Optimal. Leaf size=54 \[ \frac {c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]
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Rubi [A] time = 0.01, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {379} \[ \frac {c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]
Antiderivative was successfully verified.
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Rule 379
Rubi steps
\begin {align*} \int \frac {\left (c+d x^n\right )^{1-\frac {1}{n}}}{\left (a+b x^n\right )^2} \, dx &=\frac {c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{a^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 53, normalized size = 0.98 \[ \frac {c x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (2,\frac {1}{n};1+\frac {1}{n};\frac {(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{a^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.16, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (d x^{n} + c\right )}^{\frac {n - 1}{n}}}{b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} + 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.92, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{n}+c \right )^{-\frac {1}{n}+1}}{\left (b \,x^{n}+a \right )^{2}}\, dx \]
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 \[ \int \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} + 1}}{{\left (b x^{n} + a\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,x^n\right )}^{1-\frac {1}{n}}}{{\left (a+b\,x^n\right )}^2} \,d x \]
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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