3.1282 \(\int \frac{1}{x^{11} \left (a+b x^5\right )^2} \, dx\)

Optimal. Leaf size=69 \[ -\frac{3 b^2 \log \left (a+b x^5\right )}{5 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{b^2}{5 a^3 \left (a+b x^5\right )}+\frac{2 b}{5 a^3 x^5}-\frac{1}{10 a^2 x^{10}} \]

[Out]

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

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Rubi [A]  time = 0.100333, antiderivative size = 69, 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 \[ -\frac{3 b^2 \log \left (a+b x^5\right )}{5 a^4}+\frac{3 b^2 \log (x)}{a^4}+\frac{b^2}{5 a^3 \left (a+b x^5\right )}+\frac{2 b}{5 a^3 x^5}-\frac{1}{10 a^2 x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^11*(a + b*x^5)^2),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.6734, size = 70, normalized size = 1.01 \[ - \frac{1}{10 a^{2} x^{10}} + \frac{b^{2}}{5 a^{3} \left (a + b x^{5}\right )} + \frac{2 b}{5 a^{3} x^{5}} + \frac{3 b^{2} \log{\left (x^{5} \right )}}{5 a^{4}} - \frac{3 b^{2} \log{\left (a + b x^{5} \right )}}{5 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**11/(b*x**5+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.108411, size = 57, normalized size = 0.83 \[ \frac{-6 b^2 \log \left (a+b x^5\right )+a \left (\frac{2 b^2}{a+b x^5}-\frac{a}{x^{10}}+\frac{4 b}{x^5}\right )+30 b^2 \log (x)}{10 a^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^11*(a + b*x^5)^2),x]

[Out]

(a*(-(a/x^10) + (4*b)/x^5 + (2*b^2)/(a + b*x^5)) + 30*b^2*Log[x] - 6*b^2*Log[a +
b*x^5])/(10*a^4)

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Maple [A]  time = 0.014, size = 62, normalized size = 0.9 \[ -{\frac{1}{10\,{a}^{2}{x}^{10}}}+{\frac{2\,b}{5\,{a}^{3}{x}^{5}}}+{\frac{{b}^{2}}{5\,{a}^{3} \left ( b{x}^{5}+a \right ) }}+3\,{\frac{{b}^{2}\ln \left ( x \right ) }{{a}^{4}}}-{\frac{3\,{b}^{2}\ln \left ( b{x}^{5}+a \right ) }{5\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^11/(b*x^5+a)^2,x)

[Out]

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

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Maxima [A]  time = 1.43829, size = 95, normalized size = 1.38 \[ \frac{6 \, b^{2} x^{10} + 3 \, a b x^{5} - a^{2}}{10 \,{\left (a^{3} b x^{15} + a^{4} x^{10}\right )}} - \frac{3 \, b^{2} \log \left (b x^{5} + a\right )}{5 \, a^{4}} + \frac{3 \, b^{2} \log \left (x^{5}\right )}{5 \, a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)^2*x^11),x, algorithm="maxima")

[Out]

1/10*(6*b^2*x^10 + 3*a*b*x^5 - a^2)/(a^3*b*x^15 + a^4*x^10) - 3/5*b^2*log(b*x^5
+ a)/a^4 + 3/5*b^2*log(x^5)/a^4

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Fricas [A]  time = 0.221131, size = 122, normalized size = 1.77 \[ \frac{6 \, a b^{2} x^{10} + 3 \, a^{2} b x^{5} - a^{3} - 6 \,{\left (b^{3} x^{15} + a b^{2} x^{10}\right )} \log \left (b x^{5} + a\right ) + 30 \,{\left (b^{3} x^{15} + a b^{2} x^{10}\right )} \log \left (x\right )}{10 \,{\left (a^{4} b x^{15} + a^{5} x^{10}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)^2*x^11),x, algorithm="fricas")

[Out]

1/10*(6*a*b^2*x^10 + 3*a^2*b*x^5 - a^3 - 6*(b^3*x^15 + a*b^2*x^10)*log(b*x^5 + a
) + 30*(b^3*x^15 + a*b^2*x^10)*log(x))/(a^4*b*x^15 + a^5*x^10)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**11/(b*x**5+a)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.228444, size = 115, normalized size = 1.67 \[ -\frac{3 \, b^{2}{\rm ln}\left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{4}} + \frac{3 \, b^{2}{\rm ln}\left ({\left | x \right |}\right )}{a^{4}} + \frac{3 \, b^{3} x^{5} + 4 \, a b^{2}}{5 \,{\left (b x^{5} + a\right )} a^{4}} - \frac{9 \, b^{2} x^{10} - 4 \, a b x^{5} + a^{2}}{10 \, a^{4} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^5 + a)^2*x^11),x, algorithm="giac")

[Out]

-3/5*b^2*ln(abs(b*x^5 + a))/a^4 + 3*b^2*ln(abs(x))/a^4 + 1/5*(3*b^3*x^5 + 4*a*b^
2)/((b*x^5 + a)*a^4) - 1/10*(9*b^2*x^10 - 4*a*b*x^5 + a^2)/(a^4*x^10)