Optimal. Leaf size=158 \[ \frac{(d x)^{m+1} \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1)} \]
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Rubi [A] time = 0.129094, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {1385, 510} \[ \frac{(d x)^{m+1} \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{d (m+1)} \]
Antiderivative was successfully verified.
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Rule 1385
Rule 510
Rubi steps
\begin{align*} \int (d x)^m \left (a+b x^n+c x^{2 n}\right )^p \, dx &=\left (\left (1+\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (1+\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p\right ) \int (d x)^m \left (1+\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )^p \left (1+\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )^p \, dx\\ &=\frac{(d x)^{1+m} \left (1+\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (1+\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{1+m}{n};-p,-p;\frac{1+m+n}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{d (1+m)}\\ \end{align*}
Mathematica [A] time = 0.377937, size = 181, normalized size = 1.15 \[ \frac{x (d x)^m \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}\right )^{-p} \left (a+x^n \left (b+c x^n\right )\right )^p F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )}{m+1} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b{x}^{n}+c{x}^{2\,n} \right ) ^{p}\, 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{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (d x\right )^{m}\,{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 ({\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (d x\right )^{m}, 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{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (d x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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