3.2485 \(\int \frac{1}{(d+e x) \sqrt [3]{c^2 d^2-b c d e+b^2 e^2+3 b c e^2 x+3 c^2 e^2 x^2}} \, dx\)

Optimal. Leaf size=242 \[ -\frac{\tan ^{-1}\left (\frac{2 (-b e+c d-c e x)}{\sqrt{3} \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} e (2 c d-b e)^{2/3}}+\frac{\log \left (-3 c e^2 \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}+3 c e^2 (c d-b e)-3 c^2 e^3 x\right )}{2 e (2 c d-b e)^{2/3}}-\frac{\log (d+e x)}{2 e (2 c d-b e)^{2/3}} \]

[Out]

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

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Rubi [A]  time = 0.356178, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 52, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.019 \[ -\frac{\tan ^{-1}\left (\frac{2 (-b e+c d-c e x)}{\sqrt{3} \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}}+\frac{1}{\sqrt{3}}\right )}{\sqrt{3} e (2 c d-b e)^{2/3}}+\frac{\log \left (-3 c e^2 \sqrt [3]{2 c d-b e} \sqrt [3]{b^2 e^2-b c d e+3 b c e^2 x+c^2 d^2+3 c^2 e^2 x^2}+3 c e^2 (c d-b e)-3 c^2 e^3 x\right )}{2 e (2 c d-b e)^{2/3}}-\frac{\log (d+e x)}{2 e (2 c d-b e)^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 86.9851, size = 238, normalized size = 0.98 \[ - \frac{\log{\left (d + e x \right )}}{2 e \left (b e - 2 c d\right )^{\frac{2}{3}}} + \frac{\log{\left (- 3 c^{2} e^{3} x + 3 c e^{2} \left (- b e + c d\right ) + 3 c e^{2} \sqrt [3]{b e - 2 c d} \sqrt [3]{b^{2} e^{2} - b c d e + 3 b c e^{2} x + c^{2} d^{2} + 3 c^{2} e^{2} x^{2}} \right )}}{2 e \left (b e - 2 c d\right )^{\frac{2}{3}}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} - \frac{2 \sqrt{3} \left (- 3 b e + 3 c d - 3 c e x\right )}{9 \sqrt [3]{b e - 2 c d} \sqrt [3]{b^{2} e^{2} - b c d e + 3 b c e^{2} x + c^{2} d^{2} + 3 c^{2} e^{2} x^{2}}} \right )}}{3 e \left (b e - 2 c d\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-log(d + e*x)/(2*e*(b*e - 2*c*d)**(2/3)) + log(-3*c**2*e**3*x + 3*c*e**2*(-b*e +
c*d) + 3*c*e**2*(b*e - 2*c*d)**(1/3)*(b**2*e**2 - b*c*d*e + 3*b*c*e**2*x + c**2
*d**2 + 3*c**2*e**2*x**2)**(1/3))/(2*e*(b*e - 2*c*d)**(2/3)) - sqrt(3)*atan(sqrt
(3)/3 - 2*sqrt(3)*(-3*b*e + 3*c*d - 3*c*e*x)/(9*(b*e - 2*c*d)**(1/3)*(b**2*e**2
- b*c*d*e + 3*b*c*e**2*x + c**2*d**2 + 3*c**2*e**2*x**2)**(1/3)))/(3*e*(b*e - 2*
c*d)**(2/3))

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Mathematica [C]  time = 0.925177, size = 317, normalized size = 1.31 \[ -\frac{\sqrt [3]{3} \sqrt [3]{\frac{-\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}+3 b c e^2+6 c^2 e^2 x}{c^2 e (d+e x)}} \sqrt [3]{\frac{\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}+3 b c e^2+6 c^2 e^2 x}{c^2 e (d+e x)}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{-6 d e c^2+3 b e^2 c+\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}}{6 c^2 e (d+e x)},\frac{6 d e c^2-3 b e^2 c+\sqrt{3} \sqrt{-c^2 e^2 (b e-2 c d)^2}}{6 c^2 e (d+e x)}\right )}{2\ 2^{2/3} e \sqrt [3]{b^2 e^2+b c e (3 e x-d)+c^2 \left (d^2+3 e^2 x^2\right )}} \]

Warning: Unable to verify antiderivative.

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

[Out]

-(3^(1/3)*((3*b*c*e^2 - Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)] + 6*c^2*e^2*x)
/(c^2*e*(d + e*x)))^(1/3)*((3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)
] + 6*c^2*e^2*x)/(c^2*e*(d + e*x)))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3, -(-6*c^2*
d*e + 3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)])/(6*c^2*e*(d + e*x))
, (6*c^2*d*e - 3*b*c*e^2 + Sqrt[3]*Sqrt[-(c^2*e^2*(-2*c*d + b*e)^2)])/(6*c^2*e*(
d + e*x))])/(2*2^(2/3)*e*(b^2*e^2 + b*c*e*(-d + 3*e*x) + c^2*(d^2 + 3*e^2*x^2))^
(1/3))

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Maple [F]  time = 0.42, size = 0, normalized size = 0. \[ \int{\frac{1}{ex+d}{\frac{1}{\sqrt [3]{3\,{c}^{2}{e}^{2}{x}^{2}+3\,bc{e}^{2}x+{b}^{2}{e}^{2}-bcde+{c}^{2}{d}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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