Optimal. Leaf size=33 \[ \frac {a}{b^2 n \left (a+b x^n\right )}+\frac {\log \left (a+b x^n\right )}{b^2 n} \]
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Rubi [A] time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.00, 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, 43} \[ \frac {a}{b^2 n \left (a+b x^n\right )}+\frac {\log \left (a+b x^n\right )}{b^2 n} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x}{(a+b x)^2} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^2}+\frac {1}{b (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=\frac {a}{b^2 n \left (a+b x^n\right )}+\frac {\log \left (a+b x^n\right )}{b^2 n}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 27, normalized size = 0.82 \[ \frac {\frac {a}{a+b x^n}+\log \left (a+b x^n\right )}{b^2 n} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 36, normalized size = 1.09 \[ \frac {{\left (b x^{n} + a\right )} \log \left (b x^{n} + a\right ) + a}{b^{3} n x^{n} + a b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 38, normalized size = 1.15 \[ \frac {a}{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right ) b^{2} n}+\frac {\ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 39, normalized size = 1.18 \[ \frac {a}{b^{3} n x^{n} + a b^{2} n} + \frac {\log \left (\frac {b x^{n} + a}{b}\right )}{b^{2} n} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^{2\,n-1}}{{\left (a+b\,x^n\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 99.17, size = 95, normalized size = 2.88 \[ \begin {cases} \frac {\log {\relax (x )}}{a^{2}} & \text {for}\: b = 0 \wedge n = 0 \\\frac {x^{2 n}}{2 a^{2} n} & \text {for}\: b = 0 \\\frac {\log {\relax (x )}}{\left (a + b\right )^{2}} & \text {for}\: n = 0 \\\frac {a \log {\left (\frac {a}{b} + x^{n} \right )}}{a b^{2} n + b^{3} n x^{n}} + \frac {a}{a b^{2} n + b^{3} n x^{n}} + \frac {b x^{n} \log {\left (\frac {a}{b} + x^{n} \right )}}{a b^{2} n + b^{3} n x^{n}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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