3.17.74 \(\int \frac {(1+x^3)^{2/3} (4+6 x^3+3 x^6)}{x^6 (8+6 x^3+3 x^6)} \, dx\)

Optimal. Leaf size=112 \[ \frac {1}{16} \text {RootSum}\left [8 \text {$\#$1}^6-10 \text {$\#$1}^3+5\& ,\frac {-6 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+6 \text {$\#$1}^3 \log (x)+5 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-5 \log (x)}{8 \text {$\#$1}^4-5 \text {$\#$1}}\& \right ]+\frac {\left (x^3+1\right )^{2/3} \left (-23 x^3-8\right )}{80 x^5} \]

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Rubi [C]  time = 0.66, antiderivative size = 203, normalized size of antiderivative = 1.81, number of steps used = 9, number of rules used = 5, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {6728, 264, 277, 239, 429} \begin {gather*} -\frac {3 \left (-\sqrt {15}+15 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {3 x^3}{3-i \sqrt {15}}\right )}{80 \left (\sqrt {15}+3 i\right )}-\frac {3 \left (\sqrt {15}+15 i\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {3 x^3}{3+i \sqrt {15}}\right )}{80 \left (-\sqrt {15}+3 i\right )}-\frac {3}{16} \log \left (\sqrt [3]{x^3+1}-x\right )+\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {\frac {2 x}{\sqrt [3]{x^3+1}}+1}{\sqrt {3}}\right )-\frac {\left (x^3+1\right )^{5/3}}{10 x^5}-\frac {3 \left (x^3+1\right )^{2/3}}{16 x^2} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((1 + x^3)^(2/3)*(4 + 6*x^3 + 3*x^6))/(x^6*(8 + 6*x^3 + 3*x^6)),x]

[Out]

(-3*(1 + x^3)^(2/3))/(16*x^2) - (1 + x^3)^(5/3)/(10*x^5) - (3*(15*I - Sqrt[15])*x*AppellF1[1/3, -2/3, 1, 4/3,
-x^3, (-3*x^3)/(3 - I*Sqrt[15])])/(80*(3*I + Sqrt[15])) - (3*(15*I + Sqrt[15])*x*AppellF1[1/3, -2/3, 1, 4/3, -
x^3, (-3*x^3)/(3 + I*Sqrt[15])])/(80*(3*I - Sqrt[15])) + (Sqrt[3]*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]])
/8 - (3*Log[-x + (1 + x^3)^(1/3)])/16

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 (4+6 x^3+3 x^6\right )}{x^6 \left (8+6 x^3+3 x^6\right )} \, dx &=\int \left (\frac {\left (1+x^3\right )^{2/3}}{2 x^6}+\frac {3 \left (1+x^3\right )^{2/3}}{8 x^3}-\frac {3 \left (1+x^3\right )^{2/3} \left (2+3 x^3\right )}{8 \left (8+6 x^3+3 x^6\right )}\right ) \, dx\\ &=\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, dx-\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3} \left (2+3 x^3\right )}{8+6 x^3+3 x^6} \, dx+\frac {1}{2} \int \frac {\left (1+x^3\right )^{2/3}}{x^6} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {3}{8} \int \frac {1}{\sqrt [3]{1+x^3}} \, dx-\frac {3}{8} \int \left (\frac {\left (3+i \sqrt {\frac {3}{5}}\right ) \left (1+x^3\right )^{2/3}}{6-2 i \sqrt {15}+6 x^3}+\frac {\left (3-i \sqrt {\frac {3}{5}}\right ) \left (1+x^3\right )^{2/3}}{6+2 i \sqrt {15}+6 x^3}\right ) \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}+\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{16} \log \left (-x+\sqrt [3]{1+x^3}\right )-\frac {1}{40} \left (3 \left (15-i \sqrt {15}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{6+2 i \sqrt {15}+6 x^3} \, dx-\frac {1}{40} \left (3 \left (15+i \sqrt {15}\right )\right ) \int \frac {\left (1+x^3\right )^{2/3}}{6-2 i \sqrt {15}+6 x^3} \, dx\\ &=-\frac {3 \left (1+x^3\right )^{2/3}}{16 x^2}-\frac {\left (1+x^3\right )^{5/3}}{10 x^5}-\frac {3 \left (15 i-\sqrt {15}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {3 x^3}{3-i \sqrt {15}}\right )}{80 \left (3 i+\sqrt {15}\right )}-\frac {3 \left (15 i+\sqrt {15}\right ) x F_1\left (\frac {1}{3};-\frac {2}{3},1;\frac {4}{3};-x^3,-\frac {3 x^3}{3+i \sqrt {15}}\right )}{80 \left (3 i-\sqrt {15}\right )}+\frac {1}{8} \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+x^3}}}{\sqrt {3}}\right )-\frac {3}{16} \log \left (-x+\sqrt [3]{1+x^3}\right )\\ \end {align*}

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Mathematica [C]  time = 2.41, size = 446, normalized size = 3.98 \begin {gather*} \left (x^3+1\right )^{2/3} \left (-\frac {1}{10 x^5}-\frac {23}{80 x^2}\right )+\frac {-\frac {6 \sqrt [3]{5-i \sqrt {15}} \left (\sqrt {15}-i\right ) \log \left (-\frac {\sqrt [3]{5} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{5-i \sqrt {15}}\right )}{\sqrt {15}-3 i}-\frac {6 \sqrt [3]{5+i \sqrt {15}} \left (\sqrt {15}+i\right ) \log \left (-\frac {\sqrt [3]{5} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{5+i \sqrt {15}}\right )}{\sqrt {15}+3 i}+\frac {\left (15-i \sqrt {15}\right ) \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1-i \sqrt {\frac {3}{5}}} \sqrt [3]{x^3+1}}}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{25-5 i \sqrt {15}} x}{\sqrt [3]{x^3+1}}+\frac {5^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (5-i \sqrt {15}\right )^{2/3}\right )\right )}{\left (5-i \sqrt {15}\right )^{2/3}}+\frac {\left (15+i \sqrt {15}\right ) \left (2 \sqrt {3} \tan ^{-1}\left (\frac {1+\frac {2 x}{\sqrt [3]{1+i \sqrt {\frac {3}{5}}} \sqrt [3]{x^3+1}}}{\sqrt {3}}\right )+\log \left (\frac {\sqrt [3]{25+5 i \sqrt {15}} x}{\sqrt [3]{x^3+1}}+\frac {5^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+\left (5+i \sqrt {15}\right )^{2/3}\right )\right )}{\left (5+i \sqrt {15}\right )^{2/3}}}{96 \sqrt [3]{5}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 + x^3)^(2/3)*(4 + 6*x^3 + 3*x^6))/(x^6*(8 + 6*x^3 + 3*x^6)),x]

[Out]

(-1/10*1/x^5 - 23/(80*x^2))*(1 + x^3)^(2/3) + ((-6*(5 - I*Sqrt[15])^(1/3)*(-I + Sqrt[15])*Log[(5 - I*Sqrt[15])
^(1/3) - (5^(1/3)*x)/(1 + x^3)^(1/3)])/(-3*I + Sqrt[15]) - (6*(5 + I*Sqrt[15])^(1/3)*(I + Sqrt[15])*Log[(5 + I
*Sqrt[15])^(1/3) - (5^(1/3)*x)/(1 + x^3)^(1/3)])/(3*I + Sqrt[15]) + ((15 - I*Sqrt[15])*(2*Sqrt[3]*ArcTan[(1 +
(2*x)/((1 - I*Sqrt[3/5])^(1/3)*(1 + x^3)^(1/3)))/Sqrt[3]] + Log[(5 - I*Sqrt[15])^(2/3) + (5^(2/3)*x^2)/(1 + x^
3)^(2/3) + ((25 - (5*I)*Sqrt[15])^(1/3)*x)/(1 + x^3)^(1/3)]))/(5 - I*Sqrt[15])^(2/3) + ((15 + I*Sqrt[15])*(2*S
qrt[3]*ArcTan[(1 + (2*x)/((1 + I*Sqrt[3/5])^(1/3)*(1 + x^3)^(1/3)))/Sqrt[3]] + Log[(5 + I*Sqrt[15])^(2/3) + (5
^(2/3)*x^2)/(1 + x^3)^(2/3) + ((25 + (5*I)*Sqrt[15])^(1/3)*x)/(1 + x^3)^(1/3)]))/(5 + I*Sqrt[15])^(2/3))/(96*5
^(1/3))

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IntegrateAlgebraic [A]  time = 0.23, size = 112, normalized size = 1.00 \begin {gather*} \frac {\left (-8-23 x^3\right ) \left (1+x^3\right )^{2/3}}{80 x^5}+\frac {1}{16} \text {RootSum}\left [5-10 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+6 \log (x) \text {$\#$1}^3-6 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+8 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((1 + x^3)^(2/3)*(4 + 6*x^3 + 3*x^6))/(x^6*(8 + 6*x^3 + 3*x^6)),x]

[Out]

((-8 - 23*x^3)*(1 + x^3)^(2/3))/(80*x^5) + RootSum[5 - 10*#1^3 + 8*#1^6 & , (-5*Log[x] + 5*Log[(1 + x^3)^(1/3)
- x*#1] + 6*Log[x]*#1^3 - 6*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 8*#1^4) & ]/16

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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)*(3*x^6+6*x^3+4)/x^6/(3*x^6+6*x^3+8),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^{6} + 6 \, x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, x^{6} + 6 \, x^{3} + 8\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(3*x^6+6*x^3+4)/x^6/(3*x^6+6*x^3+8),x, algorithm="giac")

[Out]

integrate((3*x^6 + 6*x^3 + 4)*(x^3 + 1)^(2/3)/((3*x^6 + 6*x^3 + 8)*x^6), x)

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maple [B]  time = 222.17, size = 6679, normalized size = 59.63

method	result	size
risch	\(\text {Expression too large to display}\)	\(6679\)
trager	\(\text {Expression too large to display}\)	\(9175\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(3*x^6+6*x^3+4)/x^6/(3*x^6+6*x^3+8),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 (3 \, x^{6} + 6 \, x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (3 \, x^{6} + 6 \, x^{3} + 8\right )} x^{6}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^3+1)^(2/3)*(3*x^6+6*x^3+4)/x^6/(3*x^6+6*x^3+8),x, algorithm="maxima")

[Out]

integrate((3*x^6 + 6*x^3 + 4)*(x^3 + 1)^(2/3)/((3*x^6 + 6*x^3 + 8)*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 (3\,x^6+6\,x^3+4\right )}{x^6\,\left (3\,x^6+6\,x^3+8\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^3 + 1)^(2/3)*(6*x^3 + 3*x^6 + 4))/(x^6*(6*x^3 + 3*x^6 + 8)),x)

[Out]

int(((x^3 + 1)^(2/3)*(6*x^3 + 3*x^6 + 4))/(x^6*(6*x^3 + 3*x^6 + 8)), 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)*(3*x**6+6*x**3+4)/x**6/(3*x**6+6*x**3+8),x)

[Out]

Timed out

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