3.5.7 \(\int \frac {x^9}{(1-x^3)^{4/3} (1+x^3)} \, dx\)

Optimal. Leaf size=174 \[ \frac {5}{6} \left (1-x^3\right )^{2/3} x+\frac {\log \left (x^3+1\right )}{12 \sqrt [3]{2}}-\frac {\log \left (-\sqrt [3]{1-x^3}-\sqrt [3]{2} x\right )}{4 \sqrt [3]{2}}-\frac {1}{6} \log \left (\sqrt [3]{1-x^3}+x\right )+\frac {\tan ^{-1}\left (\frac {1-\frac {2 x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{3 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {1-\frac {2 \sqrt [3]{2} x}{\sqrt [3]{1-x^3}}}{\sqrt {3}}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {x^4}{2 \sqrt [3]{1-x^3}} \]

________________________________________________________________________________________

Rubi [C]  time = 0.02, antiderivative size = 26, normalized size of antiderivative = 0.15, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {510} \begin {gather*} \frac {1}{10} x^{10} F_1\left (\frac {10}{3};\frac {4}{3},1;\frac {13}{3};x^3,-x^3\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[x^9/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

(x^10*AppellF1[10/3, 4/3, 1, 13/3, x^3, -x^3])/10

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rubi steps

\begin {align*} \int \frac {x^9}{\left (1-x^3\right )^{4/3} \left (1+x^3\right )} \, dx &=\frac {1}{10} x^{10} F_1\left (\frac {10}{3};\frac {4}{3},1;\frac {13}{3};x^3,-x^3\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.25, size = 152, normalized size = 0.87 \begin {gather*} \frac {1}{72} \left (-6 x^4 F_1\left (\frac {4}{3};\frac {1}{3},1;\frac {7}{3};x^3,-x^3\right )-\frac {12 \left (2 x^3-5\right ) x}{\sqrt [3]{1-x^3}}-5\ 2^{2/3} \left (2 \log \left (\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+1\right )+2 \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 \sqrt [3]{2} x}{\sqrt [3]{x^3-1}}-1}{\sqrt {3}}\right )-\log \left (-\frac {\sqrt [3]{2} x}{\sqrt [3]{x^3-1}}+\frac {2^{2/3} x^2}{\left (x^3-1\right )^{2/3}}+1\right )\right )\right ) \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[x^9/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

((-12*x*(-5 + 2*x^3))/(1 - x^3)^(1/3) - 6*x^4*AppellF1[4/3, 1/3, 1, 7/3, x^3, -x^3] - 5*2^(2/3)*(2*Sqrt[3]*Arc
Tan[(-1 + (2*2^(1/3)*x)/(-1 + x^3)^(1/3))/Sqrt[3]] - Log[1 + (2^(2/3)*x^2)/(-1 + x^3)^(2/3) - (2^(1/3)*x)/(-1
+ x^3)^(1/3)] + 2*Log[1 + (2^(1/3)*x)/(-1 + x^3)^(1/3)]))/72

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.57, size = 248, normalized size = 1.43 \begin {gather*} -\frac {1}{9} \log \left (\sqrt [3]{1-x^3}+x\right )-\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3}+2 x\right )}{6 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{1-x^3}-x}\right )}{3 \sqrt {3}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{1-x^3}-x}\right )}{2 \sqrt [3]{2} \sqrt {3}}+\frac {\left (1-x^3\right )^{2/3} \left (2 x^4-5 x\right )}{6 \left (x^3-1\right )}+\frac {1}{18} \log \left (-\sqrt [3]{1-x^3} x+\left (1-x^3\right )^{2/3}+x^2\right )+\frac {\log \left (2^{2/3} \sqrt [3]{1-x^3} x-\sqrt [3]{2} \left (1-x^3\right )^{2/3}-2 x^2\right )}{12 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^9/((1 - x^3)^(4/3)*(1 + x^3)),x]

[Out]

((1 - x^3)^(2/3)*(-5*x + 2*x^4))/(6*(-1 + x^3)) - ArcTan[(Sqrt[3]*x)/(-x + 2*(1 - x^3)^(1/3))]/(3*Sqrt[3]) - A
rcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(1 - x^3)^(1/3))]/(2*2^(1/3)*Sqrt[3]) - Log[x + (1 - x^3)^(1/3)]/9 - Log[2*x +
2^(2/3)*(1 - x^3)^(1/3)]/(6*2^(1/3)) + Log[x^2 - x*(1 - x^3)^(1/3) + (1 - x^3)^(2/3)]/18 + Log[-2*x^2 + 2^(2/
3)*x*(1 - x^3)^(1/3) - 2^(1/3)*(1 - x^3)^(2/3)]/(12*2^(1/3))

________________________________________________________________________________________

fricas [B]  time = 0.49, size = 271, normalized size = 1.56 \begin {gather*} \frac {6 \, \sqrt {6} 2^{\frac {1}{6}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - 1\right )} \arctan \left (\frac {2^{\frac {1}{6}} {\left (\sqrt {6} 2^{\frac {1}{3}} x + 2 \, \sqrt {6} \left (-1\right )^{\frac {1}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, x}\right ) + 6 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - 1\right )} \log \left (\frac {2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) - 3 \cdot 2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{3} - 1\right )} \log \left (-\frac {2^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{2} + 2^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x - {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 8 \, \sqrt {3} {\left (x^{3} - 1\right )} \arctan \left (-\frac {\sqrt {3} x - 2 \, \sqrt {3} {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{3 \, x}\right ) - 8 \, {\left (x^{3} - 1\right )} \log \left (\frac {x + {\left (-x^{3} + 1\right )}^{\frac {1}{3}}}{x}\right ) + 4 \, {\left (x^{3} - 1\right )} \log \left (\frac {x^{2} - {\left (-x^{3} + 1\right )}^{\frac {1}{3}} x + {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 12 \, {\left (2 \, x^{4} - 5 \, x\right )} {\left (-x^{3} + 1\right )}^{\frac {2}{3}}}{72 \, {\left (x^{3} - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="fricas")

[Out]

1/72*(6*sqrt(6)*2^(1/6)*(-1)^(1/3)*(x^3 - 1)*arctan(1/6*2^(1/6)*(sqrt(6)*2^(1/3)*x + 2*sqrt(6)*(-1)^(1/3)*(-x^
3 + 1)^(1/3))/x) + 6*2^(2/3)*(-1)^(1/3)*(x^3 - 1)*log((2^(1/3)*(-1)^(2/3)*x + (-x^3 + 1)^(1/3))/x) - 3*2^(2/3)
*(-1)^(1/3)*(x^3 - 1)*log(-(2^(2/3)*(-1)^(1/3)*x^2 + 2^(1/3)*(-1)^(2/3)*(-x^3 + 1)^(1/3)*x - (-x^3 + 1)^(2/3))
/x^2) + 8*sqrt(3)*(x^3 - 1)*arctan(-1/3*(sqrt(3)*x - 2*sqrt(3)*(-x^3 + 1)^(1/3))/x) - 8*(x^3 - 1)*log((x + (-x
^3 + 1)^(1/3))/x) + 4*(x^3 - 1)*log((x^2 - (-x^3 + 1)^(1/3)*x + (-x^3 + 1)^(2/3))/x^2) + 12*(2*x^4 - 5*x)*(-x^
3 + 1)^(2/3))/(x^3 - 1)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{9}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="giac")

[Out]

integrate(x^9/((x^3 + 1)*(-x^3 + 1)^(4/3)), x)

________________________________________________________________________________________

maple [F]  time = 1.46, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{9}}{\left (-x^{3}+1\right )^{\frac {4}{3}} \left (x^{3}+1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(-x^3+1)^(4/3)/(x^3+1),x)

[Out]

int(x^9/(-x^3+1)^(4/3)/(x^3+1),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{9}}{{\left (x^{3} + 1\right )} {\left (-x^{3} + 1\right )}^{\frac {4}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(-x^3+1)^(4/3)/(x^3+1),x, algorithm="maxima")

[Out]

integrate(x^9/((x^3 + 1)*(-x^3 + 1)^(4/3)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^9}{{\left (1-x^3\right )}^{4/3}\,\left (x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/((1 - x^3)^(4/3)*(x^3 + 1)),x)

[Out]

int(x^9/((1 - x^3)^(4/3)*(x^3 + 1)), x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{9}}{\left (- \left (x - 1\right ) \left (x^{2} + x + 1\right )\right )^{\frac {4}{3}} \left (x + 1\right ) \left (x^{2} - x + 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(-x**3+1)**(4/3)/(x**3+1),x)

[Out]

Integral(x**9/((-(x - 1)*(x**2 + x + 1))**(4/3)*(x + 1)*(x**2 - x + 1)), x)

________________________________________________________________________________________