\(\int (d+e x)^2 (a d e+(c d^2+a e^2) x+c d e x^2)^p \, dx\) [390]

Optimal result

Integrand size = 35, antiderivative size = 89 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=-\frac {(d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,2 (2+p),4+p,\frac {c d (d+e x)}{c d^2-a e^2}\right )}{\left (c d^2-a e^2\right ) (3+p)} \] Output:

-(e*x+d)^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(p+1)*hypergeom([1, 4+2*p],[4 
+p],c*d*(e*x+d)/(-a*e^2+c*d^2))/(-a*e^2+c*d^2)/(3+p)
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.26 \[ \int (d+e x)^2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^p \, dx=\frac {\left (c d^2-a e^2\right )^2 (a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p} ((a e+c d x) (d+e x))^p \operatorname {Hypergeometric2F1}\left (-2-p,1+p,2+p,\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c^3 d^3 (1+p)} \] Input:

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

Output:

((c*d^2 - a*e^2)^2*(a*e + c*d*x)*((a*e + c*d*x)*(d + e*x))^p*Hypergeometri 
c2F1[-2 - p, 1 + p, 2 + p, (e*(a*e + c*d*x))/(-(c*d^2) + a*e^2)])/(c^3*d^3 
*(1 + p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p)
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.39, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {1138, 80, 79}

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.

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

\(\Big \downarrow \) 1138

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

\(\Big \downarrow \) 80

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

\(\Big \downarrow \) 79

\(\displaystyle \frac {\left (c d^2-a e^2\right )^2 (a e+c d x) \left (\frac {c d (d+e x)}{c d^2-a e^2}\right )^{-p} \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^p \operatorname {Hypergeometric2F1}\left (-p-2,p+1,p+2,-\frac {e (a e+c d x)}{c d^2-a e^2}\right )}{c^3 d^3 (p+1)}\)

Input:

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

Output:

((c*d^2 - a*e^2)^2*(a*e + c*d*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^p 
*Hypergeometric2F1[-2 - p, 1 + p, 2 + p, -((e*(a*e + c*d*x))/(c*d^2 - a*e^ 
2))])/(c^3*d^3*(1 + p)*((c*d*(d + e*x))/(c*d^2 - a*e^2))^p)
 

Defintions of rubi rules used

rule 79

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(( 
a + b*x)^(m + 1)/(b*(m + 1)*(b/(b*c - a*d))^n))*Hypergeometric2F1[-n, m + 1 
, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m, n}, x] 
&&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[b/(b*c - a*d), 0] && (RationalQ[m] 
 ||  !(RationalQ[n] && GtQ[-d/(b*c - a*d), 0]))
 

rule 80

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(c 
 + d*x)^FracPart[n]/((b/(b*c - a*d))^IntPart[n]*(b*((c + d*x)/(b*c - a*d))) 
^FracPart[n])   Int[(a + b*x)^m*Simp[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d) 
), x]^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] &&  !IntegerQ[m] &&  !Integ 
erQ[n] && (RationalQ[m] ||  !SimplerQ[n + 1, m + 1])
 

rule 1138

Int[((d_) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Sy 
mbol] :> Simp[d^m*((a + b*x + c*x^2)^FracPart[p]/((1 + e*(x/d))^FracPart[p] 
*(a/d + (c*x)/e)^FracPart[p]))   Int[(1 + e*(x/d))^(m + p)*(a/d + (c/e)*x)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] 
&& (IntegerQ[m] || GtQ[d, 0]) &&  !(IGtQ[m, 0] && (IntegerQ[3*p] || Integer 
Q[4*p]))
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

int((e*x+d)^2*(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**3*d*e**5*p + 2*(a*d*e + 
a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a**3*d*e**5 - (a*d*e + a*e**2*x + c*d 
**2*x + c*d*e*x**2)**p*a**3*e**6*p**2*x - 2*(a*d*e + a*e**2*x + c*d**2*x + 
 c*d*e*x**2)**p*a**3*e**6*p*x - 6*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x** 
2)**p*a**2*c*d**3*e**3*p - 6*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p 
*a**2*c*d**3*e**3 + 5*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a**2*c 
*d**2*e**4*p**2*x + 4*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a**2*c 
*d**2*e**4*p*x + 2*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a**2*c*d* 
e**5*p**2*x**2 + (a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a**2*c*d*e* 
*5*p*x**2 + 8*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**5*e* 
p**2 + 17*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**5*e*p + 
8*(a*d*e + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**5*e + 13*(a*d*e 
+ a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**4*e**2*p**2*x + 20*(a*d*e 
 + a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**4*e**2*p*x + 6*(a*d*e + 
a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**4*e**2*x + 12*(a*d*e + a*e* 
*2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**3*e**3*p**2*x**2 + 18*(a*d*e + 
a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**3*e**3*p*x**2 + 6*(a*d*e + 
a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**3*e**3*x**2 + 4*(a*d*e + a* 
e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**2*e**4*p**2*x**3 + 6*(a*d*e + 
 a*e**2*x + c*d**2*x + c*d*e*x**2)**p*a*c**2*d**2*e**4*p*x**3 + 2*(a*d*...