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

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 = 20.33, size = 87, normalized size = 1.00 \begin {gather*} \frac {3 \sqrt [3]{-1+x^8}}{x}+\sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^8}}\right )+\log \left (-x+\sqrt [3]{-1+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^8}+\left (-1+x^8\right )^{2/3}\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 = 39.15, 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 = 197.67, size = 606, normalized size = 6.97

method	result	size
trager	\(\frac {3 \left (x^{8}-1\right )^{\frac {1}{3}}}{x}-6 \ln \left (\frac {5141860766708129562144283736588098527161696987675808 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{8}-31108581467491893886477983948838281630881262490114194 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{8}+20147370141885701559053156928011918590840844728896840 x^{8}-163896811938821629793349044103745640553279091482166380 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}+88061685794919605613792863040370530405109906887374910 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{8}-1\right )^{\frac {2}{3}} x +88061685794919605613792863040370530405109906887374910 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{8}-1\right )^{\frac {1}{3}} x^{2}+92711108733725916128403386928015888364971603628407342 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+45336674395944826278015303299258325827740741623155051 \left (x^{8}-1\right )^{\frac {2}{3}} x +45336674395944826278015303299258325827740741623155051 \left (x^{8}-1\right )^{\frac {1}{3}} x^{2}+20779444499278194156984236361047586625063302602744584 x^{3}-5141860766708129562144283736588098527161696987675808 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}+31108581467491893886477983948838281630881262490114194 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-20147370141885701559053156928011918590840844728896840}{x^{8}-x^{3}-1}\right ) \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )+6 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \ln \left (\frac {13140237860954654571358816426666205541798616800476611104 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{8}+72969596556035543676622386529356559272725556718096737834 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{8}+5325987117630295715001316896928139382907509781721573335 x^{8}-418845081817929614462062273599985301644830910515191978940 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}+111793230762796870634677068939366581324372951945784600474 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{8}-1\right )^{\frac {2}{3}} x +111793230762796870634677068939366581324372951945784600474 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{8}-1\right )^{\frac {1}{3}} x^{2}-28793839786067833023729227118876493965524653724764698666 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+16757573329138738851585113739228332287150734968044551246 \left (x^{8}-1\right )^{\frac {2}{3}} x +16757573329138738851585113739228332287150734968044551246 \left (x^{8}-1\right )^{\frac {1}{3}} x^{2}+162007212703583139619811920819106901381217027580884360 x^{3}-13140237860954654571358816426666205541798616800476611104 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}-72969596556035543676622386529356559272725556718096737834 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-5325987117630295715001316896928139382907509781721573335}{x^{8}-x^{3}-1}\right )-\ln \left (\frac {5141860766708129562144283736588098527161696987675808 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{8}-31108581467491893886477983948838281630881262490114194 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{8}+20147370141885701559053156928011918590840844728896840 x^{8}-163896811938821629793349044103745640553279091482166380 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{3}+88061685794919605613792863040370530405109906887374910 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{8}-1\right )^{\frac {2}{3}} x +88061685794919605613792863040370530405109906887374910 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \left (x^{8}-1\right )^{\frac {1}{3}} x^{2}+92711108733725916128403386928015888364971603628407342 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+45336674395944826278015303299258325827740741623155051 \left (x^{8}-1\right )^{\frac {2}{3}} x +45336674395944826278015303299258325827740741623155051 \left (x^{8}-1\right )^{\frac {1}{3}} x^{2}+20779444499278194156984236361047586625063302602744584 x^{3}-5141860766708129562144283736588098527161696987675808 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}+31108581467491893886477983948838281630881262490114194 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )-20147370141885701559053156928011918590840844728896840}{x^{8}-x^{3}-1}\right )\)	\(606\)
risch	\(\frac {3 \left (x^{8}-1\right )^{\frac {1}{3}}}{x}+\frac {\left (\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (\frac {-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{16}-x^{16}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{11}+\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{11}-3 \left (x^{16}-2 x^{8}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{9}-3 \left (x^{16}-2 x^{8}+1\right )^{\frac {1}{3}} x^{9}+4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}+2 x^{8}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-3 \left (x^{16}-2 x^{8}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-\RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{16}-2 x^{8}+1\right )^{\frac {2}{3}} x^{2}+3 \left (x^{16}-2 x^{8}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x +3 \left (x^{16}-2 x^{8}+1\right )^{\frac {1}{3}} x -2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (x^{4}+1\right ) \left (x^{8}-x^{3}-1\right )}\right )-\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{16}+x^{16}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{11}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{11}+3 \left (x^{16}-2 x^{8}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{9}+x^{11}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}-2 x^{8}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{16}-2 x^{8}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{16}-2 x^{8}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (x^{4}+1\right ) \left (x^{8}-x^{3}-1\right )}\right ) \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-\ln \left (\frac {2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{16}+x^{16}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{11}+3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{11}+3 \left (x^{16}-2 x^{8}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{9}+x^{11}-4 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{8}-2 x^{8}-2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}+3 \left (x^{16}-2 x^{8}+1\right )^{\frac {2}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}-3 \left (x^{16}-2 x^{8}+1\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x -x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )+1}{\left (-1+x \right ) \left (1+x \right ) \left (x^{2}+1\right ) \left (x^{4}+1\right ) \left (x^{8}-x^{3}-1\right )}\right )\right ) \left (\left (x^{8}-1\right )^{2}\right )^{\frac {1}{3}}}{\left (x^{8}-1\right )^{\frac {2}{3}}}\)	\(726\)

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,method=_RETURNVERBOSE)

[Out]

3*(x^8-1)^(1/3)/x-6*ln((5141860766708129562144283736588098527161696987675808*RootOf(36*_Z^2+6*_Z+1)^2*x^8-3110
8581467491893886477983948838281630881262490114194*RootOf(36*_Z^2+6*_Z+1)*x^8+201473701418857015590531569280119
18590840844728896840*x^8-163896811938821629793349044103745640553279091482166380*RootOf(36*_Z^2+6*_Z+1)^2*x^3+8
8061685794919605613792863040370530405109906887374910*RootOf(36*_Z^2+6*_Z+1)*(x^8-1)^(2/3)*x+880616857949196056
13792863040370530405109906887374910*RootOf(36*_Z^2+6*_Z+1)*(x^8-1)^(1/3)*x^2+927111087337259161284033869280158
88364971603628407342*RootOf(36*_Z^2+6*_Z+1)*x^3+45336674395944826278015303299258325827740741623155051*(x^8-1)^
(2/3)*x+45336674395944826278015303299258325827740741623155051*(x^8-1)^(1/3)*x^2+207794444992781941569842363610
47586625063302602744584*x^3-5141860766708129562144283736588098527161696987675808*RootOf(36*_Z^2+6*_Z+1)^2+3110
8581467491893886477983948838281630881262490114194*RootOf(36*_Z^2+6*_Z+1)-2014737014188570155905315692801191859
0840844728896840)/(x^8-x^3-1))*RootOf(36*_Z^2+6*_Z+1)+6*RootOf(36*_Z^2+6*_Z+1)*ln((131402378609546545713588164
26666205541798616800476611104*RootOf(36*_Z^2+6*_Z+1)^2*x^8+729695965560355436766223865293565592727255567180967
37834*RootOf(36*_Z^2+6*_Z+1)*x^8+5325987117630295715001316896928139382907509781721573335*x^8-41884508181792961
4462062273599985301644830910515191978940*RootOf(36*_Z^2+6*_Z+1)^2*x^3+1117932307627968706346770689393665813243
72951945784600474*RootOf(36*_Z^2+6*_Z+1)*(x^8-1)^(2/3)*x+11179323076279687063467706893936658132437295194578460
0474*RootOf(36*_Z^2+6*_Z+1)*(x^8-1)^(1/3)*x^2-28793839786067833023729227118876493965524653724764698666*RootOf(
36*_Z^2+6*_Z+1)*x^3+16757573329138738851585113739228332287150734968044551246*(x^8-1)^(2/3)*x+16757573329138738
851585113739228332287150734968044551246*(x^8-1)^(1/3)*x^2+1620072127035831396198119208191069013812170275808843
60*x^3-13140237860954654571358816426666205541798616800476611104*RootOf(36*_Z^2+6*_Z+1)^2-729695965560355436766
22386529356559272725556718096737834*RootOf(36*_Z^2+6*_Z+1)-532598711763029571500131689692813938290750978172157
3335)/(x^8-x^3-1))-ln((5141860766708129562144283736588098527161696987675808*RootOf(36*_Z^2+6*_Z+1)^2*x^8-31108
581467491893886477983948838281630881262490114194*RootOf(36*_Z^2+6*_Z+1)*x^8+2014737014188570155905315692801191
8590840844728896840*x^8-163896811938821629793349044103745640553279091482166380*RootOf(36*_Z^2+6*_Z+1)^2*x^3+88
061685794919605613792863040370530405109906887374910*RootOf(36*_Z^2+6*_Z+1)*(x^8-1)^(2/3)*x+8806168579491960561
3792863040370530405109906887374910*RootOf(36*_Z^2+6*_Z+1)*(x^8-1)^(1/3)*x^2+9271110873372591612840338692801588
8364971603628407342*RootOf(36*_Z^2+6*_Z+1)*x^3+45336674395944826278015303299258325827740741623155051*(x^8-1)^(
2/3)*x+45336674395944826278015303299258325827740741623155051*(x^8-1)^(1/3)*x^2+2077944449927819415698423636104
7586625063302602744584*x^3-5141860766708129562144283736588098527161696987675808*RootOf(36*_Z^2+6*_Z+1)^2+31108
581467491893886477983948838281630881262490114194*RootOf(36*_Z^2+6*_Z+1)-20147370141885701559053156928011918590
840844728896840)/(x^8-x^3-1))

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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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