Integrand size = 22, antiderivative size = 238 \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\frac {x (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{2},-p,2,\frac {3+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2 (1+m)}-\frac {2 e x^2 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {2+m}{2},-p,2,\frac {4+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3 (2+m)}+\frac {e^2 x^3 (g x)^m \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {3+m}{2},-p,2,\frac {5+m}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^4 (3+m)} \]
x*(g*x)^m*(c*x^2+a)^p*AppellF1(1/2+1/2*m,2,-p,3/2+1/2*m,e^2*x^2/d^2,-c*x^2 /a)/d^2/(1+m)/((1+c*x^2/a)^p)-2*e*x^2*(g*x)^m*(c*x^2+a)^p*AppellF1(1+1/2*m ,2,-p,2+1/2*m,e^2*x^2/d^2,-c*x^2/a)/d^3/(2+m)/((1+c*x^2/a)^p)+e^2*x^3*(g*x )^m*(c*x^2+a)^p*AppellF1(3/2+1/2*m,2,-p,5/2+1/2*m,e^2*x^2/d^2,-c*x^2/a)/d^ 4/(3+m)/((1+c*x^2/a)^p)
\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx \]
Time = 0.45 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {623, 622, 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.
\(\displaystyle \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx\) |
\(\Big \downarrow \) 623 |
\(\displaystyle x^{-m} (g x)^m \int \frac {x^m \left (c x^2+a\right )^p}{(d+e x)^2}dx\) |
\(\Big \downarrow \) 622 |
\(\displaystyle x^{-m} (g x)^m \int \left (\frac {d^2 \left (c x^2+a\right )^p x^m}{\left (d^2-e^2 x^2\right )^2}-\frac {2 d e \left (c x^2+a\right )^p x^{m+1}}{\left (d^2-e^2 x^2\right )^2}+\frac {e^2 \left (c x^2+a\right )^p x^{m+2}}{\left (e^2 x^2-d^2\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle x^{-m} (g x)^m \left (\frac {x^{m+1} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+1}{2},-p,2,\frac {m+3}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^2 (m+1)}+\frac {e^2 x^{m+3} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+3}{2},-p,2,\frac {m+5}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^4 (m+3)}-\frac {2 e x^{m+2} \left (a+c x^2\right )^p \left (\frac {c x^2}{a}+1\right )^{-p} \operatorname {AppellF1}\left (\frac {m+2}{2},-p,2,\frac {m+4}{2},-\frac {c x^2}{a},\frac {e^2 x^2}{d^2}\right )}{d^3 (m+2)}\right )\) |
((g*x)^m*((x^(1 + m)*(a + c*x^2)^p*AppellF1[(1 + m)/2, -p, 2, (3 + m)/2, - ((c*x^2)/a), (e^2*x^2)/d^2])/(d^2*(1 + m)*(1 + (c*x^2)/a)^p) - (2*e*x^(2 + m)*(a + c*x^2)^p*AppellF1[(2 + m)/2, -p, 2, (4 + m)/2, -((c*x^2)/a), (e^2 *x^2)/d^2])/(d^3*(2 + m)*(1 + (c*x^2)/a)^p) + (e^2*x^(3 + m)*(a + c*x^2)^p *AppellF1[(3 + m)/2, -p, 2, (5 + m)/2, -((c*x^2)/a), (e^2*x^2)/d^2])/(d^4* (3 + m)*(1 + (c*x^2)/a)^p)))/x^m
3.5.35.3.1 Defintions of rubi rules used
Int[(x_)^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbo l] :> Int[ExpandIntegrand[x^m*(a + b*x^2)^p, (c/(c^2 - d^2*x^2) - d*(x/(c^2 - d^2*x^2)))^(-n), x], x] /; FreeQ[{a, b, c, d, m, p}, x] && ILtQ[n, -1]
Int[((e_)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*x)^m/x^m Int[x^m*(c + d*x)^n*(a + b*x^2)^p, x], x] / ; FreeQ[{a, b, c, d, e, m, p}, x] && ILtQ[n, 0]
\[\int \frac {\left (g x \right )^{m} \left (c \,x^{2}+a \right )^{p}}{\left (e x +d \right )^{2}}d x\]
\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\text {Timed out} \]
\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
\[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p} \left (g x\right )^{m}}{{\left (e x + d\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(g x)^m \left (a+c x^2\right )^p}{(d+e x)^2} \, dx=\int \frac {{\left (g\,x\right )}^m\,{\left (c\,x^2+a\right )}^p}{{\left (d+e\,x\right )}^2} \,d x \]