3.1281 \(\int \frac{1}{x^6 \left (a+b x^5\right )^2} \, dx\)

Optimal. Leaf size=52 \[ \frac{2 b \log \left (a+b x^5\right )}{5 a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{5 a^2 \left (a+b x^5\right )}-\frac{1}{5 a^2 x^5} \]

[Out]

-1/(5*a^2*x^5) - b/(5*a^2*(a + b*x^5)) - (2*b*Log[x])/a^3 + (2*b*Log[a + b*x^5])
/(5*a^3)

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Rubi [A]  time = 0.0800447, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 b \log \left (a+b x^5\right )}{5 a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{5 a^2 \left (a+b x^5\right )}-\frac{1}{5 a^2 x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^5)^2),x]

[Out]

-1/(5*a^2*x^5) - b/(5*a^2*(a + b*x^5)) - (2*b*Log[x])/a^3 + (2*b*Log[a + b*x^5])
/(5*a^3)

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Rubi in Sympy [A]  time = 11.4603, size = 53, normalized size = 1.02 \[ - \frac{b}{5 a^{2} \left (a + b x^{5}\right )} - \frac{1}{5 a^{2} x^{5}} - \frac{2 b \log{\left (x^{5} \right )}}{5 a^{3}} + \frac{2 b \log{\left (a + b x^{5} \right )}}{5 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**5+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0751835, size = 41, normalized size = 0.79 \[ -\frac{a \left (\frac{b}{a+b x^5}+\frac{1}{x^5}\right )-2 b \log \left (a+b x^5\right )+10 b \log (x)}{5 a^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^5)^2),x]

[Out]

-(a*(x^(-5) + b/(a + b*x^5)) + 10*b*Log[x] - 2*b*Log[a + b*x^5])/(5*a^3)

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Maple [A]  time = 0.014, size = 47, normalized size = 0.9 \[ -{\frac{1}{5\,{x}^{5}{a}^{2}}}-{\frac{b}{5\,{a}^{2} \left ( b{x}^{5}+a \right ) }}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}+{\frac{2\,b\ln \left ( b{x}^{5}+a \right ) }{5\,{a}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^5+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.4396, size = 72, normalized size = 1.38 \[ -\frac{2 \, b x^{5} + a}{5 \,{\left (a^{2} b x^{10} + a^{3} x^{5}\right )}} + \frac{2 \, b \log \left (b x^{5} + a\right )}{5 \, a^{3}} - \frac{2 \, b \log \left (x^{5}\right )}{5 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)^2*x^6),x, algorithm="maxima")

[Out]

-1/5*(2*b*x^5 + a)/(a^2*b*x^10 + a^3*x^5) + 2/5*b*log(b*x^5 + a)/a^3 - 2/5*b*log
(x^5)/a^3

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Fricas [A]  time = 0.220529, size = 99, normalized size = 1.9 \[ -\frac{2 \, a b x^{5} + a^{2} - 2 \,{\left (b^{2} x^{10} + a b x^{5}\right )} \log \left (b x^{5} + a\right ) + 10 \,{\left (b^{2} x^{10} + a b x^{5}\right )} \log \left (x\right )}{5 \,{\left (a^{3} b x^{10} + a^{4} x^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)^2*x^6),x, algorithm="fricas")

[Out]

-1/5*(2*a*b*x^5 + a^2 - 2*(b^2*x^10 + a*b*x^5)*log(b*x^5 + a) + 10*(b^2*x^10 + a
*b*x^5)*log(x))/(a^3*b*x^10 + a^4*x^5)

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Sympy [A]  time = 43.2649, size = 53, normalized size = 1.02 \[ - \frac{a + 2 b x^{5}}{5 a^{3} x^{5} + 5 a^{2} b x^{10}} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{2 b \log{\left (\frac{a}{b} + x^{5} \right )}}{5 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**5+a)**2,x)

[Out]

-(a + 2*b*x**5)/(5*a**3*x**5 + 5*a**2*b*x**10) - 2*b*log(x)/a**3 + 2*b*log(a/b +
x**5)/(5*a**3)

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GIAC/XCAS [A]  time = 0.226623, size = 69, normalized size = 1.33 \[ \frac{2 \, b{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{3}} - \frac{2 \, b{\rm ln}\left ({\left | x \right |}\right )}{a^{3}} - \frac{2 \, b x^{5} + a}{5 \,{\left (b x^{10} + a x^{5}\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)^2*x^6),x, algorithm="giac")

[Out]

2/5*b*ln(abs(b*x^5 + a))/a^3 - 2*b*ln(abs(x))/a^3 - 1/5*(2*b*x^5 + a)/((b*x^10 +
a*x^5)*a^2)