3.189 \(\int \frac{1}{x^5 \left (b x^2+c x^4\right )} \, dx\)

Optimal. Leaf size=63 \[ \frac{c^3 \log \left (b+c x^2\right )}{2 b^4}-\frac{c^3 \log (x)}{b^4}-\frac{c^2}{2 b^3 x^2}+\frac{c}{4 b^2 x^4}-\frac{1}{6 b x^6} \]

[Out]

-1/(6*b*x^6) + c/(4*b^2*x^4) - c^2/(2*b^3*x^2) - (c^3*Log[x])/b^4 + (c^3*Log[b +
c*x^2])/(2*b^4)

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Rubi [A]  time = 0.0929339, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{c^3 \log \left (b+c x^2\right )}{2 b^4}-\frac{c^3 \log (x)}{b^4}-\frac{c^2}{2 b^3 x^2}+\frac{c}{4 b^2 x^4}-\frac{1}{6 b x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(b*x^2 + c*x^4)),x]

[Out]

-1/(6*b*x^6) + c/(4*b^2*x^4) - c^2/(2*b^3*x^2) - (c^3*Log[x])/b^4 + (c^3*Log[b +
c*x^2])/(2*b^4)

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Rubi in Sympy [A]  time = 16.5699, size = 60, normalized size = 0.95 \[ - \frac{1}{6 b x^{6}} + \frac{c}{4 b^{2} x^{4}} - \frac{c^{2}}{2 b^{3} x^{2}} - \frac{c^{3} \log{\left (x^{2} \right )}}{2 b^{4}} + \frac{c^{3} \log{\left (b + c x^{2} \right )}}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(c*x**4+b*x**2),x)

[Out]

-1/(6*b*x**6) + c/(4*b**2*x**4) - c**2/(2*b**3*x**2) - c**3*log(x**2)/(2*b**4) +
c**3*log(b + c*x**2)/(2*b**4)

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Mathematica [A]  time = 0.0122656, size = 63, normalized size = 1. \[ \frac{c^3 \log \left (b+c x^2\right )}{2 b^4}-\frac{c^3 \log (x)}{b^4}-\frac{c^2}{2 b^3 x^2}+\frac{c}{4 b^2 x^4}-\frac{1}{6 b x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(b*x^2 + c*x^4)),x]

[Out]

-1/(6*b*x^6) + c/(4*b^2*x^4) - c^2/(2*b^3*x^2) - (c^3*Log[x])/b^4 + (c^3*Log[b +
c*x^2])/(2*b^4)

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Maple [A]  time = 0.01, size = 56, normalized size = 0.9 \[ -{\frac{1}{6\,b{x}^{6}}}+{\frac{c}{4\,{b}^{2}{x}^{4}}}-{\frac{{c}^{2}}{2\,{b}^{3}{x}^{2}}}-{\frac{{c}^{3}\ln \left ( x \right ) }{{b}^{4}}}+{\frac{{c}^{3}\ln \left ( c{x}^{2}+b \right ) }{2\,{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(c*x^4+b*x^2),x)

[Out]

-1/6/b/x^6+1/4*c/b^2/x^4-1/2*c^2/b^3/x^2-c^3*ln(x)/b^4+1/2*c^3*ln(c*x^2+b)/b^4

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Maxima [A]  time = 0.70294, size = 78, normalized size = 1.24 \[ \frac{c^{3} \log \left (c x^{2} + b\right )}{2 \, b^{4}} - \frac{c^{3} \log \left (x^{2}\right )}{2 \, b^{4}} - \frac{6 \, c^{2} x^{4} - 3 \, b c x^{2} + 2 \, b^{2}}{12 \, b^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)*x^5),x, algorithm="maxima")

[Out]

1/2*c^3*log(c*x^2 + b)/b^4 - 1/2*c^3*log(x^2)/b^4 - 1/12*(6*c^2*x^4 - 3*b*c*x^2
+ 2*b^2)/(b^3*x^6)

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Fricas [A]  time = 0.260537, size = 78, normalized size = 1.24 \[ \frac{6 \, c^{3} x^{6} \log \left (c x^{2} + b\right ) - 12 \, c^{3} x^{6} \log \left (x\right ) - 6 \, b c^{2} x^{4} + 3 \, b^{2} c x^{2} - 2 \, b^{3}}{12 \, b^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)*x^5),x, algorithm="fricas")

[Out]

1/12*(6*c^3*x^6*log(c*x^2 + b) - 12*c^3*x^6*log(x) - 6*b*c^2*x^4 + 3*b^2*c*x^2 -
2*b^3)/(b^4*x^6)

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Sympy [A]  time = 2.19135, size = 56, normalized size = 0.89 \[ - \frac{2 b^{2} - 3 b c x^{2} + 6 c^{2} x^{4}}{12 b^{3} x^{6}} - \frac{c^{3} \log{\left (x \right )}}{b^{4}} + \frac{c^{3} \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(c*x**4+b*x**2),x)

[Out]

-(2*b**2 - 3*b*c*x**2 + 6*c**2*x**4)/(12*b**3*x**6) - c**3*log(x)/b**4 + c**3*lo
g(b/c + x**2)/(2*b**4)

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GIAC/XCAS [A]  time = 0.270705, size = 95, normalized size = 1.51 \[ -\frac{c^{3}{\rm ln}\left (x^{2}\right )}{2 \, b^{4}} + \frac{c^{3}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4}} + \frac{11 \, c^{3} x^{6} - 6 \, b c^{2} x^{4} + 3 \, b^{2} c x^{2} - 2 \, b^{3}}{12 \, b^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + b*x^2)*x^5),x, algorithm="giac")

[Out]

-1/2*c^3*ln(x^2)/b^4 + 1/2*c^3*ln(abs(c*x^2 + b))/b^4 + 1/12*(11*c^3*x^6 - 6*b*c
^2*x^4 + 3*b^2*c*x^2 - 2*b^3)/(b^4*x^6)