3.2630 \(\int \frac{x^{-1-n}}{\left (a+b x^n\right )^3} \, dx\)

Optimal. Leaf size=77 \[ \frac{3 b \log \left (a+b x^n\right )}{a^4 n}-\frac{3 b \log (x)}{a^4}-\frac{2 b}{a^3 n \left (a+b x^n\right )}-\frac{x^{-n}}{a^3 n}-\frac{b}{2 a^2 n \left (a+b x^n\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.114486, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{3 b \log \left (a+b x^n\right )}{a^4 n}-\frac{3 b \log (x)}{a^4}-\frac{2 b}{a^3 n \left (a+b x^n\right )}-\frac{x^{-n}}{a^3 n}-\frac{b}{2 a^2 n \left (a+b x^n\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n)/(a + b*x^n)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 16.9118, size = 71, normalized size = 0.92 \[ - \frac{b}{2 a^{2} n \left (a + b x^{n}\right )^{2}} - \frac{2 b}{a^{3} n \left (a + b x^{n}\right )} - \frac{x^{- n}}{a^{3} n} - \frac{3 b \log{\left (x^{n} \right )}}{a^{4} n} + \frac{3 b \log{\left (a + b x^{n} \right )}}{a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.065776, size = 79, normalized size = 1.03 \[ -\frac{b^3}{2 a^4 n \left (a x^{-n}+b\right )^2}+\frac{3 b^2}{a^4 n \left (a x^{-n}+b\right )}+\frac{3 b \log \left (a x^{-n}+b\right )}{a^4 n}-\frac{x^{-n}}{a^3 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.05, size = 132, normalized size = 1.7 \[{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( -{\frac{1}{an}}-3\,{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}}}-6\,{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}}+6\,{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-3\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}}+{\frac{9\,{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,{a}^{4}n}} \right ) }+3\,{\frac{b\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{4}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-n)/(a+b*x^n)^3,x)

[Out]

(-1/a/n-3*b/a^2*ln(x)*exp(n*ln(x))-6*b^2/a^3*ln(x)*exp(n*ln(x))^2+6*b^2/a^3/n*ex
p(n*ln(x))^2-3*b^3/a^4*ln(x)*exp(n*ln(x))^3+9/2*b^3/a^4/n*exp(n*ln(x))^3)/exp(n*
ln(x))/(a+b*exp(n*ln(x)))^2+3*b/a^4/n*ln(a+b*exp(n*ln(x)))

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Maxima [A]  time = 1.45285, size = 123, normalized size = 1.6 \[ -\frac{6 \, b^{2} x^{2 \, n} + 9 \, a b x^{n} + 2 \, a^{2}}{2 \,{\left (a^{3} b^{2} n x^{3 \, n} + 2 \, a^{4} b n x^{2 \, n} + a^{5} n x^{n}\right )}} - \frac{3 \, b \log \left (x\right )}{a^{4}} + \frac{3 \, b \log \left (\frac{b x^{n} + a}{b}\right )}{a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(6*b^2*x^(2*n) + 9*a*b*x^n + 2*a^2)/(a^3*b^2*n*x^(3*n) + 2*a^4*b*n*x^(2*n)
+ a^5*n*x^n) - 3*b*log(x)/a^4 + 3*b*log((b*x^n + a)/b)/(a^4*n)

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Fricas [A]  time = 0.232108, size = 188, normalized size = 2.44 \[ -\frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) + 2 \, a^{3} + 6 \,{\left (2 \, a b^{2} n \log \left (x\right ) + a b^{2}\right )} x^{2 \, n} + 3 \,{\left (2 \, a^{2} b n \log \left (x\right ) + 3 \, a^{2} b\right )} x^{n} - 6 \,{\left (b^{3} x^{3 \, n} + 2 \, a b^{2} x^{2 \, n} + a^{2} b x^{n}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{4} b^{2} n x^{3 \, n} + 2 \, a^{5} b n x^{2 \, n} + a^{6} n x^{n}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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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(x**(-1-n)/(a+b*x**n)**3,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(-n - 1)/(b*x^n + a)^3, x)