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

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 = 0.0466631, 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, Rules used = {2043, 1631, 629} \[ \frac{6}{5} \left (x-3 \sqrt{x}\right )^{5/3} \]

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

Rule 2043

Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{d = Denominator[n]}, Dist[d, Subst[Int
[x^(d - 1)*(SubstFor[x^n, Pq, x] /. x -> x^(d*n))*(a*x^(d*j) + b*x^(d*n))^p, x], x, x^(1/d)], x]] /; FreeQ[{a,
b, j, n, p}, x] && PolyQ[Pq, x^n] &&  !IntegerQ[p] && NeQ[n, j] && RationalQ[j, n] && IntegerQ[j/n] && LtQ[-1
, n, 1]

Rule 1631

Int[(Pq_)*((e_.)*(x_))^(m_.)*((b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[e, Int[(e*x)^(m - 1)*Polynom
ialQuotient[Pq, b + c*x, x]*(b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{b, c, e, m, p}, x] && PolyQ[Pq, x] && EqQ[
PolynomialRemainder[Pq, b + c*x, x], 0]

Rule 629

Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d*(a + b*x + c*x^2)^(p +
1))/(b*(p + 1)), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{9-9 \sqrt{x}+2 x}{\sqrt [3]{-3 \sqrt{x}+x}} \, dx &=2 \operatorname{Subst}\left (\int \frac{x \left (9-9 x+2 x^2\right )}{\sqrt [3]{-3 x+x^2}} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int (-3+2 x) \left (-3 x+x^2\right )^{2/3} \, dx,x,\sqrt{x}\right )\\ &=\frac{6}{5} \left (-3 \sqrt{x}+x\right )^{5/3}\\ \end{align*}

Mathematica [A]  time = 0.0247953, 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 = 0.059, size = 125, normalized size = 7.4 \begin{align*}{\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 ) }}}} \end{align*}

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([1/3,1
1/3],[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)*hype
rgeom([1/3,8/3],[11/3],1/3*x^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{2 \, x - 9 \, \sqrt{x} + 9}{{\left (x - 3 \, \sqrt{x}\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.36866, size = 36, normalized size = 2.12 \begin{align*} \frac{6}{5} \,{\left (x - 3 \, \sqrt{x}\right )}^{\frac{5}{3}} \end{align*}

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, algorithm="fricas")

[Out]

6/5*(x - 3*sqrt(x))^(5/3)

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

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

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, algorithm="giac")

[Out]

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