\(\int \frac {(b x+c x^2)^p}{d+e x} \, dx\) [252]

Optimal result

Integrand size = 19, antiderivative size = 58 \[ \int \frac {\left (b x+c x^2\right )^p}{d+e x} \, dx=\frac {x \left (1+\frac {c x}{b}\right )^{-p} \left (b x+c x^2\right )^p \operatorname {AppellF1}\left (1+p,-p,1,2+p,-\frac {c x}{b},-\frac {e x}{d}\right )}{d (1+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.98 \[ \int \frac {\left (b x+c x^2\right )^p}{d+e x} \, dx=\frac {x \left (\frac {b+c x}{b}\right )^{-p} (x (b+c x))^p \operatorname {AppellF1}\left (1+p,-p,1,2+p,-\frac {c x}{b},-\frac {e x}{d}\right )}{d (1+p)} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.67, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1178, 150}

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.

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

\(\Big \downarrow \) 1178

\(\displaystyle -\frac {\left (b x+c x^2\right )^p \left (\frac {1}{d+e x}\right )^{2 p} \left (\frac {e x}{d+e x}\right )^{-p} \left (\frac {e (b+c x)}{c (d+e x)}\right )^{-p} \int \left (\frac {1}{d+e x}\right )^{-2 p-1} \left (1-\frac {d}{d+e x}\right )^p \left (1-\frac {d-\frac {b e}{c}}{d+e x}\right )^pd\frac {1}{d+e x}}{e}\)

\(\Big \downarrow \) 150

\(\displaystyle \frac {\left (b x+c x^2\right )^p \left (\frac {e x}{d+e x}\right )^{-p} \left (\frac {e (b+c x)}{c (d+e x)}\right )^{-p} \operatorname {AppellF1}\left (-2 p,-p,-p,1-2 p,\frac {d}{d+e x},\frac {d-\frac {b e}{c}}{d+e x}\right )}{2 e p}\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 150

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((e_) + (f_.)*(x_))^(p_), x_ 
] :> Simp[c^n*e^p*((b*x)^(m + 1)/(b*(m + 1)))*AppellF1[m + 1, -n, -p, m + 2 
, (-d)*(x/c), (-f)*(x/e)], x] /; FreeQ[{b, c, d, e, f, m, n, p}, x] &&  !In 
tegerQ[m] &&  !IntegerQ[n] && GtQ[c, 0] && (IntegerQ[p] || GtQ[e, 0])
 

rule 1178

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(-(1/(d + e*x))^(2*p))*((a + 
b*x + c*x^2)^p/(e*(e*((b - q + 2*c*x)/(2*c*(d + e*x))))^p*(e*((b + q + 2*c* 
x)/(2*c*(d + e*x))))^p))   Subst[Int[x^(-m - 2*(p + 1))*Simp[1 - (d - e*((b 
 - q)/(2*c)))*x, x]^p*Simp[1 - (d - e*((b + q)/(2*c)))*x, x]^p, x], x, 1/(d 
 + e*x)], x]] /; FreeQ[{a, b, c, d, e, p}, x] && ILtQ[m, 0]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (b x+c x^2\right )^p}{d+e x} \, dx=\frac {\left (c \,x^{2}+b x \right )^{p} b -\left (\int \frac {\left (c \,x^{2}+b x \right )^{p}}{b c \,e^{2} x^{3}+2 c^{2} d e \,x^{3}+b^{2} e^{2} x^{2}+3 b c d e \,x^{2}+2 c^{2} d^{2} x^{2}+b^{2} d e x +2 b c \,d^{2} x}d x \right ) b^{3} d e p -2 \left (\int \frac {\left (c \,x^{2}+b x \right )^{p}}{b c \,e^{2} x^{3}+2 c^{2} d e \,x^{3}+b^{2} e^{2} x^{2}+3 b c d e \,x^{2}+2 c^{2} d^{2} x^{2}+b^{2} d e x +2 b c \,d^{2} x}d x \right ) b^{2} c \,d^{2} p -\left (\int \frac {\left (c \,x^{2}+b x \right )^{p} x}{b c \,e^{2} x^{2}+2 c^{2} d e \,x^{2}+b^{2} e^{2} x +3 b c d e x +2 c^{2} d^{2} x +b^{2} d e +2 b c \,d^{2}}d x \right ) b^{2} c \,e^{2} p +4 \left (\int \frac {\left (c \,x^{2}+b x \right )^{p} x}{b c \,e^{2} x^{2}+2 c^{2} d e \,x^{2}+b^{2} e^{2} x +3 b c d e x +2 c^{2} d^{2} x +b^{2} d e +2 b c \,d^{2}}d x \right ) c^{3} d^{2} p}{p \left (b e +2 c d \right )} \] Input:

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

Output:

((b*x + c*x**2)**p*b - int((b*x + c*x**2)**p/(b**2*d*e*x + b**2*e**2*x**2 
+ 2*b*c*d**2*x + 3*b*c*d*e*x**2 + b*c*e**2*x**3 + 2*c**2*d**2*x**2 + 2*c** 
2*d*e*x**3),x)*b**3*d*e*p - 2*int((b*x + c*x**2)**p/(b**2*d*e*x + b**2*e** 
2*x**2 + 2*b*c*d**2*x + 3*b*c*d*e*x**2 + b*c*e**2*x**3 + 2*c**2*d**2*x**2 
+ 2*c**2*d*e*x**3),x)*b**2*c*d**2*p - int(((b*x + c*x**2)**p*x)/(b**2*d*e 
+ b**2*e**2*x + 2*b*c*d**2 + 3*b*c*d*e*x + b*c*e**2*x**2 + 2*c**2*d**2*x + 
 2*c**2*d*e*x**2),x)*b**2*c*e**2*p + 4*int(((b*x + c*x**2)**p*x)/(b**2*d*e 
 + b**2*e**2*x + 2*b*c*d**2 + 3*b*c*d*e*x + b*c*e**2*x**2 + 2*c**2*d**2*x 
+ 2*c**2*d*e*x**2),x)*c**3*d**2*p)/(p*(b*e + 2*c*d))