3.22.40 \(\int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x (-1+x^3)} \, dx\) [2140]

3.22.40.1 Optimal result

Integrand size = 25, antiderivative size = 156 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\frac {2}{3} \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )-\frac {1}{3} \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{x^3+x^5}}{-x^2+\sqrt {x^3+x^5}}\right )-\frac {2}{3} \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{2} x}{\sqrt [4]{x^3+x^5}}\right )+\frac {1}{3} \sqrt {2} \text {arctanh}\left (\frac {\frac {x^2}{\sqrt {2}}+\frac {\sqrt {x^3+x^5}}{\sqrt {2}}}{x \sqrt [4]{x^3+x^5}}\right ) \]

output

2/3*2^(1/4)*arctan(2^(1/4)*x/(x^5+x^3)^(1/4))-1/3*2^(1/2)*arctan(2^(1/2)*x 
*(x^5+x^3)^(1/4)/(-x^2+(x^5+x^3)^(1/2)))-2/3*2^(1/4)*arctanh(2^(1/4)*x/(x^ 
5+x^3)^(1/4))+1/3*2^(1/2)*arctanh((1/2*2^(1/2)*x^2+1/2*(x^5+x^3)^(1/2)*2^( 
1/2))/x/(x^5+x^3)^(1/4))
 

3.22.40.2 Mathematica [A] (verified)

Time = 1.84 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.10 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\frac {\sqrt [4]{2} \sqrt [4]{x^3+x^5} \left (2 \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\sqrt [4]{2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}-\sqrt {1+x^2}}\right )-2 \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{x}}{\sqrt [4]{1+x^2}}\right )+\sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{x} \sqrt [4]{1+x^2}}{\sqrt {x}+\sqrt {1+x^2}}\right )\right )}{3 x^{3/4} \sqrt [4]{1+x^2}} \]

input

Integrate[((1 + x)*(x^3 + x^5)^(1/4))/(x*(-1 + x^3)),x]
 

output

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

3.22.40.3 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.

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

\(\Big \downarrow \) 2467

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2035

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

\(\Big \downarrow \) 7239

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

\(\Big \downarrow \) 7276

\(\displaystyle -\frac {4 \sqrt [4]{x^5+x^3} \int \left (\frac {\sqrt [4]{x^2+1} \left (1-\sqrt [4]{x}\right )}{12 \left (\sqrt {x}+\sqrt [4]{x}+1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}-1\right )}-\frac {\sqrt [4]{x^2+1}}{3 \left (\sqrt {x}+1\right )}+\frac {\left (\sqrt [4]{x}+1\right ) \sqrt [4]{x^2+1}}{12 \left (\sqrt {x}-\sqrt [4]{x}+1\right )}+\frac {\left (2 \sqrt {x}-1\right ) \sqrt [4]{x^2+1}}{6 \left (x-\sqrt {x}+1\right )}\right )d\sqrt [4]{x}}{x^{3/4} \sqrt [4]{x^2+1}}\)

input

Int[((1 + x)*(x^3 + x^5)^(1/4))/(x*(-1 + x^3)),x]
 

output

$Aborted
 

3.22.40.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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]
 

3.22.40.4 Maple [A] (verified)

Time = 12.35 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.30

method	result	size
pseudoelliptic	\(-\frac {\ln \left (\frac {-2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2^{\frac {1}{4}} x -\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}}{3}-\frac {2 \arctan \left (\frac {2^{\frac {3}{4}} \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{2 x}\right ) 2^{\frac {1}{4}}}{3}+\frac {\ln \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}{-\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}\, x +x^{2}+\sqrt {x^{3} \left (x^{2}+1\right )}}\right ) \sqrt {2}}{6}+\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}+x}{x}\right ) \sqrt {2}}{3}+\frac {\arctan \left (\frac {\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}} \sqrt {2}-x}{x}\right ) \sqrt {2}}{3}\)	\(203\)
trager	\(\text {Expression too large to display}\)	\(737\)

input

int((1+x)*(x^5+x^3)^(1/4)/x/(x^3-1),x,method=_RETURNVERBOSE)
 

output

-1/3*ln((-2^(1/4)*x-(x^3*(x^2+1))^(1/4))/(2^(1/4)*x-(x^3*(x^2+1))^(1/4)))* 
2^(1/4)-2/3*arctan(1/2*2^(3/4)/x*(x^3*(x^2+1))^(1/4))*2^(1/4)+1/6*ln(((x^3 
*(x^2+1))^(1/4)*2^(1/2)*x+x^2+(x^3*(x^2+1))^(1/2))/(-(x^3*(x^2+1))^(1/4)*2 
^(1/2)*x+x^2+(x^3*(x^2+1))^(1/2)))*2^(1/2)+1/3*arctan(((x^3*(x^2+1))^(1/4) 
*2^(1/2)+x)/x)*2^(1/2)+1/3*arctan(((x^3*(x^2+1))^(1/4)*2^(1/2)-x)/x)*2^(1/ 
2)
 

3.22.40.5 Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 3.00 (sec) , antiderivative size = 656, normalized size of antiderivative = 4.21 \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\left (\frac {1}{12} i - \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} - \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (\left (i + 1\right ) \, x^{4} - \left (i + 1\right ) \, x^{3} + \left (i + 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) - \left (\frac {1}{12} i - \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + \left (2 i - 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (-\left (i + 1\right ) \, x^{4} + \left (i + 1\right ) \, x^{3} - \left (i + 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) - \left (\frac {1}{12} i + \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (-\left (i - 1\right ) \, x^{4} + \left (i - 1\right ) \, x^{3} - \left (i - 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) + \left (\frac {1}{12} i + \frac {1}{12}\right ) \, \sqrt {2} \log \left (\frac {-4 i \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} - \left (2 i + 2\right ) \, \sqrt {2} \sqrt {x^{5} + x^{3}} x + \sqrt {2} {\left (\left (i - 1\right ) \, x^{4} - \left (i - 1\right ) \, x^{3} + \left (i - 1\right ) \, x^{2}\right )} - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} + x^{3} + x^{2}}\right ) - \frac {1}{6} \cdot 2^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} + 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x + 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) + \frac {1}{6} \cdot 2^{\frac {1}{4}} \log \left (-\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} - 2^{\frac {3}{4}} {\left (x^{4} + 2 \, x^{3} + x^{2}\right )} - 4 \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x + 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) - \frac {1}{6} i \cdot 2^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (i \, x^{4} + 2 i \, x^{3} + i \, x^{2}\right )} - 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) + \frac {1}{6} i \cdot 2^{\frac {1}{4}} \log \left (\frac {4 \, \sqrt {2} {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x^{2} + 2^{\frac {3}{4}} {\left (-i \, x^{4} - 2 i \, x^{3} - i \, x^{2}\right )} + 4 i \cdot 2^{\frac {1}{4}} \sqrt {x^{5} + x^{3}} x - 4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{x^{4} - 2 \, x^{3} + x^{2}}\right ) \]

input

integrate((1+x)*(x^5+x^3)^(1/4)/x/(x^3-1),x, algorithm="fricas")
 

output

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

3.22.40.6 Sympy [F]

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

input

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

output

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

3.22.40.7 Maxima [F]

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

input

integrate((1+x)*(x^5+x^3)^(1/4)/x/(x^3-1),x, algorithm="maxima")
 

output

integrate((x^5 + x^3)^(1/4)*(x + 1)/((x^3 - 1)*x), x)
 

3.22.40.8 Giac [F]

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

input

integrate((1+x)*(x^5+x^3)^(1/4)/x/(x^3-1),x, algorithm="giac")
 

output

integrate((x^5 + x^3)^(1/4)*(x + 1)/((x^3 - 1)*x), x)
 

3.22.40.9 Mupad [F(-1)]

Timed out. \[ \int \frac {(1+x) \sqrt [4]{x^3+x^5}}{x \left (-1+x^3\right )} \, dx=\int \frac {{\left (x^5+x^3\right )}^{1/4}\,\left (x+1\right )}{x\,\left (x^3-1\right )} \,d x \]

input

int(((x^3 + x^5)^(1/4)*(x + 1))/(x*(x^3 - 1)),x)
 

output

int(((x^3 + x^5)^(1/4)*(x + 1))/(x*(x^3 - 1)), x)