3.313 \(\int \frac{-2 x^5+3 x^8-x^2 (-1+3 x^3)^{2/3}}{(-1+3 x^3)^{3/4}} \, dx\)

Optimal. Leaf size=46 \[ \frac{4}{243} \left (3 x^3-1\right )^{9/4}-\frac{4}{33} \left (3 x^3-1\right )^{11/12}-\frac{4}{27} \sqrt [4]{3 x^3-1} \]

[Out]

(-4*(-1 + 3*x^3)^(1/4))/27 - (4*(-1 + 3*x^3)^(11/12))/33 + (4*(-1 + 3*x^3)^(9/4))/243

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Rubi [A]  time = 0.193269, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {6742, 266, 43, 261} \[ \frac{4}{243} \left (3 x^3-1\right )^{9/4}-\frac{4}{33} \left (3 x^3-1\right )^{11/12}-\frac{4}{27} \sqrt [4]{3 x^3-1} \]

Antiderivative was successfully verified.

[In]

Int[(-2*x^5 + 3*x^8 - x^2*(-1 + 3*x^3)^(2/3))/(-1 + 3*x^3)^(3/4),x]

[Out]

(-4*(-1 + 3*x^3)^(1/4))/27 - (4*(-1 + 3*x^3)^(11/12))/33 + (4*(-1 + 3*x^3)^(9/4))/243

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{-2 x^5+3 x^8-x^2 \left (-1+3 x^3\right )^{2/3}}{\left (-1+3 x^3\right )^{3/4}} \, dx &=\int \left (-\frac{2 x^5}{\left (-1+3 x^3\right )^{3/4}}+\frac{3 x^8}{\left (-1+3 x^3\right )^{3/4}}-\frac{x^2}{\sqrt [12]{-1+3 x^3}}\right ) \, dx\\ &=-\left (2 \int \frac{x^5}{\left (-1+3 x^3\right )^{3/4}} \, dx\right )+3 \int \frac{x^8}{\left (-1+3 x^3\right )^{3/4}} \, dx-\int \frac{x^2}{\sqrt [12]{-1+3 x^3}} \, dx\\ &=-\frac{4}{33} \left (-1+3 x^3\right )^{11/12}-\frac{2}{3} \operatorname{Subst}\left (\int \frac{x}{(-1+3 x)^{3/4}} \, dx,x,x^3\right )+\operatorname{Subst}\left (\int \frac{x^2}{(-1+3 x)^{3/4}} \, dx,x,x^3\right )\\ &=-\frac{4}{33} \left (-1+3 x^3\right )^{11/12}-\frac{2}{3} \operatorname{Subst}\left (\int \left (\frac{1}{3 (-1+3 x)^{3/4}}+\frac{1}{3} \sqrt [4]{-1+3 x}\right ) \, dx,x,x^3\right )+\operatorname{Subst}\left (\int \left (\frac{1}{9 (-1+3 x)^{3/4}}+\frac{2}{9} \sqrt [4]{-1+3 x}+\frac{1}{9} (-1+3 x)^{5/4}\right ) \, dx,x,x^3\right )\\ &=-\frac{4}{27} \sqrt [4]{-1+3 x^3}-\frac{4}{33} \left (-1+3 x^3\right )^{11/12}+\frac{4}{243} \left (-1+3 x^3\right )^{9/4}\\ \end{align*}

Mathematica [A]  time = 0.0805153, size = 40, normalized size = 0.87 \[ -\frac{4 \sqrt [4]{3 x^3-1} \left (-99 x^6+66 x^3+81 \left (3 x^3-1\right )^{2/3}+88\right )}{2673} \]

Antiderivative was successfully verified.

[In]

Integrate[(-2*x^5 + 3*x^8 - x^2*(-1 + 3*x^3)^(2/3))/(-1 + 3*x^3)^(3/4),x]

[Out]

(-4*(-1 + 3*x^3)^(1/4)*(88 + 66*x^3 - 99*x^6 + 81*(-1 + 3*x^3)^(2/3)))/2673

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Maple [C]  time = 0.067, size = 116, normalized size = 2.5 \begin{align*} -{\frac{{x}^{6}}{3} \left ( -{\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{{\frac{3}{4}}}{\mbox{$_2$F$_1$}({\frac{3}{4}},2;\,3;\,3\,{x}^{3})} \left ({\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{-{\frac{3}{4}}}}+{\frac{{x}^{9}}{3} \left ( -{\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{{\frac{3}{4}}}{\mbox{$_2$F$_1$}({\frac{3}{4}},3;\,4;\,3\,{x}^{3})} \left ({\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{-{\frac{3}{4}}}}-{\frac{{x}^{3}}{3} \left ( -{\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{{\frac{1}{12}}}{\mbox{$_2$F$_1$}({\frac{1}{12}},1;\,2;\,3\,{x}^{3})} \left ({\it signum} \left ( 3\,{x}^{3}-1 \right ) \right ) ^{-{\frac{1}{12}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-2*x^5+3*x^8-x^2*(3*x^3-1)^(2/3))/(3*x^3-1)^(3/4),x)

[Out]

-1/3/signum(3*x^3-1)^(3/4)*(-signum(3*x^3-1))^(3/4)*x^6*hypergeom([3/4,2],[3],3*x^3)+1/3/signum(3*x^3-1)^(3/4)
*(-signum(3*x^3-1))^(3/4)*x^9*hypergeom([3/4,3],[4],3*x^3)-1/3/signum(3*x^3-1)^(1/12)*(-signum(3*x^3-1))^(1/12
)*x^3*hypergeom([1/12,1],[2],3*x^3)

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Maxima [A]  time = 0.972361, size = 46, normalized size = 1. \begin{align*} \frac{4}{243} \,{\left (3 \, x^{3} - 1\right )}^{\frac{9}{4}} - \frac{4}{33} \,{\left (3 \, x^{3} - 1\right )}^{\frac{11}{12}} - \frac{4}{27} \,{\left (3 \, x^{3} - 1\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5+3*x^8-x^2*(3*x^3-1)^(2/3))/(3*x^3-1)^(3/4),x, algorithm="maxima")

[Out]

4/243*(3*x^3 - 1)^(9/4) - 4/33*(3*x^3 - 1)^(11/12) - 4/27*(3*x^3 - 1)^(1/4)

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Fricas [A]  time = 2.03497, size = 97, normalized size = 2.11 \begin{align*} \frac{4}{243} \,{\left (9 \, x^{6} - 6 \, x^{3} - 8\right )}{\left (3 \, x^{3} - 1\right )}^{\frac{1}{4}} - \frac{4}{33} \,{\left (3 \, x^{3} - 1\right )}^{\frac{11}{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5+3*x^8-x^2*(3*x^3-1)^(2/3))/(3*x^3-1)^(3/4),x, algorithm="fricas")

[Out]

4/243*(9*x^6 - 6*x^3 - 8)*(3*x^3 - 1)^(1/4) - 4/33*(3*x^3 - 1)^(11/12)

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Sympy [C]  time = 3.08764, size = 221, normalized size = 4.8 \begin{align*} - \frac{4 \left (3 x^{3} - 1\right )^{\frac{11}{12}}}{33} - 2 \left (\begin{cases} \frac{4 x^{3} \sqrt [4]{3 x^{3} - 1}}{45} + \frac{16 \sqrt [4]{3 x^{3} - 1}}{135} & \text{for}\: 3 \left |{x^{3}}\right | > 1 \\- \frac{4 x^{3} \sqrt [4]{1 - 3 x^{3}} e^{- \frac{3 i \pi }{4}}}{45} - \frac{16 \sqrt [4]{1 - 3 x^{3}} e^{- \frac{3 i \pi }{4}}}{135} & \text{otherwise} \end{cases}\right ) + 3 \left (\begin{cases} \frac{4 x^{6} \sqrt [4]{3 x^{3} - 1}}{81} + \frac{32 x^{3} \sqrt [4]{3 x^{3} - 1}}{1215} + \frac{128 \sqrt [4]{3 x^{3} - 1}}{3645} & \text{for}\: 3 \left |{x^{3}}\right | > 1 \\\frac{4 x^{6} \sqrt [4]{1 - 3 x^{3}} e^{\frac{i \pi }{4}}}{81} + \frac{32 x^{3} \sqrt [4]{1 - 3 x^{3}} e^{\frac{i \pi }{4}}}{1215} + \frac{128 \sqrt [4]{1 - 3 x^{3}} e^{\frac{i \pi }{4}}}{3645} & \text{otherwise} \end{cases}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x**5+3*x**8-x**2*(3*x**3-1)**(2/3))/(3*x**3-1)**(3/4),x)

[Out]

-4*(3*x**3 - 1)**(11/12)/33 - 2*Piecewise((4*x**3*(3*x**3 - 1)**(1/4)/45 + 16*(3*x**3 - 1)**(1/4)/135, 3*Abs(x
**3) > 1), (-4*x**3*(1 - 3*x**3)**(1/4)*exp(-3*I*pi/4)/45 - 16*(1 - 3*x**3)**(1/4)*exp(-3*I*pi/4)/135, True))
+ 3*Piecewise((4*x**6*(3*x**3 - 1)**(1/4)/81 + 32*x**3*(3*x**3 - 1)**(1/4)/1215 + 128*(3*x**3 - 1)**(1/4)/3645
, 3*Abs(x**3) > 1), (4*x**6*(1 - 3*x**3)**(1/4)*exp(I*pi/4)/81 + 32*x**3*(1 - 3*x**3)**(1/4)*exp(I*pi/4)/1215
+ 128*(1 - 3*x**3)**(1/4)*exp(I*pi/4)/3645, True))

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Giac [A]  time = 1.07351, size = 46, normalized size = 1. \begin{align*} \frac{4}{243} \,{\left (3 \, x^{3} - 1\right )}^{\frac{9}{4}} - \frac{4}{33} \,{\left (3 \, x^{3} - 1\right )}^{\frac{11}{12}} - \frac{4}{27} \,{\left (3 \, x^{3} - 1\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-2*x^5+3*x^8-x^2*(3*x^3-1)^(2/3))/(3*x^3-1)^(3/4),x, algorithm="giac")

[Out]

4/243*(3*x^3 - 1)^(9/4) - 4/33*(3*x^3 - 1)^(11/12) - 4/27*(3*x^3 - 1)^(1/4)