\(\int \frac {(b x+c x^2)^p}{d+e x+f x^2} \, dx\) [175]

Optimal result

Integrand size = 24, antiderivative size = 201 \[ \int \frac {\left (b x+c x^2\right )^p}{d+e x+f x^2} \, dx=\frac {2 f x \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \operatorname {AppellF1}\left (1+p,-p,1,2+p,-\frac {c x}{b},-\frac {2 f x}{e-\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f} \left (e-\sqrt {e^2-4 d f}\right ) (1+p)}-\frac {2 f x \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \operatorname {AppellF1}\left (1+p,-p,1,2+p,-\frac {c x}{b},-\frac {2 f x}{e+\sqrt {e^2-4 d f}}\right )}{\sqrt {e^2-4 d f} \left (e+\sqrt {e^2-4 d f}\right ) (1+p)} \] Output:

2*f*x*(c*x^2+b*x)^p*AppellF1(p+1,-p,1,2+p,-c*x/b,-2*f*x/(e-(-4*d*f+e^2)^(1 
/2)))/(-4*d*f+e^2)^(1/2)/(e-(-4*d*f+e^2)^(1/2))/(p+1)/((1+c*x/b)^p)-2*f*x* 
(c*x^2+b*x)^p*AppellF1(p+1,-p,1,2+p,-c*x/b,-2*f*x/(e+(-4*d*f+e^2)^(1/2)))/ 
(-4*d*f+e^2)^(1/2)/(e+(-4*d*f+e^2)^(1/2))/(p+1)/((1+c*x/b)^p)
 

Mathematica [F]

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

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

Output:

Integrate[(b*x + c*x^2)^p/(d + e*x + f*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.

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

\(\Big \downarrow \) 1325

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 1325

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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