Optimal. Leaf size=57 \[ \frac{x \left (a+b x^n\right )^{-\frac{b c}{n (b c-a d)}} \left (c+d x^n\right )^{\frac{a d}{n (b c-a d)}}}{a c} \]
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Rubi [A] time = 0.0271107, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 69, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.014, Rules used = {381} \[ \frac{x \left (a+b x^n\right )^{-\frac{b c}{n (b c-a d)}} \left (c+d x^n\right )^{\frac{a d}{n (b c-a d)}}}{a c} \]
Antiderivative was successfully verified.
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Rule 381
Rubi steps
\begin{align*} \int \left (a+b x^n\right )^{\frac{a d n-b c (1+n)}{(b c-a d) n}} \left (c+d x^n\right )^{\frac{a d-b c n+a d n}{b c n-a d n}} \, dx &=\frac{x \left (a+b x^n\right )^{-\frac{b c}{(b c-a d) n}} \left (c+d x^n\right )^{\frac{a d}{(b c-a d) n}}}{a c}\\ \end{align*}
Mathematica [A] time = 0.0492145, size = 55, normalized size = 0.96 \[ \frac{x \left (a+b x^n\right )^{-\frac{b c}{b c n-a d n}} \left (c+d x^n\right )^{\frac{a d}{b c n-a d n}}}{a c} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.817, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b{x}^{n} \right ) ^{{\frac{adn-bc \left ( 1+n \right ) }{ \left ( -ad+bc \right ) n}}} \left ( c+d{x}^{n} \right ) ^{{\frac{adn-bcn+ad}{-adn+bcn}}}\, 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{1}{{\left (b x^{n} + a\right )}^{\frac{b c{\left (n + 1\right )} - a d n}{{\left (b c - a d\right )} n}}{\left (d x^{n} + c\right )}^{\frac{b c n - a d n - a d}{b c n - a d n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.71863, size = 209, normalized size = 3.67 \begin{align*} \frac{{\left (b d x x^{2 \, n} + a c x +{\left (b c + a d\right )} x x^{n}\right )}{\left (d x^{n} + c\right )}^{\frac{a d -{\left (b c - a d\right )} n}{{\left (b c - a d\right )} n}}}{{\left (b x^{n} + a\right )}^{\frac{b c +{\left (b c - a d\right )} n}{{\left (b c - a d\right )} n}} a c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{1}{{\left (b x^{n} + a\right )}^{\frac{b c{\left (n + 1\right )} - a d n}{{\left (b c - a d\right )} n}}{\left (d x^{n} + c\right )}^{\frac{b c n - a d n - a d}{b c n - a d n}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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