3.2622 \(\int \frac{x^{-1-n}}{\left (a+b x^n\right )^2} \, dx\)

Optimal. Leaf size=57 \[ \frac{2 b \log \left (a+b x^n\right )}{a^3 n}-\frac{2 b \log (x)}{a^3}-\frac{b}{a^2 n \left (a+b x^n\right )}-\frac{x^{-n}}{a^2 n} \]

[Out]

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

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Rubi [A]  time = 0.0880273, antiderivative size = 57, 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 \[ \frac{2 b \log \left (a+b x^n\right )}{a^3 n}-\frac{2 b \log (x)}{a^3}-\frac{b}{a^2 n \left (a+b x^n\right )}-\frac{x^{-n}}{a^2 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n)/(a + b*x^n)^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 13.1157, size = 53, normalized size = 0.93 \[ - \frac{b}{a^{2} n \left (a + b x^{n}\right )} - \frac{x^{- n}}{a^{2} n} - \frac{2 b \log{\left (x^{n} \right )}}{a^{3} n} + \frac{2 b \log{\left (a + b x^{n} \right )}}{a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0792345, size = 45, normalized size = 0.79 \[ \frac{\frac{b^2 x^n}{a+b x^n}+2 b \log \left (a x^{-n}+b\right )-a x^{-n}}{a^3 n} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.033, size = 97, normalized size = 1.7 \[{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) } \left ( 2\,{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-{\frac{1}{an}}-2\,{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}}}-2\,{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}} \right ) }+2\,{\frac{b\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{3}n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-n)/(a+b*x^n)^2,x)

[Out]

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

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Maxima [A]  time = 1.45245, size = 84, normalized size = 1.47 \[ -\frac{2 \, b x^{n} + a}{a^{2} b n x^{2 \, n} + a^{3} n x^{n}} - \frac{2 \, b \log \left (x\right )}{a^{3}} + \frac{2 \, b \log \left (\frac{b x^{n} + a}{b}\right )}{a^{3} n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.230616, size = 111, normalized size = 1.95 \[ -\frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) + a^{2} + 2 \,{\left (a b n \log \left (x\right ) + a b\right )} x^{n} - 2 \,{\left (b^{2} x^{2 \, n} + a b x^{n}\right )} \log \left (b x^{n} + a\right )}{a^{3} b n x^{2 \, n} + a^{4} n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{-n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^(-n - 1)/(b*x^n + a)^2, x)