Optimal. Leaf size=288 \[ d x \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )+\frac{e x^{n+1} \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{n+1} \]
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Rubi [A] time = 0.286561, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {1432, 1348, 429, 1385, 510} \[ d x \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )+\frac{e x^{n+1} \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{n+1} \]
Antiderivative was successfully verified.
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Rule 1432
Rule 1348
Rule 429
Rule 1385
Rule 510
Rubi steps
\begin{align*} \int \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^p \, dx &=\int \left (d \left (a+b x^n+c x^{2 n}\right )^p+e x^n \left (a+b x^n+c x^{2 n}\right )^p\right ) \, dx\\ &=d \int \left (a+b x^n+c x^{2 n}\right )^p \, dx+e \int x^n \left (a+b x^n+c x^{2 n}\right )^p \, dx\\ &=\left (d \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 \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+\left (e \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 x^n \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{e x^{1+n} \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 (1+\frac{1}{n};-p,-p;2+\frac{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 )}{1+n}+d x \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}{n};-p,-p;1+\frac{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 )\\ \end{align*}
Mathematica [A] time = 0.688071, size = 243, normalized size = 0.84 \[ \frac{x \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}\right )^{-p} \left (a+x^n \left (b+c x^n\right )\right )^p \left (d (n+1) F_1\left (\frac{1}{n};-p,-p;1+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+e x^n F_1\left (1+\frac{1}{n};-p,-p;2+\frac{1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )}{n+1} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.085, size = 0, normalized size = 0. \begin{align*} \int \left ( d+e{x}^{n} \right ) \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 (e x^{n} + d\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}\,{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 (e x^{n} + d\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}, 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 (e x^{n} + d\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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