3.538 \(\int \frac{5+\sqrt [4]{x}}{-6+x} \, dx\)

Optimal. Leaf size=54 \[ 4 \sqrt [4]{x}+5 \log (6-x)-2 \sqrt [4]{6} \tan ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right )-2 \sqrt [4]{6} \tanh ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right ) \]

[Out]

4*x^(1/4) - 2*6^(1/4)*ArcTan[x^(1/4)/6^(1/4)] - 2*6^(1/4)*ArcTanh[x^(1/4)/6^(1/4
)] + 5*Log[6 - x]

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Rubi [A]  time = 0.135973, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462 \[ 4 \sqrt [4]{x}+5 \log (6-x)-2 \sqrt [4]{6} \tan ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right )-2 \sqrt [4]{6} \tanh ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(5 + x^(1/4))/(-6 + x),x]

[Out]

4*x^(1/4) - 2*6^(1/4)*ArcTan[x^(1/4)/6^(1/4)] - 2*6^(1/4)*ArcTanh[x^(1/4)/6^(1/4
)] + 5*Log[6 - x]

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Rubi in Sympy [A]  time = 4.28562, size = 53, normalized size = 0.98 \[ 4 \sqrt [4]{x} + 5 \log{\left (- x + 6 \right )} - 2 \sqrt [4]{6} \operatorname{atan}{\left (\frac{6^{\frac{3}{4}} \sqrt [4]{x}}{6} \right )} - 2 \sqrt [4]{6} \operatorname{atanh}{\left (\frac{6^{\frac{3}{4}} \sqrt [4]{x}}{6} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((5+x**(1/4))/(-6+x),x)

[Out]

4*x**(1/4) + 5*log(-x + 6) - 2*6**(1/4)*atan(6**(3/4)*x**(1/4)/6) - 2*6**(1/4)*a
tanh(6**(3/4)*x**(1/4)/6)

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Mathematica [A]  time = 0.0461681, size = 77, normalized size = 1.43 \[ 4 \sqrt [4]{x}+\sqrt [4]{6} \log \left (6-6^{3/4} \sqrt [4]{x}\right )-\sqrt [4]{6} \log \left (6^{3/4} \sqrt [4]{x}+6\right )+5 \log (6-x)-2 \sqrt [4]{6} \tan ^{-1}\left (\frac{\sqrt [4]{x}}{\sqrt [4]{6}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(5 + x^(1/4))/(-6 + x),x]

[Out]

4*x^(1/4) - 2*6^(1/4)*ArcTan[x^(1/4)/6^(1/4)] + 6^(1/4)*Log[6 - 6^(3/4)*x^(1/4)]
- 6^(1/4)*Log[6 + 6^(3/4)*x^(1/4)] + 5*Log[6 - x]

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Maple [A]  time = 0.008, size = 52, normalized size = 1. \[ 4\,\sqrt [4]{x}-2\,\sqrt [4]{6}\arctan \left ( 1/6\,\sqrt [4]{x}{6}^{3/4} \right ) -\sqrt [4]{6}\ln \left ({1 \left ( \sqrt [4]{x}+\sqrt [4]{6} \right ) \left ( \sqrt [4]{x}-\sqrt [4]{6} \right ) ^{-1}} \right ) +5\,\ln \left ( -6+x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((5+x^(1/4))/(-6+x),x)

[Out]

4*x^(1/4)-2*6^(1/4)*arctan(1/6*x^(1/4)*6^(3/4))-6^(1/4)*ln((x^(1/4)+6^(1/4))/(x^
(1/4)-6^(1/4)))+5*ln(-6+x)

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Maxima [A]  time = 0.81135, size = 90, normalized size = 1.67 \[ -2 \cdot 6^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 6^{\frac{3}{4}} x^{\frac{1}{4}}\right ) + 6^{\frac{1}{4}} \log \left (-\frac{6^{\frac{1}{4}} - x^{\frac{1}{4}}}{6^{\frac{1}{4}} + x^{\frac{1}{4}}}\right ) + 4 \, x^{\frac{1}{4}} + 5 \, \log \left (\sqrt{6} + \sqrt{x}\right ) + 5 \, \log \left (-\sqrt{6} + \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(1/4) + 5)/(x - 6),x, algorithm="maxima")

[Out]

-2*6^(1/4)*arctan(1/6*6^(3/4)*x^(1/4)) + 6^(1/4)*log(-(6^(1/4) - x^(1/4))/(6^(1/
4) + x^(1/4))) + 4*x^(1/4) + 5*log(sqrt(6) + sqrt(x)) + 5*log(-sqrt(6) + sqrt(x)
)

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Fricas [A]  time = 0.281884, size = 111, normalized size = 2.06 \[ -{\left (6^{\frac{1}{4}} - 5\right )} \log \left (2 \cdot 6^{\frac{1}{4}} + 2 \, x^{\frac{1}{4}}\right ) +{\left (6^{\frac{1}{4}} + 5\right )} \log \left (-2 \cdot 6^{\frac{1}{4}} + 2 \, x^{\frac{1}{4}}\right ) + 4 \cdot 6^{\frac{1}{4}} \arctan \left (\frac{6^{\frac{1}{4}}}{\sqrt{\sqrt{6} + \sqrt{x}} + x^{\frac{1}{4}}}\right ) + 4 \, x^{\frac{1}{4}} + 5 \, \log \left (4 \, \sqrt{6} + 4 \, \sqrt{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(1/4) + 5)/(x - 6),x, algorithm="fricas")

[Out]

-(6^(1/4) - 5)*log(2*6^(1/4) + 2*x^(1/4)) + (6^(1/4) + 5)*log(-2*6^(1/4) + 2*x^(
1/4)) + 4*6^(1/4)*arctan(6^(1/4)/(sqrt(sqrt(6) + sqrt(x)) + x^(1/4))) + 4*x^(1/4
) + 5*log(4*sqrt(6) + 4*sqrt(x))

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Sympy [A]  time = 4.12421, size = 182, normalized size = 3.37 \[ \frac{5 \sqrt [4]{x} \Gamma \left (\frac{5}{4}\right )}{\Gamma \left (\frac{9}{4}\right )} + 5 \log{\left (x - 6 \right )} + \frac{5 \sqrt [4]{6} \log{\left (- \frac{6^{\frac{3}{4}} \sqrt [4]{x}}{6} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} - \frac{5 \sqrt [4]{6} i \log{\left (- \frac{6^{\frac{3}{4}} \sqrt [4]{x} e^{\frac{i \pi }{2}}}{6} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} - \frac{5 \sqrt [4]{6} \log{\left (- \frac{6^{\frac{3}{4}} \sqrt [4]{x} e^{i \pi }}{6} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{5 \sqrt [4]{6} i \log{\left (- \frac{6^{\frac{3}{4}} \sqrt [4]{x} e^{\frac{3 i \pi }{2}}}{6} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((5+x**(1/4))/(-6+x),x)

[Out]

5*x**(1/4)*gamma(5/4)/gamma(9/4) + 5*log(x - 6) + 5*6**(1/4)*log(-6**(3/4)*x**(1
/4)/6 + 1)*gamma(5/4)/(4*gamma(9/4)) - 5*6**(1/4)*I*log(-6**(3/4)*x**(1/4)*exp_p
olar(I*pi/2)/6 + 1)*gamma(5/4)/(4*gamma(9/4)) - 5*6**(1/4)*log(-6**(3/4)*x**(1/4
)*exp_polar(I*pi)/6 + 1)*gamma(5/4)/(4*gamma(9/4)) + 5*6**(1/4)*I*log(-6**(3/4)*
x**(1/4)*exp_polar(3*I*pi/2)/6 + 1)*gamma(5/4)/(4*gamma(9/4))

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GIAC/XCAS [A]  time = 0.304351, size = 74, normalized size = 1.37 \[ -2 \cdot 6^{\frac{1}{4}} \arctan \left (\frac{1}{6} \cdot 6^{\frac{3}{4}} x^{\frac{1}{4}}\right ) - 6^{\frac{1}{4}}{\rm ln}\left (6^{\frac{1}{4}} + x^{\frac{1}{4}}\right ) + 6^{\frac{1}{4}}{\rm ln}\left ({\left | -6^{\frac{1}{4}} + x^{\frac{1}{4}} \right |}\right ) + 4 \, x^{\frac{1}{4}} + 5 \,{\rm ln}\left ({\left | x - 6 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((x^(1/4) + 5)/(x - 6),x, algorithm="giac")

[Out]

-2*6^(1/4)*arctan(1/6*6^(3/4)*x^(1/4)) - 6^(1/4)*ln(6^(1/4) + x^(1/4)) + 6^(1/4)
*ln(abs(-6^(1/4) + x^(1/4))) + 4*x^(1/4) + 5*ln(abs(x - 6))