3.17.68 \(\int \frac {(1+x^3)^{2/3} (2+x^3) (4+3 x^3)}{x^6 (4+2 x^3+x^6)} \, dx\)

Optimal. Leaf size=112 \[ \frac {1}{12} \text {RootSum}\left [4 \text {$\#$1}^6-6 \text {$\#$1}^3+3\& ,\frac {-8 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+8 \text {$\#$1}^3 \log (x)+9 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-9 \log (x)}{4 \text {$\#$1}^4-3 \text {$\#$1}}\& \right ]+\frac {\left (x^3+1\right )^{2/3} \left (-23 x^3-8\right )}{20 x^5} \]

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Rubi [C]  time = 0.85, antiderivative size = 203, normalized size of antiderivative = 1.81, number of steps used = 9, number of rules used = 5, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.135, Rules used = {6728, 264, 277, 239, 429} \begin {gather*} -\frac {\left (\sqrt {3}+9 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{1-i \sqrt {3}}\right )}{12 \left (\sqrt {3}+i\right )}-\frac {\left (-\sqrt {3}+9 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{1+i \sqrt {3}}\right )}{12 \left (-\sqrt {3}+i\right )}-\frac {3}{4} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {2 \left (x^3+1\right )^{5/3}}{5 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{4 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((1 + x^3)^(2/3)*(2 + x^3)*(4 + 3*x^3))/(x^6*(4 + 2*x^3 + x^6)),x]

[Out]

(-3*(1 + x^3)^(2/3))/(4*x^2) - (2*(1 + x^3)^(5/3))/(5*x^5) - ((9*I + Sqrt[3])*x*AppellF1[1/3, -2/3, 1, 4/3, -x
^3, -(x^3/(1 - I*Sqrt[3]))])/(12*(I + Sqrt[3])) - ((9*I - Sqrt[3])*x*AppellF1[1/3, -2/3, 1, 4/3, -x^3, -(x^3/(
1 + I*Sqrt[3]))])/(12*(I - Sqrt[3])) + (Sqrt[3]*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])/2 - (3*Log[-x + (
1 + x^3)^(1/3)])/4

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 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 (1+x^3\right )^{2/3} \left (2+x^3\right ) \left (4+3 x^3\right )}{x^6 \left (4+2 x^3+x^6\right )} \, dx &=\int \left (\frac {2 \left (1+x^3\right )^{2/3}}{x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{2 x^3}+\frac {\left (-4-3 x^3\right ) \left (1+x^3\right )^{2/3}}{2 \left (4+2 x^3+x^6\right )}\right ) \, dx\\ &=\frac {1}{2} \int \frac {\left (-4-3 x^3\right ) \left (1+x^3\right )^{2/3}}{4+2 x^3+x^6} \, dx+\frac {3}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx+2 \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{2} \int \left (\frac {\left (-3+\frac {i}{\sqrt {3}}\right ) \left (1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3}+\frac {\left (-3-\frac {i}{\sqrt {3}}\right ) \left (1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3}\right ) \, dx+\frac {3}{2} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{4} \log \left (-x+\sqrt [3]{1+x^3}\right )+\frac {1}{6} \left (-9+i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{2-2 i \sqrt {3}+2 x^3} \, dx-\frac {1}{6} \left (9+i \sqrt {3}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{2+2 i \sqrt {3}+2 x^3} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{4 x^2}-\frac {2 \left (1+x^3\right )^{5/3}}{5 x^5}-\frac {\left (9 i+\sqrt {3}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{1-i \sqrt {3}}\right )}{12 \left (i+\sqrt {3}\right )}-\frac {\left (9 i-\sqrt {3}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {x^3}{1+i \sqrt {3}}\right )}{12 \left (i-\sqrt {3}\right )}+\frac {1}{2} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{4} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 2.55, size = 464, normalized size = 4.14 \begin {gather*} \left (x^3+1\right )^{2/3} \left (-\frac {2}{5 x^5}-\frac {23}{20 x^2}\right )+\frac {1}{72} \left (-\frac {2\ 3^{2/3} \sqrt [3]{3-i \sqrt {3}} \left (3 \sqrt {3}+i\right ) \log \left (-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{3-i \sqrt {3}}\right )}{\sqrt {3}-i}+\frac {2\ 3^{2/3} \left (-3 \sqrt {3}+i\right ) \sqrt [3]{3+i \sqrt {3}} \log \left (-\frac {\sqrt [3]{3} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{3+i \sqrt {3}}\right )}{\sqrt {3}+i}+\frac {\left (3 \sqrt {3}+i\right ) \left (6 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 x}{\sqrt [6]{3} \sqrt [3]{3-i \sqrt {3}} \sqrt [3]{x^3+1}}\right )+\sqrt {3} \log \left (\frac {\sqrt [3]{9-3 i \sqrt {3}} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (3-i \sqrt {3}\right )^{2/3}\right )\right )}{\left (1-\frac {i}{\sqrt {3}}\right )^{2/3}}+\frac {\sqrt [6]{3} \left (9-i \sqrt {3}\right ) \left (6 \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2 x}{\sqrt [6]{3} \sqrt [3]{3+i \sqrt {3}} \sqrt [3]{x^3+1}}\right )+\sqrt {3} \log \left (\frac {\sqrt [3]{9+3 i \sqrt {3}} x}{\sqrt [3]{x^3+1}}+\frac {3^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (3+i \sqrt {3}\right )^{2/3}\right )\right )}{\left (3+i \sqrt {3}\right )^{2/3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)^(2/3)*(2 + x^3)*(4 + 3*x^3))/(x^6*(4 + 2*x^3 + x^6)),x]

[Out]

(-2/(5*x^5) - 23/(20*x^2))*(1 + x^3)^(2/3) + ((-2*3^(2/3)*(3 - I*Sqrt[3])^(1/3)*(I + 3*Sqrt[3])*Log[(3 - I*Sqr
t[3])^(1/3) - (3^(1/3)*x)/(1 + x^3)^(1/3)])/(-I + Sqrt[3]) + (2*3^(2/3)*(I - 3*Sqrt[3])*(3 + I*Sqrt[3])^(1/3)*
Log[(3 + I*Sqrt[3])^(1/3) - (3^(1/3)*x)/(1 + x^3)^(1/3)])/(I + Sqrt[3]) + ((I + 3*Sqrt[3])*(6*ArcTan[1/Sqrt[3]
+ (2*x)/(3^(1/6)*(3 - I*Sqrt[3])^(1/3)*(1 + x^3)^(1/3))] + Sqrt[3]*Log[(3 - I*Sqrt[3])^(2/3) + (3^(2/3)*x^2)/
(1 + x^3)^(2/3) + ((9 - (3*I)*Sqrt[3])^(1/3)*x)/(1 + x^3)^(1/3)]))/(1 - I/Sqrt[3])^(2/3) + (3^(1/6)*(9 - I*Sqr
t[3])*(6*ArcTan[1/Sqrt[3] + (2*x)/(3^(1/6)*(3 + I*Sqrt[3])^(1/3)*(1 + x^3)^(1/3))] + Sqrt[3]*Log[(3 + I*Sqrt[3
])^(2/3) + (3^(2/3)*x^2)/(1 + x^3)^(2/3) + ((9 + (3*I)*Sqrt[3])^(1/3)*x)/(1 + x^3)^(1/3)]))/(3 + I*Sqrt[3])^(2
/3))/72

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IntegrateAlgebraic [A]  time = 0.24, size = 112, normalized size = 1.00 \begin {gather*} \frac {\left (-8-23 x^3\right ) \left (1+x^3\right )^{2/3}}{20 x^5}+\frac {1}{12} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-9 \log (x)+9 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+8 \log (x) \text {$\#$1}^3-8 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^3)^(2/3)*(2 + x^3)*(4 + 3*x^3))/(x^6*(4 + 2*x^3 + x^6)),x]

[Out]

((-8 - 23*x^3)*(1 + x^3)^(2/3))/(20*x^5) + RootSum[3 - 6*#1^3 + 4*#1^6 & , (-9*Log[x] + 9*Log[(1 + x^3)^(1/3)
- x*#1] + 8*Log[x]*#1^3 - 8*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1 + 4*#1^4) & ]/12

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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)*(x^3+2)*(3*x^3+4)/x^6/(x^6+2*x^3+4),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 (3 \, x^{3} + 4\right )} {\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^3+2)*(3*x^3+4)/x^6/(x^6+2*x^3+4),x, algorithm="giac")

[Out]

integrate((3*x^3 + 4)*(x^3 + 2)*(x^3 + 1)^(2/3)/((x^6 + 2*x^3 + 4)*x^6), x)

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{3}+1\right )^{\frac {2}{3}} \left (x^{3}+2\right ) \left (3 x^{3}+4\right )}{x^{6} \left (x^{6}+2 x^{3}+4\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(x^3+2)*(3*x^3+4)/x^6/(x^6+2*x^3+4),x)

[Out]

int((x^3+1)^(2/3)*(x^3+2)*(3*x^3+4)/x^6/(x^6+2*x^3+4),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (3 \, x^{3} + 4\right )} {\left (x^{3} + 2\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} + 2 \, x^{3} + 4\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(x^3+2)*(3*x^3+4)/x^6/(x^6+2*x^3+4),x, algorithm="maxima")

[Out]

integrate((3*x^3 + 4)*(x^3 + 2)*(x^3 + 1)^(2/3)/((x^6 + 2*x^3 + 4)*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 (x^3+2\right )\,\left (3\,x^3+4\right )}{x^6\,\left (x^6+2\,x^3+4\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)*(3*x^3 + 4))/(x^6*(2*x^3 + x^6 + 4)),x)

[Out]

int(((x^3 + 1)^(2/3)*(x^3 + 2)*(3*x^3 + 4))/(x^6*(2*x^3 + x^6 + 4)), x)

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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+1)**(2/3)*(x**3+2)*(3*x**3+4)/x**6/(x**6+2*x**3+4),x)

[Out]

Timed out

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