3.2580 \(\int (b e-c e x)^p \left (b^2+b c x+c^2 x^2\right )^p \, dx\)

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 = 0.0818606, 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 \[ x \left (b^2+b c x+c^2 x^2\right )^p \left (1-\frac{c^3 x^3}{b^3}\right )^{-p} (b e-c e x)^p \, _2F_1\left (\frac{1}{3},-p;\frac{4}{3};\frac{c^3 x^3}{b^3}\right ) \]

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

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Rubi in Sympy [A]  time = 30.1578, size = 58, normalized size = 0.87 \[ x \left (1 - \frac{c^{3} x^{3}}{b^{3}}\right )^{- p} \left (b e - c e x\right )^{p} \left (b^{2} + b c x + c^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{c^{3} x^{3}}{b^{3}}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-c*e*x+b*e)**p*(c**2*x**2+b*c*x+b**2)**p,x)

[Out]

x*(1 - c**3*x**3/b**3)**(-p)*(b*e - c*e*x)**p*(b**2 + b*c*x + c**2*x**2)**p*hype
r((-p, 1/3), (4/3,), c**3*x**3/b**3)

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Mathematica [C]  time = 0.579403, 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]*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)

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Maple [F]  time = 0.261, size = 0, normalized size = 0. \[ \int \left ( -xec+be \right ) ^{p} \left ({c}^{2}{x}^{2}+bxc+{b}^{2} \right ) ^{p}\, dx \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c^2*x^2 + b*c*x + b^2)^p*(-c*e*x + b*e)^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. \[{\rm integral}\left ({\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p}{\left (-c e x + b e\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c^2*x^2 + b*c*x + b^2)^p*(-c*e*x + b*e)^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(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{p}{\left (-c e x + b e\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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