3.17.99 \(\int \frac {(1+x^2) \sqrt [3]{-x+2 x^3}}{x^2 (1+x^4)} \, dx\) [1699]

3.17.99.1 Optimal result

Integrand size = 29, antiderivative size = 114 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=-\frac {3 \sqrt [3]{-x+2 x^3}}{2 x}+\frac {1}{4} \text {RootSum}\left [5-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-5 \log (x)+5 \log \left (\sqrt [3]{-x+2 x^3}-x \text {$\#$1}\right )+\log (x) \text {$\#$1}^3-\log \left (\sqrt [3]{-x+2 x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ] \]

output

Unintegrable
 

3.17.99.2 Mathematica [A] (verified)

Time = 1.78 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.18 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\frac {18-36 x^2+x^{2/3} \left (-1+2 x^2\right )^{2/3} \text {RootSum}\left [5-4 \text {$\#$1}^3+\text {$\#$1}^6\&,\frac {-10 \log (x)+15 \log \left (\sqrt [3]{-1+2 x^2}-x^{2/3} \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^3-3 \log \left (\sqrt [3]{-1+2 x^2}-x^{2/3} \text {$\#$1}\right ) \text {$\#$1}^3}{-2 \text {$\#$1}^2+\text {$\#$1}^5}\&\right ]}{12 \left (x \left (-1+2 x^2\right )\right )^{2/3}} \]

input

Integrate[((1 + x^2)*(-x + 2*x^3)^(1/3))/(x^2*(1 + x^4)),x]
 

output

(18 - 36*x^2 + x^(2/3)*(-1 + 2*x^2)^(2/3)*RootSum[5 - 4*#1^3 + #1^6 & , (- 
10*Log[x] + 15*Log[(-1 + 2*x^2)^(1/3) - x^(2/3)*#1] + 2*Log[x]*#1^3 - 3*Lo 
g[(-1 + 2*x^2)^(1/3) - x^(2/3)*#1]*#1^3)/(-2*#1^2 + #1^5) & ])/(12*(x*(-1 
+ 2*x^2))^(2/3))
 

3.17.99.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 2035

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7276

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \left (\frac {\sqrt [3]{x} \sqrt [3]{2 x^2-1} \left (x^{2/3}-1\right )}{3 \left (x^{4/3}+1\right )}+\frac {\sqrt [3]{2 x^2-1}}{x}+\frac {\sqrt [3]{x} \left (-x^2-2 x^{4/3}+2 x^{2/3}+1\right ) \sqrt [3]{2 x^2-1}}{3 \left (x^{8/3}-x^{4/3}+1\right )}\right )d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

\(\Big \downarrow \) 7239

\(\displaystyle \frac {3 \sqrt [3]{2 x^3-x} \int \frac {\left (x^2+1\right ) \sqrt [3]{2 x^2-1}}{x \left (x^4+1\right )}d\sqrt [3]{x}}{\sqrt [3]{x} \sqrt [3]{2 x^2-1}}\)

input

Int[((1 + x^2)*(-x + 2*x^3)^(1/3))/(x^2*(1 + x^4)),x]
 

output

$Aborted
 

3.17.99.3.1 Defintions of rubi rules used

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 7239

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl 
erIntegrandQ[v, u, x]]
 

rule 7276

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE 
xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ 
[n, 0]
 

3.17.99.4 Maple [N/A] (verified)

Time = 168.42 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.65

method	result	size
pseudoelliptic	\(\frac {-\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6}-4 \textit {\_Z}^{3}+5\right )}{\sum }\frac {\left (\textit {\_R}^{3}-5\right ) \ln \left (\frac {-\textit {\_R} x +\left (2 x^{3}-x \right )^{\frac {1}{3}}}{x}\right )}{\textit {\_R}^{2} \left (\textit {\_R}^{3}-2\right )}\right ) x -6 \left (2 x^{3}-x \right )^{\frac {1}{3}}}{4 x}\)	\(74\)
trager	\(\text {Expression too large to display}\)	\(10722\)
risch	\(\text {Expression too large to display}\)	\(11848\)

input

int((x^2+1)*(2*x^3-x)^(1/3)/x^2/(x^4+1),x,method=_RETURNVERBOSE)
 

output

1/4*(-sum((_R^3-5)*ln((-_R*x+(2*x^3-x)^(1/3))/x)/_R^2/(_R^3-2),_R=RootOf(_ 
Z^6-4*_Z^3+5))*x-6*(2*x^3-x)^(1/3))/x
 

3.17.99.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\text {Exception raised: TypeError} \]

input

integrate((x^2+1)*(2*x^3-x)^(1/3)/x^2/(x^4+1),x, algorithm="fricas")
 

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (trace 0)
 

3.17.99.6 Sympy [N/A]

Not integrable

Time = 1.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.23 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\int \frac {\sqrt [3]{x \left (2 x^{2} - 1\right )} \left (x^{2} + 1\right )}{x^{2} \left (x^{4} + 1\right )}\, dx \]

input

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

output

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

3.17.99.7 Maxima [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{{\left (x^{4} + 1\right )} x^{2}} \,d x } \]

input

integrate((x^2+1)*(2*x^3-x)^(1/3)/x^2/(x^4+1),x, algorithm="maxima")
 

output

integrate((2*x^3 - x)^(1/3)*(x^2 + 1)/((x^4 + 1)*x^2), x)
 

3.17.99.8 Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 1, normalized size of antiderivative = 0.01 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\int { \frac {{\left (2 \, x^{3} - x\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )}}{{\left (x^{4} + 1\right )} x^{2}} \,d x } \]

input

integrate((x^2+1)*(2*x^3-x)^(1/3)/x^2/(x^4+1),x, algorithm="giac")
 

output

undef
 

3.17.99.9 Mupad [N/A]

Not integrable

Time = 5.65 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.25 \[ \int \frac {\left (1+x^2\right ) \sqrt [3]{-x+2 x^3}}{x^2 \left (1+x^4\right )} \, dx=\int \frac {{\left (2\,x^3-x\right )}^{1/3}\,\left (x^2+1\right )}{x^2\,\left (x^4+1\right )} \,d x \]

input

int(((2*x^3 - x)^(1/3)*(x^2 + 1))/(x^2*(x^4 + 1)),x)
 

output

int(((2*x^3 - x)^(1/3)*(x^2 + 1))/(x^2*(x^4 + 1)), x)