Optimal. Leaf size=262 \[ \frac{d^2 (f x)^{m+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right )}{f (m+1)}+\frac{2 d e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+n+1}+\frac{e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2 n+1}{2 n},-p;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+2 n+1} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.159533, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {1561, 365, 364, 20} \[ \frac{d^2 (f x)^{m+1} \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right )}{f (m+1)}+\frac{2 d e x^{n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+n+1}+\frac{e^2 x^{2 n+1} (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+2 n+1}{2 n},-p;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+2 n+1} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1561
Rule 365
Rule 364
Rule 20
Rubi steps
\begin{align*} \int (f x)^m \left (d+e x^n\right )^2 \left (a+c x^{2 n}\right )^p \, dx &=\int \left (d^2 (f x)^m \left (a+c x^{2 n}\right )^p+2 d e x^n (f x)^m \left (a+c x^{2 n}\right )^p+e^2 x^{2 n} (f x)^m \left (a+c x^{2 n}\right )^p\right ) \, dx\\ &=d^2 \int (f x)^m \left (a+c x^{2 n}\right )^p \, dx+(2 d e) \int x^n (f x)^m \left (a+c x^{2 n}\right )^p \, dx+e^2 \int x^{2 n} (f x)^m \left (a+c x^{2 n}\right )^p \, dx\\ &=\left (2 d e x^{-m} (f x)^m\right ) \int x^{m+n} \left (a+c x^{2 n}\right )^p \, dx+\left (e^2 x^{-m} (f x)^m\right ) \int x^{m+2 n} \left (a+c x^{2 n}\right )^p \, dx+\left (d^2 \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int (f x)^m \left (1+\frac{c x^{2 n}}{a}\right )^p \, dx\\ &=\frac{d^2 (f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac{1+m}{2 n},-p;1+\frac{1+m}{2 n};-\frac{c x^{2 n}}{a}\right )}{f (1+m)}+\left (2 d e x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int x^{m+n} \left (1+\frac{c x^{2 n}}{a}\right )^p \, dx+\left (e^2 x^{-m} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p}\right ) \int x^{m+2 n} \left (1+\frac{c x^{2 n}}{a}\right )^p \, dx\\ &=\frac{d^2 (f x)^{1+m} \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac{1+m}{2 n},-p;1+\frac{1+m}{2 n};-\frac{c x^{2 n}}{a}\right )}{f (1+m)}+\frac{2 d e x^{1+n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac{1+m+n}{2 n},-p;\frac{1+m+3 n}{2 n};-\frac{c x^{2 n}}{a}\right )}{1+m+n}+\frac{e^2 x^{1+2 n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac{c x^{2 n}}{a}\right )^{-p} \, _2F_1\left (\frac{1+m+2 n}{2 n},-p;\frac{1+m+4 n}{2 n};-\frac{c x^{2 n}}{a}\right )}{1+m+2 n}\\ \end{align*}
Mathematica [A] time = 0.168455, size = 189, normalized size = 0.72 \[ x (f x)^m \left (a+c x^{2 n}\right )^p \left (\frac{c x^{2 n}}{a}+1\right )^{-p} \left (\frac{d^2 \, _2F_1\left (\frac{m+1}{2 n},-p;\frac{m+1}{2 n}+1;-\frac{c x^{2 n}}{a}\right )}{m+1}+e x^n \left (\frac{2 d \, _2F_1\left (\frac{m+n+1}{2 n},-p;\frac{m+3 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+n+1}+\frac{e x^n \, _2F_1\left (\frac{m+2 n+1}{2 n},-p;\frac{m+4 n+1}{2 n};-\frac{c x^{2 n}}{a}\right )}{m+2 n+1}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.075, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( d+e{x}^{n} \right ) ^{2} \left ( a+c{x}^{2\,n} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (e x^{n} + d\right )}^{2}{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (e^{2} x^{2 \, n} + 2 \, d e x^{n} + d^{2}\right )}{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]