Optimal. Leaf size=57 \[ -\frac{b}{a^2 n \left (a+b x^n\right )}+\frac{2 b \log \left (a+b x^n\right )}{a^3 n}-\frac{2 b \log (x)}{a^3}-\frac{x^{-n}}{a^2 n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0330685, 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, Rules used = {266, 44} \[ -\frac{b}{a^2 n \left (a+b x^n\right )}+\frac{2 b \log \left (a+b x^n\right )}{a^3 n}-\frac{2 b \log (x)}{a^3}-\frac{x^{-n}}{a^2 n} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{x^{-1-n}}{\left (a+b x^n\right )^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^2 (a+b x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^2}-\frac{2 b}{a^3 x}+\frac{b^2}{a^2 (a+b x)^2}+\frac{2 b^2}{a^3 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-n}}{a^2 n}-\frac{b}{a^2 n \left (a+b x^n\right )}-\frac{2 b \log (x)}{a^3}+\frac{2 b \log \left (a+b x^n\right )}{a^3 n}\\ \end{align*}
Mathematica [A] time = 0.104938, size = 45, normalized size = 0.79 \[ -\frac{a \left (\frac{b}{a+b x^n}+x^{-n}\right )-2 b \log \left (a+b x^n\right )+2 b n \log (x)}{a^3 n} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.018, size = 97, normalized size = 1.7 \begin{align*}{\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.99635, size = 84, normalized size = 1.47 \begin{align*} -\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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.02976, size = 182, normalized size = 3.19 \begin{align*} -\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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]