Integrand size = 33, antiderivative size = 359 \[ \int \frac {\sqrt {c-d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}+\frac {\sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a b \sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}}-\frac {\sqrt {c} \sqrt {d} (b e+a f) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{2 a b^2 \sqrt {c-d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {c} \left (b^2 c e+a^2 d f\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (-\frac {b c}{a d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{2 a^2 b^2 \sqrt {d} \sqrt {c-d x^2} \sqrt {e+f x^2}} \]
1/2*x*(-d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/a/(b*x^2+a)+1/2*EllipticE(x*d^(1/2) /c^(1/2),(-c*f/d/e)^(1/2))*c^(1/2)*d^(1/2)*(1-d*x^2/c)^(1/2)*(f*x^2+e)^(1/ 2)/a/b/(-d*x^2+c)^(1/2)/(1+f*x^2/e)^(1/2)+1/2*(a^2*d*f+b^2*c*e)*EllipticPi (x*d^(1/2)/c^(1/2),-b*c/a/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^2/b^2/d^(1/2)/(-d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)-1/2*(a*f+ b*e)*EllipticF(x*d^(1/2)/c^(1/2),(-c*f/d/e)^(1/2))*c^(1/2)*d^(1/2)*(1-d*x^ 2/c)^(1/2)*(1+f*x^2/e)^(1/2)/a/b^2/(-d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)
Result contains complex when optimal does not.
Time = 3.32 (sec) , antiderivative size = 422, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {c-d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\frac {\frac {c e x}{a+b x^2}-\frac {d e x^3}{a+b x^2}+\frac {c f x^3}{a+b x^2}-\frac {d f x^5}{a+b x^2}+\frac {i c \sqrt {-\frac {d}{c}} e \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} E\left (i \text {arcsinh}\left (\sqrt {-\frac {d}{c}} x\right )|-\frac {c f}{d e}\right )}{b}-\frac {i c \sqrt {-\frac {d}{c}} (b e+a f) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {d}{c}} x\right ),-\frac {c f}{d e}\right )}{b^2}+\frac {i d e \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (-\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {-\frac {d}{c}} x\right ),-\frac {c f}{d e}\right )}{a \left (-\frac {d}{c}\right )^{3/2}}+\frac {i a c \sqrt {-\frac {d}{c}} f \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \operatorname {EllipticPi}\left (-\frac {b c}{a d},i \text {arcsinh}\left (\sqrt {-\frac {d}{c}} x\right ),-\frac {c f}{d e}\right )}{b^2}}{2 a \sqrt {c-d x^2} \sqrt {e+f x^2}} \]
((c*e*x)/(a + b*x^2) - (d*e*x^3)/(a + b*x^2) + (c*f*x^3)/(a + b*x^2) - (d* f*x^5)/(a + b*x^2) + (I*c*Sqrt[-(d/c)]*e*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x ^2)/e]*EllipticE[I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/b - (I*c*Sqrt [-(d/c)]*(b*e + a*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*A rcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/b^2 + (I*d*e*Sqrt[1 - (d*x^2)/c]* Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c)]*x], -((c*f)/(d*e))])/(a*(-(d/c))^(3/2)) + (I*a*c*Sqrt[-(d/c)]*f*Sqrt[1 - (d*x^ 2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), I*ArcSinh[Sqrt[-(d/c) ]*x], -((c*f)/(d*e))])/b^2)/(2*a*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])
Time = 0.60 (sec) , antiderivative size = 360, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {423, 399, 323, 323, 321, 331, 330, 327, 413, 413, 412}
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.
| \(\displaystyle \int \frac {\sqrt {c-d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 423 |
| \(\displaystyle -\frac {d f \int \frac {a-b x^2}{\sqrt {c-d x^2} \sqrt {f x^2+e}}dx}{2 a b^2}+\frac {1}{2} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {c-d x^2} \sqrt {f x^2+e}}dx+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 399 |
| \(\displaystyle \frac {1}{2} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {c-d x^2} \sqrt {f x^2+e}}dx-\frac {d f \left (\frac {(a f+b e) \int \frac {1}{\sqrt {c-d x^2} \sqrt {f x^2+e}}dx}{f}-\frac {b \int \frac {\sqrt {f x^2+e}}{\sqrt {c-d x^2}}dx}{f}\right )}{2 a b^2}+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {1}{2} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {c-d x^2} \sqrt {f x^2+e}}dx-\frac {d f \left (\frac {\sqrt {\frac {f x^2}{e}+1} (a f+b e) \int \frac {1}{\sqrt {c-d x^2} \sqrt {\frac {f x^2}{e}+1}}dx}{f \sqrt {e+f x^2}}-\frac {b \int \frac {\sqrt {f x^2+e}}{\sqrt {c-d x^2}}dx}{f}\right )}{2 a b^2}+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 323 |
| \(\displaystyle \frac {1}{2} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {c-d x^2} \sqrt {f x^2+e}}dx-\frac {d f \left (\frac {\sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} (a f+b e) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1}}dx}{f \sqrt {c-d x^2} \sqrt {e+f x^2}}-\frac {b \int \frac {\sqrt {f x^2+e}}{\sqrt {c-d x^2}}dx}{f}\right )}{2 a b^2}+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 321 |
| \(\displaystyle -\frac {d f \left (\frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} (a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {e+f x^2}}-\frac {b \int \frac {\sqrt {f x^2+e}}{\sqrt {c-d x^2}}dx}{f}\right )}{2 a b^2}+\frac {1}{2} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {c-d x^2} \sqrt {f x^2+e}}dx+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 331 |
| \(\displaystyle -\frac {d f \left (\frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} (a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {e+f x^2}}-\frac {b \sqrt {1-\frac {d x^2}{c}} \int \frac {\sqrt {f x^2+e}}{\sqrt {1-\frac {d x^2}{c}}}dx}{f \sqrt {c-d x^2}}\right )}{2 a b^2}+\frac {1}{2} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {c-d x^2} \sqrt {f x^2+e}}dx+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 330 |
| \(\displaystyle -\frac {d f \left (\frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} (a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {e+f x^2}}-\frac {b \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} \int \frac {\sqrt {\frac {f x^2}{e}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{f \sqrt {c-d x^2} \sqrt {\frac {f x^2}{e}+1}}\right )}{2 a b^2}+\frac {1}{2} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {c-d x^2} \sqrt {f x^2+e}}dx+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 327 |
| \(\displaystyle \frac {1}{2} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {c-d x^2} \sqrt {f x^2+e}}dx-\frac {d f \left (\frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} (a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {e+f x^2}}-\frac {b \sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {\frac {f x^2}{e}+1}}\right )}{2 a b^2}+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {\sqrt {1-\frac {d x^2}{c}} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {f x^2+e}}dx}{2 \sqrt {c-d x^2}}-\frac {d f \left (\frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} (a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {e+f x^2}}-\frac {b \sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {\frac {f x^2}{e}+1}}\right )}{2 a b^2}+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 413 |
| \(\displaystyle \frac {\sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \int \frac {1}{\left (b x^2+a\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1}}dx}{2 \sqrt {c-d x^2} \sqrt {e+f x^2}}-\frac {d f \left (\frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} (a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {e+f x^2}}-\frac {b \sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {\frac {f x^2}{e}+1}}\right )}{2 a b^2}+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
| \(\Big \downarrow \) 412 |
| \(\displaystyle \frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \left (\frac {a d f}{b^2}+\frac {c e}{a}\right ) \operatorname {EllipticPi}\left (-\frac {b c}{a d},\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{2 a \sqrt {d} \sqrt {c-d x^2} \sqrt {e+f x^2}}-\frac {d f \left (\frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} (a f+b e) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {e+f x^2}}-\frac {b \sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{\sqrt {d} f \sqrt {c-d x^2} \sqrt {\frac {f x^2}{e}+1}}\right )}{2 a b^2}+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}\) |
(x*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)) - (d*f*(-((b*Sqrt[c] *Sqrt[1 - (d*x^2)/c]*Sqrt[e + f*x^2]*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]] , -((c*f)/(d*e))])/(Sqrt[d]*f*Sqrt[c - d*x^2]*Sqrt[1 + (f*x^2)/e])) + (Sqr t[c]*(b*e + a*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[ArcSin[ (Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(Sqrt[d]*f*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])))/(2*a*b^2) + (Sqrt[c]*((c*e)/a + (a*d*f)/b^2)*Sqrt[1 - (d*x^2)/c ]*Sqrt[1 + (f*x^2)/e]*EllipticPi[-((b*c)/(a*d)), ArcSin[(Sqrt[d]*x)/Sqrt[c ]], -((c*f)/(d*e))])/(2*a*Sqrt[d]*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])
3.1.99.3.1 Defintions of rubi rules used
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c /(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0] && !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S imp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[1/(Sqrt[a + b*x^2]*Sqrt[1 + ( d/c)*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && !GtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ (Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) )], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2] Int[Sqrt[a + b*x^2]/Sqrt[1 + (d/c)*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && !GtQ[c, 0]
Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) ^2]), x_Symbol] :> Simp[f/b Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/b Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr eeQ[{a, b, c, d, e, f}, x] && !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
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[(Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2])/((a_) + (b_.)*(x_ )^2)^2, x_Symbol] :> Simp[x*Sqrt[c + d*x^2]*(Sqrt[e + f*x^2]/(2*a*(a + b*x^ 2))), x] + (Simp[(b^2*c*e - a^2*d*f)/(2*a*b^2) Int[1/((a + b*x^2)*Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Simp[d*(f/(2*a*b^2)) Int[(a - b*x^2)/ (Sqrt[c + d*x^2]*Sqrt[e + f*x^2]), x], x]) /; FreeQ[{a, b, c, d, e, f}, x]
Time = 3.40 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.60
| method | result | size |
| elliptic | \(\frac {\sqrt {\left (-d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {x \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}{2 a \left (b \,x^{2}+a \right )}-\frac {d f \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {c f -d e}{e d}}\right )}{2 b^{2} \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}-\frac {d e \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {c f -d e}{e d}}\right )}{2 a b \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}+\frac {d e \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, E\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {c f -d e}{e d}}\right )}{2 a b \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}+\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {\frac {d}{c}}, -\frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {\frac {d}{c}}}\right ) d f}{2 b^{2} \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}+\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \Pi \left (x \sqrt {\frac {d}{c}}, -\frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {\frac {d}{c}}}\right ) c e}{2 a^{2} \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}\right )}{\sqrt {-d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\) | \(574\) |
| default | \(-\frac {\sqrt {-d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}\, \left (\sqrt {\frac {d}{c}}\, a \,b^{2} d f \,x^{5}+\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {c f}{d e}}\right ) a^{2} b d f \,x^{2}+\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {c f}{d e}}\right ) a \,b^{2} d e \,x^{2}-\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {c f}{d e}}\right ) a \,b^{2} d e \,x^{2}-\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \Pi \left (x \sqrt {\frac {d}{c}}, -\frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {\frac {d}{c}}}\right ) a^{2} b d f \,x^{2}-\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \Pi \left (x \sqrt {\frac {d}{c}}, -\frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {\frac {d}{c}}}\right ) b^{3} c e \,x^{2}-\sqrt {\frac {d}{c}}\, a \,b^{2} c f \,x^{3}+\sqrt {\frac {d}{c}}\, a \,b^{2} d e \,x^{3}+\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {c f}{d e}}\right ) a^{3} d f +\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {c f}{d e}}\right ) a^{2} b d e -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, E\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {c f}{d e}}\right ) a^{2} b d e -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \Pi \left (x \sqrt {\frac {d}{c}}, -\frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {\frac {d}{c}}}\right ) a^{3} d f -\sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {f \,x^{2}+e}{e}}\, \Pi \left (x \sqrt {\frac {d}{c}}, -\frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {\frac {d}{c}}}\right ) a \,b^{2} c e -\sqrt {\frac {d}{c}}\, a \,b^{2} c e x \right )}{2 \left (-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e \right ) a^{2} \left (b \,x^{2}+a \right ) b^{2} \sqrt {\frac {d}{c}}}\) | \(773\) |
((-d*x^2+c)*(f*x^2+e))^(1/2)/(-d*x^2+c)^(1/2)/(f*x^2+e)^(1/2)*(1/2*x/a*(-d *f*x^4+c*f*x^2-d*e*x^2+c*e)^(1/2)/(b*x^2+a)-1/2*d*f/b^2/(d/c)^(1/2)*(1-d*x ^2/c)^(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)*Ellipti cF(x*(d/c)^(1/2),(-1-(c*f-d*e)/e/d)^(1/2))-1/2*d/a/b*e/(d/c)^(1/2)*(1-d*x^ 2/c)^(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)*Elliptic F(x*(d/c)^(1/2),(-1-(c*f-d*e)/e/d)^(1/2))+1/2*d/a/b*e/(d/c)^(1/2)*(1-d*x^2 /c)^(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)*EllipticE (x*(d/c)^(1/2),(-1-(c*f-d*e)/e/d)^(1/2))+1/2/b^2/(d/c)^(1/2)*(1-d*x^2/c)^( 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*c/a/d,(-f/e)^(1/2)/(d/c)^(1/2))*d*f+1/2/a^2/(d/c)^(1/2)*(1-d *x^2/c)^(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)*Ellip ticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1/2)/(d/c)^(1/2))*c*e)
Timed out. \[ \int \frac {\sqrt {c-d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {c-d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {c - d x^{2}} \sqrt {e + f x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \]
\[ \int \frac {\sqrt {c-d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c} \sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
\[ \int \frac {\sqrt {c-d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\int { \frac {\sqrt {-d x^{2} + c} \sqrt {f x^{2} + e}}{{\left (b x^{2} + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {c-d x^2} \sqrt {e+f x^2}}{\left (a+b x^2\right )^2} \, dx=\int \frac {\sqrt {c-d\,x^2}\,\sqrt {f\,x^2+e}}{{\left (b\,x^2+a\right )}^2} \,d x \]