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

Optimal. Leaf size=112 \[ \frac {1}{96} \text {RootSum}\left [8 \text {$\#$1}^6-20 \text {$\#$1}^3+13\& ,\frac {-32 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )+32 \text {$\#$1}^3 \log (x)+39 \log \left (\sqrt [3]{x^3+1}-\text {$\#$1} x\right )-39 \log (x)}{4 \text {$\#$1}^4-5 \text {$\#$1}}\& \right ]+\frac {\left (x^3+1\right )^{2/3} \left (-23 x^3-8\right )}{80 x^5} \]

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Rubi [F]  time = 1.30, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (8-4 x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-3*(1 + x^3)^(2/3))/(16*x^2) - (1 + x^3)^(5/3)/(10*x^5) + (Sqrt[3]*ArcTan[(1 + (2*x)/(1 + x^3)^(1/3))/Sqrt[3]
])/8 - (3*Log[-x + (1 + x^3)^(1/3)])/16 + (1/48 - I/16)*Defer[Int][(1 + x^3)^(2/3)/((2 - 2*I) + 2*x), x] + (1/
48 + I/16)*Defer[Int][(1 + x^3)^(2/3)/((2 + 2*I) + 2*x), x] + Defer[Int][(1 + x^3)^(2/3)/(4 - 4*x + 2*x^2 - 2*
x^3 + x^4), x]/3 - (5*Defer[Int][(x*(1 + x^3)^(2/3))/(4 - 4*x + 2*x^2 - 2*x^3 + x^4), x])/24 - Defer[Int][(x^3
*(1 + x^3)^(2/3))/(4 - 4*x + 2*x^2 - 2*x^3 + x^4), x]/48

Rubi steps

\begin {align*} \int \frac {\left (1+x^3\right )^{2/3} \left (4+x^3\right )}{x^6 \left (8-4 x^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 {(4+x) \left (1+x^3\right )^{2/3}}{48 \left (2+2 x+x^2\right )}+\frac {\left (16-10 x-x^3\right ) \left (1+x^3\right )^{2/3}}{48 \left (4-4 x+2 x^2-2 x^3+x^4\right )}\right ) \, dx\\ &=\frac {1}{48} \int \frac {(4+x) \left (1+x^3\right )^{2/3}}{2+2 x+x^2} \, dx+\frac {1}{48} \int \frac {\left (16-10 x-x^3\right ) \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4} \, dx+\frac {3}{8} \int \frac {\left (1+x^3\right )^{2/3}}{x^3} \, 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 {1}{48} \int \left (\frac {(1-3 i) \left (1+x^3\right )^{2/3}}{(2-2 i)+2 x}+\frac {(1+3 i) \left (1+x^3\right )^{2/3}}{(2+2 i)+2 x}\right ) \, dx+\frac {1}{48} \int \left (\frac {16 \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4}-\frac {10 x \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4}-\frac {x^3 \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4}\right ) \, dx+\frac {3}{8} \int \frac {1}{\sqrt [3]{1+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 {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}{48} \int \frac {x^3 \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4} \, dx+\left (\frac {1}{48}-\frac {i}{16}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{(2-2 i)+2 x} \, dx+\left (\frac {1}{48}+\frac {i}{16}\right ) \int \frac {\left (1+x^3\right )^{2/3}}{(2+2 i)+2 x} \, dx-\frac {5}{24} \int \frac {x \left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4} \, dx+\frac {1}{3} \int \frac {\left (1+x^3\right )^{2/3}}{4-4 x+2 x^2-2 x^3+x^4} \, dx\\ \end {align*}

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Mathematica [C]  time = 0.68, size = 337, normalized size = 3.01 \begin {gather*} \left (x^3+1\right )^{2/3} \left (-\frac {1}{10 x^5}-\frac {23}{80 x^2}\right )+\frac {-(16-2 i) \sqrt [3]{5-i} \log \left (-\frac {\sqrt [3]{13} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{10-2 i}\right )-(16+2 i) \sqrt [3]{5+i} \log \left (-\frac {\sqrt [3]{13} x}{\sqrt [3]{x^3+1}}+\sqrt [3]{10+2 i}\right )+(16-2 i) \sqrt {3} \sqrt [3]{5-i} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} x}{\sqrt [3]{\frac {5}{13}-\frac {i}{13}} \sqrt {3} \sqrt [3]{x^3+1}}\right )+(16+2 i) \sqrt {3} \sqrt [3]{5+i} \tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} x}{\sqrt [3]{\frac {5}{13}+\frac {i}{13}} \sqrt {3} \sqrt [3]{x^3+1}}\right )+(8-i) \sqrt [3]{5-i} \log \left (\frac {\sqrt [3]{130-26 i} x}{\sqrt [3]{x^3+1}}+\frac {13^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+(10-2 i)^{2/3}\right )+(8+i) \sqrt [3]{5+i} \log \left (\frac {\sqrt [3]{130+26 i} x}{\sqrt [3]{x^3+1}}+\frac {13^{2/3} x^2}{\left (x^3+1\right )^{2/3}}+(10+2 i)^{2/3}\right )}{96\ 2^{2/3} \sqrt [3]{13}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1/10*1/x^5 - 23/(80*x^2))*(1 + x^3)^(2/3) + ((16 - 2*I)*Sqrt[3]*(5 - I)^(1/3)*ArcTan[1/Sqrt[3] + (2^(2/3)*x)
/((5/13 - I/13)^(1/3)*Sqrt[3]*(1 + x^3)^(1/3))] + (16 + 2*I)*Sqrt[3]*(5 + I)^(1/3)*ArcTan[1/Sqrt[3] + (2^(2/3)
*x)/((5/13 + I/13)^(1/3)*Sqrt[3]*(1 + x^3)^(1/3))] - (16 - 2*I)*(5 - I)^(1/3)*Log[(10 - 2*I)^(1/3) - (13^(1/3)
*x)/(1 + x^3)^(1/3)] - (16 + 2*I)*(5 + I)^(1/3)*Log[(10 + 2*I)^(1/3) - (13^(1/3)*x)/(1 + x^3)^(1/3)] + (8 - I)
*(5 - I)^(1/3)*Log[(10 - 2*I)^(2/3) + (13^(2/3)*x^2)/(1 + x^3)^(2/3) + ((130 - 26*I)^(1/3)*x)/(1 + x^3)^(1/3)]
+ (8 + I)*(5 + I)^(1/3)*Log[(10 + 2*I)^(2/3) + (13^(2/3)*x^2)/(1 + x^3)^(2/3) + ((130 + 26*I)^(1/3)*x)/(1 + x
^3)^(1/3)])/(96*2^(2/3)*13^(1/3))

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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}}{80 x^5}+\frac {1}{96} \text {RootSum}\left [13-20 \text {$\#$1}^3+8 \text {$\#$1}^6\&,\frac {-39 \log (x)+39 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right )+32 \log (x) \text {$\#$1}^3-32 \log \left (\sqrt [3]{1+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-5 \text {$\#$1}+4 \text {$\#$1}^4}\&\right ] \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-8 - 23*x^3)*(1 + x^3)^(2/3))/(80*x^5) + RootSum[13 - 20*#1^3 + 8*#1^6 & , (-39*Log[x] + 39*Log[(1 + x^3)^(1
/3) - x*#1] + 32*Log[x]*#1^3 - 32*Log[(1 + x^3)^(1/3) - x*#1]*#1^3)/(-5*#1 + 4*#1^4) & ]/96

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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+4)/x^6/(x^6-4*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 (x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 4 \, 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)*(x^3+4)/x^6/(x^6-4*x^3+8),x, algorithm="giac")

[Out]

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

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maple [B]  time = 240.52, size = 9666, normalized size = 86.30

method	result	size
risch	\(\text {Expression too large to display}\)	\(9666\)
trager	\(\text {Expression too large to display}\)	\(12545\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^3+1)^(2/3)*(x^3+4)/x^6/(x^6-4*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 (x^{3} + 4\right )} {\left (x^{3} + 1\right )}^{\frac {2}{3}}}{{\left (x^{6} - 4 \, 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)*(x^3+4)/x^6/(x^6-4*x^3+8),x, algorithm="maxima")

[Out]

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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