3.13.43 \(\int \frac {(1+x^5)^{2/3} (-3+2 x^5)}{x^3 (1-x^3+x^5)} \, dx\) [1243]

3.13.43.1 Optimal result

Integrand size = 32, antiderivative size = 90 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right )}{x^3 \left (1-x^3+x^5\right )} \, dx=\frac {3 \left (1+x^5\right )^{2/3}}{2 x^2}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^5}}\right )+\log \left (-x+\sqrt [3]{1+x^5}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^5}+\left (1+x^5\right )^{2/3}\right ) \]

output

3/2*(x^5+1)^(2/3)/x^2-3^(1/2)*arctan(3^(1/2)*x/(x+2*(x^5+1)^(1/3)))+ln(-x+ 
(x^5+1)^(1/3))-1/2*ln(x^2+x*(x^5+1)^(1/3)+(x^5+1)^(2/3))
 

3.13.43.2 Mathematica [A] (verified)

Time = 1.15 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.00 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right )}{x^3 \left (1-x^3+x^5\right )} \, dx=\frac {3 \left (1+x^5\right )^{2/3}}{2 x^2}-\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{1+x^5}}\right )+\log \left (-x+\sqrt [3]{1+x^5}\right )-\frac {1}{2} \log \left (x^2+x \sqrt [3]{1+x^5}+\left (1+x^5\right )^{2/3}\right ) \]

input

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

output

(3*(1 + x^5)^(2/3))/(2*x^2) - Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*(1 + x^5)^ 
(1/3))] + Log[-x + (1 + x^5)^(1/3)] - Log[x^2 + x*(1 + x^5)^(1/3) + (1 + x 
^5)^(2/3)]/2
 

3.13.43.3 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.

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

\(\Big \downarrow \) 7293

\(\displaystyle \int \left (\frac {\left (5 x^2-3\right ) \left (x^5+1\right )^{2/3}}{x^5-x^3+1}-\frac {3 \left (x^5+1\right )^{2/3}}{x^3}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -3 \int \frac {\left (x^5+1\right )^{2/3}}{x^5-x^3+1}dx+5 \int \frac {x^2 \left (x^5+1\right )^{2/3}}{x^5-x^3+1}dx+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {2}{5},\frac {3}{5},-x^5\right )}{2 x^2}\)

input

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

output

$Aborted
 

3.13.43.3.1 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] 
]
 

3.13.43.4 Maple [A] (verified)

Time = 11.45 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{5}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2 \ln \left (\frac {-x +\left (x^{5}+1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-\ln \left (\frac {x^{2}+x \left (x^{5}+1\right )^{\frac {1}{3}}+\left (x^{5}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{5}+1\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(95\)
risch	\(\frac {3 \left (x^{5}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-2 x^{5}+3 \left (x^{5}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}-3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}+1\right )^{\frac {2}{3}} x -2 x^{3}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-2}{x^{5}-x^{3}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) \ln \left (-\frac {-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{5}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )^{2} x^{3}-x^{5}+3 \left (x^{5}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right ) x^{3}+3 \left (x^{5}+1\right )^{\frac {2}{3}} x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\textit {\_Z} +1\right )-1}{x^{5}-x^{3}+1}\right )\)	\(222\)
trager	\(\frac {3 \left (x^{5}+1\right )^{\frac {2}{3}}}{2 x^{2}}+\ln \left (\frac {-444836727 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}-26958924 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}[phone] x^{5}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}} x -1469304801 \left (x^{5}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}-305544126 \left (x^{5}+1\right )^{\frac {2}{3}} x -184224141 \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}[phone] x^{3}-444836727 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-26958924 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )[phone]}{x^{5}-x^{3}+1}\right )+3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (-\frac {-970550226 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{5}-1118829135 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{5}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{3}-148278909 x^{5}[phone] \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{5}+1\right )^{\frac {2}{3}} x [phone] \left (x^{5}+1\right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-26958924 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{3}[phone] \left (x^{5}+1\right )^{\frac {2}{3}} x -184224141 \left (x^{5}+1\right )^{\frac {1}{3}} x^{2}-49426303 x^{3}-970550226 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-1118829135 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-148278909}{x^{5}-x^{3}+1}\right )\)	\(404\)

input

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

output

1/2*(2*3^(1/2)*arctan(1/3*3^(1/2)/x*(x+2*(x^5+1)^(1/3)))*x^2+2*ln((-x+(x^5 
+1)^(1/3))/x)*x^2-ln((x^2+x*(x^5+1)^(1/3)+(x^5+1)^(2/3))/x^2)*x^2+3*(x^5+1 
)^(2/3))/x^2
 

3.13.43.5 Fricas [A] (verification not implemented)

Time = 3.28 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.50 \[ \int \frac {\left (1+x^5\right )^{2/3} \left (-3+2 x^5\right )}{x^3 \left (1-x^3+x^5\right )} \, dx=-\frac {2 \, \sqrt {3} x^{2} \arctan \left (\frac {67616276 \, \sqrt {3} {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} + 10249526 \, \sqrt {3} {\left (x^{5} + 1\right )}^{\frac {2}{3}} x + \sqrt {3} {\left (1423013 \, x^{5} + 37509888 \, x^{3} + 1423013\right )}}{300763 \, x^{5} - 86350888 \, x^{3} + 300763}\right ) - x^{2} \log \left (\frac {x^{5} - x^{3} + 3 \, {\left (x^{5} + 1\right )}^{\frac {1}{3}} x^{2} - 3 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}} x + 1}{x^{5} - x^{3} + 1}\right ) - 3 \, {\left (x^{5} + 1\right )}^{\frac {2}{3}}}{2 \, x^{2}} \]

input

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

output

-1/2*(2*sqrt(3)*x^2*arctan((67616276*sqrt(3)*(x^5 + 1)^(1/3)*x^2 + 1024952 
6*sqrt(3)*(x^5 + 1)^(2/3)*x + sqrt(3)*(1423013*x^5 + 37509888*x^3 + 142301 
3))/(300763*x^5 - 86350888*x^3 + 300763)) - x^2*log((x^5 - x^3 + 3*(x^5 + 
1)^(1/3)*x^2 - 3*(x^5 + 1)^(2/3)*x + 1)/(x^5 - x^3 + 1)) - 3*(x^5 + 1)^(2/ 
3))/x^2
 

3.13.43.6 Sympy [F]

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

input

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

output

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

3.13.43.7 Maxima [F]

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

input

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

output

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

3.13.43.8 Giac [F]

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

input

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

output

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

3.13.43.9 Mupad [F(-1)]

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

input

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

output

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