2 a^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a*x + b*x^3)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 12.6329, size = 46, normalized size = 0.94 \[ - \frac{b}{2 a^{2} \left (a + b x^{2}\right )} - \frac{1}{2 a^{2} x^{2}} - \frac{b \log{\left (x^{2} \right )}}{a^{3}} + \frac{b \log{\left (a + b x^{2} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0593991, size = 41, normalized size = 0.84 \[ -\frac{a \left (\frac{b}{a+b x^2}+\frac{1}{x^2}\right )-2 b \log \left (a+b x^2\right )+4 b \log (x)}{2 a^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 46, normalized size = 0.9 \[ -{\frac{1}{2\,{a}^{2}{x}^{2}}}-{\frac{b}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) }{{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x^3+a*x)^2,x)

[Out]

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

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Maxima [A]  time = 1.39546, size = 68, normalized size = 1.39 \[ -\frac{2 \, b x^{2} + a}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac{b \log \left (b x^{2} + a\right )}{a^{3}} - \frac{2 \, b \log \left (x\right )}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.20666, size = 99, normalized size = 2.02 \[ -\frac{2 \, a b x^{2} + a^{2} - 2 \,{\left (b^{2} x^{4} + a b x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left (b^{2} x^{4} + a b x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.04594, size = 49, normalized size = 1. \[ - \frac{a + 2 b x^{2}}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{b \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.218026, size = 69, normalized size = 1.41 \[ -\frac{b{\rm ln}\left (x^{2}\right )}{a^{3}} + \frac{b{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{3}} - \frac{2 \, b x^{2} + a}{2 \,{\left (b x^{4} + a x^{2}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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