\(\int (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [392]

Optimal result

Integrand size = 27, antiderivative size = 113 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=-\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-1-p} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (1+p)} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.85 \[ \int \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {(a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p} ((a e+c d x) (d+e x))^p \operatorname {Hypergeometric2F1}\left (-p,1+p,2+p,\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d (1+p)} \] Input:

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

Output:

((a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometric2F1[-p, 1 + p, 2 
+ p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c*d*(1 + p)*((c*d*(d + e*x))/ 
(c*d^2 - a*e^2))^p)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {1096}

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.

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

\(\Big \downarrow \) 1096

\(\displaystyle -\frac {\left (-\frac {e (a e+c d x)}{c d^2-a e^2}\right )^{-p-1} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-p,p+1,p+2,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{(p+1) \left (c d^2-a e^2\right )}\)

Input:

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

Output:

-(((-((e*(a*e + c*d*x))/(c*d^2 - a*e^2)))^(-1 - p)*(a*d*e + (c*d^2 + a*e^2 
)*x + c*d*e*x^2)^(1 + p)*Hypergeometric2F1[-p, 1 + p, 2 + p, (c*d*(d + e*x 
))/(c*d^2 - a*e^2)])/((c*d^2 - a*e^2)*(1 + p)))
 

Defintions of rubi rules used

rule 1096

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Rt[b^2 
 - 4*a*c, 2]}, Simp[(-(a + b*x + c*x^2)^(p + 1)/(q*(p + 1)*((q - b - 2*c*x) 
/(2*q))^(p + 1)))*Hypergeometric2F1[-p, p + 1, p + 2, (b + q + 2*c*x)/(2*q) 
], x]] /; FreeQ[{a, b, c, p}, x] &&  !IntegerQ[4*p] &&  !IntegerQ[3*p]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**p, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*d*e + (a*d*e + a*e**2*x 
 + c*d**2*x + c*d*e*x**2)**p*a*e**2*x + (a*d*e + a*e**2*x + c*d**2*x + c*d 
*e*x**2)**p*c*d**2*x + 2*int(((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)** 
p*x)/(2*a**2*d*e**3*p + a**2*d*e**3 + 2*a**2*e**4*p*x + a**2*e**4*x + 2*a* 
c*d**3*e*p + a*c*d**3*e + 4*a*c*d**2*e**2*p*x + 2*a*c*d**2*e**2*x + 2*a*c* 
d*e**3*p*x**2 + a*c*d*e**3*x**2 + 2*c**2*d**4*p*x + c**2*d**4*x + 2*c**2*d 
**3*e*p*x**2 + c**2*d**3*e*x**2),x)*a**3*e**6*p**2 + int(((a*d*e + a*e**2* 
x + c*d**2*x + c*d*e*x**2)**p*x)/(2*a**2*d*e**3*p + a**2*d*e**3 + 2*a**2*e 
**4*p*x + a**2*e**4*x + 2*a*c*d**3*e*p + a*c*d**3*e + 4*a*c*d**2*e**2*p*x 
+ 2*a*c*d**2*e**2*x + 2*a*c*d*e**3*p*x**2 + a*c*d*e**3*x**2 + 2*c**2*d**4* 
p*x + c**2*d**4*x + 2*c**2*d**3*e*p*x**2 + c**2*d**3*e*x**2),x)*a**3*e**6* 
p - 2*int(((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*x)/(2*a**2*d*e**3 
*p + a**2*d*e**3 + 2*a**2*e**4*p*x + a**2*e**4*x + 2*a*c*d**3*e*p + a*c*d* 
*3*e + 4*a*c*d**2*e**2*p*x + 2*a*c*d**2*e**2*x + 2*a*c*d*e**3*p*x**2 + a*c 
*d*e**3*x**2 + 2*c**2*d**4*p*x + c**2*d**4*x + 2*c**2*d**3*e*p*x**2 + c**2 
*d**3*e*x**2),x)*a**2*c*d**2*e**4*p**2 - int(((a*d*e + a*e**2*x + c*d**2*x 
 + c*d*e*x**2)**p*x)/(2*a**2*d*e**3*p + a**2*d*e**3 + 2*a**2*e**4*p*x + a* 
*2*e**4*x + 2*a*c*d**3*e*p + a*c*d**3*e + 4*a*c*d**2*e**2*p*x + 2*a*c*d**2 
*e**2*x + 2*a*c*d*e**3*p*x**2 + a*c*d*e**3*x**2 + 2*c**2*d**4*p*x + c**2*d 
**4*x + 2*c**2*d**3*e*p*x**2 + c**2*d**3*e*x**2),x)*a**2*c*d**2*e**4*p ...