3.150 \(\int \frac{e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx\)

Optimal. Leaf size=234 \[ -\frac{3 (d e-c f) \log \left (2^{2/3} d \sqrt [3]{d^3 x^3-c^3}+d (c-d x)\right )}{4 \sqrt [3]{2} c d^2}+\frac{\sqrt{3} (d e-c f) \tan ^{-1}\left (\frac{1-\frac{\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} c d^2}-\frac{f \log \left (\sqrt [3]{d^3 x^3-c^3}-d x\right )}{2 d^2}+\frac{f \tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{d^3 x^3-c^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d^2}+\frac{(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2} \]

[Out]

(f*ArcTan[(1 + (2*d*x)/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^2) + (Sqrt[3
]*(d*e - c*f)*ArcTan[(1 - (2^(1/3)*(c - d*x))/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/
(2*2^(1/3)*c*d^2) + ((d*e - c*f)*Log[(c - d*x)*(c + d*x)^2])/(4*2^(1/3)*c*d^2) -
(f*Log[-(d*x) + (-c^3 + d^3*x^3)^(1/3)])/(2*d^2) - (3*(d*e - c*f)*Log[d*(c - d*
x) + 2^(2/3)*d*(-c^3 + d^3*x^3)^(1/3)])/(4*2^(1/3)*c*d^2)

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Rubi [A]  time = 0.428321, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{3 (d e-c f) \log \left (2^{2/3} d \sqrt [3]{d^3 x^3-c^3}+d (c-d x)\right )}{4 \sqrt [3]{2} c d^2}+\frac{\sqrt{3} (d e-c f) \tan ^{-1}\left (\frac{1-\frac{\sqrt [3]{2} (c-d x)}{\sqrt [3]{d^3 x^3-c^3}}}{\sqrt{3}}\right )}{2 \sqrt [3]{2} c d^2}-\frac{f \log \left (\sqrt [3]{d^3 x^3-c^3}-d x\right )}{2 d^2}+\frac{f \tan ^{-1}\left (\frac{\frac{2 d x}{\sqrt [3]{d^3 x^3-c^3}}+1}{\sqrt{3}}\right )}{\sqrt{3} d^2}+\frac{(d e-c f) \log \left ((c-d x) (c+d x)^2\right )}{4 \sqrt [3]{2} c d^2} \]

Antiderivative was successfully verified.

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

[Out]

(f*ArcTan[(1 + (2*d*x)/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/(Sqrt[3]*d^2) + (Sqrt[3
]*(d*e - c*f)*ArcTan[(1 - (2^(1/3)*(c - d*x))/(-c^3 + d^3*x^3)^(1/3))/Sqrt[3]])/
(2*2^(1/3)*c*d^2) + ((d*e - c*f)*Log[(c - d*x)*(c + d*x)^2])/(4*2^(1/3)*c*d^2) -
(f*Log[-(d*x) + (-c^3 + d^3*x^3)^(1/3)])/(2*d^2) - (3*(d*e - c*f)*Log[d*(c - d*
x) + 2^(2/3)*d*(-c^3 + d^3*x^3)^(1/3)])/(4*2^(1/3)*c*d^2)

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Mathematica [A]  time = 0.204517, size = 0, normalized size = 0. \[ \int \frac{e+f x}{(c+d x) \sqrt [3]{-c^3+d^3 x^3}} \, dx \]

Verification is Not applicable to the result.

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

[Out]

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

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Maple [F]  time = 0.061, size = 0, normalized size = 0. \[ \int{\frac{fx+e}{dx+c}{\frac{1}{\sqrt [3]{{d}^{3}{x}^{3}-{c}^{3}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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