Integrand size = 17, antiderivative size = 189 \[ \int \sqrt {a+\frac {b}{c+d x^2}} \, dx=x \sqrt {a+\frac {b}{c+d x^2}}-\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x^2}} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}}+\frac {\sqrt {c} \sqrt {a+\frac {b}{c+d x^2}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {\frac {c \left (b+a c+a d x^2\right )}{(b+a c) \left (c+d x^2\right )}}} \] Output:
x*(a+b/(d*x^2+c))^(1/2)-c^(1/2)*(a+b/(d*x^2+c))^(1/2)*EllipticE(d^(1/2)*x/ c^(1/2)/(1+d*x^2/c)^(1/2),(b/(a*c+b))^(1/2))/d^(1/2)/(c*(a*d*x^2+a*c+b)/(a *c+b)/(d*x^2+c))^(1/2)+c^(1/2)*(a+b/(d*x^2+c))^(1/2)*InverseJacobiAM(arcta n(d^(1/2)*x/c^(1/2)),(b/(a*c+b))^(1/2))/d^(1/2)/(c*(a*d*x^2+a*c+b)/(a*c+b) /(d*x^2+c))^(1/2)
Time = 10.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \sqrt {a+\frac {b}{c+d x^2}} \, dx=\frac {\sqrt {\frac {c+d x^2}{c}} \sqrt {\frac {b+a c+a d x^2}{c+d x^2}} E\left (\arcsin \left (\sqrt {-\frac {d}{c}} x\right )|\frac {a c}{b+a c}\right )}{\sqrt {-\frac {d}{c}} \sqrt {\frac {b+a c+a d x^2}{b+a c}}} \] Input:
Integrate[Sqrt[a + b/(c + d*x^2)],x]
Output:
(Sqrt[(c + d*x^2)/c]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*EllipticE[ArcSi n[Sqrt[-(d/c)]*x], (a*c)/(b + a*c)])/(Sqrt[-(d/c)]*Sqrt[(b + a*c + a*d*x^2 )/(b + a*c)])
Time = 0.67 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.49, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2057, 2058, 324, 320, 388, 313}
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.
\(\displaystyle \int \sqrt {a+\frac {b}{c+d x^2}} \, dx\) |
\(\Big \downarrow \) 2057 |
\(\displaystyle \int \sqrt {\frac {a c+a d x^2+b}{c+d x^2}}dx\) |
\(\Big \downarrow \) 2058 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \int \frac {\sqrt {a d x^2+b+a c}}{\sqrt {d x^2+c}}dx}{\sqrt {a c+a d x^2+b}}\) |
\(\Big \downarrow \) 324 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left ((a c+b) \int \frac {1}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+a d \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx\right )}{\sqrt {a c+a d x^2+b}}\) |
\(\Big \downarrow \) 320 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (a d \int \frac {x^2}{\sqrt {d x^2+c} \sqrt {a d x^2+b+a c}}dx+\frac {\sqrt {c} \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{\sqrt {a c+a d x^2+b}}\) |
\(\Big \downarrow \) 388 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (a d \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {c \int \frac {\sqrt {a d x^2+b+a c}}{\left (d x^2+c\right )^{3/2}}dx}{a d}\right )+\frac {\sqrt {c} \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{\sqrt {a c+a d x^2+b}}\) |
\(\Big \downarrow \) 313 |
\(\displaystyle \frac {\sqrt {c+d x^2} \sqrt {\frac {a c+a d x^2+b}{c+d x^2}} \left (a d \left (\frac {x \sqrt {a c+a d x^2+b}}{a d \sqrt {c+d x^2}}-\frac {\sqrt {c} \sqrt {a c+a d x^2+b} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|\frac {b}{b+a c}\right )}{a d^{3/2} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )+\frac {\sqrt {c} \sqrt {a c+a d x^2+b} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),\frac {b}{b+a c}\right )}{\sqrt {d} \sqrt {c+d x^2} \sqrt {\frac {c \left (a c+a d x^2+b\right )}{(a c+b) \left (c+d x^2\right )}}}\right )}{\sqrt {a c+a d x^2+b}}\) |
Input:
Int[Sqrt[a + b/(c + d*x^2)],x]
Output:
(Sqrt[c + d*x^2]*Sqrt[(b + a*c + a*d*x^2)/(c + d*x^2)]*(a*d*((x*Sqrt[b + a *c + a*d*x^2])/(a*d*Sqrt[c + d*x^2]) - (Sqrt[c]*Sqrt[b + a*c + a*d*x^2]*El lipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/(a*d^(3/2)*Sqrt[c + d*x ^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^2))])) + (Sqrt[c]*Sqr t[b + a*c + a*d*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[c]], b/(b + a*c)])/ (Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[(c*(b + a*c + a*d*x^2))/((b + a*c)*(c + d*x^ 2))])))/Sqrt[b + a*c + a*d*x^2]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Sim p[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; FreeQ [{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*( c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /; Fre eQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] && !SimplerSqrtQ[b/a, d/c]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ a Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] + Simp[b Int[x^2/(Sqr t[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c ] && PosQ[b/a]
Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt[c + d*x^2])), x] - Simp[c/b Int[Sqrt[ a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] && !SimplerSqrtQ[b/a, d/c]
Int[(u_.)*((a_) + (b_.)/((c_) + (d_.)*(x_)^(n_)))^(p_), x_Symbol] :> Int[u* ((b + a*c + a*d*x^n)/(c + d*x^n))^p, x] /; FreeQ[{a, b, c, d, n, p}, x]
Int[(u_.)*((e_.)*((a_.) + (b_.)*(x_)^(n_.))^(q_.)*((c_) + (d_.)*(x_)^(n_))^ (r_.))^(p_), x_Symbol] :> Simp[Simp[(e*(a + b*x^n)^q*(c + d*x^n)^r)^p/((a + b*x^n)^(p*q)*(c + d*x^n)^(p*r))] Int[u*(a + b*x^n)^(p*q)*(c + d*x^n)^(p* r), x], x] /; FreeQ[{a, b, c, d, e, n, p, q, r}, x]
Time = 1.14 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.05
method | result | size |
default | \(\frac {\left (a c \operatorname {EllipticE}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right )+\operatorname {EllipticF}\left (x \sqrt {-\frac {a d}{a c +b}}, \sqrt {\frac {a c +b}{a c}}\right ) b \right ) \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {a d \,x^{2}+a c +b}{a c +b}}\, \left (d \,x^{2}+c \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{\sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {-\frac {a d}{a c +b}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}\) | \(199\) |
Input:
int((a+b/(d*x^2+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
(a*c*EllipticE(x*(-a*d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))+EllipticF(x*(-a *d/(a*c+b))^(1/2),((a*c+b)/a/c)^(1/2))*b)*((d*x^2+c)/c)^(1/2)*((a*d*x^2+a* c+b)/(a*c+b))^(1/2)*(d*x^2+c)*((a*d*x^2+a*c+b)/(d*x^2+c))^(1/2)/(a*d^2*x^4 +2*a*c*d*x^2+b*d*x^2+a*c^2+b*c)^(1/2)/(-a*d/(a*c+b))^(1/2)/((d*x^2+c)*(a*d *x^2+a*c+b))^(1/2)
Time = 0.09 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.78 \[ \int \sqrt {a+\frac {b}{c+d x^2}} \, dx=-\frac {a^{\frac {3}{2}} c^{2} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a c^{2} + {\left (a c + b\right )} d\right )} \sqrt {a} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a c + b}{a c}) - {\left (a c d x^{2} + a c^{2}\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{a c d x} \] Input:
integrate((a+b/(d*x^2+c))^(1/2),x, algorithm="fricas")
Output:
-(a^(3/2)*c^2*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), (a*c + b)/(a*c )) - (a*c^2 + (a*c + b)*d)*sqrt(a)*x*sqrt(-c/d)*elliptic_f(arcsin(sqrt(-c/ d)/x), (a*c + b)/(a*c)) - (a*c*d*x^2 + a*c^2)*sqrt((a*d*x^2 + a*c + b)/(d* x^2 + c)))/(a*c*d*x)
\[ \int \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int \sqrt {a + \frac {b}{c + d x^{2}}}\, dx \] Input:
integrate((a+b/(d*x**2+c))**(1/2),x)
Output:
Integral(sqrt(a + b/(c + d*x**2)), x)
\[ \int \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int { \sqrt {a + \frac {b}{d x^{2} + c}} \,d x } \] Input:
integrate((a+b/(d*x^2+c))^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(a + b/(d*x^2 + c)), x)
\[ \int \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int { \sqrt {a + \frac {b}{d x^{2} + c}} \,d x } \] Input:
integrate((a+b/(d*x^2+c))^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(a + b/(d*x^2 + c)), x)
Timed out. \[ \int \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int \sqrt {a+\frac {b}{d\,x^2+c}} \,d x \] Input:
int((a + b/(c + d*x^2))^(1/2),x)
Output:
int((a + b/(c + d*x^2))^(1/2), x)
\[ \int \sqrt {a+\frac {b}{c+d x^2}} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}}{d \,x^{2}+c}d x \] Input:
int((a+b/(d*x^2+c))^(1/2),x)
Output:
int((sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b))/(c + d*x**2),x)