Optimal. Leaf size=50 \[ -\frac {(c x)^{n-1} \, _2F_1\left (1,-\frac {1-n}{n};2-\frac {1}{n};-\frac {b x^n}{a}\right )}{a c (1-n)} \]
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Rubi [A] time = 0.01, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {364} \[ -\frac {(c x)^{n-1} \, _2F_1\left (1,-\frac {1-n}{n};2-\frac {1}{n};-\frac {b x^n}{a}\right )}{a c (1-n)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin {align*} \int \frac {(c x)^{-2+n}}{a+b x^n} \, dx &=-\frac {(c x)^{-1+n} \, _2F_1\left (1,-\frac {1-n}{n};2-\frac {1}{n};-\frac {b x^n}{a}\right )}{a c (1-n)}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 44, normalized size = 0.88 \[ \frac {x (c x)^{n-2} \, _2F_1\left (1,\frac {n-1}{n};\frac {n-1}{n}+1;-\frac {b x^n}{a}\right )}{a (n-1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c x\right )^{n - 2}}{b x^{n} + a}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{n - 2}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{n -2}}{b \,x^{n}+a}\, dx \]
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 \[ -a c^{n} \int \frac {1}{b^{2} c^{2} x^{2} x^{n} + a b c^{2} x^{2}}\,{d x} - \frac {c^{n - 2}}{b x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c\,x\right )}^{n-2}}{a+b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 11.52, size = 41, normalized size = 0.82 \[ \frac {c^{n} \Phi \left (\frac {a x^{- n} e^{i \pi }}{b}, 1, \frac {1}{n}\right ) \Gamma \left (- \frac {1}{n}\right )}{b c^{2} n^{2} x \Gamma \left (1 - \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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