Optimal. Leaf size=245 \[ \frac{2 c (f x)^{m+1} \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (\frac{m+1}{n};1,-q;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{f (m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c (f x)^{m+1} \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (\frac{m+1}{n};1,-q;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{f (m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]
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Rubi [A] time = 0.539941, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {1556, 511, 510} \[ \frac{2 c (f x)^{m+1} \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (\frac{m+1}{n};1,-q;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{f (m+1) \sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c (f x)^{m+1} \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (\frac{m+1}{n};1,-q;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{f (m+1) \sqrt{b^2-4 a c} \left (\sqrt{b^2-4 a c}+b\right )} \]
Antiderivative was successfully verified.
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Rule 1556
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{(f x)^m \left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx &=\frac{(2 c) \int \frac{(f x)^m \left (d+e x^n\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^n} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c) \int \frac{(f x)^m \left (d+e x^n\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{\left (2 c \left (d+e x^n\right )^q \left (1+\frac{e x^n}{d}\right )^{-q}\right ) \int \frac{(f x)^m \left (1+\frac{e x^n}{d}\right )^q}{b-\sqrt{b^2-4 a c}+2 c x^n} \, dx}{\sqrt{b^2-4 a c}}-\frac{\left (2 c \left (d+e x^n\right )^q \left (1+\frac{e x^n}{d}\right )^{-q}\right ) \int \frac{(f x)^m \left (1+\frac{e x^n}{d}\right )^q}{b+\sqrt{b^2-4 a c}+2 c x^n} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{2 c (f x)^{1+m} \left (d+e x^n\right )^q \left (1+\frac{e x^n}{d}\right )^{-q} F_1\left (\frac{1+m}{n};1,-q;\frac{1+m+n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{\sqrt{b^2-4 a c} \left (b-\sqrt{b^2-4 a c}\right ) f (1+m)}-\frac{2 c (f x)^{1+m} \left (d+e x^n\right )^q \left (1+\frac{e x^n}{d}\right )^{-q} F_1\left (\frac{1+m}{n};1,-q;\frac{1+m+n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{\sqrt{b^2-4 a c} \left (b+\sqrt{b^2-4 a c}\right ) f (1+m)}\\ \end{align*}
Mathematica [F] time = 0.174706, size = 0, normalized size = 0. \[ \int \frac{(f x)^m \left (d+e x^n\right )^q}{a+b x^n+c x^{2 n}} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx \right ) ^{m} \left ( d+e{x}^{n} \right ) ^{q}}{a+b{x}^{n}+c{x}^{2\,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 (e x^{n} + d\right )}^{q} \left (f x\right )^{m}}{c x^{2 \, n} + b x^{n} + a}\,{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 (e x^{n} + d\right )}^{q} \left (f x\right )^{m}}{c x^{2 \, n} + b x^{n} + a}, x\right ) \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{{\left (e x^{n} + d\right )}^{q} \left (f x\right )^{m}}{c x^{2 \, n} + b x^{n} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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