3.682 \(\int \frac{9-9 \sqrt{x}+2 x}{\sqrt [3]{-3 \sqrt{x}+x}} \, dx\)

Optimal. Leaf size=17 \[ \frac{6}{5} \left (x-3 \sqrt{x}\right )^{5/3} \]

[Out]

(6*(-3*Sqrt[x] + x)^(5/3))/5

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Rubi [A]  time = 0.0720455, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.115 \[ \frac{6}{5} \left (x-3 \sqrt{x}\right )^{5/3} \]

Antiderivative was successfully verified.

[In]  Int[(9 - 9*Sqrt[x] + 2*x)/(-3*Sqrt[x] + x)^(1/3),x]

[Out]

(6*(-3*Sqrt[x] + x)^(5/3))/5

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Rubi in Sympy [A]  time = 37.6662, size = 36, normalized size = 2.12 \[ - \frac{18 \sqrt{x} \left (- 3 \sqrt{x} + x\right )^{\frac{2}{3}}}{5} + \frac{6 x \left (- 3 \sqrt{x} + x\right )^{\frac{2}{3}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((9+2*x-9*x**(1/2))/(x-3*x**(1/2))**(1/3),x)

[Out]

-18*sqrt(x)*(-3*sqrt(x) + x)**(2/3)/5 + 6*x*(-3*sqrt(x) + x)**(2/3)/5

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Mathematica [A]  time = 0.0157377, size = 17, normalized size = 1. \[ \frac{6}{5} \left (x-3 \sqrt{x}\right )^{5/3} \]

Antiderivative was successfully verified.

[In]  Integrate[(9 - 9*Sqrt[x] + 2*x)/(-3*Sqrt[x] + x)^(1/3),x]

[Out]

(6*(-3*Sqrt[x] + x)^(5/3))/5

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Maple [C]  time = 0.123, size = 125, normalized size = 7.4 \[{\frac{18\,{3}^{2/3}}{5}\sqrt [3]{-{\it signum} \left ( -1+{\frac{1}{3}\sqrt{x}} \right ) }{x}^{{\frac{5}{6}}}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{5}{3}};\,{\frac{8}{3}};\,{\frac{1}{3}\sqrt{x}})}{\frac{1}{\sqrt [3]{{\it signum} \left ( -1+{\frac{1}{3}\sqrt{x}} \right ) }}}}+{\frac{4\,{3}^{2/3}}{11}\sqrt [3]{-{\it signum} \left ( -1+{\frac{1}{3}\sqrt{x}} \right ) }{x}^{{\frac{11}{6}}}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{11}{3}};\,{\frac{14}{3}};\,{\frac{1}{3}\sqrt{x}})}{\frac{1}{\sqrt [3]{{\it signum} \left ( -1+{\frac{1}{3}\sqrt{x}} \right ) }}}}-{\frac{9\,{3}^{2/3}}{4}\sqrt [3]{-{\it signum} \left ( -1+{\frac{1}{3}\sqrt{x}} \right ) }{x}^{{\frac{4}{3}}}{\mbox{$_2$F$_1$}({\frac{1}{3}},{\frac{8}{3}};\,{\frac{11}{3}};\,{\frac{1}{3}\sqrt{x}})}{\frac{1}{\sqrt [3]{{\it signum} \left ( -1+{\frac{1}{3}\sqrt{x}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((9+2*x-9*x^(1/2))/(x-3*x^(1/2))^(1/3),x)

[Out]

18/5*3^(2/3)/signum(-1+1/3*x^(1/2))^(1/3)*(-signum(-1+1/3*x^(1/2)))^(1/3)*x^(5/6
)*hypergeom([1/3,5/3],[8/3],1/3*x^(1/2))+4/11*3^(2/3)/signum(-1+1/3*x^(1/2))^(1/
3)*(-signum(-1+1/3*x^(1/2)))^(1/3)*x^(11/6)*hypergeom([1/3,11/3],[14/3],1/3*x^(1
/2))-9/4*3^(2/3)/signum(-1+1/3*x^(1/2))^(1/3)*(-signum(-1+1/3*x^(1/2)))^(1/3)*x^
(4/3)*hypergeom([1/3,8/3],[11/3],1/3*x^(1/2))

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Maxima [A]  time = 0.97544, size = 31, normalized size = 1.82 \[ \frac{6 \,{\left (x^{\frac{11}{6}} - 6 \, x^{\frac{4}{3}} + 9 \, x^{\frac{5}{6}}\right )}}{5 \,{\left (\sqrt{x} - 3\right )}^{\frac{1}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x - 9*sqrt(x) + 9)/(x - 3*sqrt(x))^(1/3),x, algorithm="maxima")

[Out]

6/5*(x^(11/6) - 6*x^(4/3) + 9*x^(5/6))/(sqrt(x) - 3)^(1/3)

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Fricas [A]  time = 0.301829, size = 15, normalized size = 0.88 \[ \frac{6}{5} \,{\left (x - 3 \, \sqrt{x}\right )}^{\frac{5}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x - 9*sqrt(x) + 9)/(x - 3*sqrt(x))^(1/3),x, algorithm="fricas")

[Out]

6/5*(x - 3*sqrt(x))^(5/3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{- 9 \sqrt{x} + 2 x + 9}{\sqrt [3]{- 3 \sqrt{x} + x}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((9+2*x-9*x**(1/2))/(x-3*x**(1/2))**(1/3),x)

[Out]

Integral((-9*sqrt(x) + 2*x + 9)/(-3*sqrt(x) + x)**(1/3), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{2 \, x - 9 \, \sqrt{x} + 9}{{\left (x - 3 \, \sqrt{x}\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*x - 9*sqrt(x) + 9)/(x - 3*sqrt(x))^(1/3),x, algorithm="giac")

[Out]

integrate((2*x - 9*sqrt(x) + 9)/(x - 3*sqrt(x))^(1/3), x)