\(\int \frac {(a+c x^2)^p}{e x+f x^2} \, dx\) [179]

Optimal result

Integrand size = 21, antiderivative size = 177 \[ \int \frac {\left (a+c x^2\right )^p}{e x+f x^2} \, dx=-\frac {f x \left (a+c x^2\right )^p \left (1+\frac {c x^2}{a}\right )^{-p} \operatorname {AppellF1}\left (\frac {1}{2},-p,1,\frac {3}{2},-\frac {c x^2}{a},\frac {f^2 x^2}{e^2}\right )}{e^2}-\frac {\left (a+c x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {a+c x^2}{a}\right )}{2 a e (1+p)}+\frac {f^2 \left (a+c x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,\frac {f^2 \left (a+c x^2\right )}{c e^2+a f^2}\right )}{2 e \left (c e^2+a f^2\right ) (1+p)} \] Output:

-f*x*(c*x^2+a)^p*AppellF1(1/2,1,-p,3/2,f^2*x^2/e^2,-c*x^2/a)/e^2/((1+c*x^2 
/a)^p)-1/2*(c*x^2+a)^(p+1)*hypergeom([1, p+1],[2+p],(c*x^2+a)/a)/a/e/(p+1) 
+1/2*f^2*(c*x^2+a)^(p+1)*hypergeom([1, p+1],[2+p],f^2*(c*x^2+a)/(a*f^2+c*e 
^2))/e/(a*f^2+c*e^2)/(p+1)
 

Mathematica [A] (warning: unable to verify)

Time = 0.71 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.96 \[ \int \frac {\left (a+c x^2\right )^p}{e x+f x^2} \, dx=\frac {\left (a+c x^2\right )^p \left (-\left (\frac {f \left (-\sqrt {-\frac {a}{c}}+x\right )}{e+f x}\right )^{-p} \left (\frac {f \left (\sqrt {-\frac {a}{c}}+x\right )}{e+f x}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {e-\sqrt {-\frac {a}{c}} f}{e+f x},\frac {e+\sqrt {-\frac {a}{c}} f}{e+f x}\right )+\left (1+\frac {a}{c x^2}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-p,-p,1-p,-\frac {a}{c x^2}\right )\right )}{2 e p} \] Input:

Integrate[(a + c*x^2)^p/(e*x + f*x^2),x]
 

Output:

((a + c*x^2)^p*(-(AppellF1[-2*p, -p, -p, 1 - 2*p, (e - Sqrt[-(a/c)]*f)/(e 
+ f*x), (e + Sqrt[-(a/c)]*f)/(e + f*x)]/(((f*(-Sqrt[-(a/c)] + x))/(e + f*x 
))^p*((f*(Sqrt[-(a/c)] + x))/(e + f*x))^p)) + Hypergeometric2F1[-p, -p, 1 
- p, -(a/(c*x^2))]/(1 + a/(c*x^2))^p))/(2*e*p)
 

Rubi [F]

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.

\(\displaystyle \int \frac {\left (a+c x^2\right )^p}{e x+f x^2} \, dx\)

\(\Big \downarrow \) 1326

\(\displaystyle \int \frac {\left (a+c x^2\right )^p}{e x+f x^2}dx\)

Input:

Int[(a + c*x^2)^p/(e*x + f*x^2),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1326

Int[((a_) + (c_.)*(x_)^2)^(p_)*((d_.) + (e_.)*(x_) + (f_.)*(x_)^2)^(q_), x_ 
Symbol] :> Unintegrable[(a + c*x^2)^p*(d + e*x + f*x^2)^q, x] /; FreeQ[{a, 
c, d, e, f, p, q}, x] &&  !IGtQ[p, 0] &&  !IGtQ[q, 0]
 

Maple [F]

\[\int \frac {\left (c \,x^{2}+a \right )^{p}}{f \,x^{2}+e x}d x\]

Input:

int((c*x^2+a)^p/(f*x^2+e*x),x)
 

Output:

int((c*x^2+a)^p/(f*x^2+e*x),x)
 

Fricas [F]

\[ \int \frac {\left (a+c x^2\right )^p}{e x+f x^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{f x^{2} + e x} \,d x } \] Input:

integrate((c*x^2+a)^p/(f*x^2+e*x),x, algorithm="fricas")
 

Output:

integral((c*x^2 + a)^p/(f*x^2 + e*x), x)
 

Sympy [F]

\[ \int \frac {\left (a+c x^2\right )^p}{e x+f x^2} \, dx=\int \frac {\left (a + c x^{2}\right )^{p}}{x \left (e + f x\right )}\, dx \] Input:

integrate((c*x**2+a)**p/(f*x**2+e*x),x)
 

Output:

Integral((a + c*x**2)**p/(x*(e + f*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a+c x^2\right )^p}{e x+f x^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{f x^{2} + e x} \,d x } \] Input:

integrate((c*x^2+a)^p/(f*x^2+e*x),x, algorithm="maxima")
 

Output:

integrate((c*x^2 + a)^p/(f*x^2 + e*x), x)
 

Giac [F]

\[ \int \frac {\left (a+c x^2\right )^p}{e x+f x^2} \, dx=\int { \frac {{\left (c x^{2} + a\right )}^{p}}{f x^{2} + e x} \,d x } \] Input:

integrate((c*x^2+a)^p/(f*x^2+e*x),x, algorithm="giac")
 

Output:

integrate((c*x^2 + a)^p/(f*x^2 + e*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+c x^2\right )^p}{e x+f x^2} \, dx=\int \frac {{\left (c\,x^2+a\right )}^p}{f\,x^2+e\,x} \,d x \] Input:

int((a + c*x^2)^p/(e*x + f*x^2),x)
 

Output:

int((a + c*x^2)^p/(e*x + f*x^2), x)
 

Reduce [F]

\[ \int \frac {\left (a+c x^2\right )^p}{e x+f x^2} \, dx=\int \frac {\left (c \,x^{2}+a \right )^{p}}{f \,x^{2}+e x}d x \] Input:

int((c*x^2+a)^p/(f*x^2+e*x),x)
 

Output:

int((a + c*x**2)**p/(e*x + f*x**2),x)