\(\int (b e-c e x)^p (b^2+b c x+c^2 x^2)^p \, dx\) [796]

Optimal result

Integrand size = 29, antiderivative size = 67 \[ \int (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \, dx=x (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac {c^3 x^3}{b^3}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},-p,\frac {4}{3},\frac {c^3 x^3}{b^3}\right ) \] Output:

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

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 5 in optimal.

Time = 0.33 (sec) , antiderivative size = 243, normalized size of antiderivative = 3.63 \[ \int (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \, dx=\frac {(e (b-c x))^p (-b+c x) \left (\frac {b c-\sqrt {3} \sqrt {-b^2 c^2}+2 c^2 x}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}\right )^{-p} \left (\frac {b c+\sqrt {3} \sqrt {-b^2 c^2}+2 c^2 x}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}}\right )^{-p} \left (b^2+b c x+c^2 x^2\right )^p \operatorname {AppellF1}\left (1+p,-p,-p,2+p,\frac {2 c (b-c x)}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}},\frac {2 c (b-c x)}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}\right )}{c (1+p)} \] Input:

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

Output:

((e*(b - c*x))^p*(-b + c*x)*(b^2 + b*c*x + c^2*x^2)^p*AppellF1[1 + p, -p, 
-p, 2 + p, (2*c*(b - c*x))/(3*b*c + Sqrt[3]*Sqrt[-(b^2*c^2)]), (2*c*(b - c 
*x))/(3*b*c - Sqrt[3]*Sqrt[-(b^2*c^2)])])/(c*(1 + p)*((b*c - Sqrt[3]*Sqrt[ 
-(b^2*c^2)] + 2*c^2*x)/(3*b*c - Sqrt[3]*Sqrt[-(b^2*c^2)]))^p*((b*c + Sqrt[ 
3]*Sqrt[-(b^2*c^2)] + 2*c^2*x)/(3*b*c + Sqrt[3]*Sqrt[-(b^2*c^2)]))^p)
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {1151, 779, 778}

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.

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

\(\Big \downarrow \) 1151

\(\displaystyle \left (b^2+b c x+c^2 x^2\right )^p (b e-c e x)^p \left (b^3 e-c^3 e x^3\right )^{-p} \int \left (b^3 e-c^3 e x^3\right )^pdx\)

\(\Big \downarrow \) 779

\(\displaystyle \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac {c^3 x^3}{b^3}\right )^{-p} (b e-c e x)^p \int \left (1-\frac {c^3 x^3}{b^3}\right )^pdx\)

\(\Big \downarrow \) 778

\(\displaystyle x \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac {c^3 x^3}{b^3}\right )^{-p} (b e-c e x)^p \operatorname {Hypergeometric2F1}\left (\frac {1}{3},-p,\frac {4}{3},\frac {c^3 x^3}{b^3}\right )\)

Input:

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

Output:

(x*(b*e - c*e*x)^p*(b^2 + b*c*x + c^2*x^2)^p*Hypergeometric2F1[1/3, -p, 4/ 
3, (c^3*x^3)/b^3])/(1 - (c^3*x^3)/b^3)^p
 

Defintions of rubi rules used

rule 778

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 
1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p 
, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 
GtQ[a, 0])
 

rule 779

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x 
^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p])   Int[(1 + b*(x^n/a))^p, x], x 
] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILtQ[Si 
mplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 1151

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F]

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

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

Output:

Integral((-e*(-b + c*x))**p*(b**2 + b*c*x + c**2*x**2)**p, x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \, dx=\frac {\left (-c e x +b e \right )^{p} \left (c^{2} x^{2}+b c x +b^{2}\right )^{p} x +9 \left (\int \frac {\left (-c e x +b e \right )^{p} \left (c^{2} x^{2}+b c x +b^{2}\right )^{p}}{-3 c^{3} p \,x^{3}-c^{3} x^{3}+3 b^{3} p +b^{3}}d x \right ) b^{3} p^{2}+3 \left (\int \frac {\left (-c e x +b e \right )^{p} \left (c^{2} x^{2}+b c x +b^{2}\right )^{p}}{-3 c^{3} p \,x^{3}-c^{3} x^{3}+3 b^{3} p +b^{3}}d x \right ) b^{3} p}{3 p +1} \] Input:

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

Output:

((b*e - c*e*x)**p*(b**2 + b*c*x + c**2*x**2)**p*x + 9*int(((b*e - c*e*x)** 
p*(b**2 + b*c*x + c**2*x**2)**p)/(3*b**3*p + b**3 - 3*c**3*p*x**3 - c**3*x 
**3),x)*b**3*p**2 + 3*int(((b*e - c*e*x)**p*(b**2 + b*c*x + c**2*x**2)**p) 
/(3*b**3*p + b**3 - 3*c**3*p*x**3 - c**3*x**3),x)*b**3*p)/(3*p + 1)