3.143 \(\int \frac{1}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2}} \, dx\)

Optimal. Leaf size=204 \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c+3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c}}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}} \]

[Out]

-ArcTan[(Sqrt[3]*Sqrt[d]*x)/Sqrt[c]]/(2*2^(2/3)*Sqrt[3]*c^(5/6)*Sqrt[d]) + (Sqrt
[3]*ArcTan[(Sqrt[3]*Sqrt[d]*x)/(c^(1/6)*(c^(1/3) + 2^(1/3)*(c + 3*d*x^2)^(1/3)))
])/(2*2^(2/3)*c^(5/6)*Sqrt[d]) - ArcTanh[Sqrt[c]/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)
*Sqrt[d]) - ArcTanh[(c^(1/6)*(c^(1/3) - 2^(1/3)*(c + 3*d*x^2)^(1/3)))/(Sqrt[d]*x
)]/(2*2^(2/3)*c^(5/6)*Sqrt[d])

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Rubi [A]  time = 0.122218, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ \frac{\sqrt{3} \tan ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt [6]{c} \left (\sqrt [3]{2} \sqrt [3]{c+3 d x^2}+\sqrt [3]{c}\right )}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{c} \left (\sqrt [3]{c}-\sqrt [3]{2} \sqrt [3]{c+3 d x^2}\right )}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}}-\frac{\tan ^{-1}\left (\frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}}\right )}{2\ 2^{2/3} \sqrt{3} c^{5/6} \sqrt{d}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{c}}{\sqrt{d} x}\right )}{2\ 2^{2/3} c^{5/6} \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

-ArcTan[(Sqrt[3]*Sqrt[d]*x)/Sqrt[c]]/(2*2^(2/3)*Sqrt[3]*c^(5/6)*Sqrt[d]) + (Sqrt
[3]*ArcTan[(Sqrt[3]*Sqrt[d]*x)/(c^(1/6)*(c^(1/3) + 2^(1/3)*(c + 3*d*x^2)^(1/3)))
])/(2*2^(2/3)*c^(5/6)*Sqrt[d]) - ArcTanh[Sqrt[c]/(Sqrt[d]*x)]/(2*2^(2/3)*c^(5/6)
*Sqrt[d]) - ArcTanh[(c^(1/6)*(c^(1/3) - 2^(1/3)*(c + 3*d*x^2)^(1/3)))/(Sqrt[d]*x
)]/(2*2^(2/3)*c^(5/6)*Sqrt[d])

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Rubi in Sympy [A]  time = 38.6386, size = 376, normalized size = 1.84 \[ \frac{\sqrt [3]{2} \log{\left (\sqrt{3} - \frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{5}{6}} \sqrt{d}} - \frac{\sqrt [3]{2} \log{\left (\sqrt{3} + \frac{\sqrt{3} \sqrt{d} x}{\sqrt{c}} \right )}}{8 c^{\frac{5}{6}} \sqrt{d}} + \frac{\sqrt [3]{2} \log{\left (3 \sqrt{3} d - \frac{3 \sqrt{3} d^{\frac{3}{2}} x}{\sqrt{c}} - \frac{3 \sqrt [3]{2} \sqrt{3} d \sqrt [3]{c + 3 d x^{2}}}{\sqrt [3]{c}} \right )}}{8 c^{\frac{5}{6}} \sqrt{d}} - \frac{\sqrt [3]{2} \log{\left (3 \sqrt{3} d + \frac{3 \sqrt{3} d^{\frac{3}{2}} x}{\sqrt{c}} - \frac{3 \sqrt [3]{2} \sqrt{3} d \sqrt [3]{c + 3 d x^{2}}}{\sqrt [3]{c}} \right )}}{8 c^{\frac{5}{6}} \sqrt{d}} - \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \left (\sqrt{c} - \sqrt{d} x\right )}{3 \sqrt [6]{c} \sqrt [3]{c + 3 d x^{2}}} \right )}}{12 c^{\frac{5}{6}} \sqrt{d}} + \frac{\sqrt [3]{2} \sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3}}{3} + \frac{2^{\frac{2}{3}} \sqrt{3} \left (\sqrt{c} + \sqrt{d} x\right )}{3 \sqrt [6]{c} \sqrt [3]{c + 3 d x^{2}}} \right )}}{12 c^{\frac{5}{6}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2**(1/3)*log(sqrt(3) - sqrt(3)*sqrt(d)*x/sqrt(c))/(8*c**(5/6)*sqrt(d)) - 2**(1/3
)*log(sqrt(3) + sqrt(3)*sqrt(d)*x/sqrt(c))/(8*c**(5/6)*sqrt(d)) + 2**(1/3)*log(3
*sqrt(3)*d - 3*sqrt(3)*d**(3/2)*x/sqrt(c) - 3*2**(1/3)*sqrt(3)*d*(c + 3*d*x**2)*
*(1/3)/c**(1/3))/(8*c**(5/6)*sqrt(d)) - 2**(1/3)*log(3*sqrt(3)*d + 3*sqrt(3)*d**
(3/2)*x/sqrt(c) - 3*2**(1/3)*sqrt(3)*d*(c + 3*d*x**2)**(1/3)/c**(1/3))/(8*c**(5/
6)*sqrt(d)) - 2**(1/3)*sqrt(3)*atan(sqrt(3)/3 + 2**(2/3)*sqrt(3)*(sqrt(c) - sqrt
(d)*x)/(3*c**(1/6)*(c + 3*d*x**2)**(1/3)))/(12*c**(5/6)*sqrt(d)) + 2**(1/3)*sqrt
(3)*atan(sqrt(3)/3 + 2**(2/3)*sqrt(3)*(sqrt(c) + sqrt(d)*x)/(3*c**(1/6)*(c + 3*d
*x**2)**(1/3)))/(12*c**(5/6)*sqrt(d))

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Mathematica [C]  time = 0.231827, size = 153, normalized size = 0.75 \[ \frac{3 c x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 d x^2}{c},\frac{d x^2}{c}\right )}{\left (c-d x^2\right ) \sqrt [3]{c+3 d x^2} \left (2 d x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{3 d x^2}{c},\frac{d x^2}{c}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{3 d x^2}{c},\frac{d x^2}{c}\right )\right )+3 c F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 d x^2}{c},\frac{d x^2}{c}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(3*c*x*AppellF1[1/2, 1/3, 1, 3/2, (-3*d*x^2)/c, (d*x^2)/c])/((c - d*x^2)*(c + 3*
d*x^2)^(1/3)*(3*c*AppellF1[1/2, 1/3, 1, 3/2, (-3*d*x^2)/c, (d*x^2)/c] + 2*d*x^2*
(AppellF1[3/2, 1/3, 2, 5/2, (-3*d*x^2)/c, (d*x^2)/c] - AppellF1[3/2, 4/3, 1, 5/2
, (-3*d*x^2)/c, (d*x^2)/c])))

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Maple [F]  time = 0.067, size = 0, normalized size = 0. \[ \int{\frac{1}{-d{x}^{2}+c}{\frac{1}{\sqrt [3]{3\,d{x}^{2}+c}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-integrate(1/((3*d*x^2 + c)^(1/3)*(d*x^2 - c)), 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*d*x^2 + c)^(1/3)*(d*x^2 - c)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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