Optimal. Leaf size=40 \[ \frac {(c x)^{n+1} \, _2F_1\left (1,1+\frac {1}{n};2+\frac {1}{n};-\frac {b x^n}{a}\right )}{a c (n+1)} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {364} \[ \frac {(c x)^{n+1} \, _2F_1\left (1,1+\frac {1}{n};2+\frac {1}{n};-\frac {b x^n}{a}\right )}{a c (n+1)} \]
Antiderivative was successfully verified.
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Rule 364
Rubi steps
\begin {align*} \int \frac {(c x)^n}{a+b x^n} \, dx &=\frac {(c x)^{1+n} \, _2F_1\left (1,1+\frac {1}{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 = 36, normalized size = 0.90 \[ \frac {x (c x)^n \, _2F_1\left (1,1+\frac {1}{n};2+\frac {1}{n};-\frac {b x^n}{a}\right )}{a (n+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\left (c x\right )^{n}}{b x^{n} + a}, x\right ) \]
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 \[ \int \frac {\left (c x\right )^{n}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (c x \right )^{n}}{b \,x^{n}+a}\, dx \]
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 \[ -a c^{n} \int \frac {1}{b^{2} x^{n} + a b}\,{d x} + \frac {c^{n} x}{b} \]
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 \[ \int \frac {{\left (c\,x\right )}^n}{a+b\,x^n} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.22, size = 41, normalized size = 1.02 \[ - \frac {c^{n} x \Phi \left (\frac {a x^{- n} e^{i \pi }}{b}, 1, \frac {e^{i \pi }}{n}\right ) \Gamma \left (\frac {1}{n}\right )}{b n^{2} \Gamma \left (1 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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