3.2578 \(\int \frac{1}{\sqrt [3]{b e-c e x} \sqrt [3]{b^2+b c x+c^2 x^2}} \, dx\)

Optimal. Leaf size=196 \[ \frac{\sqrt [3]{b^3 e-c^3 e x^3} \log \left (\sqrt [3]{b^3 e-c^3 e x^3}+c \sqrt [3]{e} x\right )}{2 c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}}-\frac{\sqrt [3]{b^3 e-c^3 e x^3} \tan ^{-1}\left (\frac{1-\frac{2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt{3}}\right )}{\sqrt{3} c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}} \]

[Out]

-(((b^3*e - c^3*e*x^3)^(1/3)*ArcTan[(1 - (2*c*e^(1/3)*x)/(b^3*e - c^3*e*x^3)^(1/
3))/Sqrt[3]])/(Sqrt[3]*c*e^(1/3)*(b*e - c*e*x)^(1/3)*(b^2 + b*c*x + c^2*x^2)^(1/
3))) + ((b^3*e - c^3*e*x^3)^(1/3)*Log[c*e^(1/3)*x + (b^3*e - c^3*e*x^3)^(1/3)])/
(2*c*e^(1/3)*(b*e - c*e*x)^(1/3)*(b^2 + b*c*x + c^2*x^2)^(1/3))

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Rubi [A]  time = 0.166967, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \frac{\sqrt [3]{b^3 e-c^3 e x^3} \log \left (\sqrt [3]{b^3 e-c^3 e x^3}+c \sqrt [3]{e} x\right )}{2 c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}}-\frac{\sqrt [3]{b^3 e-c^3 e x^3} \tan ^{-1}\left (\frac{1-\frac{2 c \sqrt [3]{e} x}{\sqrt [3]{b^3 e-c^3 e x^3}}}{\sqrt{3}}\right )}{\sqrt{3} c \sqrt [3]{e} \sqrt [3]{b^2+b c x+c^2 x^2} \sqrt [3]{b e-c e x}} \]

Antiderivative was successfully verified.

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

[Out]

-(((b^3*e - c^3*e*x^3)^(1/3)*ArcTan[(1 - (2*c*e^(1/3)*x)/(b^3*e - c^3*e*x^3)^(1/
3))/Sqrt[3]])/(Sqrt[3]*c*e^(1/3)*(b*e - c*e*x)^(1/3)*(b^2 + b*c*x + c^2*x^2)^(1/
3))) + ((b^3*e - c^3*e*x^3)^(1/3)*Log[c*e^(1/3)*x + (b^3*e - c^3*e*x^3)^(1/3)])/
(2*c*e^(1/3)*(b*e - c*e*x)^(1/3)*(b^2 + b*c*x + c^2*x^2)^(1/3))

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Rubi in Sympy [A]  time = 63.674, size = 298, normalized size = 1.52 \[ \frac{\left (b e - c e x\right )^{\frac{2}{3}} \left (b^{2} + b c x + c^{2} x^{2}\right )^{\frac{2}{3}} \log{\left (\frac{c \sqrt [3]{e} x}{\sqrt [3]{b^{3} e - c^{3} e x^{3}}} + 1 \right )}}{3 c \sqrt [3]{e} \left (b^{3} e - c^{3} e x^{3}\right )^{\frac{2}{3}}} - \frac{\left (b e - c e x\right )^{\frac{2}{3}} \left (b^{2} + b c x + c^{2} x^{2}\right )^{\frac{2}{3}} \log{\left (\frac{c^{2} e^{\frac{2}{3}} x^{2}}{\left (b^{3} e - c^{3} e x^{3}\right )^{\frac{2}{3}}} - \frac{c \sqrt [3]{e} x}{\sqrt [3]{b^{3} e - c^{3} e x^{3}}} + 1 \right )}}{6 c \sqrt [3]{e} \left (b^{3} e - c^{3} e x^{3}\right )^{\frac{2}{3}}} - \frac{\sqrt{3} \left (b e - c e x\right )^{\frac{2}{3}} \left (b^{2} + b c x + c^{2} x^{2}\right )^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (- \frac{2 c \sqrt [3]{e} x}{3 \sqrt [3]{b^{3} e - c^{3} e x^{3}}} + \frac{1}{3}\right ) \right )}}{3 c \sqrt [3]{e} \left (b^{3} e - c^{3} e x^{3}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b*e - c*e*x)**(2/3)*(b**2 + b*c*x + c**2*x**2)**(2/3)*log(c*e**(1/3)*x/(b**3*e
- c**3*e*x**3)**(1/3) + 1)/(3*c*e**(1/3)*(b**3*e - c**3*e*x**3)**(2/3)) - (b*e -
c*e*x)**(2/3)*(b**2 + b*c*x + c**2*x**2)**(2/3)*log(c**2*e**(2/3)*x**2/(b**3*e
- c**3*e*x**3)**(2/3) - c*e**(1/3)*x/(b**3*e - c**3*e*x**3)**(1/3) + 1)/(6*c*e**
(1/3)*(b**3*e - c**3*e*x**3)**(2/3)) - sqrt(3)*(b*e - c*e*x)**(2/3)*(b**2 + b*c*
x + c**2*x**2)**(2/3)*atan(sqrt(3)*(-2*c*e**(1/3)*x/(3*(b**3*e - c**3*e*x**3)**(
1/3)) + 1/3))/(3*c*e**(1/3)*(b**3*e - c**3*e*x**3)**(2/3))

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Mathematica [C]  time = 0.338495, size = 241, normalized size = 1.23 \[ -\frac{3 \sqrt [3]{\frac{-\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}} \sqrt [3]{\frac{\sqrt{3} \sqrt{-b^2 c^2}+b c+2 c^2 x}{\sqrt{3} \sqrt{-b^2 c^2}+3 b c}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{2 c (b-c x)}{3 b c+\sqrt{3} \sqrt{-b^2 c^2}},\frac{2 c (b-c x)}{3 b c-\sqrt{3} \sqrt{-b^2 c^2}}\right ) (e (b-c x))^{2/3}}{2 c e \sqrt [3]{b^2+b c x+c^2 x^2}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-3*(e*(b - c*x))^(2/3)*((b*c - Sqrt[3]*Sqrt[-(b^2*c^2)] + 2*c^2*x)/(3*b*c - Sqr
t[3]*Sqrt[-(b^2*c^2)]))^(1/3)*((b*c + Sqrt[3]*Sqrt[-(b^2*c^2)] + 2*c^2*x)/(3*b*c
+ Sqrt[3]*Sqrt[-(b^2*c^2)]))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, (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)])])/(2*c*e*(b^2 + b*c*x + c^2*x^2)^(1/3))

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Maple [F]  time = 0.283, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [3]{-xec+be}}}{\frac{1}{\sqrt [3]{{c}^{2}{x}^{2}+bxc+{b}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 54.7356, size = 271, normalized size = 1.38 \[ -\frac{2 \, \sqrt{3} \arctan \left (-\frac{\sqrt{3}{\left (2 \, c e^{\frac{1}{3}} x -{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{1}{3}}{\left (-c e x + b e\right )}^{\frac{1}{3}}\right )}}{3 \,{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{1}{3}}{\left (-c e x + b e\right )}^{\frac{1}{3}}}\right ) + \log \left (c^{2} e^{\frac{2}{3}} x^{2} -{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{1}{3}}{\left (-c e x + b e\right )}^{\frac{1}{3}} c e^{\frac{1}{3}} x +{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{2}{3}}{\left (-c e x + b e\right )}^{\frac{2}{3}}\right ) - 2 \, \log \left (c e^{\frac{1}{3}} x +{\left (c^{2} x^{2} + b c x + b^{2}\right )}^{\frac{1}{3}}{\left (-c e x + b e\right )}^{\frac{1}{3}}\right )}{6 \, c e^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*(2*sqrt(3)*arctan(-1/3*sqrt(3)*(2*c*e^(1/3)*x - (c^2*x^2 + b*c*x + b^2)^(1/
3)*(-c*e*x + b*e)^(1/3))/((c^2*x^2 + b*c*x + b^2)^(1/3)*(-c*e*x + b*e)^(1/3))) +
log(c^2*e^(2/3)*x^2 - (c^2*x^2 + b*c*x + b^2)^(1/3)*(-c*e*x + b*e)^(1/3)*c*e^(1
/3)*x + (c^2*x^2 + b*c*x + b^2)^(2/3)*(-c*e*x + b*e)^(2/3)) - 2*log(c*e^(1/3)*x
+ (c^2*x^2 + b*c*x + b^2)^(1/3)*(-c*e*x + b*e)^(1/3)))/(c*e^(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out