Integrand size = 30, antiderivative size = 191 \[ \int \frac {1}{(a+b x) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=-\frac {b \text {arctanh}\left (\frac {\sqrt {b^2 e+a^2 f} \sqrt {c+d x^2}}{\sqrt {b^2 c+a^2 d} \sqrt {e+f x^2}}\right )}{\sqrt {b^2 c+a^2 d} \sqrt {b^2 e+a^2 f}}+\frac {\sqrt {-c} \sqrt {1+\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (-\frac {b^2 c}{a^2 d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
-b*arctanh((a^2*f+b^2*e)^(1/2)*(d*x^2+c)^(1/2)/(a^2*d+b^2*c)^(1/2)/(f*x^2+ e)^(1/2))/(a^2*d+b^2*c)^(1/2)/(a^2*f+b^2*e)^(1/2)+EllipticPi(x*d^(1/2)/(-c )^(1/2),-b^2*c/a^2/d,(c*f/d/e)^(1/2))*(-c)^(1/2)*(1+d*x^2/c)^(1/2)*(1+f*x^ 2/e)^(1/2)/a/d^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)
Result contains complex when optimal does not.
Time = 4.19 (sec) , antiderivative size = 772, normalized size of antiderivative = 4.04 \[ \int \frac {1}{(a+b x) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\frac {2 \sqrt {d} \left (\sqrt {c}+i \sqrt {d} x\right ) \sqrt {\frac {\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right ) \left (i \sqrt {c}+\sqrt {d} x\right )}{\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (-i \sqrt {c}+\sqrt {d} x\right )}} \left (\sqrt {e}+i \sqrt {f} x\right ) \sqrt {\frac {\sqrt {c} \sqrt {d} \left (i \sqrt {e}+\sqrt {f} x\right )}{\left (-\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (-i \sqrt {c}+\sqrt {d} x\right )}} \left (\left (b \sqrt {c}+i a \sqrt {d}\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right ) \left (i \sqrt {c}+\sqrt {d} x\right )}{\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (-i \sqrt {c}+\sqrt {d} x\right )}}\right ),\frac {\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right )^2}{\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right )^2}\right )-2 b \sqrt {c} \operatorname {EllipticPi}\left (\frac {\left (b \sqrt {c}-i a \sqrt {d}\right ) \left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right )}{\left (b \sqrt {c}+i a \sqrt {d}\right ) \left (-\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right )},\arcsin \left (\sqrt {\frac {\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right ) \left (i \sqrt {c}+\sqrt {d} x\right )}{\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (-i \sqrt {c}+\sqrt {d} x\right )}}\right ),\frac {\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right )^2}{\left (\sqrt {d} \sqrt {e}-\sqrt {c} \sqrt {f}\right )^2}\right )\right )}{\left (b \sqrt {c}-i a \sqrt {d}\right ) \left (b \sqrt {c}+i a \sqrt {d}\right ) \left (-\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \sqrt {\frac {\sqrt {c} \sqrt {d} \left (\sqrt {e}+i \sqrt {f} x\right )}{\left (\sqrt {d} \sqrt {e}+\sqrt {c} \sqrt {f}\right ) \left (\sqrt {c}+i \sqrt {d} x\right )}} \sqrt {c+d x^2} \sqrt {e+f x^2}} \]
(2*Sqrt[d]*(Sqrt[c] + I*Sqrt[d]*x)*Sqrt[((Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f ])*(I*Sqrt[c] + Sqrt[d]*x))/((Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])*((-I)*Sqr t[c] + Sqrt[d]*x))]*(Sqrt[e] + I*Sqrt[f]*x)*Sqrt[(Sqrt[c]*Sqrt[d]*(I*Sqrt[ e] + Sqrt[f]*x))/((-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])*((-I)*Sqrt[c] + S qrt[d]*x))]*((b*Sqrt[c] + I*a*Sqrt[d])*EllipticF[ArcSin[Sqrt[((Sqrt[d]*Sqr t[e] - Sqrt[c]*Sqrt[f])*(I*Sqrt[c] + Sqrt[d]*x))/((Sqrt[d]*Sqrt[e] + Sqrt[ c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x))]], (Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt [f])^2/(Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])^2] - 2*b*Sqrt[c]*EllipticPi[((b *Sqrt[c] - I*a*Sqrt[d])*(Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f]))/((b*Sqrt[c] + I*a*Sqrt[d])*(-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])), ArcSin[Sqrt[((Sqrt[ d]*Sqrt[e] - Sqrt[c]*Sqrt[f])*(I*Sqrt[c] + Sqrt[d]*x))/((Sqrt[d]*Sqrt[e] + Sqrt[c]*Sqrt[f])*((-I)*Sqrt[c] + Sqrt[d]*x))]], (Sqrt[d]*Sqrt[e] + Sqrt[c ]*Sqrt[f])^2/(Sqrt[d]*Sqrt[e] - Sqrt[c]*Sqrt[f])^2]))/((b*Sqrt[c] - I*a*Sq rt[d])*(b*Sqrt[c] + I*a*Sqrt[d])*(-(Sqrt[d]*Sqrt[e]) + Sqrt[c]*Sqrt[f])*Sq rt[(Sqrt[c]*Sqrt[d]*(Sqrt[e] + I*Sqrt[f]*x))/((Sqrt[d]*Sqrt[e] + Sqrt[c]*S qrt[f])*(Sqrt[c] + I*Sqrt[d]*x))]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])
Time = 0.70 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2538, 413, 413, 412, 435, 104, 221}
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.
| \(\displaystyle \int \frac {1}{(a+b x) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx\) |
| \(\Big \downarrow \) 2538 |
| \(\displaystyle a \int \frac {1}{\left (a^2-b^2 x^2\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx-b \int \frac {x}{\left (a^2-b^2 x^2\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {a \sqrt {\frac {d x^2}{c}+1} \int \frac {1}{\left (a^2-b^2 x^2\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {f x^2+e}}dx}{\sqrt {c+d x^2}}-b \int \frac {x}{\left (a^2-b^2 x^2\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {a \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \int \frac {1}{\left (a^2-b^2 x^2\right ) \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1}}dx}{\sqrt {c+d x^2} \sqrt {e+f x^2}}-b \int \frac {x}{\left (a^2-b^2 x^2\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \operatorname {EllipticPi}\left (-\frac {b^2 c}{a^2 d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2}}-b \int \frac {x}{\left (a^2-b^2 x^2\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx\) |
| \(\Big \downarrow \) 435 |
| \(\displaystyle \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \operatorname {EllipticPi}\left (-\frac {b^2 c}{a^2 d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {1}{2} b \int \frac {1}{\left (a^2-b^2 x^2\right ) \sqrt {d x^2+c} \sqrt {f x^2+e}}dx^2\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \operatorname {EllipticPi}\left (-\frac {b^2 c}{a^2 d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2}}-b \int \frac {1}{-\left (\left (f a^2+b^2 e\right ) x^4\right )+b^2 c+a^2 d}d\frac {\sqrt {d x^2+c}}{\sqrt {f x^2+e}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\sqrt {-c} \sqrt {\frac {d x^2}{c}+1} \sqrt {\frac {f x^2}{e}+1} \operatorname {EllipticPi}\left (-\frac {b^2 c}{a^2 d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {-c}}\right ),\frac {c f}{d e}\right )}{a \sqrt {d} \sqrt {c+d x^2} \sqrt {e+f x^2}}-\frac {b \text {arctanh}\left (\frac {\sqrt {c+d x^2} \sqrt {a^2 f+b^2 e}}{\sqrt {e+f x^2} \sqrt {a^2 d+b^2 c}}\right )}{\sqrt {a^2 d+b^2 c} \sqrt {a^2 f+b^2 e}}\) |
-((b*ArcTanh[(Sqrt[b^2*e + a^2*f]*Sqrt[c + d*x^2])/(Sqrt[b^2*c + a^2*d]*Sq rt[e + f*x^2])])/(Sqrt[b^2*c + a^2*d]*Sqrt[b^2*e + a^2*f])) + (Sqrt[-c]*Sq rt[1 + (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b^2*c)/(a^2*d)), ArcSi n[(Sqrt[d]*x)/Sqrt[-c]], (c*f)/(d*e)])/(a*Sqrt[d]*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])
3.6.23.3.1 Defintions of rubi rules used
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
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]
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* (c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] && !( !GtQ[f/e, 0] && S implerSqrtQ[-f/e, -d/c])
Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x _)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/((a + b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && !GtQ[c, 0]
Int[(x_)^(m_.)*((a_.) + (b_.)*(x_)^2)^(p_.)*((c_.) + (d_.)*(x_)^2)^(q_.)*(( e_.) + (f_.)*(x_)^2)^(r_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2) *(a + b*x)^p*(c + d*x)^q*(e + f*x)^r, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] && IntegerQ[(m - 1)/2]
Int[1/(((a_) + (b_.)*(x_))*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_) ^2]), x_Symbol] :> Simp[a Int[1/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] - Simp[b Int[x/((a^2 - b^2*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f}, x]
Time = 4.54 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.42
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (-\frac {\operatorname {arctanh}\left (\frac {\frac {\left (c f +d e \right ) a^{2}}{b^{2}}+2 c e +\left (c f +d e \right ) x^{2}+\frac {2 d f \,x^{2} a^{2}}{b^{2}}}{2 \sqrt {\frac {d f \,a^{4}}{b^{4}}+\frac {\left (c f +d e \right ) a^{2}}{b^{2}}+c e}\, \sqrt {d f \,x^{4}+c f \,x^{2}+e d \,x^{2}+c e}}\right )}{2 \sqrt {\frac {d f \,a^{4}}{b^{4}}+\frac {\left (c f +d e \right ) a^{2}}{b^{2}}+c e}}+\frac {b \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, -\frac {b^{2} c}{a^{2} d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right )}{\sqrt {-\frac {d}{c}}\, a \sqrt {d f \,x^{4}+c f \,x^{2}+e d \,x^{2}+c e}}\right )}{\sqrt {d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}\, b}\) | \(272\) |
| default | \(\frac {\left (2 b \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \Pi \left (x \sqrt {-\frac {d}{c}}, -\frac {b^{2} c}{a^{2} d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {-\frac {d}{c}}}\right ) \sqrt {\frac {a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} e c}{b^{4}}}-\operatorname {arctanh}\left (\frac {2 a^{2} d f \,x^{2}+b^{2} c f \,x^{2}+b^{2} d e \,x^{2}+a^{2} c f +a^{2} d e +2 b^{2} c e}{2 b^{2} \sqrt {\frac {a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} e c}{b^{4}}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}}\right ) \sqrt {-\frac {d}{c}}\, a \sqrt {\left (d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\right ) \sqrt {f \,x^{2}+e}\, \sqrt {d \,x^{2}+c}}{2 b a \sqrt {\frac {a^{4} d f +a^{2} b^{2} c f +a^{2} b^{2} d e +b^{4} e c}{b^{4}}}\, \sqrt {-\frac {d}{c}}\, \left (d f \,x^{4}+c f \,x^{2}+e d \,x^{2}+c e \right )}\) | \(339\) |
((d*x^2+c)*(f*x^2+e))^(1/2)/(d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)/b*(-1/2/(d*f*a ^4/b^4+(c*f+d*e)*a^2/b^2+c*e)^(1/2)*arctanh(1/2*((c*f+d*e)*a^2/b^2+2*c*e+( c*f+d*e)*x^2+2*d*f*x^2*a^2/b^2)/(d*f*a^4/b^4+(c*f+d*e)*a^2/b^2+c*e)^(1/2)/ (d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2))+1/(-d/c)^(1/2)/a*b*(1+1/c*d*x^2)^(1/2 )*(1+f*x^2/e)^(1/2)/(d*f*x^4+c*f*x^2+d*e*x^2+c*e)^(1/2)*EllipticPi(x*(-d/c )^(1/2),-b^2*c/a^2/d,(-f/e)^(1/2)/(-d/c)^(1/2)))
Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(a+b x) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\left (a + b x\right ) \sqrt {c + d x^{2}} \sqrt {e + f x^{2}}}\, dx \]
\[ \int \frac {1}{(a+b x) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{\sqrt {d x^{2} + c} \sqrt {f x^{2} + e} {\left (b x + a\right )}} \,d x } \]
\[ \int \frac {1}{(a+b x) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int { \frac {1}{\sqrt {d x^{2} + c} \sqrt {f x^{2} + e} {\left (b x + a\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(a+b x) \sqrt {c+d x^2} \sqrt {e+f x^2}} \, dx=\int \frac {1}{\sqrt {d\,x^2+c}\,\sqrt {f\,x^2+e}\,\left (a+b\,x\right )} \,d x \]