Optimal. Leaf size=24 \[ \frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2} \]
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Rubi [A]
time = 0.00, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {251}
\begin {gather*} \frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 251
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b x^n\right )^2} \, dx &=\frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} \frac {x \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )}{a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \,x^{n}\right )^{2}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.39, size = 24, normalized size = 1.00 \begin {gather*} {\rm integral}\left (\frac {1}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.52, size = 257, normalized size = 10.71 \begin {gather*} \frac {n x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {n x \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {x \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} + \frac {b n x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} - \frac {b x x^{n} \Phi \left (\frac {b x^{n} e^{i \pi }}{a}, 1, \frac {1}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{a^{2} \left (a n^{3} \Gamma \left (1 + \frac {1}{n}\right ) + b n^{3} x^{n} \Gamma \left (1 + \frac {1}{n}\right )\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.17, size = 25, normalized size = 1.04 \begin {gather*} \frac {x\,{{}}_2{\mathrm {F}}_1\left (2,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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