\(\int (d+e x) (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [391]

Optimal result

Integrand size = 33, antiderivative size = 87 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=-\frac {(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,3+2 p,3+p,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (2+p)} \] Output:

-(e*x+d)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(p+1)*hypergeom([1, 3+2*p],[3+p 
],c*d*(e*x+d)/(-a*e^2+c*d^2))/(-a*e^2+c*d^2)/(2+p)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.08 \[ \int (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {\left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-1-p} ((a e+c d x) (d+e x))^{1+p} \operatorname {Hypergeometric2F1}\left (-1-p,1+p,2+p,\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d (1+p)} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(176\) vs. \(2(87)=174\).

Time = 0.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {1160, 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.

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

\(\Big \downarrow \) 1160

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

\(\Big \downarrow \) 1096

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

Input:

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

Output:

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

rule 1160

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol 
] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b 
*e)/(2*c)   Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] 
 && NeQ[p, -1]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - (a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a**2*d*e**3 + (a*d*e + a 
*e**2*x + c*d**2*x + c*d*e*x**2)**p*a**2*e**4*p*x + 4*(a*d*e + a*e**2*x + 
c*d**2*x + c*d*e*x**2)**p*a*c*d**3*e*p + 3*(a*d*e + a*e**2*x + c*d**2*x + 
c*d*e*x**2)**p*a*c*d**3*e + 4*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)** 
p*a*c*d**2*e**2*p*x + 2*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c* 
d**2*e**2*x + 2*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c*d*e**3*p 
*x**2 + (a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c*d*e**3*x**2 + 3* 
(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*c**2*d**4*p*x + 2*(a*d*e + a 
*e**2*x + c*d**2*x + c*d*e*x**2)**p*c**2*d**4*x + 2*(a*d*e + a*e**2*x + c* 
d**2*x + c*d*e*x**2)**p*c**2*d**3*e*p*x**2 + (a*d*e + a*e**2*x + c*d**2*x 
+ c*d*e*x**2)**p*c**2*d**3*e*x**2 - 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**4*e**8*p**3 - 3*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**4*e**8*p**2 - int(((a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**...