3.12.92 \(\int \frac {\sqrt [3]{-1+x^8} (3+5 x^8)}{x^2 (-1-x^3+x^8)} \, dx\) [1192]

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

[Out]

3*(x^8-1)^(1/3)/x+3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^8-1)^(1/3)))+ln(-x+(x^8-1)^(1/3))-1/2*ln(x^2+x*(x^8-1)^(1/3
)+(x^8-1)^(2/3))

________________________________________________________________________________________

Rubi [F]
time = 0.52, 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*}

________________________________________________________________________________________

Mathematica [F]
time = 20.14, 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]

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 121.37, size = 404, normalized size = 4.64

method	result	size
trager	\(\frac {3 \left (x^{8}-1\right )^{\frac {1}{3}}}{x}+\ln \left (\frac {-7307978203625661545045587277178357092074803807564774144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{8}-42231670217524712977455416360650782490626298662361426848 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{8}-11999592213302340806483919167406986310739709092101905279 x^{8}+232941805240567961748328094460060132309884371366127175840 \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 -100545439974832433109510682435369993722904409808267307476 \left (x^{8}-1\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}+68589517269050658819502781053801157425459351117937867466 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+16757573329138738851585113739228332287150734968044551246 \left (x^{8}-1\right )^{\frac {2}{3}} x +1874631797994072920861064417332764600244757022919548833 \left (x^{8}-1\right )^{\frac {1}{3}} x^{2}-365006607248740404759967122962950153938850466679905864 x^{3}+7307978203625661545045587277178357092074803807564774144 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}+42231670217524712977455416360650782490626298662361426848 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )+11999592213302340806483919167406986310739709092101905279}{x^{8}-x^{3}-1}\right )+6 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) \ln \left (-\frac {5832259657328993026313229149487848449723812992911836960 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2} x^{8}-28793839786067833023729227118876493965524653724764698666 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{8}-11634585606053600401723952044444036156800858625421999415 x^{8}-185903276577361652713734179139925169334946539149064803100 \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 -11247790787964437525166386503996587601468542137517292998 \left (x^{8}-1\right )^{\frac {1}{3}} \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{2}-101763436342103376700351613648233053238250210442861436500 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right ) x^{3}+1874631797994072920861064417332764600244757022919548833 \left (x^{8}-1\right )^{\frac {2}{3}} x +16757573329138738851585113739228332287150734968044551246 \left (x^{8}-1\right )^{\frac {1}{3}} x^{2}-11999592213302340806483919167406986310739709092101905279 x^{3}-5832259657328993026313229149487848449723812992911836960 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )^{2}+28793839786067833023729227118876493965524653724764698666 \RootOf \left (36 \textit {\_Z}^{2}+6 \textit {\_Z} +1\right )+11634585606053600401723952044444036156800858625421999415}{x^{8}-x^{3}-1}\right )\)	\(404\)
risch	\(\text {Expression too large to display}\)	\(726\)

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+ln((-7307978203625661545045587277178357092074803807564774144*RootOf(36*_Z^2+6*_Z+1)^2*x^8-42
231670217524712977455416360650782490626298662361426848*RootOf(36*_Z^2+6*_Z+1)*x^8-1199959221330234080648391916
7406986310739709092101905279*x^8+232941805240567961748328094460060132309884371366127175840*RootOf(36*_Z^2+6*_Z
+1)^2*x^3+111793230762796870634677068939366581324372951945784600474*RootOf(36*_Z^2+6*_Z+1)*(x^8-1)^(2/3)*x-100
545439974832433109510682435369993722904409808267307476*(x^8-1)^(1/3)*RootOf(36*_Z^2+6*_Z+1)*x^2+68589517269050
658819502781053801157425459351117937867466*RootOf(36*_Z^2+6*_Z+1)*x^3+1675757332913873885158511373922833228715
0734968044551246*(x^8-1)^(2/3)*x+1874631797994072920861064417332764600244757022919548833*(x^8-1)^(1/3)*x^2-365
006607248740404759967122962950153938850466679905864*x^3+730797820362566154504558727717835709207480380756477414
4*RootOf(36*_Z^2+6*_Z+1)^2+42231670217524712977455416360650782490626298662361426848*RootOf(36*_Z^2+6*_Z+1)+119
99592213302340806483919167406986310739709092101905279)/(x^8-x^3-1))+6*RootOf(36*_Z^2+6*_Z+1)*ln(-(583225965732
8993026313229149487848449723812992911836960*RootOf(36*_Z^2+6*_Z+1)^2*x^8-2879383978606783302372922711887649396
5524653724764698666*RootOf(36*_Z^2+6*_Z+1)*x^8-11634585606053600401723952044444036156800858625421999415*x^8-18
5903276577361652713734179139925169334946539149064803100*RootOf(36*_Z^2+6*_Z+1)^2*x^3+1117932307627968706346770
68939366581324372951945784600474*RootOf(36*_Z^2+6*_Z+1)*(x^8-1)^(2/3)*x-11247790787964437525166386503996587601
468542137517292998*(x^8-1)^(1/3)*RootOf(36*_Z^2+6*_Z+1)*x^2-10176343634210337670035161364823305323825021044286
1436500*RootOf(36*_Z^2+6*_Z+1)*x^3+1874631797994072920861064417332764600244757022919548833*(x^8-1)^(2/3)*x+167
57573329138738851585113739228332287150734968044551246*(x^8-1)^(1/3)*x^2-11999592213302340806483919167406986310
739709092101905279*x^3-5832259657328993026313229149487848449723812992911836960*RootOf(36*_Z^2+6*_Z+1)^2+287938
39786067833023729227118876493965524653724764698666*RootOf(36*_Z^2+6*_Z+1)+116345856060536004017239520444440361
56800858625421999415)/(x^8-x^3-1))

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \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)

________________________________________________________________________________________

Fricas [A]
time = 11.07, 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

________________________________________________________________________________________

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)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)

________________________________________________________________________________________

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)

________________________________________________________________________________________