Optimal. Leaf size=33 \[ \frac {x^2 \, _2F_1\left (2,\frac {2}{n};\frac {n+2}{n};-\frac {b x^n}{a}\right )}{2 a^2} \]
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Rubi [A] time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {364} \[ \frac {x^2 \, _2F_1\left (2,\frac {2}{n};\frac {n+2}{n};-\frac {b x^n}{a}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin {align*} \int \frac {x}{\left (a+b x^n\right )^2} \, dx &=\frac {x^2 \, _2F_1\left (2,\frac {2}{n};\frac {2+n}{n};-\frac {b x^n}{a}\right )}{2 a^2}\\ \end {align*}
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Mathematica [A] time = 0.00, size = 33, normalized size = 1.00 \[ \frac {x^2 \, _2F_1\left (2,\frac {2}{n};1+\frac {2}{n};-\frac {b x^n}{a}\right )}{2 a^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]
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 \[ \int \frac {x}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (b \,x^{n}+a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ {\left (n - 2\right )} \int \frac {x}{a b n x^{n} + a^{2} n}\,{d x} + \frac {x^{2}}{a b n x^{n} + a^{2} n} \]
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 \[ \int \frac {x}{{\left (a+b\,x^n\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.22, size = 274, normalized size = 8.30 \[ \frac {2 n x^{2} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {2}{n}\right ) \Gamma \left (\frac {2}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {2}{n}\right )\right )} + \frac {2 n x^{2} \Gamma \left (\frac {2}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {2}{n}\right )\right )} - \frac {4 x^{2} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {2}{n}\right ) \Gamma \left (\frac {2}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {2}{n}\right )\right )} + \frac {2 b n x^{2} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {2}{n}\right ) \Gamma \left (\frac {2}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {2}{n}\right )\right )} - \frac {4 b x^{2} x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {2}{n}\right ) \Gamma \left (\frac {2}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {2}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {2}{n}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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