3.11.58 \(\int \frac {\sqrt [3]{-1+x^8} (3+5 x^8)}{x^2 (-1-x^3+x^8)} \, dx\)

Optimal. Leaf size=87 \[ \frac {3 \sqrt [3]{x^8-1}}{x}+\log \left (\sqrt [3]{x^8-1}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^8-1}+x}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^8-1} x+\left (x^8-1\right )^{2/3}+x^2\right ) \]

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Rubi [F]  time = 0.79, 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 {\sqrt [3]{-1+x^8} \left (3+5 x^8\right )}{x^2 \left (-1-x^3+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((-1 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 - x^3 + x^8)),x]

[Out]

(3*(-1 + x^8)^(1/3)*Hypergeometric2F1[-1/3, -1/8, 7/8, x^8])/(x*(1 - x^8)^(1/3)) - 3*Defer[Int][(x*(-1 + x^8)^
(1/3))/(-1 - x^3 + x^8), x] + 8*Defer[Int][(x^6*(-1 + x^8)^(1/3))/(-1 - x^3 + x^8), x]

Rubi steps

\begin {align*} \int \frac {\sqrt [3]{-1+x^8} \left (3+5 x^8\right )}{x^2 \left (-1-x^3+x^8\right )} \, dx &=\int \left (-\frac {3 \sqrt [3]{-1+x^8}}{x^2}+\frac {x \left (3-8 x^5\right ) \sqrt [3]{-1+x^8}}{1+x^3-x^8}\right ) \, dx\\ &=-\left (3 \int \frac {\sqrt [3]{-1+x^8}}{x^2} \, dx\right )+\int \frac {x \left (3-8 x^5\right ) \sqrt [3]{-1+x^8}}{1+x^3-x^8} \, dx\\ &=-\frac {\left (3 \sqrt [3]{-1+x^8}\right ) \int \frac {\sqrt [3]{1-x^8}}{x^2} \, dx}{\sqrt [3]{1-x^8}}+\int \left (-\frac {3 x \sqrt [3]{-1+x^8}}{-1-x^3+x^8}+\frac {8 x^6 \sqrt [3]{-1+x^8}}{-1-x^3+x^8}\right ) \, dx\\ &=\frac {3 \sqrt [3]{-1+x^8} \, _2F_1\left (-\frac {1}{3},-\frac {1}{8};\frac {7}{8};x^8\right )}{x \sqrt [3]{1-x^8}}-3 \int \frac {x \sqrt [3]{-1+x^8}}{-1-x^3+x^8} \, dx+8 \int \frac {x^6 \sqrt [3]{-1+x^8}}{-1-x^3+x^8} \, dx\\ \end {align*}

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Mathematica [F]  time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{-1+x^8} \left (3+5 x^8\right )}{x^2 \left (-1-x^3+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-1 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 - x^3 + x^8)),x]

[Out]

Integrate[((-1 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 - x^3 + x^8)), x]

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IntegrateAlgebraic [A]  time = 20.31, size = 87, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{x^8-1}}{x}+\log \left (\sqrt [3]{x^8-1}-x\right )+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{2 \sqrt [3]{x^8-1}+x}\right )-\frac {1}{2} \log \left (\sqrt [3]{x^8-1} x+\left (x^8-1\right )^{2/3}+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-1 + x^8)^(1/3)*(3 + 5*x^8))/(x^2*(-1 - x^3 + x^8)),x]

[Out]

(3*(-1 + x^8)^(1/3))/x + Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(-1 + x^8)^(1/3))] + Log[-x + (-1 + x^8)^(1/3)] - L
og[x^2 + x*(-1 + x^8)^(1/3) + (-1 + x^8)^(2/3)]/2

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fricas [A]  time = 38.50, size = 131, normalized size = 1.51 \begin {gather*} \frac {2 \, \sqrt {3} x \arctan \left (-\frac {31069389038531798383012393094747362616575064091434751962020601837507558239516138425325377239789317495328857903057957141206059288722620160721093489516063746612973182 \, \sqrt {3} {\left (x^{8} - 1\right )}^{\frac {1}{3}} x^{2} - 24620142163963087452447726858369178030030967023250856622849105390649652817268567947362178503080085821866784600572345611200568455939022999883192079164797236311980480 \, \sqrt {3} {\left (x^{8} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (14098730908269987597917744450355902431760205999000820135495290627669890741173905802396636062023876418322337000958016148565005886294703209808664629857632230121011200 \, x^{8} - 10874107470985632132635411332166810138488157464908872465909542404240938030050120563415036693669260581591300349715210383562260469902904629389713924681998974970514849 \, x^{3} - 14098730908269987597917744450355902431760205999000820135495290627669890741173905802396636062023876418322337000958016148565005886294703209808664629857632230121011200\right )}}{3 \, {\left (9251742523290005295394971478800280999715753799405283223501747806428870154589708393514732281743754536574942347080177746431157381208775803010963333365470079627264000 \, x^{8} + 18593023077957437622335088497757989323587261757937521068933105807649735373802644792829045589690947122022878904734973629772156491122045777291179450974960411835212831 \, x^{3} - 9251742523290005295394971478800280999715753799405283223501747806428870154589708393514732281743754536574942347080177746431157381208775803010963333365470079627264000\right )}}\right ) + x \log \left (\frac {x^{8} - x^{3} + 3 \, {\left (x^{8} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{8} - 1\right )}^{\frac {2}{3}} x - 1}{x^{8} - x^{3} - 1}\right ) + 6 \, {\left (x^{8} - 1\right )}^{\frac {1}{3}}}{2 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8-1)^(1/3)*(5*x^8+3)/x^2/(x^8-x^3-1),x, algorithm="fricas")

[Out]

1/2*(2*sqrt(3)*x*arctan(-1/3*(31069389038531798383012393094747362616575064091434751962020601837507558239516138
425325377239789317495328857903057957141206059288722620160721093489516063746612973182*sqrt(3)*(x^8 - 1)^(1/3)*x
^2 - 246201421639630874524477268583691780300309670232508566228491053906496528172685679473621785030800858218667
84600572345611200568455939022999883192079164797236311980480*sqrt(3)*(x^8 - 1)^(2/3)*x + sqrt(3)*(1409873090826
99875979177444503559024317602059990008201354952906276698907411739058023966360620238764183223370009580161485650
05886294703209808664629857632230121011200*x^8 - 10874107470985632132635411332166810138488157464908872465909542
404240938030050120563415036693669260581591300349715210383562260469902904629389713924681998974970514849*x^3 - 1
40987309082699875979177444503559024317602059990008201354952906276698907411739058023966360620238764183223370009
58016148565005886294703209808664629857632230121011200))/(92517425232900052953949714788002809997157537994052832
23501747806428870154589708393514732281743754536574942347080177746431157381208775803010963333365470079627264000
*x^8 + 1859302307795743762233508849775798932358726175793752106893310580764973537380264479282904558969094712202
2878904734973629772156491122045777291179450974960411835212831*x^3 - 925174252329000529539497147880028099971575
37994052832235017478064288701545897083935147322817437545365749423470801777464311573812087758030109633333654700
79627264000)) + x*log((x^8 - x^3 + 3*(x^8 - 1)^(1/3)*x^2 - 3*(x^8 - 1)^(2/3)*x - 1)/(x^8 - x^3 - 1)) + 6*(x^8
- 1)^(1/3))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{8} + 3\right )} {\left (x^{8} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} - x^{3} - 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8-1)^(1/3)*(5*x^8+3)/x^2/(x^8-x^3-1),x, algorithm="giac")

[Out]

integrate((5*x^8 + 3)*(x^8 - 1)^(1/3)/((x^8 - x^3 - 1)*x^2), x)

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maple [C]  time = 6.78, size = 725, normalized size = 8.33

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^8-1)^(1/3)*(5*x^8+3)/x^2/(x^8-x^3-1),x)

[Out]

3*(x^8-1)^(1/3)/x+(RootOf(_Z^2+_Z+1)*ln(-(2*x^16*RootOf(_Z^2+_Z+1)+x^16-2*RootOf(_Z^2+_Z+1)^2*x^11-x^11*RootOf
(_Z^2+_Z+1)+3*(x^16-2*x^8+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^9+3*(x^16-2*x^8+1)^(1/3)*x^9-4*RootOf(_Z^2+_Z+1)*x^8-2*
x^8+3*(x^16-2*x^8+1)^(2/3)*RootOf(_Z^2+_Z+1)*x^2+2*RootOf(_Z^2+_Z+1)^2*x^3+3*(x^16-2*x^8+1)^(2/3)*x^2+RootOf(_
Z^2+_Z+1)*x^3-3*(x^16-2*x^8+1)^(1/3)*RootOf(_Z^2+_Z+1)*x-3*(x^16-2*x^8+1)^(1/3)*x+2*RootOf(_Z^2+_Z+1)+1)/(-1+x
)/(1+x)/(x^2+1)/(x^4+1)/(x^8-x^3-1))-ln((2*x^16*RootOf(_Z^2+_Z+1)+x^16+2*RootOf(_Z^2+_Z+1)^2*x^11+3*x^11*RootO
f(_Z^2+_Z+1)+3*(x^16-2*x^8+1)^(1/3)*RootOf(_Z^2+_Z+1)*x^9+x^11-4*RootOf(_Z^2+_Z+1)*x^8-2*x^8+3*(x^16-2*x^8+1)^
(2/3)*RootOf(_Z^2+_Z+1)*x^2-2*RootOf(_Z^2+_Z+1)^2*x^3-3*RootOf(_Z^2+_Z+1)*x^3-3*(x^16-2*x^8+1)^(1/3)*RootOf(_Z
^2+_Z+1)*x-x^3+2*RootOf(_Z^2+_Z+1)+1)/(-1+x)/(1+x)/(x^2+1)/(x^4+1)/(x^8-x^3-1))*RootOf(_Z^2+_Z+1)-ln((2*x^16*R
ootOf(_Z^2+_Z+1)+x^16+2*RootOf(_Z^2+_Z+1)^2*x^11+3*x^11*RootOf(_Z^2+_Z+1)+3*(x^16-2*x^8+1)^(1/3)*RootOf(_Z^2+_
Z+1)*x^9+x^11-4*RootOf(_Z^2+_Z+1)*x^8-2*x^8+3*(x^16-2*x^8+1)^(2/3)*RootOf(_Z^2+_Z+1)*x^2-2*RootOf(_Z^2+_Z+1)^2
*x^3-3*RootOf(_Z^2+_Z+1)*x^3-3*(x^16-2*x^8+1)^(1/3)*RootOf(_Z^2+_Z+1)*x-x^3+2*RootOf(_Z^2+_Z+1)+1)/(-1+x)/(1+x
)/(x^2+1)/(x^4+1)/(x^8-x^3-1)))/(x^8-1)^(2/3)*((x^8-1)^2)^(1/3)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (5 \, x^{8} + 3\right )} {\left (x^{8} - 1\right )}^{\frac {1}{3}}}{{\left (x^{8} - x^{3} - 1\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^8-1)^(1/3)*(5*x^8+3)/x^2/(x^8-x^3-1),x, algorithm="maxima")

[Out]

integrate((5*x^8 + 3)*(x^8 - 1)^(1/3)/((x^8 - x^3 - 1)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int -\frac {{\left (x^8-1\right )}^{1/3}\,\left (5\,x^8+3\right )}{x^2\,\left (-x^8+x^3+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((x^8 - 1)^(1/3)*(5*x^8 + 3))/(x^2*(x^3 - x^8 + 1)),x)

[Out]

int(-((x^8 - 1)^(1/3)*(5*x^8 + 3))/(x^2*(x^3 - x^8 + 1)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )} \left (5 x^{8} + 3\right )}{x^{2} \left (x^{8} - x^{3} - 1\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**8-1)**(1/3)*(5*x**8+3)/x**2/(x**8-x**3-1),x)

[Out]

Integral(((x - 1)*(x + 1)*(x**2 + 1)*(x**4 + 1))**(1/3)*(5*x**8 + 3)/(x**2*(x**8 - x**3 - 1)), x)

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