\(\int \frac {\sqrt {x}}{a+c x^4} \, dx\) [137]

Optimal result

Integrand size = 15, antiderivative size = 228 \[ \int \frac {\sqrt {x}}{a+c x^4} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{5/8} c^{3/8}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt {2} (-a)^{5/8} c^{3/8}}+\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{5/8} c^{3/8}}-\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 (-a)^{5/8} c^{3/8}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{-a}+\sqrt [4]{c} x}\right )}{2 \sqrt {2} (-a)^{5/8} c^{3/8}} \] Output:

-1/4*arctan(-1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8))*2^(1/2)/(-a)^(5/8)/c^(3 
/8)-1/4*arctan(1+2^(1/2)*c^(1/8)*x^(1/2)/(-a)^(1/8))*2^(1/2)/(-a)^(5/8)/c^ 
(3/8)+1/2*arctan(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(5/8)/c^(3/8)-1/2*arctan 
h(c^(1/8)*x^(1/2)/(-a)^(1/8))/(-a)^(5/8)/c^(3/8)+1/4*arctanh(2^(1/2)*(-a)^ 
(1/8)*c^(1/8)*x^(1/2)/((-a)^(1/4)+c^(1/4)*x))*2^(1/2)/(-a)^(5/8)/c^(3/8)
 

Mathematica [A] (verified)

Time = 0.68 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt {x}}{a+c x^4} \, dx=\frac {\sqrt {2-\sqrt {2}} \arctan \left (\frac {\sqrt {1-\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )-\sqrt {2+\sqrt {2}} \arctan \left (\frac {\sqrt {1+\frac {1}{\sqrt {2}}} \left (\sqrt [4]{a}-\sqrt [4]{c} x\right )}{\sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}\right )+\sqrt {2-\sqrt {2}} \text {arctanh}\left (\frac {\sqrt {2+\sqrt {2}} \sqrt [8]{a} \sqrt [8]{c} \sqrt {x}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )-\sqrt {2+\sqrt {2}} \text {arctanh}\left (\frac {\sqrt [8]{a} \sqrt [8]{c} \sqrt {-\left (\left (-2+\sqrt {2}\right ) x\right )}}{\sqrt [4]{a}+\sqrt [4]{c} x}\right )}{4 a^{5/8} c^{3/8}} \] Input:

Integrate[Sqrt[x]/(a + c*x^4),x]
 

Output:

(Sqrt[2 - Sqrt[2]]*ArcTan[(Sqrt[1 - 1/Sqrt[2]]*(a^(1/4) - c^(1/4)*x))/(a^( 
1/8)*c^(1/8)*Sqrt[x])] - Sqrt[2 + Sqrt[2]]*ArcTan[(Sqrt[1 + 1/Sqrt[2]]*(a^ 
(1/4) - c^(1/4)*x))/(a^(1/8)*c^(1/8)*Sqrt[x])] + Sqrt[2 - Sqrt[2]]*ArcTanh 
[(Sqrt[2 + Sqrt[2]]*a^(1/8)*c^(1/8)*Sqrt[x])/(a^(1/4) + c^(1/4)*x)] - Sqrt 
[2 + Sqrt[2]]*ArcTanh[(a^(1/8)*c^(1/8)*Sqrt[-((-2 + Sqrt[2])*x)])/(a^(1/4) 
 + c^(1/4)*x)])/(4*a^(5/8)*c^(3/8))
 

Rubi [A] (verified)

Time = 0.94 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.44, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.867, Rules used = {851, 829, 826, 827, 218, 221, 1476, 1082, 217, 1479, 25, 27, 1103}

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.

\(\displaystyle \int \frac {\sqrt {x}}{a+c x^4} \, dx\)

\(\Big \downarrow \) 851

\(\displaystyle 2 \int \frac {x}{c x^4+a}d\sqrt {x}\)

\(\Big \downarrow \) 829

\(\displaystyle 2 \left (-\frac {\int \frac {x}{\sqrt {-a}-\sqrt {c} x^2}d\sqrt {x}}{2 \sqrt {-a}}-\frac {\int \frac {x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt {-a}}\right )\)

\(\Big \downarrow \) 826

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

\(\Big \downarrow \) 827

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

\(\Big \downarrow \) 218

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

\(\Big \downarrow \) 221

\(\displaystyle 2 \left (-\frac {\frac {\int \frac {\sqrt [4]{c} x+\sqrt [4]{-a}}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}-\frac {\int \frac {\sqrt [4]{-a}-\sqrt [4]{c} x}{\sqrt {c} x^2+\sqrt {-a}}d\sqrt {x}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}\right )\)

\(\Big \downarrow \) 1476

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

\(\Big \downarrow \) 1082

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

\(\Big \downarrow \) 217

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

\(\Big \downarrow \) 1479

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1103

\(\displaystyle 2 \left (-\frac {\frac {\text {arctanh}\left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}-\frac {\arctan \left (\frac {\sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{2 \sqrt [8]{-a} c^{3/8}}}{2 \sqrt {-a}}-\frac {\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}+1\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [8]{c} \sqrt {x}}{\sqrt [8]{-a}}\right )}{\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}-\frac {\frac {\log \left (\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}-\frac {\log \left (-\sqrt {2} \sqrt [8]{-a} \sqrt [8]{c} \sqrt {x}+\sqrt [4]{-a}+\sqrt [4]{c} x\right )}{2 \sqrt {2} \sqrt [8]{-a} \sqrt [8]{c}}}{2 \sqrt [4]{c}}}{2 \sqrt {-a}}\right )\)

Input:

Int[Sqrt[x]/(a + c*x^4),x]
 

Output:

2*(-1/2*(-1/2*ArcTan[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/((-a)^(1/8)*c^(3/8)) + 
ArcTanh[(c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(2*(-a)^(1/8)*c^(3/8)))/Sqrt[-a] - ( 
(-(ArcTan[1 - (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(Sqrt[2]*(-a)^(1/8)*c^ 
(1/8))) + ArcTan[1 + (Sqrt[2]*c^(1/8)*Sqrt[x])/(-a)^(1/8)]/(Sqrt[2]*(-a)^( 
1/8)*c^(1/8)))/(2*c^(1/4)) - (-1/2*Log[(-a)^(1/4) - Sqrt[2]*(-a)^(1/8)*c^( 
1/8)*Sqrt[x] + c^(1/4)*x]/(Sqrt[2]*(-a)^(1/8)*c^(1/8)) + Log[(-a)^(1/4) + 
Sqrt[2]*(-a)^(1/8)*c^(1/8)*Sqrt[x] + c^(1/4)*x]/(2*Sqrt[2]*(-a)^(1/8)*c^(1 
/8)))/(2*c^(1/4)))/(2*Sqrt[-a]))
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

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

rule 218

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

rule 221

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

rule 826

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

rule 827

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

rule 829

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

rule 851

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = 
 Denominator[m]}, Simp[k/c   Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^ 
n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && 
FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1476

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
2*(d/e), 2]}, Simp[e/(2*c)   Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 
e/(2*c)   Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] 
 && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
 

rule 1479

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 
-2*(d/e), 2]}, Simp[e/(2*c*q)   Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], 
 x] + Simp[e/(2*c*q)   Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F 
reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.45 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.13

method	result	size
derivativedivides	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{5}}}{4 c}\)	\(29\)
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{8}+a \right )}{\sum }\frac {\ln \left (\sqrt {x}-\textit {\_R} \right )}{\textit {\_R}^{5}}}{4 c}\)	\(29\)

Input:

int(x^(1/2)/(c*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.09 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.26 \[ \int \frac {\sqrt {x}}{a+c x^4} \, dx=-\left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{2} c \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{2} c \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) - \left (\frac {1}{8} i + \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {1}{8}} \log \left (\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} a^{2} c \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + \left (\frac {1}{8} i - \frac {1}{8}\right ) \, \sqrt {2} \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {1}{8}} \log \left (-\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} a^{2} c \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) - \frac {1}{4} \, \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {1}{8}} \log \left (a^{2} c \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + \frac {1}{4} i \, \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {1}{8}} \log \left (i \, a^{2} c \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) - \frac {1}{4} i \, \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {1}{8}} \log \left (-i \, a^{2} c \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) + \frac {1}{4} \, \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {1}{8}} \log \left (-a^{2} c \left (-\frac {1}{a^{5} c^{3}}\right )^{\frac {3}{8}} + \sqrt {x}\right ) \] Input:

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

Output:

-(1/8*I - 1/8)*sqrt(2)*(-1/(a^5*c^3))^(1/8)*log((1/2*I + 1/2)*sqrt(2)*a^2* 
c*(-1/(a^5*c^3))^(3/8) + sqrt(x)) + (1/8*I + 1/8)*sqrt(2)*(-1/(a^5*c^3))^( 
1/8)*log(-(1/2*I - 1/2)*sqrt(2)*a^2*c*(-1/(a^5*c^3))^(3/8) + sqrt(x)) - (1 
/8*I + 1/8)*sqrt(2)*(-1/(a^5*c^3))^(1/8)*log((1/2*I - 1/2)*sqrt(2)*a^2*c*( 
-1/(a^5*c^3))^(3/8) + sqrt(x)) + (1/8*I - 1/8)*sqrt(2)*(-1/(a^5*c^3))^(1/8 
)*log(-(1/2*I + 1/2)*sqrt(2)*a^2*c*(-1/(a^5*c^3))^(3/8) + sqrt(x)) - 1/4*( 
-1/(a^5*c^3))^(1/8)*log(a^2*c*(-1/(a^5*c^3))^(3/8) + sqrt(x)) + 1/4*I*(-1/ 
(a^5*c^3))^(1/8)*log(I*a^2*c*(-1/(a^5*c^3))^(3/8) + sqrt(x)) - 1/4*I*(-1/( 
a^5*c^3))^(1/8)*log(-I*a^2*c*(-1/(a^5*c^3))^(3/8) + sqrt(x)) + 1/4*(-1/(a^ 
5*c^3))^(1/8)*log(-a^2*c*(-1/(a^5*c^3))^(3/8) + sqrt(x))
 

Sympy [A] (verification not implemented)

Time = 8.58 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {x}}{a+c x^4} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge c = 0 \\- \frac {2}{5 c x^{\frac {5}{2}}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a} & \text {for}\: c = 0 \\- \frac {\left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (\sqrt {x} - \sqrt [8]{- \frac {a}{c}} \right )}}{4 a} + \frac {\left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (\sqrt {x} + \sqrt [8]{- \frac {a}{c}} \right )}}{4 a} + \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (- 4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a} - \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \log {\left (4 \sqrt {2} \sqrt {x} \sqrt [8]{- \frac {a}{c}} + 4 x + 4 \sqrt [4]{- \frac {a}{c}} \right )}}{8 a} - \frac {\left (- \frac {a}{c}\right )^{\frac {3}{8}} \operatorname {atan}{\left (\frac {\sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} \right )}}{2 a} + \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} - 1 \right )}}{4 a} + \frac {\sqrt {2} \left (- \frac {a}{c}\right )^{\frac {3}{8}} \operatorname {atan}{\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt [8]{- \frac {a}{c}}} + 1 \right )}}{4 a} & \text {otherwise} \end {cases} \] Input:

integrate(x**(1/2)/(c*x**4+a),x)
 

Output:

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(c, 0)), (-2/(5*c*x**(5/2)), Eq(a, 0 
)), (2*x**(3/2)/(3*a), Eq(c, 0)), (-(-a/c)**(3/8)*log(sqrt(x) - (-a/c)**(1 
/8))/(4*a) + (-a/c)**(3/8)*log(sqrt(x) + (-a/c)**(1/8))/(4*a) + sqrt(2)*(- 
a/c)**(3/8)*log(-4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 4*(-a/c)**(1/4))/ 
(8*a) - sqrt(2)*(-a/c)**(3/8)*log(4*sqrt(2)*sqrt(x)*(-a/c)**(1/8) + 4*x + 
4*(-a/c)**(1/4))/(8*a) - (-a/c)**(3/8)*atan(sqrt(x)/(-a/c)**(1/8))/(2*a) + 
 sqrt(2)*(-a/c)**(3/8)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) - 1)/(4*a) + sqr 
t(2)*(-a/c)**(3/8)*atan(sqrt(2)*sqrt(x)/(-a/c)**(1/8) + 1)/(4*a), True))
 

Maxima [F]

\[ \int \frac {\sqrt {x}}{a+c x^4} \, dx=\int { \frac {\sqrt {x}}{c x^{4} + a} \,d x } \] Input:

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

Output:

integrate(sqrt(x)/(c*x^4 + a), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (151) = 302\).

Time = 0.22 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.95 \[ \int \frac {\sqrt {x}}{a+c x^4} \, dx=-\frac {\left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (-\frac {\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {3}{8}} \arctan \left (-\frac {\sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} - 2 \, \sqrt {x}}{\sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}}}\right )}{2 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (-\sqrt {x} \sqrt {\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {2 \, \sqrt {2} + 4}} - \frac {\left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} + \frac {\left (\frac {a}{c}\right )^{\frac {3}{8}} \log \left (-\sqrt {x} \sqrt {-\sqrt {2} + 2} \left (\frac {a}{c}\right )^{\frac {1}{8}} + x + \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}{4 \, a \sqrt {-2 \, \sqrt {2} + 4}} \] Input:

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

Output:

-1/2*(a/c)^(3/8)*arctan((sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt 
(sqrt(2) + 2)*(a/c)^(1/8)))/(a*sqrt(2*sqrt(2) + 4)) - 1/2*(a/c)^(3/8)*arct 
an(-(sqrt(-sqrt(2) + 2)*(a/c)^(1/8) - 2*sqrt(x))/(sqrt(sqrt(2) + 2)*(a/c)^ 
(1/8)))/(a*sqrt(2*sqrt(2) + 4)) + 1/2*(a/c)^(3/8)*arctan((sqrt(sqrt(2) + 2 
)*(a/c)^(1/8) + 2*sqrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a*sqrt(-2*sq 
rt(2) + 4)) + 1/2*(a/c)^(3/8)*arctan(-(sqrt(sqrt(2) + 2)*(a/c)^(1/8) - 2*s 
qrt(x))/(sqrt(-sqrt(2) + 2)*(a/c)^(1/8)))/(a*sqrt(-2*sqrt(2) + 4)) + 1/4*( 
a/c)^(3/8)*log(sqrt(x)*sqrt(sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a 
*sqrt(2*sqrt(2) + 4)) - 1/4*(a/c)^(3/8)*log(-sqrt(x)*sqrt(sqrt(2) + 2)*(a/ 
c)^(1/8) + x + (a/c)^(1/4))/(a*sqrt(2*sqrt(2) + 4)) - 1/4*(a/c)^(3/8)*log( 
sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x + (a/c)^(1/4))/(a*sqrt(-2*sqrt( 
2) + 4)) + 1/4*(a/c)^(3/8)*log(-sqrt(x)*sqrt(-sqrt(2) + 2)*(a/c)^(1/8) + x 
 + (a/c)^(1/4))/(a*sqrt(-2*sqrt(2) + 4))
 

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.52 \[ \int \frac {\sqrt {x}}{a+c x^4} \, dx=\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}}{{\left (-a\right )}^{1/8}}\right )}{2\,{\left (-a\right )}^{5/8}\,c^{3/8}}+\frac {\mathrm {atan}\left (\frac {c^{1/8}\,\sqrt {x}\,1{}\mathrm {i}}{{\left (-a\right )}^{1/8}}\right )\,1{}\mathrm {i}}{2\,{\left (-a\right )}^{5/8}\,c^{3/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}-\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{4}+\frac {1}{4}{}\mathrm {i}\right )}{{\left (-a\right )}^{5/8}\,c^{3/8}}+\frac {\sqrt {2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,c^{1/8}\,\sqrt {x}\,\left (\frac {1}{2}+\frac {1}{2}{}\mathrm {i}\right )}{{\left (-a\right )}^{1/8}}\right )\,\left (-\frac {1}{4}-\frac {1}{4}{}\mathrm {i}\right )}{{\left (-a\right )}^{5/8}\,c^{3/8}} \] Input:

int(x^(1/2)/(a + c*x^4),x)
 

Output:

atan((c^(1/8)*x^(1/2))/(-a)^(1/8))/(2*(-a)^(5/8)*c^(3/8)) + (atan((c^(1/8) 
*x^(1/2)*1i)/(-a)^(1/8))*1i)/(2*(-a)^(5/8)*c^(3/8)) - (2^(1/2)*atan((2^(1/ 
2)*c^(1/8)*x^(1/2)*(1/2 - 1i/2))/(-a)^(1/8))*(1/4 - 1i/4))/((-a)^(5/8)*c^( 
3/8)) - (2^(1/2)*atan((2^(1/2)*c^(1/8)*x^(1/2)*(1/2 + 1i/2))/(-a)^(1/8))*( 
1/4 + 1i/4))/((-a)^(5/8)*c^(3/8))
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 649, normalized size of antiderivative = 2.85 \[ \int \frac {\sqrt {x}}{a+c x^4} \, dx =\text {Too large to display} \] Input:

int(x^(1/2)/(c*x^4+a),x)
 

Output:

(c**(5/8)*a**(3/8)*(2*sqrt(sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sq 
rt( - sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 
 2))) - 2*sqrt(sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) - 
 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2))) - 2*sqrt(sqrt( 
2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*sqrt(x)*c 
**(1/4))/(c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2))) + 2*sqrt(sqrt(2) + 2)*atan 
((c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + 2*sqrt(x)*c**(1/4))/(c**(1/8)*a 
**(1/8)*sqrt(sqrt(2) + 2))) - 2*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((c**(1/8 
)*a**(1/8)*sqrt(sqrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt 
( - sqrt(2) + 2))) - 2*sqrt( - sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt(s 
qrt(2) + 2) - 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2)) 
) + 2*sqrt( - sqrt(2) + 2)*sqrt(2)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 
2) + 2*sqrt(x)*c**(1/4))/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2))) + 2*sqr 
t( - sqrt(2) + 2)*atan((c**(1/8)*a**(1/8)*sqrt(sqrt(2) + 2) + 2*sqrt(x)*c* 
*(1/4))/(c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2))) + sqrt( - sqrt(2) + 2)*s 
qrt(2)*log( - sqrt(x)*c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + a**(1/4) + 
c**(1/4)*x) - sqrt( - sqrt(2) + 2)*sqrt(2)*log(sqrt(x)*c**(1/8)*a**(1/8)*s 
qrt( - sqrt(2) + 2) + a**(1/4) + c**(1/4)*x) + sqrt( - sqrt(2) + 2)*log( - 
 sqrt(x)*c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2) + a**(1/4) + c**(1/4)*x) - 
 sqrt( - sqrt(2) + 2)*log(sqrt(x)*c**(1/8)*a**(1/8)*sqrt( - sqrt(2) + 2...