Optimal. Leaf size=95 \[ \frac {b x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a (b c-a d)}-\frac {d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)} \]
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Rubi [A] time = 0.03, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {382, 379} \[ \frac {b x \left (c+d x^n\right )^{-1/n} \, _2F_1\left (1,\frac {1}{n};1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{a (b c-a d)}-\frac {d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 379
Rule 382
Rubi steps
\begin {align*} \int \frac {\left (c+d x^n\right )^{-1-\frac {1}{n}}}{a+b x^n} \, dx &=-\frac {d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)}+\frac {b \int \frac {\left (c+d x^n\right )^{-1/n}}{a+b x^n} \, dx}{b c-a d}\\ &=-\frac {d x \left (c+d x^n\right )^{-1/n}}{c (b c-a d)}+\frac {b 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 (b c-a d)}\\ \end {align*}
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Mathematica [C] time = 6.94, size = 153, normalized size = 1.61 \[ \frac {x \left (c+d x^n\right )^{-\frac {n+1}{n}} \left (\frac {b n x^{2 n} (a d-b c) \, _2F_1\left (2,2+\frac {1}{n};3+\frac {1}{n};\frac {(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{a^2 (2 n+1) \left (c+d x^n\right )}+\frac {b x^n \Phi \left (\frac {(a d-b c) x^n}{a \left (d x^n+c\right )},1,1+\frac {1}{n}\right )}{a}+\frac {a \left (c+d x^n\right )}{c \left (a+b x^n\right )}\right )}{a} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.06, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (b x^{n} + a\right )} {\left (d x^{n} + c\right )}^{\frac {n + 1}{n}}}, x\right ) \]
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 \[ \int \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.84, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \,x^{n}+c \right )^{-\frac {1}{n}-1}}{b \,x^{n}+a}\, dx \]
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 \[ \int \frac {{\left (d x^{n} + c\right )}^{-\frac {1}{n} - 1}}{b x^{n} + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\left (a+b\,x^n\right )\,{\left (c+d\,x^n\right )}^{\frac {1}{n}+1}} \,d x \]
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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