Integrand size = 23, antiderivative size = 53 \[ \int \frac {\left (b x+c x^2\right )^p}{e x+f x^2} \, dx=\frac {\left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \operatorname {AppellF1}\left (p,-p,1,1+p,-\frac {c x}{b},-\frac {f x}{e}\right )}{e p} \] Output:
(c*x^2+b*x)^p*AppellF1(p,-p,1,p+1,-c*x/b,-f*x/e)/e/p/((1+c*x/b)^p)
Time = 0.29 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.58 \[ \int \frac {\left (b x+c x^2\right )^p}{e x+f x^2} \, dx=\frac {(x (b+c x))^p \left (1+\frac {c x}{b}\right )^{-p} \left (-f p x \operatorname {AppellF1}\left (1+p,-p,1,2+p,-\frac {c x}{b},-\frac {f x}{e}\right )+e (1+p) \operatorname {Hypergeometric2F1}\left (-p,p,1+p,-\frac {c x}{b}\right )\right )}{e^2 p (1+p)} \] Input:
Integrate[(b*x + c*x^2)^p/(e*x + f*x^2),x]
Output:
((x*(b + c*x))^p*(-(f*p*x*AppellF1[1 + p, -p, 1, 2 + p, -((c*x)/b), -((f*x )/e)]) + e*(1 + p)*Hypergeometric2F1[-p, p, 1 + p, -((c*x)/b)]))/(e^2*p*(1 + p)*(1 + (c*x)/b)^p)
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.
\(\displaystyle \int \frac {\left (b x+c x^2\right )^p}{e x+f x^2} \, dx\) |
\(\Big \downarrow \) 1325 |
\(\displaystyle \int \frac {\left (b x+c x^2\right )^p}{e x+f x^2}dx\) |
Input:
Int[(b*x + c*x^2)^p/(e*x + f*x^2),x]
Output:
$Aborted
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 ]
\[\int \frac {\left (c \,x^{2}+b x \right )^{p}}{f \,x^{2}+e x}d x\]
Input:
int((c*x^2+b*x)^p/(f*x^2+e*x),x)
Output:
int((c*x^2+b*x)^p/(f*x^2+e*x),x)
\[ \int \frac {\left (b x+c x^2\right )^p}{e x+f x^2} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{p}}{f x^{2} + e x} \,d x } \] Input:
integrate((c*x^2+b*x)^p/(f*x^2+e*x),x, algorithm="fricas")
Output:
integral((c*x^2 + b*x)^p/(f*x^2 + e*x), x)
\[ \int \frac {\left (b x+c x^2\right )^p}{e x+f x^2} \, dx=\int \frac {\left (x \left (b + c x\right )\right )^{p}}{x \left (e + f x\right )}\, dx \] Input:
integrate((c*x**2+b*x)**p/(f*x**2+e*x),x)
Output:
Integral((x*(b + c*x))**p/(x*(e + f*x)), x)
\[ \int \frac {\left (b x+c x^2\right )^p}{e x+f x^2} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{p}}{f x^{2} + e x} \,d x } \] Input:
integrate((c*x^2+b*x)^p/(f*x^2+e*x),x, algorithm="maxima")
Output:
integrate((c*x^2 + b*x)^p/(f*x^2 + e*x), x)
\[ \int \frac {\left (b x+c x^2\right )^p}{e x+f x^2} \, dx=\int { \frac {{\left (c x^{2} + b x\right )}^{p}}{f x^{2} + e x} \,d x } \] Input:
integrate((c*x^2+b*x)^p/(f*x^2+e*x),x, algorithm="giac")
Output:
integrate((c*x^2 + b*x)^p/(f*x^2 + e*x), x)
Timed out. \[ \int \frac {\left (b x+c x^2\right )^p}{e x+f x^2} \, dx=\int \frac {{\left (c\,x^2+b\,x\right )}^p}{f\,x^2+e\,x} \,d x \] Input:
int((b*x + c*x^2)^p/(e*x + f*x^2),x)
Output:
int((b*x + c*x^2)^p/(e*x + f*x^2), x)
\[ \int \frac {\left (b x+c x^2\right )^p}{e x+f x^2} \, dx=\int \frac {\left (c \,x^{2}+b x \right )^{p}}{f \,x^{2}+e x}d x \] Input:
int((c*x^2+b*x)^p/(f*x^2+e*x),x)
Output:
int((b*x + c*x**2)**p/(e*x + f*x**2),x)