Optimal. Leaf size=323 \[ \frac{d (f x)^{m+1} \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1)}+\frac{e x^{n+1} (f x)^m \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{m+n+1} \]
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Rubi [A] time = 0.37063, antiderivative size = 323, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1560, 1385, 510, 20} \[ \frac{d (f x)^{m+1} \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{f (m+1)}+\frac{e x^{n+1} (f x)^m \left (\frac{2 c x^n}{b-\sqrt{b^2-4 a c}}+1\right )^{-p} \left (\frac{2 c x^n}{\sqrt{b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b-\sqrt{b^2-4 a c}},-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}}\right )}{m+n+1} \]
Antiderivative was successfully verified.
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Rule 1560
Rule 1385
Rule 510
Rule 20
Rubi steps
\begin{align*} \int (f x)^m \left (d+e x^n\right ) \left (a+b x^n+c x^{2 n}\right )^p \, dx &=\int \left (d (f x)^m \left (a+b x^n+c x^{2 n}\right )^p+e x^n (f x)^m \left (a+b x^n+c x^{2 n}\right )^p\right ) \, dx\\ &=d \int (f x)^m \left (a+b x^n+c x^{2 n}\right )^p \, dx+e \int x^n (f x)^m \left (a+b x^n+c x^{2 n}\right )^p \, dx\\ &=\left (e x^{-m} (f x)^m\right ) \int x^{m+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 (f 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 (f 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 )}{f (1+m)}+\left (e x^{-m} (f 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} \left (a+b x^n+c x^{2 n}\right )^p\right ) \int x^{m+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{d (f 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 )}{f (1+m)}+\frac{e x^{1+n} (f 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} \left (a+b x^n+c x^{2 n}\right )^p F_1\left (\frac{1+m+n}{n};-p,-p;\frac{1+m+2 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 )}{1+m+n}\\ \end{align*}
Mathematica [A] time = 0.633243, size = 273, normalized size = 0.85 \[ \frac{x (f x)^m \left (\frac{-\sqrt{b^2-4 a c}+b+2 c x^n}{b-\sqrt{b^2-4 a c}}\right )^{-p} \left (\frac{\sqrt{b^2-4 a c}+b+2 c x^n}{\sqrt{b^2-4 a c}+b}\right )^{-p} \left (a+x^n \left (b+c x^n\right )\right )^p \left (d (m+n+1) F_1\left (\frac{m+1}{n};-p,-p;\frac{m+n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )+e (m+1) x^n F_1\left (\frac{m+n+1}{n};-p,-p;\frac{m+2 n+1}{n};-\frac{2 c x^n}{b+\sqrt{b^2-4 a c}},\frac{2 c x^n}{\sqrt{b^2-4 a c}-b}\right )\right )}{(m+1) (m+n+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.049, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \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} \left (f 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 (e x^{n} + d\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f 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 (e x^{n} + d\right )}{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} \left (f x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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