### 3.2590 $$\int (b e-c e x)^p (b^2+b c x+c^2 x^2)^p \, dx$$

Optimal. Leaf size=67 $x \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac{c^3 x^3}{b^3}\right )^{-p} (b e-c e x)^p \, _2F_1\left (\frac{1}{3},-p;\frac{4}{3};\frac{c^3 x^3}{b^3}\right )$

[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

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Rubi [A]  time = 0.0334323, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.103, Rules used = {713, 246, 245} $x \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac{c^3 x^3}{b^3}\right )^{-p} (b e-c e x)^p \, _2F_1\left (\frac{1}{3},-p;\frac{4}{3};\frac{c^3 x^3}{b^3}\right )$

Antiderivative was successfully veriﬁed.

[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 713

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]

Rule 246

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^Fr
acPart[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 245

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])

Rubi steps

\begin{align*} \int (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \, dx &=\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]  time = 0.306608, size = 243, normalized size = 3.63 $\frac{(c x-b) \left (\frac{-\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}\right )^{-p} \left (\frac{\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{\sqrt{3} \sqrt{-b^2 c^2}+3 b c}\right )^{-p} \left (b^2+b c x+c^2 x^2\right )^p F_1\left (p+1;-p,-p;p+2;\frac{2 c (b-c x)}{3 b c+\sqrt{3} \sqrt{-b^2 c^2}},\frac{2 c (b-c x)}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}\right ) (e (b-c x))^p}{c (p+1)}$

Warning: Unable to verify antiderivative.

[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)

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Maple [F]  time = 3.016, size = 0, normalized size = 0. \begin{align*} \int \left ( -cex+be \right ) ^{p} \left ({c}^{2}{x}^{2}+bcx+{b}^{2} \right ) ^{p}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p}{\left (-c e x + b e\right )}^{p}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p}{\left (-c e x + b e\right )}^{p}, x\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (- e \left (- b + c x\right )\right )^{p} \left (b^{2} + b c x + c^{2} x^{2}\right )^{p}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p}{\left (-c e x + b e\right )}^{p}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)