Optimal. Leaf size=59 \[ x \left (a+b x^n\right )^{-\frac {1-n}{n}} \left (\frac {b x^n}{a}+1\right )^{\frac {1}{n}-1} \, _2F_1\left (\frac {1}{n}-1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {246, 245} \[ x \left (a+b x^n\right )^{-\frac {1-n}{n}} \left (\frac {b x^n}{a}+1\right )^{\frac {1}{n}-1} \, _2F_1\left (\frac {1}{n}-1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right ) \]
Antiderivative was successfully verified.
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Rule 245
Rule 246
Rubi steps
\begin {align*} \int \left (a+b x^n\right )^{-\frac {1-n}{n}} \, dx &=\left (\left (a+b x^n\right )^{-\frac {1-n}{n}} \left (1+\frac {b x^n}{a}\right )^{\frac {1-n}{n}}\right ) \int \left (1+\frac {b x^n}{a}\right )^{-\frac {1-n}{n}} \, dx\\ &=x \left (a+b x^n\right )^{-\frac {1-n}{n}} \left (1+\frac {b x^n}{a}\right )^{-1+\frac {1}{n}} \, _2F_1\left (-1+\frac {1}{n},\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right )\\ \end {align*}
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Mathematica [A] time = 0.00, size = 53, normalized size = 0.90 \[ a x \left (a+b x^n\right )^{-1/n} \left (\frac {b x^n}{a}+1\right )^{\frac {1}{n}} \, _2F_1\left (\frac {1}{n}-1,\frac {1}{n};1+\frac {1}{n};-\frac {b x^n}{a}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.74, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x^{n} + a\right )}^{\frac {n - 1}{n}}, x\right ) \]
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 \[ \int {\left (b x^{n} + a\right )}^{\frac {n - 1}{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.21, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+a \right )^{-\frac {-n +1}{n}}\, dx \]
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 \[ \int {\left (b x^{n} + a\right )}^{\frac {n - 1}{n}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.51, size = 54, normalized size = 0.92 \[ \frac {a\,x\,{\left (\frac {b\,x^n}{a}+1\right )}^{1/n}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{n}-1,\frac {1}{n};\ \frac {1}{n}+1;\ -\frac {b\,x^n}{a}\right )}{{\left (a+b\,x^n\right )}^{1/n}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 12.22, size = 42, normalized size = 0.71 \[ \frac {a a^{- \frac {1}{n}} x \Gamma \left (\frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{n}, -1 + \frac {1}{n} \\ 1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (1 + \frac {1}{n}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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