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

Optimal. Leaf size=73 \[ \frac {\left (x^3+1\right )^{2/3}}{2 x^2}-\frac {2}{3} \text {RootSum}\left [2 \text {$\#$1}^6-3 \text {$\#$1}^3+2\& ,\frac {\log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-\log (x)}{4 \text {$\#$1}^4-3 \text {$\#$1}}\& \right ] \]

________________________________________________________________________________________

Rubi [C]  time = 0.47, antiderivative size = 187, normalized size of antiderivative = 2.56, number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6728, 277, 239, 429} \begin {gather*} \frac {\left (3 \sqrt {7}+7 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {2 x^3}{1-i \sqrt {7}}\right )}{7 \left (\sqrt {7}+i\right )}+\frac {\left (-3 \sqrt {7}+7 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {2 x^3}{1+i \sqrt {7}}\right )}{7 \left (-\sqrt {7}+i\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 )^{2/3}}{2 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(1 + x^3)^(2/3)/(2*x^2) + ((7*I + 3*Sqrt[7])*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, (-2*x^3)/(1 - I*Sqrt[7])])/(7
*(I + Sqrt[7])) + ((7*I - 3*Sqrt[7])*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, (-2*x^3)/(1 + I*Sqrt[7])])/(7*(I - Sq
rt[7])) - 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 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 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{2/3}}{x^3 \left (2+x^3+x^6\right )} \, dx &=\int \left (-\frac {\left (1+x^3\right )^{2/3}}{x^3}+\frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{2+x^3+x^6}\right ) \, dx\\ &=-\int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+\int \frac {\left (1+x^3\right )^{2/3} \left (2+x^3\right )}{2+x^3+x^6} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\int \frac {1}{\sqrt [3]{1+x^3}} \, dx+\int \left (\frac {\left (1-\frac {3 i}{\sqrt {7}}\right ) \left (1+x^3\right )^{2/3}}{1-i \sqrt {7}+2 x^3}+\frac {\left (1+\frac {3 i}{\sqrt {7}}\right ) \left (1+x^3\right )^{2/3}}{1+i \sqrt {7}+2 x^3}\right ) \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\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}{7} \left (7-3 i \sqrt {7}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1-i \sqrt {7}+2 x^3} \, dx+\frac {1}{7} \left (7+3 i \sqrt {7}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{1+i \sqrt {7}+2 x^3} \, dx\\ &=\frac {\left (1+x^3\right )^{2/3}}{2 x^2}+\frac {\left (7 i+3 \sqrt {7}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {2 x^3}{1-i \sqrt {7}}\right )}{7 \left (i+\sqrt {7}\right )}+\frac {\left (7 i-3 \sqrt {7}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {2 x^3}{1+i \sqrt {7}}\right )}{7 \left (i-\sqrt {7}\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*}

________________________________________________________________________________________

Mathematica [C]  time = 0.21, size = 381, normalized size = 5.22 \begin {gather*} \frac {\left (x^3+1\right )^{2/3}}{2 x^2}+\frac {i \left (-\frac {2 \log \left (\sqrt [3]{\sqrt {7}+i}-\frac {\sqrt [3]{\sqrt {7}-i} x}{\sqrt [3]{x^3+1}}\right )}{\sqrt [3]{\frac {\sqrt {7}-i}{\sqrt {7}+i}}}+2 \sqrt [3]{\frac {\sqrt {7}-i}{\sqrt {7}+i}} \log \left (\sqrt [3]{\sqrt {7}-i}-\frac {\sqrt [3]{\sqrt {7}+i} x}{\sqrt [3]{x^3+1}}\right )+\frac {2 \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\left (\sqrt {7}-i\right )^{2/3} x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )+\log \left (\frac {2 x}{\sqrt [3]{x^3+1}}+\frac {\left (\sqrt {7}-i\right )^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (\sqrt {7}+i\right )^{2/3}\right )\right )}{\left (\sqrt {7}-i\right )^{2/3}}-\frac {2 \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {\left (\sqrt {7}+i\right )^{2/3} x}{\sqrt [3]{x^3+1}}}{\sqrt {3}}\right )+\log \left (\frac {2 x}{\sqrt [3]{x^3+1}}+\frac {\left (\sqrt {7}+i\right )^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (\sqrt {7}-i\right )^{2/3}\right )\right )}{\left (\sqrt {7}+i\right )^{2/3}}\right )}{3 \sqrt {7}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(1 + x^3)^(2/3)/(2*x^2) + ((I/3)*((2*(2*Sqrt[3]*ArcTan[(1 + ((-I + Sqrt[7])^(2/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]]
+ Log[(I + Sqrt[7])^(2/3) + ((-I + Sqrt[7])^(2/3)*x^2)/(1 + x^3)^(2/3) + (2*x)/(1 + x^3)^(1/3)]))/(-I + Sqrt[
7])^(2/3) - (2*(2*Sqrt[3]*ArcTan[(1 + ((I + Sqrt[7])^(2/3)*x)/(1 + x^3)^(1/3))/Sqrt[3]] + Log[(-I + Sqrt[7])^(
2/3) + ((I + Sqrt[7])^(2/3)*x^2)/(1 + x^3)^(2/3) + (2*x)/(1 + x^3)^(1/3)]))/(I + Sqrt[7])^(2/3) - (2*Log[(I +
Sqrt[7])^(1/3) - ((-I + Sqrt[7])^(1/3)*x)/(1 + x^3)^(1/3)])/((-I + Sqrt[7])/(I + Sqrt[7]))^(1/3) + 2*((-I + Sq
rt[7])/(I + Sqrt[7]))^(1/3)*Log[(-I + Sqrt[7])^(1/3) - ((I + Sqrt[7])^(1/3)*x)/(1 + x^3)^(1/3)]))/Sqrt[7]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.00, size = 73, normalized size = 1.00 \begin {gather*} \frac {\left (1+x^3\right )^{2/3}}{2 x^2}-\frac {2}{3} \text {RootSum}\left [2-3 \text {$\#$1}^3+2 \text {$\#$1}^6\&,\frac {-\log (x)+\log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [B]  time = 161.74, size = 5531, normalized size = 75.77

method	result	size
risch	\(\text {Expression too large to display}\)	\(5531\)
trager	\(\text {Expression too large to display}\)	\(9663\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________