\(\int \frac {x \sqrt {c+d x^4}}{a+b x^4} \, dx\) [225]

Optimal result

Integrand size = 22, antiderivative size = 91 \[ \int \frac {x \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {\sqrt {b c-a d} \arctan \left (\frac {\sqrt {b c-a d} x^2}{\sqrt {a} \sqrt {c+d x^4}}\right )}{2 \sqrt {a} b}+\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{2 b} \] Output:

1/2*(-a*d+b*c)^(1/2)*arctan((-a*d+b*c)^(1/2)*x^2/a^(1/2)/(d*x^4+c)^(1/2))/ 
a^(1/2)/b+1/2*d^(1/2)*arctanh(d^(1/2)*x^2/(d*x^4+c)^(1/2))/b
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.18 \[ \int \frac {x \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {\frac {\sqrt {b c-a d} \arctan \left (\frac {a \sqrt {d}+b x^2 \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{\sqrt {a} \sqrt {b c-a d}}\right )}{\sqrt {a}}+\sqrt {d} \log \left (\sqrt {d} x^2+\sqrt {c+d x^4}\right )}{2 b} \] Input:

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

Output:

((Sqrt[b*c - a*d]*ArcTan[(a*Sqrt[d] + b*x^2*(Sqrt[d]*x^2 + Sqrt[c + d*x^4] 
))/(Sqrt[a]*Sqrt[b*c - a*d])])/Sqrt[a] + Sqrt[d]*Log[Sqrt[d]*x^2 + Sqrt[c 
+ d*x^4]])/(2*b)
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {965, 301, 224, 219, 291, 218}

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.

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

\(\Big \downarrow \) 965

\(\displaystyle \frac {1}{2} \int \frac {\sqrt {d x^4+c}}{b x^4+a}dx^2\)

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{2} \left (\frac {(b c-a d) \int \frac {1}{\left (b x^4+a\right ) \sqrt {d x^4+c}}dx^2}{b}+\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{b}\right )\)

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {b c-a d} \arctan \left (\frac {x^2 \sqrt {b c-a d}}{\sqrt {a} \sqrt {c+d x^4}}\right )}{\sqrt {a} b}+\frac {\sqrt {d} \text {arctanh}\left (\frac {\sqrt {d} x^2}{\sqrt {c+d x^4}}\right )}{b}\right )\)

Input:

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

Output:

((Sqrt[b*c - a*d]*ArcTan[(Sqrt[b*c - a*d]*x^2)/(Sqrt[a]*Sqrt[c + d*x^4])]) 
/(Sqrt[a]*b) + (Sqrt[d]*ArcTanh[(Sqrt[d]*x^2)/Sqrt[c + d*x^4]])/b)/2
 

Defintions of rubi rules used

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 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 0]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 0]))
 

rule 965

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), 
 x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Subst[Int[x^((m + 1)/k - 
 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /; Free 
Q[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]
 

Maple [A] (verified)

Time = 0.37 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.85

method	result	size
pseudoelliptic	\(-\frac {\frac {\left (a d -c b \right ) \operatorname {arctanh}\left (\frac {a \sqrt {d \,x^{4}+c}}{x^{2} \sqrt {a \left (a d -c b \right )}}\right )}{\sqrt {a \left (a d -c b \right )}}-\sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {d \,x^{4}+c}}{x^{2} \sqrt {d}}\right )}{2 b}\)	\(77\)
default	\(\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 \sqrt {-a b}}-\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 \sqrt {-a b}}\)	\(689\)
elliptic	\(\frac {\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}+\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {\frac {d \sqrt {-a b}}{b}+\left (x^{2}-\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}-\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x^{2}-\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}-\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 \sqrt {-a b}}-\frac {\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}-\frac {\sqrt {d}\, \sqrt {-a b}\, \ln \left (\frac {-\frac {d \sqrt {-a b}}{b}+\left (x^{2}+\frac {\sqrt {-a b}}{b}\right ) d}{\sqrt {d}}+\sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}\right )}{b}+\frac {\left (a d -c b \right ) \ln \left (\frac {-\frac {2 \left (a d -c b \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -c b}{b}}\, \sqrt {\left (x^{2}+\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x^{2}+\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -c b}{b}}}{x^{2}+\frac {\sqrt {-a b}}{b}}\right )}{b \sqrt {-\frac {a d -c b}{b}}}}{4 \sqrt {-a b}}\)	\(689\)

Input:

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

Output:

-1/2/b*((a*d-b*c)*arctanh(a*(d*x^4+c)^(1/2)/x^2/(a*(a*d-b*c))^(1/2))/(a*(a 
*d-b*c))^(1/2)-d^(1/2)*arctanh((d*x^4+c)^(1/2)/x^2/d^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 617, normalized size of antiderivative = 6.78 \[ \int \frac {x \sqrt {c+d x^4}}{a+b x^4} \, dx=\left [\frac {2 \, \sqrt {d} \log \left (-2 \, d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right ) + \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \, {\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, b}, -\frac {4 \, \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-d}}{d x^{2}}\right ) - \sqrt {-\frac {b c - a d}{a}} \log \left (\frac {{\left (b^{2} c^{2} - 8 \, a b c d + 8 \, a^{2} d^{2}\right )} x^{8} - 2 \, {\left (3 \, a b c^{2} - 4 \, a^{2} c d\right )} x^{4} + a^{2} c^{2} + 4 \, {\left ({\left (a b c - 2 \, a^{2} d\right )} x^{6} - a^{2} c x^{2}\right )} \sqrt {d x^{4} + c} \sqrt {-\frac {b c - a d}{a}}}{b^{2} x^{8} + 2 \, a b x^{4} + a^{2}}\right )}{8 \, b}, \frac {\sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{6} + {\left (b c^{2} - a c d\right )} x^{2}\right )}}\right ) + \sqrt {d} \log \left (-2 \, d x^{4} - 2 \, \sqrt {d x^{4} + c} \sqrt {d} x^{2} - c\right )}{4 \, b}, \frac {\sqrt {\frac {b c - a d}{a}} \arctan \left (\frac {{\left ({\left (b c - 2 \, a d\right )} x^{4} - a c\right )} \sqrt {d x^{4} + c} \sqrt {\frac {b c - a d}{a}}}{2 \, {\left ({\left (b c d - a d^{2}\right )} x^{6} + {\left (b c^{2} - a c d\right )} x^{2}\right )}}\right ) - 2 \, \sqrt {-d} \arctan \left (\frac {\sqrt {d x^{4} + c} \sqrt {-d}}{d x^{2}}\right )}{4 \, b}\right ] \] Input:

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

Output:

[1/8*(2*sqrt(d)*log(-2*d*x^4 - 2*sqrt(d*x^4 + c)*sqrt(d)*x^2 - c) + sqrt(- 
(b*c - a*d)/a)*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 
 4*a^2*c*d)*x^4 + a^2*c^2 + 4*((a*b*c - 2*a^2*d)*x^6 - a^2*c*x^2)*sqrt(d*x 
^4 + c)*sqrt(-(b*c - a*d)/a))/(b^2*x^8 + 2*a*b*x^4 + a^2)))/b, -1/8*(4*sqr 
t(-d)*arctan(sqrt(d*x^4 + c)*sqrt(-d)/(d*x^2)) - sqrt(-(b*c - a*d)/a)*log( 
((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^2)*x^8 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^4 + a 
^2*c^2 + 4*((a*b*c - 2*a^2*d)*x^6 - a^2*c*x^2)*sqrt(d*x^4 + c)*sqrt(-(b*c 
- a*d)/a))/(b^2*x^8 + 2*a*b*x^4 + a^2)))/b, 1/4*(sqrt((b*c - a*d)/a)*arcta 
n(1/2*((b*c - 2*a*d)*x^4 - a*c)*sqrt(d*x^4 + c)*sqrt((b*c - a*d)/a)/((b*c* 
d - a*d^2)*x^6 + (b*c^2 - a*c*d)*x^2)) + sqrt(d)*log(-2*d*x^4 - 2*sqrt(d*x 
^4 + c)*sqrt(d)*x^2 - c))/b, 1/4*(sqrt((b*c - a*d)/a)*arctan(1/2*((b*c - 2 
*a*d)*x^4 - a*c)*sqrt(d*x^4 + c)*sqrt((b*c - a*d)/a)/((b*c*d - a*d^2)*x^6 
+ (b*c^2 - a*c*d)*x^2)) - 2*sqrt(-d)*arctan(sqrt(d*x^4 + c)*sqrt(-d)/(d*x^ 
2)))/b]
 

Sympy [F]

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

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

Output:

Integral(x*sqrt(c + d*x**4)/(a + b*x**4), x)
 

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

Exception generated. \[ \int \frac {x \sqrt {c+d x^4}}{a+b x^4} \, dx=\text {Exception raised: TypeError} \] Input:

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 218, normalized size of antiderivative = 2.40 \[ \int \frac {x \sqrt {c+d x^4}}{a+b x^4} \, dx=\frac {-\sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (-\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{4}+c}+\sqrt {d}\, \sqrt {b}\, x^{2}\right )-\sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (\sqrt {2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}-2 a d +b c}+\sqrt {b}\, \sqrt {d \,x^{4}+c}+\sqrt {d}\, \sqrt {b}\, x^{2}\right )+\sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}\, \mathrm {log}\left (2 \sqrt {d}\, \sqrt {a}\, \sqrt {a d -b c}+2 \sqrt {d}\, \sqrt {d \,x^{4}+c}\, b \,x^{2}+2 a d +2 b d \,x^{4}\right )+2 \,\mathrm {log}\left (\frac {\sqrt {d \,x^{4}+c}+\sqrt {d}\, x^{2}}{\sqrt {c}}\right ) a d}{4 \sqrt {d}\, a b} \] Input:

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

Output:

( - sqrt(d)*sqrt(a)*sqrt(a*d - b*c)*log( - sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d 
 - b*c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2) 
- sqrt(d)*sqrt(a)*sqrt(a*d - b*c)*log(sqrt(2*sqrt(d)*sqrt(a)*sqrt(a*d - b* 
c) - 2*a*d + b*c) + sqrt(b)*sqrt(c + d*x**4) + sqrt(d)*sqrt(b)*x**2) + sqr 
t(d)*sqrt(a)*sqrt(a*d - b*c)*log(2*sqrt(d)*sqrt(a)*sqrt(a*d - b*c) + 2*sqr 
t(d)*sqrt(c + d*x**4)*b*x**2 + 2*a*d + 2*b*d*x**4) + 2*log((sqrt(c + d*x** 
4) + sqrt(d)*x**2)/sqrt(c))*a*d)/(4*sqrt(d)*a*b)