\(\int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx\) [140]

Optimal result

Integrand size = 28, antiderivative size = 196 \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=-\frac {2 \sqrt {c f+\sqrt {-c} g} \sqrt {1-\frac {c (f+g x)}{c f-\sqrt {-c} g}} \sqrt {1-\frac {c (f+g x)}{c f+\sqrt {-c} g}} \operatorname {EllipticPi}\left (\frac {e \left (c f+\sqrt {-c} g\right )}{c (e f-d g)},\arcsin \left (\frac {\sqrt {c} \sqrt {f+g x}}{\sqrt {c f+\sqrt {-c} g}}\right ),\frac {c f+\sqrt {-c} g}{c f-\sqrt {-c} g}\right )}{\sqrt {c} (e f-d g) \sqrt {1+c x^2}} \] Output:

-2*(c*f+(-c)^(1/2)*g)^(1/2)*(1-c*(g*x+f)/(c*f-(-c)^(1/2)*g))^(1/2)*(1-c*(g 
*x+f)/(c*f+(-c)^(1/2)*g))^(1/2)*EllipticPi(c^(1/2)*(g*x+f)^(1/2)/(c*f+(-c) 
^(1/2)*g)^(1/2),e*(c*f+(-c)^(1/2)*g)/c/(-d*g+e*f),((c*f+(-c)^(1/2)*g)/(c*f 
-(-c)^(1/2)*g))^(1/2))/c^(1/2)/(-d*g+e*f)/(c*x^2+1)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 22.59 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.33 \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=-\frac {2 i \sqrt {\frac {g \left (\frac {i}{\sqrt {c}}+x\right )}{f+g x}} \sqrt {-\frac {\frac {i g}{\sqrt {c}}-g x}{f+g x}} (f+g x) \left (\operatorname {EllipticF}\left (i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i g}{\sqrt {c} f+i g}\right )-\operatorname {EllipticPi}\left (\frac {\sqrt {c} (e f-d g)}{e \left (\sqrt {c} f+i g\right )},i \text {arcsinh}\left (\frac {\sqrt {-f-\frac {i g}{\sqrt {c}}}}{\sqrt {f+g x}}\right ),\frac {\sqrt {c} f-i g}{\sqrt {c} f+i g}\right )\right )}{\sqrt {-f-\frac {i g}{\sqrt {c}}} (e f-d g) \sqrt {1+c x^2}} \] Input:

Integrate[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 + c*x^2]),x]
 

Output:

((-2*I)*Sqrt[(g*(I/Sqrt[c] + x))/(f + g*x)]*Sqrt[-(((I*g)/Sqrt[c] - g*x)/( 
f + g*x))]*(f + g*x)*(EllipticF[I*ArcSinh[Sqrt[-f - (I*g)/Sqrt[c]]/Sqrt[f 
+ g*x]], (Sqrt[c]*f - I*g)/(Sqrt[c]*f + I*g)] - EllipticPi[(Sqrt[c]*(e*f - 
 d*g))/(e*(Sqrt[c]*f + I*g)), I*ArcSinh[Sqrt[-f - (I*g)/Sqrt[c]]/Sqrt[f + 
g*x]], (Sqrt[c]*f - I*g)/(Sqrt[c]*f + I*g)]))/(Sqrt[-f - (I*g)/Sqrt[c]]*(e 
*f - d*g)*Sqrt[1 + c*x^2])
 

Rubi [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(826\) vs. \(2(196)=392\).

Time = 2.78 (sec) , antiderivative size = 826, normalized size of antiderivative = 4.21, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {729, 25, 1540, 1416, 2222}

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.

\(\displaystyle \int \frac {1}{\sqrt {c x^2+1} (d+e x) \sqrt {f+g x}} \, dx\)

\(\Big \downarrow \) 729

\(\displaystyle 2 \int -\frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+1}}d\sqrt {f+g x}\)

\(\Big \downarrow \) 25

\(\displaystyle -2 \int \frac {1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+1}}d\sqrt {f+g x}\)

\(\Big \downarrow \) 1540

\(\displaystyle 2 \left (\frac {e \sqrt {c f^2+g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+g^2}\right ) \int \frac {\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+g^2}}+1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+1}}d\sqrt {f+g x}}{g \left (c d (2 e f-d g)+e^2 g\right )}-\frac {\sqrt {c} \left (-\sqrt {c} \sqrt {c f^2+g^2} (e f-d g)+c e f^2+e g^2\right ) \int \frac {1}{\sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+1}}d\sqrt {f+g x}}{g \sqrt {c f^2+g^2} \left (c d (2 e f-d g)+e^2 g\right )}\right )\)

\(\Big \downarrow \) 1416

\(\displaystyle 2 \left (\frac {e \sqrt {c f^2+g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+g^2}\right ) \int \frac {\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+g^2}}+1}{(e f-d g-e (f+g x)) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+1}}d\sqrt {f+g x}}{g \left (c d (2 e f-d g)+e^2 g\right )}-\frac {\sqrt [4]{c} \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}+1}{\left (\frac {c f^2}{g^2}+1\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+g^2}}+1\right )^2}} \left (-\sqrt {c} \sqrt {c f^2+g^2} (e f-d g)+c e f^2+e g^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+g^2}}+1\right )\right )}{2 g \sqrt [4]{c f^2+g^2} \sqrt {\frac {c f^2}{g^2}-\frac {2 c f (f+g x)}{g^2}+\frac {c (f+g x)^2}{g^2}+1} \left (c d (2 e f-d g)+e^2 g\right )}\right )\)

\(\Big \downarrow \) 2222

\(\displaystyle 2 \left (\frac {e \sqrt {c f^2+g^2} \left (\sqrt {c} (e f-d g)-e \sqrt {c f^2+g^2}\right ) \left (\frac {\left (e+\frac {\sqrt {c} (e f-d g)}{\sqrt {c f^2+g^2}}\right ) \text {arctanh}\left (\frac {\sqrt {c d^2+e^2} \sqrt {f+g x}}{\sqrt {e} \sqrt {e f-d g} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+1}}\right )}{2 \sqrt {e} \sqrt {c d^2+e^2} \sqrt {e f-d g}}-\frac {\left (\frac {\sqrt {c}}{e}-\frac {\sqrt {c f^2+g^2}}{e f-d g}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+1}{\left (\frac {c f^2}{g^2}+1\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+g^2}}+1\right )^2}} \operatorname {EllipticPi}\left (\frac {\left (\sqrt {c f^2+g^2} e+\sqrt {c} (e f-d g)\right )^2}{4 \sqrt {c} e (e f-d g) \sqrt {c f^2+g^2}},2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+g^2}}+1\right )\right )}{4 \sqrt [4]{c} \sqrt [4]{c f^2+g^2} \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+1}}\right )}{g \left (g e^2+c d (2 e f-d g)\right )}-\frac {\sqrt [4]{c} \left (c e f^2+e g^2-\sqrt {c} (e f-d g) \sqrt {c f^2+g^2}\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+g^2}}+1\right ) \sqrt {\frac {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+1}{\left (\frac {c f^2}{g^2}+1\right ) \left (\frac {\sqrt {c} (f+g x)}{\sqrt {c f^2+g^2}}+1\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} \sqrt {f+g x}}{\sqrt [4]{c f^2+g^2}}\right ),\frac {1}{2} \left (\frac {\sqrt {c} f}{\sqrt {c f^2+g^2}}+1\right )\right )}{2 g \sqrt [4]{c f^2+g^2} \left (g e^2+c d (2 e f-d g)\right ) \sqrt {\frac {c f^2}{g^2}-\frac {2 c (f+g x) f}{g^2}+\frac {c (f+g x)^2}{g^2}+1}}\right )\)

Input:

Int[1/((d + e*x)*Sqrt[f + g*x]*Sqrt[1 + c*x^2]),x]
 

Output:

2*(-1/2*(c^(1/4)*(c*e*f^2 + e*g^2 - Sqrt[c]*(e*f - d*g)*Sqrt[c*f^2 + g^2]) 
*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + g^2])*Sqrt[(1 + (c*f^2)/g^2 - (2*c* 
f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((1 + (c*f^2)/g^2)*(1 + (Sqrt[c]*( 
f + g*x))/Sqrt[c*f^2 + g^2])^2)]*EllipticF[2*ArcTan[(c^(1/4)*Sqrt[f + g*x] 
)/(c*f^2 + g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt[c*f^2 + g^2])/2])/(g*(c*f^2 
+ g^2)^(1/4)*(e^2*g + c*d*(2*e*f - d*g))*Sqrt[1 + (c*f^2)/g^2 - (2*c*f*(f 
+ g*x))/g^2 + (c*(f + g*x)^2)/g^2]) + (e*Sqrt[c*f^2 + g^2]*(Sqrt[c]*(e*f - 
 d*g) - e*Sqrt[c*f^2 + g^2])*(((e + (Sqrt[c]*(e*f - d*g))/Sqrt[c*f^2 + g^2 
])*ArcTanh[(Sqrt[c*d^2 + e^2]*Sqrt[f + g*x])/(Sqrt[e]*Sqrt[e*f - d*g]*Sqrt 
[1 + (c*f^2)/g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])])/(2*Sqrt 
[e]*Sqrt[c*d^2 + e^2]*Sqrt[e*f - d*g]) - ((Sqrt[c]/e - Sqrt[c*f^2 + g^2]/( 
e*f - d*g))*(1 + (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + g^2])*Sqrt[(1 + (c*f^2)/ 
g^2 - (2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2)/((1 + (c*f^2)/g^2)*(1 + 
 (Sqrt[c]*(f + g*x))/Sqrt[c*f^2 + g^2])^2)]*EllipticPi[(Sqrt[c]*(e*f - d*g 
) + e*Sqrt[c*f^2 + g^2])^2/(4*Sqrt[c]*e*(e*f - d*g)*Sqrt[c*f^2 + g^2]), 2* 
ArcTan[(c^(1/4)*Sqrt[f + g*x])/(c*f^2 + g^2)^(1/4)], (1 + (Sqrt[c]*f)/Sqrt 
[c*f^2 + g^2])/2])/(4*c^(1/4)*(c*f^2 + g^2)^(1/4)*Sqrt[1 + (c*f^2)/g^2 - ( 
2*c*f*(f + g*x))/g^2 + (c*(f + g*x)^2)/g^2])))/(g*(e^2*g + c*d*(2*e*f - d* 
g))))
 

Defintions of rubi rules used

rule 25

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

rule 729

Int[1/(Sqrt[(c_.) + (d_.)*(x_)]*((e_.) + (f_.)*(x_))*Sqrt[(a_) + (b_.)*(x_) 
^2]), x_Symbol] :> Simp[2   Subst[Int[1/((d*e - c*f + f*x^2)*Sqrt[(b*c^2 + 
a*d^2)/d^2 - 2*b*c*(x^2/d^2) + b*(x^4/d^2)]), x], x, Sqrt[c + d*x]], x] /; 
FreeQ[{a, b, c, d, e, f}, x] && PosQ[b/a]
 

rule 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1540

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]), x_S 
ymbol] :> With[{q = Rt[c/a, 2]}, Simp[(c*d + a*e*q)/(c*d^2 - a*e^2)   Int[1 
/Sqrt[a + b*x^2 + c*x^4], x], x] - Simp[(a*e*(e + d*q))/(c*d^2 - a*e^2)   I 
nt[(1 + q*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, 
b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a]
 

rule 2222

Int[((A_) + (B_.)*(x_)^2)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + 
 (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[B/A, 2]}, Simp[(-(B*d - A*e))*(A 
rcTanh[Rt[b - c*(d/e) - a*(e/d), 2]*(x/Sqrt[a + b*x^2 + c*x^4])]/(2*d*e*Rt[ 
b - c*(d/e) - a*(e/d), 2])), x] + Simp[(B*d + A*e)*(1 + q^2*x^2)*(Sqrt[(a + 
 b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/(4*d*e*q*Sqrt[a + b*x^2 + c*x^4]))*Ell 
ipticPi[-(e - d*q^2)^2/(4*d*e*q^2), 2*ArcTan[q*x], 1/2 - b/(4*a*q^2)], x]] 
/; FreeQ[{a, b, c, d, e, A, B}, x] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] && 
 EqQ[c*A^2 - a*B^2, 0] && PosQ[B/A] && NegQ[-b + c*(d/e) + a*(e/d)]
 

Maple [A] (verified)

Time = 4.28 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.10

method	result	size
default	\(\frac {2 \left (f \sqrt {-c}+g \right ) \operatorname {EllipticPi}\left (\sqrt {\frac {\left (g x +f \right ) \sqrt {-c}}{f \sqrt {-c}+g}}, -\frac {\left (f \sqrt {-c}+g \right ) e}{\sqrt {-c}\, \left (d g -e f \right )}, \sqrt {\frac {f \sqrt {-c}+g}{f \sqrt {-c}-g}}\right ) \sqrt {-\frac {g \left (\sqrt {-c}\, x -1\right )}{f \sqrt {-c}+g}}\, \sqrt {-\frac {\left (\sqrt {-c}\, x +1\right ) g}{f \sqrt {-c}-g}}\, \sqrt {\frac {\left (g x +f \right ) \sqrt {-c}}{f \sqrt {-c}+g}}\, \sqrt {c \,x^{2}+1}\, \sqrt {g x +f}}{\sqrt {-c}\, \left (d g -e f \right ) \left (c g \,x^{3}+c f \,x^{2}+g x +f \right )}\)	\(215\)
elliptic	\(\frac {2 \sqrt {\left (g x +f \right ) \left (c \,x^{2}+1\right )}\, \left (\frac {f}{g}+\frac {1}{\sqrt {-c}}\right ) \sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}+\frac {1}{\sqrt {-c}}}}\, \sqrt {\frac {x +\frac {1}{\sqrt {-c}}}{-\frac {f}{g}+\frac {1}{\sqrt {-c}}}}\, \sqrt {\frac {x -\frac {1}{\sqrt {-c}}}{-\frac {f}{g}-\frac {1}{\sqrt {-c}}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +\frac {f}{g}}{\frac {f}{g}+\frac {1}{\sqrt {-c}}}}, \frac {-\frac {f}{g}-\frac {1}{\sqrt {-c}}}{-\frac {f}{g}+\frac {d}{e}}, \sqrt {\frac {-\frac {f}{g}-\frac {1}{\sqrt {-c}}}{-\frac {f}{g}+\frac {1}{\sqrt {-c}}}}\right )}{\sqrt {g x +f}\, \sqrt {c \,x^{2}+1}\, e \sqrt {c g \,x^{3}+c f \,x^{2}+g x +f}\, \left (-\frac {f}{g}+\frac {d}{e}\right )}\)	\(240\)

Input:

int(1/(e*x+d)/(g*x+f)^(1/2)/(c*x^2+1)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2*(f*(-c)^(1/2)+g)/(-c)^(1/2)*EllipticPi(((g*x+f)*(-c)^(1/2)/(f*(-c)^(1/2) 
+g))^(1/2),-(f*(-c)^(1/2)+g)*e/(-c)^(1/2)/(d*g-e*f),((f*(-c)^(1/2)+g)/(f*( 
-c)^(1/2)-g))^(1/2))*(-g*((-c)^(1/2)*x-1)/(f*(-c)^(1/2)+g))^(1/2)*(-((-c)^ 
(1/2)*x+1)*g/(f*(-c)^(1/2)-g))^(1/2)*((g*x+f)*(-c)^(1/2)/(f*(-c)^(1/2)+g)) 
^(1/2)*(c*x^2+1)^(1/2)*(g*x+f)^(1/2)/(d*g-e*f)/(c*g*x^3+c*f*x^2+g*x+f)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\text {Timed out} \] Input:

integrate(1/(e*x+d)/(g*x+f)^(1/2)/(c*x^2+1)^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\int \frac {1}{\left (d + e x\right ) \sqrt {f + g x} \sqrt {c x^{2} + 1}}\, dx \] Input:

integrate(1/(e*x+d)/(g*x+f)**(1/2)/(c*x**2+1)**(1/2),x)
 

Output:

Integral(1/((d + e*x)*sqrt(f + g*x)*sqrt(c*x**2 + 1)), x)
 

Maxima [F]

\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + 1} {\left (e x + d\right )} \sqrt {g x + f}} \,d x } \] Input:

integrate(1/(e*x+d)/(g*x+f)^(1/2)/(c*x^2+1)^(1/2),x, algorithm="maxima")
 

Output:

integrate(1/(sqrt(c*x^2 + 1)*(e*x + d)*sqrt(g*x + f)), x)
 

Giac [F]

\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\int { \frac {1}{\sqrt {c x^{2} + 1} {\left (e x + d\right )} \sqrt {g x + f}} \,d x } \] Input:

integrate(1/(e*x+d)/(g*x+f)^(1/2)/(c*x^2+1)^(1/2),x, algorithm="giac")
 

Output:

integrate(1/(sqrt(c*x^2 + 1)*(e*x + d)*sqrt(g*x + f)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+1}\,\left (d+e\,x\right )} \,d x \] Input:

int(1/((f + g*x)^(1/2)*(c*x^2 + 1)^(1/2)*(d + e*x)),x)
 

Output:

int(1/((f + g*x)^(1/2)*(c*x^2 + 1)^(1/2)*(d + e*x)), x)
 

Reduce [F]

\[ \int \frac {1}{(d+e x) \sqrt {f+g x} \sqrt {1+c x^2}} \, dx=\int \frac {1}{\left (e x +d \right ) \sqrt {g x +f}\, \sqrt {c \,x^{2}+1}}d x \] Input:

int(1/(e*x+d)/(g*x+f)^(1/2)/(c*x^2+1)^(1/2),x)
 

Output:

int(1/(e*x+d)/(g*x+f)^(1/2)/(c*x^2+1)^(1/2),x)