\(\int (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3)^p \, dx\) [136]

Optimal result

Integrand size = 46, antiderivative size = 151 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^p \, dx=\frac {(a+b x) \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^p \operatorname {AppellF1}\left (1+p,-p,-p,2+p,-\frac {d (a+b x)}{b c-a d},-\frac {f (a+b x)}{b e-a f}\right )}{b (1+p)} \] Output:

(b*x+a)*(a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^p* 
AppellF1(p+1,-p,-p,2+p,-d*(b*x+a)/(-a*d+b*c),-f*(b*x+a)/(-a*f+b*e))/b/(p+1 
)/((b*(d*x+c)/(-a*d+b*c))^p)/((b*(f*x+e)/(-a*f+b*e))^p)
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.80 \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^p \, dx=\frac {(a+b x) \left (\frac {b (c+d x)}{b c-a d}\right )^{-p} \left (\frac {b (e+f x)}{b e-a f}\right )^{-p} ((a+b x) (c+d x) (e+f x))^p \operatorname {AppellF1}\left (1+p,-p,-p,2+p,\frac {d (a+b x)}{-b c+a d},\frac {f (a+b x)}{-b e+a f}\right )}{b (1+p)} \] Input:

Integrate[(a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 
 + b*d*f*x^3)^p,x]
 

Output:

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

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.

\(\displaystyle \int \left (x^2 (a d f+b c f+b d e)+x (a c f+a d e+b c e)+a c e+b d f x^3\right )^p \, dx\)

\(\Big \downarrow \) 2481

\(\displaystyle \int \left (\frac {(-2 a d f+b c f+b d e) (-a d f-b c f+2 b d e) (a d f-2 b c f+b d e)}{27 b^2 d^2 f^2}+b d f \left (\frac {a d f+b c f+b d e}{3 b d f}+x\right )^3+\frac {\left (3 b d f (a c f+a d e+b c e)-(a d f+b c f+b d e)^2\right ) \left (\frac {a d f+b c f+b d e}{3 b d f}+x\right )}{3 b d f}\right )^pd\left (\frac {a d f+b c f+b d e}{3 b d f}+x\right )\)

Input:

Int[(a*c*e + (b*c*e + a*d*e + a*c*f)*x + (b*d*e + b*c*f + a*d*f)*x^2 + b*d 
*f*x^3)^p,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2481

Int[(Px_)^(p_), x_Symbol] :> With[{a = Coeff[Px, x, 0], b = Coeff[Px, x, 1] 
, c = Coeff[Px, x, 2], d = Coeff[Px, x, 3]}, Subst[Int[Simp[(2*c^3 - 9*b*c* 
d + 27*a*d^2)/(27*d^2) - (c^2 - 3*b*d)*(x/(3*d)) + d*x^3, x]^p, x], x, c/(3 
*d) + x]] /; FreeQ[p, x] && PolyQ[Px, x, 3]
 

Maple [F]

\[\int \left (a c e +\left (a c f +a d e +b c e \right ) x +\left (a d f +b c f +b d e \right ) x^{2}+b d f \,x^{3}\right )^{p}d x\]

Input:

int((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^p,x)
 

Output:

int((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^p,x)
 

Fricas [F]

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

integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^ 
p,x, algorithm="fricas")
 

Output:

integral((b*d*f*x^3 + a*c*e + (b*d*e + (b*c + a*d)*f)*x^2 + (a*c*f + (b*c 
+ a*d)*e)*x)^p, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^p \, dx=\text {Timed out} \] Input:

integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x**2+b*d*f*x**3 
)**p,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^ 
p,x, algorithm="maxima")
 

Output:

integrate((b*d*f*x^3 + a*c*e + (b*d*e + b*c*f + a*d*f)*x^2 + (b*c*e + a*d* 
e + a*c*f)*x)^p, x)
 

Giac [F]

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

integrate((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^ 
p,x, algorithm="giac")
 

Output:

integrate((b*d*f*x^3 + a*c*e + (b*d*e + b*c*f + a*d*f)*x^2 + (b*c*e + a*d* 
e + a*c*f)*x)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^p \, dx=\int {\left (b\,d\,f\,x^3+\left (a\,d\,f+b\,c\,f+b\,d\,e\right )\,x^2+\left (a\,c\,f+a\,d\,e+b\,c\,e\right )\,x+a\,c\,e\right )}^p \,d x \] Input:

int((x^2*(a*d*f + b*c*f + b*d*e) + x*(a*c*f + a*d*e + b*c*e) + a*c*e + b*d 
*f*x^3)^p,x)
 

Output:

int((x^2*(a*d*f + b*c*f + b*d*e) + x*(a*c*f + a*d*e + b*c*e) + a*c*e + b*d 
*f*x^3)^p, x)
 

Reduce [F]

\[ \int \left (a c e+(b c e+a d e+a c f) x+(b d e+b c f+a d f) x^2+b d f x^3\right )^p \, dx=\text {too large to display} \] Input:

int((a*c*e+(a*c*f+a*d*e+b*c*e)*x+(a*d*f+b*c*f+b*d*e)*x^2+b*d*f*x^3)^p,x)
 

Output:

((a*c*e + a*c*f*x + a*d*e*x + a*d*f*x**2 + b*c*e*x + b*c*f*x**2 + b*d*e*x* 
*2 + b*d*f*x**3)**p*a*c*f + (a*c*e + a*c*f*x + a*d*e*x + a*d*f*x**2 + b*c* 
e*x + b*c*f*x**2 + b*d*e*x**2 + b*d*f*x**3)**p*a*d*e + (a*c*e + a*c*f*x + 
a*d*e*x + a*d*f*x**2 + b*c*e*x + b*c*f*x**2 + b*d*e*x**2 + b*d*f*x**3)**p* 
a*d*f*x + (a*c*e + a*c*f*x + a*d*e*x + a*d*f*x**2 + b*c*e*x + b*c*f*x**2 + 
 b*d*e*x**2 + b*d*f*x**3)**p*b*c*e + (a*c*e + a*c*f*x + a*d*e*x + a*d*f*x* 
*2 + b*c*e*x + b*c*f*x**2 + b*d*e*x**2 + b*d*f*x**3)**p*b*c*f*x + (a*c*e + 
 a*c*f*x + a*d*e*x + a*d*f*x**2 + b*c*e*x + b*c*f*x**2 + b*d*e*x**2 + b*d* 
f*x**3)**p*b*d*e*x - 3*int((a*c*e + a*c*f*x + a*d*e*x + a*d*f*x**2 + b*c*e 
*x + b*c*f*x**2 + b*d*e*x**2 + b*d*f*x**3)**p/(3*a**2*c*d*e*f*p + a**2*c*d 
*e*f + 3*a**2*c*d*f**2*p*x + a**2*c*d*f**2*x + 3*a**2*d**2*e*f*p*x + a**2* 
d**2*e*f*x + 3*a**2*d**2*f**2*p*x**2 + a**2*d**2*f**2*x**2 + 3*a*b*c**2*e* 
f*p + a*b*c**2*e*f + 3*a*b*c**2*f**2*p*x + a*b*c**2*f**2*x + 3*a*b*c*d*e** 
2*p + a*b*c*d*e**2 + 9*a*b*c*d*e*f*p*x + 3*a*b*c*d*e*f*x + 6*a*b*c*d*f**2* 
p*x**2 + 2*a*b*c*d*f**2*x**2 + 3*a*b*d**2*e**2*p*x + a*b*d**2*e**2*x + 6*a 
*b*d**2*e*f*p*x**2 + 2*a*b*d**2*e*f*x**2 + 3*a*b*d**2*f**2*p*x**3 + a*b*d* 
*2*f**2*x**3 + 3*b**2*c**2*e*f*p*x + b**2*c**2*e*f*x + 3*b**2*c**2*f**2*p* 
x**2 + b**2*c**2*f**2*x**2 + 3*b**2*c*d*e**2*p*x + b**2*c*d*e**2*x + 6*b** 
2*c*d*e*f*p*x**2 + 2*b**2*c*d*e*f*x**2 + 3*b**2*c*d*f**2*p*x**3 + b**2*c*d 
*f**2*x**3 + 3*b**2*d**2*e**2*p*x**2 + b**2*d**2*e**2*x**2 + 3*b**2*d**...