Optimal. Leaf size=53 \[ \frac {x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a} \]
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Rubi [A] time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {379} \[ \frac {x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a} \]
Antiderivative was successfully verified.
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Rule 379
Rubi steps
\begin {align*} \int \frac {\left (c+d x^n\right )^{-1/n}}{a+b x^n} \, dx &=\frac {x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 52, normalized size = 0.98 \[ \frac {x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};\frac {(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.81, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{\left (\frac {1}{n}\right )}}, x\right ) \]
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 \[ \int \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{\left (\frac {1}{n}\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.98, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{n}+c \right )^{-\frac {1}{n}}}{b \,x^{n}+a}\, dx \]
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 \[ \int \frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{\left (\frac {1}{n}\right )}}\,{d x} \]
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 \[ \int \frac {1}{\left (a+b\,x^n\right )\,{\left (c+d\,x^n\right )}^{1/n}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]
Verification of antiderivative is not currently implemented for this CAS.
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