\(\int \frac {(d+f x^2)^q}{a+b x+c x^2} \, dx\) [12]

Optimal result

Integrand size = 22, antiderivative size = 412 \[ \int \frac {\left (d+f x^2\right )^q}{a+b x+c x^2} \, dx=-\frac {2 c x \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-q,1,\frac {3}{2},-\frac {f x^2}{d},\frac {4 c^2 x^2}{\left (b-\sqrt {b^2-4 a c}\right )^2}\right )}{b^2-4 a c-b \sqrt {b^2-4 a c}}-\frac {2 c x \left (d+f x^2\right )^q \left (1+\frac {f x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-q,1,\frac {3}{2},-\frac {f x^2}{d},\frac {4 c^2 x^2}{\left (b+\sqrt {b^2-4 a c}\right )^2}\right )}{b^2-4 a c+b \sqrt {b^2-4 a c}}-\frac {2 c^2 \left (d+f x^2\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,1+q,2+q,\frac {4 c^2 \left (d+f x^2\right )}{4 c^2 d+\left (b-\sqrt {b^2-4 a c}\right )^2 f}\right )}{\sqrt {b^2-4 a c} \left (4 c^2 d+\left (b-\sqrt {b^2-4 a c}\right )^2 f\right ) (1+q)}+\frac {2 c^2 \left (d+f x^2\right )^{1+q} \operatorname {Hypergeometric2F1}\left (1,1+q,2+q,\frac {4 c^2 \left (d+f x^2\right )}{4 c^2 d+\left (b+\sqrt {b^2-4 a c}\right )^2 f}\right )}{\sqrt {b^2-4 a c} \left (4 c^2 d+\left (b+\sqrt {b^2-4 a c}\right )^2 f\right ) (1+q)} \] Output:

-2*c*x*(f*x^2+d)^q*AppellF1(1/2,1,-q,3/2,4*c^2*x^2/(b-(-4*a*c+b^2)^(1/2))^ 
2,-f*x^2/d)/(b^2-4*a*c-b*(-4*a*c+b^2)^(1/2))/((1+f*x^2/d)^q)-2*c*x*(f*x^2+ 
d)^q*AppellF1(1/2,1,-q,3/2,4*c^2*x^2/(b+(-4*a*c+b^2)^(1/2))^2,-f*x^2/d)/(b 
*(-4*a*c+b^2)^(1/2)-4*a*c+b^2)/((1+f*x^2/d)^q)-2*c^2*(f*x^2+d)^(1+q)*hyper 
geom([1, 1+q],[2+q],4*c^2*(f*x^2+d)/(4*c^2*d+(b-(-4*a*c+b^2)^(1/2))^2*f))/ 
(-4*a*c+b^2)^(1/2)/(4*c^2*d+(b-(-4*a*c+b^2)^(1/2))^2*f)/(1+q)+2*c^2*(f*x^2 
+d)^(1+q)*hypergeom([1, 1+q],[2+q],4*c^2*(f*x^2+d)/(4*c^2*d+(b+(-4*a*c+b^2 
)^(1/2))^2*f))/(-4*a*c+b^2)^(1/2)/(4*c^2*d+(b+(-4*a*c+b^2)^(1/2))^2*f)/(1+ 
q)
 

Mathematica [F]

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

Integrate[(d + f*x^2)^q/(a + b*x + c*x^2),x]
 

Output:

Integrate[(d + f*x^2)^q/(a + b*x + c*x^2), x]
 

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.

\(\displaystyle \int \frac {\left (d+f x^2\right )^q}{a+b x+c x^2} \, dx\)

\(\Big \downarrow \) 1326

\(\displaystyle \int \frac {\left (d+f x^2\right )^q}{a+b x+c x^2}dx\)

Input:

Int[(d + f*x^2)^q/(a + b*x + c*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]

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

Input:

int((f*x^2+d)^q/(c*x^2+b*x+a),x)
 

Output:

int((f*x^2+d)^q/(c*x^2+b*x+a),x)
 

Fricas [F]

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

integrate((f*x^2+d)^q/(c*x^2+b*x+a),x, algorithm="fricas")
 

Output:

integral((f*x^2 + d)^q/(c*x^2 + b*x + a), x)
 

Sympy [F(-1)]

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

integrate((f*x**2+d)**q/(c*x**2+b*x+a),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((f*x^2+d)^q/(c*x^2+b*x+a),x, algorithm="maxima")
 

Output:

integrate((f*x^2 + d)^q/(c*x^2 + b*x + a), x)
 

Giac [F]

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

integrate((f*x^2+d)^q/(c*x^2+b*x+a),x, algorithm="giac")
 

Output:

integrate((f*x^2 + d)^q/(c*x^2 + b*x + a), x)
 

Mupad [F(-1)]

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

int((d + f*x^2)^q/(a + b*x + c*x^2),x)
 

Output:

int((d + f*x^2)^q/(a + b*x + c*x^2), x)
 

Reduce [F]

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

int((f*x^2+d)^q/(c*x^2+b*x+a),x)
 

Output:

int((d + f*x**2)**q/(a + b*x + c*x**2),x)