\(\int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx\) [1812]

Optimal result

Integrand size = 22, antiderivative size = 123 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=-\arctan \left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )+\text {arctanh}\left (\frac {x}{\sqrt [4]{x^3+x^4}}\right )-\frac {1}{2} \text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+\log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-2 \log \left (\sqrt [4]{x^3+x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}^3+4 \text {$\#$1}^7}\&\right ] \]

[Out]

Unintegrable

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(378\) vs. \(2(123)=246\).

Time = 0.47 (sec) , antiderivative size = 378, normalized size of antiderivative = 3.07, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {2081, 919, 65, 338, 304, 209, 212, 6860, 95} \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=-\frac {\sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \arctan \left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}}+\frac {\sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}}-\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^4+x^3} \text {arctanh}\left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{x+1}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{x+1}} \]

[In]

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

[Out]

-(((x^3 + x^4)^(1/4)*ArcTan[x^(1/4)/(1 + x)^(1/4)])/(x^(3/4)*(1 + x)^(1/4))) + (((23 - Sqrt[17])/2)^(1/4)*(x^3
+ x^4)^(1/4)*ArcTan[((2/(5 + Sqrt[17]))^(1/4)*x^(1/4))/(1 + x)^(1/4)])/(Sqrt[17]*x^(3/4)*(1 + x)^(1/4)) + (((
23 + Sqrt[17])/2)^(1/4)*(x^3 + x^4)^(1/4)*ArcTan[((5 + Sqrt[17])^(1/4)*x^(1/4))/(Sqrt[2]*(1 + x)^(1/4))])/(Sqr
t[17]*x^(3/4)*(1 + x)^(1/4)) + ((x^3 + x^4)^(1/4)*ArcTanh[x^(1/4)/(1 + x)^(1/4)])/(x^(3/4)*(1 + x)^(1/4)) - ((
(23 - Sqrt[17])/2)^(1/4)*(x^3 + x^4)^(1/4)*ArcTanh[((2/(5 + Sqrt[17]))^(1/4)*x^(1/4))/(1 + x)^(1/4)])/(Sqrt[17
]*x^(3/4)*(1 + x)^(1/4)) - (((23 + Sqrt[17])/2)^(1/4)*(x^3 + x^4)^(1/4)*ArcTanh[((5 + Sqrt[17])^(1/4)*x^(1/4))
/(Sqrt[2]*(1 + x)^(1/4))])/(Sqrt[17]*x^(3/4)*(1 + x)^(1/4))

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 304

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}
, Dist[s/(2*b), Int[1/(r + s*x^2), x], x] - Dist[s/(2*b), Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !
GtQ[a/b, 0]

Rule 338

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[a^(p + (m + 1)/n), Subst[Int[x^m/(1 - b*x^n)^(
p + (m + 1)/n + 1), x], x, x/(a + b*x^n)^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[
p, -2^(-1)] && IntegersQ[m, p + (m + 1)/n]

Rule 919

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Di
st[e*(g/c), Int[(d + e*x)^(m - 1)*(f + g*x)^(n - 1), x], x] + Dist[1/c, Int[Simp[c*d*f - a*e*g + (c*e*f + c*d*
g - b*e*g)*x, x]*(d + e*x)^(m - 1)*((f + g*x)^(n - 1)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g
}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[m] &&  !IntegerQ[n] && GtQ[m, 0] &
& GtQ[n, 0]

Rule 2081

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt [4]{x^3+x^4} \int \frac {x^{3/4} \sqrt [4]{1+x}}{-2+x+2 x^2} \, dx}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4}} \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \int \frac {2+x}{\sqrt [4]{x} (1+x)^{3/4} \left (-2+x+2 x^2\right )} \, dx}{2 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \int \left (\frac {1+\frac {7}{\sqrt {17}}}{\sqrt [4]{x} (1+x)^{3/4} \left (1-\sqrt {17}+4 x\right )}+\frac {1-\frac {7}{\sqrt {17}}}{\sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt {17}+4 x\right )}\right ) \, dx}{2 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (1+x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\left (2 \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (1+\sqrt {17}+4 x\right )} \, dx}{34 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \int \frac {1}{\sqrt [4]{x} (1+x)^{3/4} \left (1-\sqrt {17}+4 x\right )} \, dx}{34 x^{3/4} \sqrt [4]{1+x}} \\ & = \frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{x^3+x^4} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1+\sqrt {17}-\left (-3+\sqrt {17}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 x^{3/4} \sqrt [4]{1+x}}+\frac {\left (2 \left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {x^2}{1-\sqrt {17}-\left (-3-\sqrt {17}\right ) x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\sqrt {2} \left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3-\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\sqrt {2} \left (17-7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {5+\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3-\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}}-\frac {\left (\sqrt {2} \left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}-\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3+\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}}+\frac {\left (\sqrt {2} \left (17+7 \sqrt {17}\right ) \sqrt [4]{x^3+x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {5-\sqrt {17}}+\sqrt {2} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{17 \left (3+\sqrt {17}\right ) x^{3/4} \sqrt [4]{1+x}} \\ & = -\frac {\sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \arctan \left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}}+\frac {\sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{\frac {1}{2} \left (23-\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{\frac {2}{5+\sqrt {17}}} \sqrt [4]{x}}{\sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}}-\frac {\sqrt [4]{\frac {1}{2} \left (23+\sqrt {17}\right )} \sqrt [4]{x^3+x^4} \text {arctanh}\left (\frac {\sqrt [4]{5+\sqrt {17}} \sqrt [4]{x}}{\sqrt {2} \sqrt [4]{1+x}}\right )}{\sqrt {17} x^{3/4} \sqrt [4]{1+x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=-\frac {x^{9/4} (1+x)^{3/4} \left (8 \arctan \left (\sqrt [4]{\frac {x}{1+x}}\right )-8 \text {arctanh}\left (\sqrt [4]{\frac {x}{1+x}}\right )+\text {RootSum}\left [1-5 \text {$\#$1}^4+2 \text {$\#$1}^8\&,\frac {-\log (x)+4 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right )+2 \log (x) \text {$\#$1}^4-8 \log \left (\sqrt [4]{1+x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{-5 \text {$\#$1}^3+4 \text {$\#$1}^7}\&\right ]\right )}{8 \left (x^3 (1+x)\right )^{3/4}} \]

[In]

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

[Out]

-1/8*(x^(9/4)*(1 + x)^(3/4)*(8*ArcTan[(x/(1 + x))^(1/4)] - 8*ArcTanh[(x/(1 + x))^(1/4)] + RootSum[1 - 5*#1^4 +
2*#1^8 & , (-Log[x] + 4*Log[(1 + x)^(1/4) - x^(1/4)*#1] + 2*Log[x]*#1^4 - 8*Log[(1 + x)^(1/4) - x^(1/4)*#1]*#
1^4)/(-5*#1^3 + 4*#1^7) & ]))/(x^3*(1 + x))^(3/4)

Maple [N/A] (verified)

Time = 11.78 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90

method	result	size
pseudoelliptic	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \textit {\_Z}^{8}-5 \textit {\_Z}^{4}+1\right )}{\sum }\frac {\left (2 \textit {\_R}^{4}-1\right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (4 \textit {\_R}^{4}-5\right )}\right )}{2}+\frac {\ln \left (\frac {x +\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )}{2}-\frac {\ln \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}-x}{x}\right )}{2}+\arctan \left (\frac {\left (x^{3} \left (1+x \right )\right )^{\frac {1}{4}}}{x}\right )\)	\(111\)
trager	\(\text {Expression too large to display}\)	\(3016\)

[In]

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

[Out]

1/2*sum((2*_R^4-1)*ln((-_R*x+(x^3*(1+x))^(1/4))/x)/_R^3/(4*_R^4-5),_R=RootOf(2*_Z^8-5*_Z^4+1))+1/2*ln((x+(x^3*
(1+x))^(1/4))/x)-1/2*ln(((x^3*(1+x))^(1/4)-x)/x)+arctan((x^3*(1+x))^(1/4)/x)

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 3 vs. order 1.

Time = 0.28 (sec) , antiderivative size = 570, normalized size of antiderivative = 4.63 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=-\frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} \log \left (\frac {\sqrt {34} {\left (\sqrt {17} x + 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} + 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} \log \left (-\frac {\sqrt {34} {\left (\sqrt {17} x + 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {\sqrt {17} + 23}} - 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{68} \, \sqrt {34} \sqrt {-\sqrt {2} \sqrt {\sqrt {17} + 23}} \log \left (\frac {\sqrt {34} {\left (\sqrt {17} x + 17 \, x\right )} \sqrt {-\sqrt {2} \sqrt {\sqrt {17} + 23}} + 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{68} \, \sqrt {34} \sqrt {-\sqrt {2} \sqrt {\sqrt {17} + 23}} \log \left (-\frac {\sqrt {34} {\left (\sqrt {17} x + 17 \, x\right )} \sqrt {-\sqrt {2} \sqrt {\sqrt {17} + 23}} - 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} \log \left (\frac {\sqrt {34} {\left (\sqrt {17} x - 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} + 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{68} \, \sqrt {34} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} \log \left (-\frac {\sqrt {34} {\left (\sqrt {17} x - 17 \, x\right )} \sqrt {\sqrt {2} \sqrt {-\sqrt {17} + 23}} - 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{68} \, \sqrt {34} \sqrt {-\sqrt {2} \sqrt {-\sqrt {17} + 23}} \log \left (\frac {\sqrt {34} {\left (\sqrt {17} x - 17 \, x\right )} \sqrt {-\sqrt {2} \sqrt {-\sqrt {17} + 23}} + 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{68} \, \sqrt {34} \sqrt {-\sqrt {2} \sqrt {-\sqrt {17} + 23}} \log \left (-\frac {\sqrt {34} {\left (\sqrt {17} x - 17 \, x\right )} \sqrt {-\sqrt {2} \sqrt {-\sqrt {17} + 23}} - 272 \, {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \arctan \left (\frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {1}{2} \, \log \left (\frac {x + {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {1}{2} \, \log \left (-\frac {x - {\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

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

[Out]

-1/68*sqrt(34)*sqrt(sqrt(2)*sqrt(sqrt(17) + 23))*log((sqrt(34)*(sqrt(17)*x + 17*x)*sqrt(sqrt(2)*sqrt(sqrt(17)
+ 23)) + 272*(x^4 + x^3)^(1/4))/x) + 1/68*sqrt(34)*sqrt(sqrt(2)*sqrt(sqrt(17) + 23))*log(-(sqrt(34)*(sqrt(17)*
x + 17*x)*sqrt(sqrt(2)*sqrt(sqrt(17) + 23)) - 272*(x^4 + x^3)^(1/4))/x) - 1/68*sqrt(34)*sqrt(-sqrt(2)*sqrt(sqr
t(17) + 23))*log((sqrt(34)*(sqrt(17)*x + 17*x)*sqrt(-sqrt(2)*sqrt(sqrt(17) + 23)) + 272*(x^4 + x^3)^(1/4))/x)
+ 1/68*sqrt(34)*sqrt(-sqrt(2)*sqrt(sqrt(17) + 23))*log(-(sqrt(34)*(sqrt(17)*x + 17*x)*sqrt(-sqrt(2)*sqrt(sqrt(
17) + 23)) - 272*(x^4 + x^3)^(1/4))/x) + 1/68*sqrt(34)*sqrt(sqrt(2)*sqrt(-sqrt(17) + 23))*log((sqrt(34)*(sqrt(
17)*x - 17*x)*sqrt(sqrt(2)*sqrt(-sqrt(17) + 23)) + 272*(x^4 + x^3)^(1/4))/x) - 1/68*sqrt(34)*sqrt(sqrt(2)*sqrt
(-sqrt(17) + 23))*log(-(sqrt(34)*(sqrt(17)*x - 17*x)*sqrt(sqrt(2)*sqrt(-sqrt(17) + 23)) - 272*(x^4 + x^3)^(1/4
))/x) + 1/68*sqrt(34)*sqrt(-sqrt(2)*sqrt(-sqrt(17) + 23))*log((sqrt(34)*(sqrt(17)*x - 17*x)*sqrt(-sqrt(2)*sqrt
(-sqrt(17) + 23)) + 272*(x^4 + x^3)^(1/4))/x) - 1/68*sqrt(34)*sqrt(-sqrt(2)*sqrt(-sqrt(17) + 23))*log(-(sqrt(3
4)*(sqrt(17)*x - 17*x)*sqrt(-sqrt(2)*sqrt(-sqrt(17) + 23)) - 272*(x^4 + x^3)^(1/4))/x) + arctan((x^4 + x^3)^(1
/4)/x) + 1/2*log((x + (x^4 + x^3)^(1/4))/x) - 1/2*log(-(x - (x^4 + x^3)^(1/4))/x)

Sympy [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.15 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (x + 1\right )}}{2 x^{2} + x - 2}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{2 \, x^{2} + x - 2} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=\int { \frac {{\left (x^{4} + x^{3}\right )}^{\frac {1}{4}}}{2 \, x^{2} + x - 2} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 5.56 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.18 \[ \int \frac {\sqrt [4]{x^3+x^4}}{-2+x+2 x^2} \, dx=\int \frac {{\left (x^4+x^3\right )}^{1/4}}{2\,x^2+x-2} \,d x \]

[In]

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

[Out]

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