Integrand size = 26, antiderivative size = 302 \[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\frac {x (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{2 n},-p,2,1+\frac {1+m}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (1+m)}-\frac {2 e x^{1+n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m+n}{2 n},-p,2,\frac {1+m+3 n}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (1+m+n)}+\frac {e^2 x^{1+2 n} (f x)^m \left (a+c x^{2 n}\right )^p \left (1+\frac {c x^{2 n}}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m+2 n}{2 n},-p,2,\frac {1+m+4 n}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (1+m+2 n)} \]
x*(f*x)^m*(a+c*x^(2*n))^p*AppellF1(1/2*(1+m)/n,2,-p,1+1/2*(1+m)/n,e^2*x^(2 *n)/d^2,-c*x^(2*n)/a)/d^2/(1+m)/((1+c*x^(2*n)/a)^p)-2*e*x^(1+n)*(f*x)^m*(a +c*x^(2*n))^p*AppellF1(1/2*(1+m+n)/n,2,-p,1/2*(1+m+3*n)/n,e^2*x^(2*n)/d^2, -c*x^(2*n)/a)/d^3/(1+m+n)/((1+c*x^(2*n)/a)^p)+e^2*x^(1+2*n)*(f*x)^m*(a+c*x ^(2*n))^p*AppellF1(1/2*(1+m+2*n)/n,2,-p,1/2*(1+m+4*n)/n,e^2*x^(2*n)/d^2,-c *x^(2*n)/a)/d^4/(1+m+2*n)/((1+c*x^(2*n)/a)^p)
\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx \]
Time = 0.52 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.01, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {1886, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx\) |
\(\Big \downarrow \) 1886 |
\(\displaystyle x^{-m} (f x)^m \int \left (\frac {d^2 \left (c x^{2 n}+a\right )^p x^m}{\left (d^2-e^2 x^{2 n}\right )^2}-\frac {2 d e \left (c x^{2 n}+a\right )^p x^{m+n}}{\left (d^2-e^2 x^{2 n}\right )^2}+\frac {e^2 \left (c x^{2 n}+a\right )^p x^{m+2 n}}{\left (d^2-e^2 x^{2 n}\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^{-m} (f x)^m \left (\frac {x^{m+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+1}{2 n},-p,2,\frac {m+1}{2 n}+1,-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^2 (m+1)}+\frac {e^2 x^{m+2 n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+2 n+1}{2 n},-p,2,\frac {m+4 n+1}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^4 (m+2 n+1)}-\frac {2 e x^{m+n+1} \left (a+c x^{2 n}\right )^p \left (\frac {c x^{2 n}}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+n+1}{2 n},-p,2,\frac {m+3 n+1}{2 n},-\frac {c x^{2 n}}{a},\frac {e^2 x^{2 n}}{d^2}\right )}{d^3 (m+n+1)}\right )\) |
((f*x)^m*((x^(1 + m)*(a + c*x^(2*n))^p*AppellF1[(1 + m)/(2*n), -p, 2, 1 + (1 + m)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/(d^2*(1 + m)*(1 + (c* x^(2*n))/a)^p) - (2*e*x^(1 + m + n)*(a + c*x^(2*n))^p*AppellF1[(1 + m + n) /(2*n), -p, 2, (1 + m + 3*n)/(2*n), -((c*x^(2*n))/a), (e^2*x^(2*n))/d^2])/ (d^3*(1 + m + n)*(1 + (c*x^(2*n))/a)^p) + (e^2*x^(1 + m + 2*n)*(a + c*x^(2 *n))^p*AppellF1[(1 + m + 2*n)/(2*n), -p, 2, (1 + m + 4*n)/(2*n), -((c*x^(2 *n))/a), (e^2*x^(2*n))/d^2])/(d^4*(1 + m + 2*n)*(1 + (c*x^(2*n))/a)^p)))/x ^m
3.1.91.3.1 Defintions of rubi rules used
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2 _))^(p_), x_Symbol] :> Simp[(f*x)^m/x^m Int[ExpandIntegrand[x^m*(a + c*x^ (2*n))^p, (d/(d^2 - e^2*x^(2*n)) - e*(x^n/(d^2 - e^2*x^(2*n))))^(-q), x], x ], x] /; FreeQ[{a, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] && !RationalQ[ n] && !IntegerQ[p] && ILtQ[q, 0]
\[\int \frac {\left (f x \right )^{m} \left (a +c \,x^{2 n}\right )^{p}}{\left (d +e \,x^{n}\right )^{2}}d x\]
\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]
\[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int { \frac {{\left (c x^{2 \, n} + a\right )}^{p} \left (f x\right )^{m}}{{\left (e x^{n} + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(f x)^m \left (a+c x^{2 n}\right )^p}{\left (d+e x^n\right )^2} \, dx=\int \frac {{\left (a+c\,x^{2\,n}\right )}^p\,{\left (f\,x\right )}^m}{{\left (d+e\,x^n\right )}^2} \,d x \]