3.10.71 \(\int \frac {\sqrt {-1+x^3} (2+x^3) (-1-x^2+x^3)^2}{x^6 (-2-3 x^2+2 x^3)} \, dx\) [971]

3.10.71.1 Optimal result

Integrand size = 44, antiderivative size = 74 \[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx=\frac {\sqrt {-1+x^3} \left (12+10 x^2-24 x^3+15 x^4-10 x^5+12 x^6\right )}{60 x^5}-\frac {1}{4} \sqrt {\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1+x^3}}\right ) \]

output

1/60*(x^3-1)^(1/2)*(12*x^6-10*x^5+15*x^4-24*x^3+10*x^2+12)/x^5-1/8*6^(1/2) 
*arctanh(1/2*6^(1/2)*x/(x^3-1)^(1/2))
 

3.10.71.2 Mathematica [A] (verified)

Time = 1.18 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx=\frac {\sqrt {-1+x^3} \left (12+10 x^2-24 x^3+15 x^4-10 x^5+12 x^6\right )}{60 x^5}-\frac {1}{4} \sqrt {\frac {3}{2}} \text {arctanh}\left (\frac {\sqrt {\frac {3}{2}} x}{\sqrt {-1+x^3}}\right ) \]

input

Integrate[(Sqrt[-1 + x^3]*(2 + x^3)*(-1 - x^2 + x^3)^2)/(x^6*(-2 - 3*x^2 + 
 2*x^3)),x]
 

output

(Sqrt[-1 + x^3]*(12 + 10*x^2 - 24*x^3 + 15*x^4 - 10*x^5 + 12*x^6))/(60*x^5 
) - (Sqrt[3/2]*ArcTanh[(Sqrt[3/2]*x)/Sqrt[-1 + x^3]])/4
 

3.10.71.3 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.

\(\displaystyle \int \frac {\sqrt {x^3-1} \left (x^3+2\right ) \left (x^3-x^2-1\right )^2}{x^6 \left (2 x^3-3 x^2-2\right )} \, dx\)

\(\Big \downarrow \) 2465

\(\displaystyle \int \left (-\frac {2 \sqrt {x^3-1} \left (x^3+2\right )}{x \left (2 x^3-3 x^2-2\right )}+\frac {\sqrt {x^3-1} \left (x^3+2\right )}{x^2 \left (2 x^3-3 x^2-2\right )}-\frac {2 \sqrt {x^3-1} \left (x^3+2\right )}{x^3 \left (2 x^3-3 x^2-2\right )}+\frac {\sqrt {x^3-1} \left (x^3+2\right )}{2 x^3-3 x^2-2}+\frac {\sqrt {x^3-1} \left (x^3+2\right )}{x^6 \left (2 x^3-3 x^2-2\right )}+\frac {2 \sqrt {x^3-1} \left (x^3+2\right )}{x^4 \left (2 x^3-3 x^2-2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle 3 \int \frac {\sqrt {x^3-1}}{-2 x^3+3 x^2+2}dx+\frac {9}{4} \int \frac {\sqrt {x^3-1}}{2 x^3-3 x^2-2}dx+\frac {3}{4} \int \frac {x \sqrt {x^3-1}}{2 x^3-3 x^2-2}dx-\frac {2 \sqrt {2} 3^{3/4} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{\sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {9\ 3^{3/4} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right ),-7+4 \sqrt {3}\right )}{2 \sqrt {2} \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} (1-x) \sqrt {\frac {x^2+x+1}{\left (-x-\sqrt {3}+1\right )^2}} E\left (\arcsin \left (\frac {-x+\sqrt {3}+1}{-x-\sqrt {3}+1}\right )|-7+4 \sqrt {3}\right )}{8 \sqrt {-\frac {1-x}{\left (-x-\sqrt {3}+1\right )^2}} \sqrt {x^3-1}}+\frac {1}{5} \sqrt {x^3-1} x+\frac {3 \sqrt {x^3-1}}{4 \left (-x-\sqrt {3}+1\right )}-\frac {\sqrt {x^3-1}}{6}+\frac {\sqrt {x^3-1}}{4 x}+\frac {\sqrt {x^3-1}}{6 x^3}+\frac {\sqrt {x^3-1}}{5 x^5}-\frac {2 \sqrt {x^3-1}}{5 x^2}\)

input

Int[(Sqrt[-1 + x^3]*(2 + x^3)*(-1 - x^2 + x^3)^2)/(x^6*(-2 - 3*x^2 + 2*x^3 
)),x]
 

output

$Aborted
 

3.10.71.3.1 Defintions of rubi rules used

rule 2009

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

rule 2465

Int[(u_.)*(Px_)^(p_), x_Symbol] :> Int[ExpandToSum[u, Px^p, x], x] /; PolyQ 
[Px, x] && GtQ[Expon[Px, x], 2] &&  !BinomialQ[Px, x] &&  !TrinomialQ[Px, x 
] && IGtQ[p, 0]
 

3.10.71.4 Maple [A] (verified)

Time = 3.63 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {-15 \sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}-1}\, \sqrt {6}}{3 x}\right ) x^{5}+2 \sqrt {x^{3}-1}\, \left (12 x^{6}-10 x^{5}+15 x^{4}-24 x^{3}+10 x^{2}+12\right )}{120 x^{5}}\)	\(67\)
pseudoelliptic	\(\frac {-15 \sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}-1}\, \sqrt {6}}{3 x}\right ) x^{5}+2 \sqrt {x^{3}-1}\, \left (12 x^{6}-10 x^{5}+15 x^{4}-24 x^{3}+10 x^{2}+12\right )}{120 x^{5}}\)	\(67\)
risch	\(\frac {12 x^{9}-10 x^{8}+15 x^{7}-36 x^{6}+20 x^{5}-15 x^{4}+36 x^{3}-10 x^{2}-12}{60 x^{5} \sqrt {x^{3}-1}}-\frac {\sqrt {6}\, \operatorname {arctanh}\left (\frac {\sqrt {x^{3}-1}\, \sqrt {6}}{3 x}\right )}{8}\)	\(77\)
trager	\(\frac {\sqrt {x^{3}-1}\, \left (12 x^{6}-10 x^{5}+15 x^{4}-24 x^{3}+10 x^{2}+12\right )}{60 x^{5}}-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{3}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right ) x^{2}+12 \sqrt {x^{3}-1}\, x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-6\right )}{2 x^{3}-3 x^{2}-2}\right )}{16}\)	\(107\)
elliptic	\(\frac {\sqrt {x^{3}-1}\, x}{5}+\frac {\sqrt {x^{3}-1}}{5 x^{5}}-\frac {2 \sqrt {x^{3}-1}}{5 x^{2}}-\frac {\sqrt {x^{3}-1}}{6}+\frac {\sqrt {x^{3}-1}}{6 x^{3}}+\frac {\sqrt {x^{3}-1}}{4 x}+\frac {3 \left (-\frac {3}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}-\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\, \sqrt {\frac {x +\frac {1}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}+\frac {i \sqrt {3}}{2}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{8 \sqrt {x^{3}-1}}+\frac {\sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (2 \textit {\_Z}^{3}-3 \textit {\_Z}^{2}-2\right )}{\sum }\frac {\underline {\hspace {1.25 ex}}\alpha \left (-2 \underline {\hspace {1.25 ex}}\alpha ^{2}+\underline {\hspace {1.25 ex}}\alpha +1\right ) \left (-3-i \sqrt {3}\right ) \sqrt {\frac {x -1}{-3-i \sqrt {3}}}\, \sqrt {\frac {2 x +1-i \sqrt {3}}{3-i \sqrt {3}}}\, \sqrt {\frac {i \sqrt {3}+2 x +1}{i \sqrt {3}+3}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x -1}{-\frac {3}{2}-\frac {i \sqrt {3}}{2}}}, -\underline {\hspace {1.25 ex}}\alpha ^{2}+\frac {\underline {\hspace {1.25 ex}}\alpha }{2}+\frac {1}{2}-\frac {i \underline {\hspace {1.25 ex}}\alpha ^{2} \sqrt {3}}{3}+\frac {i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha }{6}+\frac {i \sqrt {3}}{6}, \sqrt {\frac {\frac {3}{2}+\frac {i \sqrt {3}}{2}}{\frac {3}{2}-\frac {i \sqrt {3}}{2}}}\right )}{\sqrt {x^{3}-1}}\right )}{16}\)	\(364\)

input

int((x^3-1)^(1/2)*(x^3+2)*(x^3-x^2-1)^2/x^6/(2*x^3-3*x^2-2),x,method=_RETU 
RNVERBOSE)
 

output

1/120*(-15*6^(1/2)*arctanh(1/3*(x^3-1)^(1/2)/x*6^(1/2))*x^5+2*(x^3-1)^(1/2 
)*(12*x^6-10*x^5+15*x^4-24*x^3+10*x^2+12))/x^5
 

3.10.71.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 141 vs. \(2 (59) = 118\).

Time = 0.30 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.91 \[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx=\frac {15 \, \sqrt {3} \sqrt {2} x^{5} \log \left (-\frac {4 \, x^{6} + 36 \, x^{5} + 9 \, x^{4} - 8 \, x^{3} - 4 \, \sqrt {3} \sqrt {2} {\left (2 \, x^{4} + 3 \, x^{3} - 2 \, x\right )} \sqrt {x^{3} - 1} - 36 \, x^{2} + 4}{4 \, x^{6} - 12 \, x^{5} + 9 \, x^{4} - 8 \, x^{3} + 12 \, x^{2} + 4}\right ) + 8 \, {\left (12 \, x^{6} - 10 \, x^{5} + 15 \, x^{4} - 24 \, x^{3} + 10 \, x^{2} + 12\right )} \sqrt {x^{3} - 1}}{480 \, x^{5}} \]

input

integrate((x^3-1)^(1/2)*(x^3+2)*(x^3-x^2-1)^2/x^6/(2*x^3-3*x^2-2),x, algor 
ithm="fricas")
 

output

1/480*(15*sqrt(3)*sqrt(2)*x^5*log(-(4*x^6 + 36*x^5 + 9*x^4 - 8*x^3 - 4*sqr 
t(3)*sqrt(2)*(2*x^4 + 3*x^3 - 2*x)*sqrt(x^3 - 1) - 36*x^2 + 4)/(4*x^6 - 12 
*x^5 + 9*x^4 - 8*x^3 + 12*x^2 + 4)) + 8*(12*x^6 - 10*x^5 + 15*x^4 - 24*x^3 
 + 10*x^2 + 12)*sqrt(x^3 - 1))/x^5
 

3.10.71.6 Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx=\text {Timed out} \]

input

integrate((x**3-1)**(1/2)*(x**3+2)*(x**3-x**2-1)**2/x**6/(2*x**3-3*x**2-2) 
,x)
 

output

Timed out
 

3.10.71.7 Maxima [F]

\[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} - x^{2} - 1\right )}^{2} {\left (x^{3} + 2\right )} \sqrt {x^{3} - 1}}{{\left (2 \, x^{3} - 3 \, x^{2} - 2\right )} x^{6}} \,d x } \]

input

integrate((x^3-1)^(1/2)*(x^3+2)*(x^3-x^2-1)^2/x^6/(2*x^3-3*x^2-2),x, algor 
ithm="maxima")
 

output

integrate((x^3 - x^2 - 1)^2*(x^3 + 2)*sqrt(x^3 - 1)/((2*x^3 - 3*x^2 - 2)*x 
^6), x)
 

3.10.71.8 Giac [F]

\[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx=\int { \frac {{\left (x^{3} - x^{2} - 1\right )}^{2} {\left (x^{3} + 2\right )} \sqrt {x^{3} - 1}}{{\left (2 \, x^{3} - 3 \, x^{2} - 2\right )} x^{6}} \,d x } \]

input

integrate((x^3-1)^(1/2)*(x^3+2)*(x^3-x^2-1)^2/x^6/(2*x^3-3*x^2-2),x, algor 
ithm="giac")
 

output

integrate((x^3 - x^2 - 1)^2*(x^3 + 2)*sqrt(x^3 - 1)/((2*x^3 - 3*x^2 - 2)*x 
^6), x)
 

3.10.71.9 Mupad [B] (verification not implemented)

Time = 7.51 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.58 \[ \int \frac {\sqrt {-1+x^3} \left (2+x^3\right ) \left (-1-x^2+x^3\right )^2}{x^6 \left (-2-3 x^2+2 x^3\right )} \, dx=\frac {x\,\sqrt {x^3-1}}{5}-\frac {\sqrt {x^3-1}}{6}+\frac {\sqrt {x^3-1}}{4\,x}-\frac {2\,\sqrt {x^3-1}}{5\,x^2}+\frac {\sqrt {x^3-1}}{6\,x^3}+\frac {\sqrt {x^3-1}}{5\,x^5}+\frac {\sqrt {2}\,\sqrt {3}\,\ln \left (\frac {3\,x^2+2\,x^3-2\,\sqrt {6}\,x\,\sqrt {x^3-1}-2}{-12\,x^3+18\,x^2+12}\right )}{16} \]

input

int(-((x^3 - 1)^(1/2)*(x^3 + 2)*(x^2 - x^3 + 1)^2)/(x^6*(3*x^2 - 2*x^3 + 2 
)),x)
 

output

(x*(x^3 - 1)^(1/2))/5 - (x^3 - 1)^(1/2)/6 + (x^3 - 1)^(1/2)/(4*x) - (2*(x^ 
3 - 1)^(1/2))/(5*x^2) + (x^3 - 1)^(1/2)/(6*x^3) + (x^3 - 1)^(1/2)/(5*x^5) 
+ (2^(1/2)*3^(1/2)*log((3*x^2 + 2*x^3 - 2*6^(1/2)*x*(x^3 - 1)^(1/2) - 2)/( 
18*x^2 - 12*x^3 + 12)))/16