Optimal. Leaf size=93 \[ \frac {x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac {1}{n}-p} \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{c} \]
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Rubi [A] time = 0.02, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {380} \[ \frac {x \left (a+b x^n\right )^p \left (c+d x^n\right )^{-\frac {1}{n}-p} \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (d x^n+c\right )}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 380
Rubi steps
\begin {align*} \int \left (a+b x^n\right )^p \left (c+d x^n\right )^{-1-\frac {1}{n}-p} \, dx &=\frac {x \left (a+b x^n\right )^p \left (\frac {c \left (a+b x^n\right )}{a \left (c+d x^n\right )}\right )^{-p} \left (c+d x^n\right )^{-\frac {1}{n}-p} \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};-\frac {(b c-a d) x^n}{a \left (c+d x^n\right )}\right )}{c}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 94, normalized size = 1.01 \[ \frac {x \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \left (c+d x^n\right )^{-\frac {n p+1}{n}} \left (\frac {d x^n}{c}+1\right )^p \, _2F_1\left (\frac {1}{n},-p;1+\frac {1}{n};\frac {(a d-b c) x^n}{a \left (d x^n+c\right )}\right )}{c} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.10, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{n} + a\right )}^{p}}{{\left (d x^{n} + c\right )}^{\frac {n p + n + 1}{n}}}, x\right ) \]
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 \[ \int {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{-p - \frac {1}{n} - 1}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.91, size = 0, normalized size = 0.00 \[ \int \left (b \,x^{n}+a \right )^{p} \left (d \,x^{n}+c \right )^{-p -\frac {1}{n}-1}\, dx \]
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 \[ \int {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{-p - \frac {1}{n} - 1}\,{d x} \]
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 \[ \int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c+d\,x^n\right )}^{p+\frac {1}{n}+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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