\(\int \frac {(b x+c x^2)^p}{d+f x^2} \, dx\) [176]

Optimal result

Integrand size = 21, antiderivative size = 138 \[ \int \frac {\left (b x+c x^2\right )^p}{d+f x^2} \, dx=\frac {x \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \operatorname {AppellF1}\left (1+p,-p,1,2+p,-\frac {c x}{b},-\frac {\sqrt {f} x}{\sqrt {-d}}\right )}{2 d (1+p)}+\frac {x \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \operatorname {AppellF1}\left (1+p,-p,1,2+p,-\frac {c x}{b},\frac {\sqrt {f} x}{\sqrt {-d}}\right )}{2 d (1+p)} \] Output:

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

Mathematica [F]

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

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

Output:

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

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

\(\Big \downarrow \) 1326

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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