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

3.12.92.1 Optimal result

Integrand size = 32, antiderivative size = 87 \[ \int \frac {\sqrt [3]{-1+x^8} \left (3+5 x^8\right )}{x^2 \left (-1-x^3+x^8\right )} \, dx=\frac {3 \sqrt [3]{-1+x^8}}{x}+\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-1+x^8}}\right )+\log \left (-x+\sqrt [3]{-1+x^8}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{-1+x^8}+\left (-1+x^8\right )^{2/3}\right ) \]

output

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))
 

3.12.92.2 Mathematica [F]

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

input

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

output

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

3.12.92.3 Rubi [F]

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -3 \int \frac {x \sqrt [3]{x^8-1}}{x^8-x^3-1}dx+8 \int \frac {x^6 \sqrt [3]{x^8-1}}{x^8-x^3-1}dx+\frac {3 \sqrt [3]{x^8-1} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},-\frac {1}{8},\frac {7}{8},x^8\right )}{x \sqrt [3]{1-x^8}}\)

input

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

output

$Aborted
 

3.12.92.3.1 Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.12.92.4 Maple [A] (verified)

Time = 221.16 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02

method	result	size
pseudoelliptic	\(\frac {-2 \sqrt {3}\, \arctan \left (\frac {\left (x +2 \left (x^{8}-1\right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 x}\right ) x +2 \ln \left (\frac {-x +\left (x^{8}-1\right )^{\frac {1}{3}}}{x}\right ) x -\ln \left (\frac {x^{2}+x \left (x^{8}-1\right )^{\frac {1}{3}}+\left (x^{8}-1\right )^{\frac {2}{3}}}{x^{2}}\right ) x +6 \left (x^{8}-1\right )^{\frac {1}{3}}}{2 x}\)	\(89\)
trager	\(\text {Expression too large to display}\)	\(606\)
risch	\(\text {Expression too large to display}\)	\(726\)

input

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

output

1/2*(-2*3^(1/2)*arctan(1/3*(x+2*(x^8-1)^(1/3))*3^(1/2)/x)*x+2*ln((-x+(x^8- 
1)^(1/3))/x)*x-ln((x^2+x*(x^8-1)^(1/3)+(x^8-1)^(2/3))/x^2)*x+6*(x^8-1)^(1/ 
3))/x
 

3.12.92.5 Fricas [A] (verification not implemented)

Time = 26.49 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.51 \[ \int \frac {\sqrt [3]{-1+x^8} \left (3+5 x^8\right )}{x^2 \left (-1-x^3+x^8\right )} \, dx=\frac {2 \, \sqrt {3} x \arctan \left (-\frac {31069389038531798383012393094747362616575064091434751962020601837507558239516138425325377239789317495328857903057957141206059288722620160721093489516063746612973182 \, \sqrt {3} {\left (x^{8} - 1\right )}^{\frac {1}{3}} x^{2} - 24620142163963087452447726858369178030030967023250856622849105390649652817268567947362178503080085821866784600572345611200568455939022999883192079164797236311980480 \, \sqrt {3} {\left (x^{8} - 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (14098730908269987597917744450355902431760205999000820135495290627669890741173905802396636062023876418322337000958016148565005886294703209808664629857632230121011200 \, x^{8} - 10874107470985632132635411332166810138488157464908872465909542404240938030050120563415036693669260581591300349715210383562260469902904629389713924681998974970514849 \, x^{3} - 14098730908269987597917744450355902431760205999000820135495290627669890741173905802396636062023876418322337000958016148565005886294703209808664629857632230121011200\right )}}{3 \, {\left (9251742523290005295394971478800280999715753799405283223501747806428870154589708393514732281743754536574942347080177746431157381208775803010963333365470079627264000 \, x^{8} + 18593023077957437622335088497757989323587261757937521068933105807649735373802644792829045589690947122022878904734973629772156491122045777291179450974960411835212831 \, x^{3} - 9251742523290005295394971478800280999715753799405283223501747806428870154589708393514732281743754536574942347080177746431157381208775803010963333365470079627264000\right )}}\right ) + x \log \left (\frac {x^{8} - x^{3} + 3 \, {\left (x^{8} - 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{8} - 1\right )}^{\frac {2}{3}} x - 1}{x^{8} - x^{3} - 1}\right ) + 6 \, {\left (x^{8} - 1\right )}^{\frac {1}{3}}}{2 \, x} \]

input

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

output

1/2*(2*sqrt(3)*x*arctan(-1/3*(31069389038531798383012393094747362616575064 
09143475196202060183750755823951613842532537723978931749532885790305795714 
1206059288722620160721093489516063746612973182*sqrt(3)*(x^8 - 1)^(1/3)*x^2 
 - 24620142163963087452447726858369178030030967023250856622849105390649652 
81726856794736217850308008582186678460057234561120056845593902299988319207 
9164797236311980480*sqrt(3)*(x^8 - 1)^(2/3)*x + sqrt(3)*(14098730908269987 
59791774445035590243176020599900082013549529062766989074117390580239663606 
2023876418322337000958016148565005886294703209808664629857632230121011200* 
x^8 - 10874107470985632132635411332166810138488157464908872465909542404240 
93803005012056341503669366926058159130034971521038356226046990290462938971 
3924681998974970514849*x^3 - 140987309082699875979177444503559024317602059 
99000820135495290627669890741173905802396636062023876418322337000958016148 
565005886294703209808664629857632230121011200))/(9251742523290005295394971 
47880028099971575379940528322350174780642887015458970839351473228174375453 
6574942347080177746431157381208775803010963333365470079627264000*x^8 + 185 
93023077957437622335088497757989323587261757937521068933105807649735373802 
64479282904558969094712202287890473497362977215649112204577729117945097496 
0411835212831*x^3 - 925174252329000529539497147880028099971575379940528322 
35017478064288701545897083935147322817437545365749423470801777464311573812 
08775803010963333365470079627264000)) + x*log((x^8 - x^3 + 3*(x^8 - 1)^...
 

3.12.92.6 Sympy [F]

\[ \int \frac {\sqrt [3]{-1+x^8} \left (3+5 x^8\right )}{x^2 \left (-1-x^3+x^8\right )} \, dx=\int \frac {\sqrt [3]{\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right ) \left (x^{4} + 1\right )} \left (5 x^{8} + 3\right )}{x^{2} \left (x^{8} - x^{3} - 1\right )}\, dx \]

input

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

output

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)
 

3.12.92.7 Maxima [F]

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

input

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

output

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

3.12.92.8 Giac [F]

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

input

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

output

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

3.12.92.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt [3]{-1+x^8} \left (3+5 x^8\right )}{x^2 \left (-1-x^3+x^8\right )} \, dx=\int -\frac {{\left (x^8-1\right )}^{1/3}\,\left (5\,x^8+3\right )}{x^2\,\left (-x^8+x^3+1\right )} \,d x \]

input

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

output

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