\(\int (b e-c e x)^p (b^2+b c x+c^2 x^2)^p \, dx\) [2590]

Optimal result

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

[Out]

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)

Rubi [A] (verified)

Time = 0.02 (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 = {727, 252, 251} \[ \int (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \, dx=x \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac {c^3 x^3}{b^3}\right )^{-p} (b e-c e x)^p \operatorname {Hypergeometric2F1}\left (\frac {1}{3},-p,\frac {4}{3},\frac {c^3 x^3}{b^3}\right ) \]

[In]

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

[Out]

(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

Rule 251

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F1[-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 252

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^Fra
cPart[p]), Int[(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, n, p}, x] &&  !IGtQ[p, 0] &&  !IntegerQ[1/n] &&  !ILt
Q[Simplify[1/n + p], 0] &&  !(IntegerQ[p] || GtQ[a, 0])

Rule 727

Int[((d_.) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[(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]
/; FreeQ[{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] &&  !Intege
rQ[p]

Rubi steps \begin{align*} \text {integral}& = \left ((b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \left (b^3 e-c^3 e x^3\right )^{-p}\right ) \int \left (b^3 e-c^3 e x^3\right )^p \, dx \\ & = \left ((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}\right ) \int \left (1-\frac {c^3 x^3}{b^3}\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} \, _2F_1\left (\frac {1}{3},-p;\frac {4}{3};\frac {c^3 x^3}{b^3}\right ) \\ \end{align*}

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 \[ \int (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \, dx=\frac {(e (b-c x))^p (-b+c x) \left (\frac {b c-\sqrt {3} \sqrt {-b^2 c^2}+2 c^2 x}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}\right )^{-p} \left (\frac {b c+\sqrt {3} \sqrt {-b^2 c^2}+2 c^2 x}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}}\right )^{-p} \left (b^2+b c x+c^2 x^2\right )^p \operatorname {AppellF1}\left (1+p,-p,-p,2+p,\frac {2 c (b-c x)}{3 b c+\sqrt {3} \sqrt {-b^2 c^2}},\frac {2 c (b-c x)}{3 b c-\sqrt {3} \sqrt {-b^2 c^2}}\right )}{c (1+p)} \]

[In]

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

[Out]

((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]*Sqr
t[-(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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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