3.2579 \(\int \frac{1}{(b e-c e x)^{2/3} \left (b^2+b c x+c^2 x^2\right )^{2/3}} \, dx\)

Optimal. Leaf size=71 \[ \frac{x \left (1-\frac{c^3 x^3}{b^3}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{c^3 x^3}{b^3}\right )}{\left (b^2+b c x+c^2 x^2\right )^{2/3} (b e-c e x)^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.100922, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x \left (1-\frac{c^3 x^3}{b^3}\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};\frac{c^3 x^3}{b^3}\right )}{\left (b^2+b c x+c^2 x^2\right )^{2/3} (b e-c e x)^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[1/((b*e - c*e*x)^(2/3)*(b^2 + b*c*x + c^2*x^2)^(2/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 30.1245, size = 68, normalized size = 0.96 \[ \frac{x \sqrt [3]{b e - c e x} \sqrt [3]{b^{2} + b c x + c^{2} x^{2}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{1}{3} \\ \frac{4}{3} \end{matrix}\middle |{\frac{c^{3} x^{3}}{b^{3}}} \right )}}{b^{3} e \sqrt [3]{1 - \frac{c^{3} x^{3}}{b^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-c*e*x+b*e)**(2/3)/(c**2*x**2+b*c*x+b**2)**(2/3),x)

[Out]

x*(b*e - c*e*x)**(1/3)*(b**2 + b*c*x + c**2*x**2)**(1/3)*hyper((2/3, 1/3), (4/3,
), c**3*x**3/b**3)/(b**3*e*(1 - c**3*x**3/b**3)**(1/3))

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Mathematica [B]  time = 0.400991, size = 232, normalized size = 3.27 \[ -\frac{3 \left (\frac{\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{\sqrt{3} \sqrt{-b^2 c^2}+3 b c}\right )^{2/3} \sqrt [3]{\frac{2 c (c x-b)}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}+1} \sqrt [3]{e (b-c x)} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{4 \sqrt{3} c \sqrt{-b^2 c^2} (c x-b)}{\left (3 b c+\sqrt{3} \sqrt{-b^2 c^2}\right ) \left (-2 x c^2-b c+\sqrt{3} \sqrt{-b^2 c^2}\right )}\right )}{c e \left (b^2+b c x+c^2 x^2\right )^{2/3}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((b*e - c*e*x)^(2/3)*(b^2 + b*c*x + c^2*x^2)^(2/3)),x]

[Out]

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

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Maple [F]  time = 0.21, size = 0, normalized size = 0. \[ \int{1 \left ( -xec+be \right ) ^{-{\frac{2}{3}}} \left ({c}^{2}{x}^{2}+bxc+{b}^{2} \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-c*e*x+b*e)^(2/3)/(c^2*x^2+b*c*x+b^2)^(2/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{2}{3}}{\left (-c e x + b e\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c^2*x^2 + b*c*x + b^2)^(2/3)*(-c*e*x + b*e)^(2/3)),x, algorithm="maxima")

[Out]

integrate(1/((c^2*x^2 + b*c*x + b^2)^(2/3)*(-c*e*x + b*e)^(2/3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{2}{3}}{\left (-c e x + b e\right )}^{\frac{2}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c^2*x^2 + b*c*x + b^2)^(2/3)*(-c*e*x + b*e)^(2/3)),x, algorithm="fricas")

[Out]

integral(1/((c^2*x^2 + b*c*x + b^2)^(2/3)*(-c*e*x + b*e)^(2/3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (- e \left (- b + c x\right )\right )^{\frac{2}{3}} \left (b^{2} + b c x + c^{2} x^{2}\right )^{\frac{2}{3}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-c*e*x+b*e)**(2/3)/(c**2*x**2+b*c*x+b**2)**(2/3),x)

[Out]

Integral(1/((-e*(-b + c*x))**(2/3)*(b**2 + b*c*x + c**2*x**2)**(2/3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{2}{3}}{\left (-c e x + b e\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c^2*x^2 + b*c*x + b^2)^(2/3)*(-c*e*x + b*e)^(2/3)),x, algorithm="giac")

[Out]

integrate(1/((c^2*x^2 + b*c*x + b^2)^(2/3)*(-c*e*x + b*e)^(2/3)), x)