3.16.39 \(\int \frac {(1+x^3)^{2/3} (1+x^3+2 x^6)}{x^6 (-1+2 x^6)} \, dx\)

Optimal. Leaf size=106 \[ \frac {\left (x^3+1\right )^{2/3} \left (7 x^3+2\right )}{10 x^5}-\frac {1}{6} \text {RootSum}\left [\text {$\#$1}^6-2 \text {$\#$1}^3-1\& ,\frac {-5 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+5 \text {$\#$1}^3 \log (x)-\log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+\log (x)}{\text {$\#$1}^4-\text {$\#$1}}\& \right ] \]

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Rubi [A]  time = 0.57, antiderivative size = 156, normalized size of antiderivative = 1.47, number of steps used = 9, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {6725, 264, 277, 239, 429} \begin {gather*} -\frac {1}{2} \left (4-\sqrt {2}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\sqrt {2} x^3\right )-\frac {1}{2} \left (4+\sqrt {2}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\sqrt {2} x^3\right )+\frac {1}{2} \log \left (\sqrt [3]{x^3+1}-x\right )-\frac {\tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {\left (x^3+1\right )^{5/3}}{5 x^5}+\frac {\left (x^3+1\right )^{2/3}}{2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((1 + x^3)^(2/3)*(1 + x^3 + 2*x^6))/(x^6*(-1 + 2*x^6)),x]

[Out]

(1 + x^3)^(2/3)/(2*x^2) + (1 + x^3)^(5/3)/(5*x^5) - ((4 - Sqrt[2])*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, -(Sqrt[
2]*x^3)])/2 - ((4 + Sqrt[2])*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, Sqrt[2]*x^3])/2 - ArcTan[(1 + (2*x)/(1 + x^3)
^(1/3))/Sqrt[3]]/Sqrt[3] + Log[-x + (1 + x^3)^(1/3)]/2

Rule 239

Int[((a_) + (b_.)*(x_)^3)^(-1/3), x_Symbol] :> Simp[ArcTan[(1 + (2*Rt[b, 3]*x)/(a + b*x^3)^(1/3))/Sqrt[3]]/(Sq
rt[3]*Rt[b, 3]), x] - Simp[Log[(a + b*x^3)^(1/3) - Rt[b, 3]*x]/(2*Rt[b, 3]), x] /; FreeQ[{a, b}, x]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rule 277

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^p)/(c*(m +
1)), x] - Dist[(b*n*p)/(c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b, c}, x] &&
IGtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1] &&  !ILtQ[(m + n*p + n + 1)/n, 0] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 429

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

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx &=\int \left (-\frac {\left (1+x^3\right )^{2/3}}{x^6}-\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {2 \left (1+x^3\right )^{2/3} \left (2+x^3\right )}{-1+2 x^6}\right ) \, dx\\ &=2 \int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{-1+2 x^6} \, dx-\int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx-\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}+2 \int \left (-\frac {\left (1+2 \sqrt {2}\right ) \left (1+x^3\right )^{2/3}}{2 \sqrt {2} \left (1-\sqrt {2} x^3\right )}+\frac {\left (1-2 \sqrt {2}\right ) \left (1+x^3\right )^{2/3}}{2 \sqrt {2} \left (1+\sqrt {2} x^3\right )}\right ) \, dx-\int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{2} \left (-4+\sqrt {2}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1+\sqrt {2} x^3} \, dx-\frac {1}{2} \left (4+\sqrt {2}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1-\sqrt {2} x^3} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (1+x^3\right )^{5/3}}{5 x^5}-\frac {1}{2} \left (4-\sqrt {2}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\sqrt {2} x^3\right )-\frac {1}{2} \left (4+\sqrt {2}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,\sqrt {2} x^3\right )-\frac {\tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {1}{2} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}

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Mathematica [F]  time = 0.61, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^3\right )^{2/3} \left (1+x^3+2 x^6\right )}{x^6 \left (-1+2 x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((1 + x^3)^(2/3)*(1 + x^3 + 2*x^6))/(x^6*(-1 + 2*x^6)),x]

[Out]

Integrate[((1 + x^3)^(2/3)*(1 + x^3 + 2*x^6))/(x^6*(-1 + 2*x^6)), x]

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IntegrateAlgebraic [A]  time = 0.22, size = 106, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3} \left (2+7 x^3\right )}{10 x^5}-\frac {1}{6} \text {RootSum}\left [-1-2 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {\log (x)-\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+5 \log (x) \text {$\#$1}^3-5 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-\text {$\#$1}+\text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^3)^(2/3)*(1 + x^3 + 2*x^6))/(x^6*(-1 + 2*x^6)),x]

[Out]

((1 + x^3)^(2/3)*(2 + 7*x^3))/(10*x^5) - RootSum[-1 - 2*#1^3 + #1^6 & , (Log[x] - Log[(1 + x^3)^(1/3) - x*#1]
+ 5*Log[x]*#1^3 - 5*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-#1 + #1^4) & ]/6

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(2*x^6+x^3+1)/x^6/(2*x^6-1),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (tr
ace 0)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(2*x^6+x^3+1)/x^6/(2*x^6-1),x, algorithm="giac")

[Out]

integrate((2*x^6 + x^3 + 1)*(x^3 + 1)^(2/3)/((2*x^6 - 1)*x^6), x)

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maple [B]  time = 144.05, size = 6315, normalized size = 59.58

method	result	size
risch	\(\text {Expression too large to display}\)	\(6315\)
trager	\(\text {Expression too large to display}\)	\(8835\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(2*x^6+x^3+1)/x^6/(2*x^6-1),x,method=_RETURNVERBOSE)

[Out]

result too large to display

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (2 \, x^{6} + x^{3} + 1\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{6} - 1\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(2*x^6+x^3+1)/x^6/(2*x^6-1),x, algorithm="maxima")

[Out]

integrate((2*x^6 + x^3 + 1)*(x^3 + 1)^(2/3)/((2*x^6 - 1)*x^6), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3+1\right )}^{2/3}\,\left (2\,x^6+x^3+1\right )}{x^6\,\left (2\,x^6-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 + 1)^(2/3)*(x^3 + 2*x^6 + 1))/(x^6*(2*x^6 - 1)),x)

[Out]

int(((x^3 + 1)^(2/3)*(x^3 + 2*x^6 + 1))/(x^6*(2*x^6 - 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (\left (x + 1\right ) \left (x^{2} - x + 1\right )\right )^{\frac {2}{3}} \left (2 x^{6} + x^{3} + 1\right )}{x^{6} \left (2 x^{6} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**3+1)**(2/3)*(2*x**6+x**3+1)/x**6/(2*x**6-1),x)

[Out]

Integral(((x + 1)*(x**2 - x + 1))**(2/3)*(2*x**6 + x**3 + 1)/(x**6*(2*x**6 - 1)), x)

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