Optimal. Leaf size=59 \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c} \]
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Rubi [A] time = 0.0289819, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {430, 429} \[ \frac{x \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c} \]
Antiderivative was successfully verified.
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Rule 430
Rule 429
Rubi steps
\begin{align*} \int \frac{\left (a+b x^n\right )^p}{c+d x^n} \, dx &=\left (\left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p}\right ) \int \frac{\left (1+\frac{b x^n}{a}\right )^p}{c+d x^n} \, dx\\ &=\frac{x \left (a+b x^n\right )^p \left (1+\frac{b x^n}{a}\right )^{-p} F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{c}\\ \end{align*}
Mathematica [B] time = 0.255761, size = 180, normalized size = 3.05 \[ \frac{a c (n+1) x \left (a+b x^n\right )^p F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )}{\left (c+d x^n\right ) \left (b c n p x^n F_1\left (1+\frac{1}{n};1-p,1;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )-a d n x^n F_1\left (1+\frac{1}{n};-p,2;2+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )+a c (n+1) F_1\left (\frac{1}{n};-p,1;1+\frac{1}{n};-\frac{b x^n}{a},-\frac{d x^n}{c}\right )\right )} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.711, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b{x}^{n} \right ) ^{p}}{c+d{x}^{n}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{n} + a\right )}^{p}}{d x^{n} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{n} + a\right )}^{p}}{d x^{n} + c}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b x^{n}\right )^{p}}{c + d x^{n}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{n} + a\right )}^{p}}{d x^{n} + c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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