Optimal. Leaf size=210 \[ \frac{c \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (-\frac{2}{n};1,-q;-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{x^2 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{c \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (-\frac{2}{n};1,-q;-\frac{2-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{x^2 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]
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Rubi [A] time = 0.475619, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {1556, 511, 510} \[ \frac{c \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (-\frac{2}{n};1,-q;-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{x^2 \left (-b \sqrt{b^2-4 a c}-4 a c+b^2\right )}+\frac{c \left (d+e x^n\right )^q \left (\frac{e x^n}{d}+1\right )^{-q} F_1\left (-\frac{2}{n};1,-q;-\frac{2-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{x^2 \left (b \sqrt{b^2-4 a c}-4 a c+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 1556
Rule 511
Rule 510
Rubi steps
\begin{align*} \int \frac{\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx &=\frac{(2 c) \int \frac{\left (d+e x^n\right )^q}{x^3 \left (b-\sqrt{b^2-4 a c}+2 c x^n\right )} \, dx}{\sqrt{b^2-4 a c}}-\frac{(2 c) \int \frac{\left (d+e x^n\right )^q}{x^3 \left (b+\sqrt{b^2-4 a c}+2 c x^n\right )} \, 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{\left (1+\frac{e x^n}{d}\right )^q}{x^3 \left (b-\sqrt{b^2-4 a c}+2 c x^n\right )} \, 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{\left (1+\frac{e x^n}{d}\right )^q}{x^3 \left (b+\sqrt{b^2-4 a c}+2 c x^n\right )} \, dx}{\sqrt{b^2-4 a c}}\\ &=\frac{c \left (d+e x^n\right )^q \left (1+\frac{e x^n}{d}\right )^{-q} F_1\left (-\frac{2}{n};1,-q;-\frac{2-n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{\left (b^2-4 a c-b \sqrt{b^2-4 a c}\right ) x^2}+\frac{c \left (d+e x^n\right )^q \left (1+\frac{e x^n}{d}\right )^{-q} F_1\left (-\frac{2}{n};1,-q;-\frac{2-n}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},-\frac{e x^n}{d}\right )}{\left (b^2-4 a c+b \sqrt{b^2-4 a c}\right ) x^2}\\ \end{align*}
Mathematica [F] time = 0.128474, size = 0, normalized size = 0. \[ \int \frac{\left (d+e x^n\right )^q}{x^3 \left (a+b x^n+c x^{2 n}\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [F] time = 0.053, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d+e{x}^{n} \right ) ^{q}}{{x}^{3} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) }}\, 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 (c x^{2 \, n} + b x^{n} + a\right )} x^{3}}\,{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}}{c x^{3} x^{2 \, n} + b x^{3} x^{n} + a x^{3}}, 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 (c x^{2 \, n} + b x^{n} + a\right )} x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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