3.446 \(\int \frac{2+\sqrt [3]{1-5 x}}{3+\sqrt [3]{1-5 x}} \, dx\)

Optimal. Leaf size=44 \[ x+\frac{3}{10} (1-5 x)^{2/3}-\frac{9}{5} \sqrt [3]{1-5 x}+\frac{27}{5} \log \left (\sqrt [3]{1-5 x}+3\right ) \]

[Out]

(-9*(1 - 5*x)^(1/3))/5 + (3*(1 - 5*x)^(2/3))/10 + x + (27*Log[3 + (1 - 5*x)^(1/3
)])/5

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Rubi [A]  time = 0.0588394, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12 \[ x+\frac{3}{10} (1-5 x)^{2/3}-\frac{9}{5} \sqrt [3]{1-5 x}+\frac{27}{5} \log \left (\sqrt [3]{1-5 x}+3\right ) \]

Antiderivative was successfully verified.

[In]  Int[(2 + (1 - 5*x)^(1/3))/(3 + (1 - 5*x)^(1/3)),x]

[Out]

(-9*(1 - 5*x)^(1/3))/5 + (3*(1 - 5*x)^(2/3))/10 + x + (27*Log[3 + (1 - 5*x)^(1/3
)])/5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ x - \frac{9 \sqrt [3]{- 5 x + 1}}{5} + \frac{27 \log{\left (\sqrt [3]{- 5 x + 1} + 3 \right )}}{5} + \frac{3 \int ^{\sqrt [3]{- 5 x + 1}} x\, dx}{5} - \frac{1}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((2+(1-5*x)**(1/3))/(3+(1-5*x)**(1/3)),x)

[Out]

x - 9*(-5*x + 1)**(1/3)/5 + 27*log((-5*x + 1)**(1/3) + 3)/5 + 3*Integral(x, (x,
(-5*x + 1)**(1/3)))/5 - 1/5

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Mathematica [A]  time = 0.0197138, size = 45, normalized size = 1.02 \[ \frac{1}{10} \left (10 x+3 (1-5 x)^{2/3}-18 \sqrt [3]{1-5 x}+54 \log \left (\sqrt [3]{1-5 x}+3\right )-2\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(2 + (1 - 5*x)^(1/3))/(3 + (1 - 5*x)^(1/3)),x]

[Out]

(-2 - 18*(1 - 5*x)^(1/3) + 3*(1 - 5*x)^(2/3) + 10*x + 54*Log[3 + (1 - 5*x)^(1/3)
])/10

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Maple [A]  time = 0.004, size = 34, normalized size = 0.8 \[ -{\frac{1}{5}}+x+{\frac{3}{10} \left ( 1-5\,x \right ) ^{{\frac{2}{3}}}}-{\frac{9}{5}\sqrt [3]{1-5\,x}}+{\frac{27}{5}\ln \left ( 3+\sqrt [3]{1-5\,x} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((2+(1-5*x)^(1/3))/(3+(1-5*x)^(1/3)),x)

[Out]

-1/5+x+3/10*(1-5*x)^(2/3)-9/5*(1-5*x)^(1/3)+27/5*ln(3+(1-5*x)^(1/3))

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Maxima [A]  time = 0.698118, size = 45, normalized size = 1.02 \[ x + \frac{3}{10} \,{\left (-5 \, x + 1\right )}^{\frac{2}{3}} - \frac{9}{5} \,{\left (-5 \, x + 1\right )}^{\frac{1}{3}} + \frac{27}{5} \, \log \left ({\left (-5 \, x + 1\right )}^{\frac{1}{3}} + 3\right ) - \frac{1}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-5*x + 1)^(1/3) + 2)/((-5*x + 1)^(1/3) + 3),x, algorithm="maxima")

[Out]

x + 3/10*(-5*x + 1)^(2/3) - 9/5*(-5*x + 1)^(1/3) + 27/5*log((-5*x + 1)^(1/3) + 3
) - 1/5

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Fricas [A]  time = 0.262665, size = 43, normalized size = 0.98 \[ x + \frac{3}{10} \,{\left (-5 \, x + 1\right )}^{\frac{2}{3}} - \frac{9}{5} \,{\left (-5 \, x + 1\right )}^{\frac{1}{3}} + \frac{27}{5} \, \log \left ({\left (-5 \, x + 1\right )}^{\frac{1}{3}} + 3\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-5*x + 1)^(1/3) + 2)/((-5*x + 1)^(1/3) + 3),x, algorithm="fricas")

[Out]

x + 3/10*(-5*x + 1)^(2/3) - 9/5*(-5*x + 1)^(1/3) + 27/5*log((-5*x + 1)^(1/3) + 3
)

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Sympy [A]  time = 0.468949, size = 39, normalized size = 0.89 \[ x + \frac{3 \left (- 5 x + 1\right )^{\frac{2}{3}}}{10} - \frac{9 \sqrt [3]{- 5 x + 1}}{5} + \frac{27 \log{\left (\sqrt [3]{- 5 x + 1} + 3 \right )}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2+(1-5*x)**(1/3))/(3+(1-5*x)**(1/3)),x)

[Out]

x + 3*(-5*x + 1)**(2/3)/10 - 9*(-5*x + 1)**(1/3)/5 + 27*log((-5*x + 1)**(1/3) +
3)/5

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GIAC/XCAS [A]  time = 0.278195, size = 45, normalized size = 1.02 \[ x + \frac{3}{10} \,{\left (-5 \, x + 1\right )}^{\frac{2}{3}} - \frac{9}{5} \,{\left (-5 \, x + 1\right )}^{\frac{1}{3}} + \frac{27}{5} \,{\rm ln}\left ({\left (-5 \, x + 1\right )}^{\frac{1}{3}} + 3\right ) - \frac{1}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((-5*x + 1)^(1/3) + 2)/((-5*x + 1)^(1/3) + 3),x, algorithm="giac")

[Out]

x + 3/10*(-5*x + 1)^(2/3) - 9/5*(-5*x + 1)^(1/3) + 27/5*ln((-5*x + 1)^(1/3) + 3)
- 1/5