3.1.99 \(\int \frac {\sqrt {c-d x^2} \sqrt {e+f x^2}}{(a+b x^2)^2} \, dx\) [99]

Optimal. Leaf size=359 \[ \frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}+\frac {\sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a b \sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}}-\frac {\sqrt {c} \sqrt {d} (b e+a f) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} F\left (\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a b^2 \sqrt {c-d x^2} \sqrt {e+f x^2}}+\frac {\sqrt {c} \left (b^2 c e+a^2 d f\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (-\frac {b c}{a d};\sin ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a^2 b^2 \sqrt {d} \sqrt {c-d x^2} \sqrt {e+f x^2}} \]

[Out]

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)*Elliptic
Pi(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)

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 359, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {562, 552, 551, 538, 438, 437, 435, 432, 430} \begin {gather*} \frac {\sqrt {c} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} \left (a^2 d f+b^2 c e\right ) \Pi \left (-\frac {b c}{a d};\text {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}}-\frac {\sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {\frac {f x^2}{e}+1} (a f+b e) F\left (\text {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} \sqrt {d} \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2} E\left (\text {ArcSin}\left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {c f}{d e}\right )}{2 a b \sqrt {c-d x^2} \sqrt {\frac {f x^2}{e}+1}}+\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2)^2,x]

[Out]

(x*Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)) + (Sqrt[c]*Sqrt[d]*Sqrt[1 - (d*x^2)/c]*Sqrt[e + f*x^2]*E
llipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((c*f)/(d*e))])/(2*a*b*Sqrt[c - d*x^2]*Sqrt[1 + (f*x^2)/e]) - (Sqrt[c]*
Sqrt[d]*(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))])/(2*a*b^2*Sqrt[c - d*x^2]*Sqrt[e + f*x^2]) + (Sqrt[c]*(b^2*c*e + a^2*d*f)*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^2*b^2*Sqrt[d]*Sqrt[c -
d*x^2]*Sqrt[e + f*x^2])

Rule 430

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(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] && Gt
Q[a, 0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])

Rule 432

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Dist[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]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[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
]

Rule 437

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[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]

Rule 438

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[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]

Rule 538

Int[((e_) + (f_.)*(x_)^(n_))/(Sqrt[(a_) + (b_.)*(x_)^(n_)]*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/
b, Int[Sqrt[a + b*x^n]/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/(Sqrt[a + b*x^n]*Sqrt[c + d*x^n]),
x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &&  !(EqQ[n, 2] && ((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && (PosQ[
d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c]))))))

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[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] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[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]

Rule 562

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] + (Dist[(b^2*c*e - a^2*d*f)/(2*a*b^2), Int[1/((a + b*x^2)*Sqr
t[c + d*x^2]*Sqrt[e + f*x^2]), x], x] + Dist[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]

Rubi steps

\begin {align*} \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 {(d f) \int \frac {a-b x^2}{\sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx}{2 a b^2}+\frac {1}{2} \left (\frac {c e}{a}+\frac {a d f}{b^2}\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx\\ &=\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}+\frac {d \int \frac {\sqrt {e+f x^2}}{\sqrt {c-d x^2}} \, dx}{2 a b}-\frac {(d (b e+a f)) \int \frac {1}{\sqrt {c-d x^2} \sqrt {e+f x^2}} \, dx}{2 a b^2}+\frac {\left (\left (\frac {c e}{a}+\frac {a d f}{b^2}\right ) \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2}} \, dx}{2 \sqrt {c-d x^2}}\\ &=\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}+\frac {\left (d \sqrt {1-\frac {d x^2}{c}}\right ) \int \frac {\sqrt {e+f x^2}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{2 a b \sqrt {c-d x^2}}-\frac {\left (d (b e+a f) \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}} \, dx}{2 a b^2 \sqrt {e+f x^2}}+\frac {\left (\left (\frac {c e}{a}+\frac {a d f}{b^2}\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\left (a+b x^2\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}} \, dx}{2 \sqrt {c-d x^2} \sqrt {e+f x^2}}\\ &=\frac {x \sqrt {c-d x^2} \sqrt {e+f x^2}}{2 a \left (a+b x^2\right )}+\frac {\sqrt {c} \left (\frac {c e}{a}+\frac {a d f}{b^2}\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (-\frac {b c}{a d};\sin ^{-1}\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 {\left (d \sqrt {1-\frac {d x^2}{c}} \sqrt {e+f x^2}\right ) \int \frac {\sqrt {1+\frac {f x^2}{e}}}{\sqrt {1-\frac {d x^2}{c}}} \, dx}{2 a b \sqrt {c-d x^2} \sqrt {1+\frac {f x^2}{e}}}-\frac {\left (d (b e+a f) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}\right ) \int \frac {1}{\sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}}} \, dx}{2 a b^2 \sqrt {c-d x^2} \sqrt {e+f x^2}}\\ &=\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 (\sin ^{-1}\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}} F\left (\sin ^{-1}\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 (\frac {c e}{a}+\frac {a d f}{b^2}\right ) \sqrt {1-\frac {d x^2}{c}} \sqrt {1+\frac {f x^2}{e}} \Pi \left (-\frac {b c}{a d};\sin ^{-1}\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}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 3.42, size = 422, normalized size = 1.18 \begin {gather*} \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 \sinh ^{-1}\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}} F\left (i \sinh ^{-1}\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}} \Pi \left (-\frac {b c}{a d};i \sinh ^{-1}\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}} \Pi \left (-\frac {b c}{a d};i \sinh ^{-1}\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}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[c - d*x^2]*Sqrt[e + f*x^2])/(a + b*x^2)^2,x]

[Out]

((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*Sq
rt[-(d/c)]*(b*e + a*f)*Sqrt[1 - (d*x^2)/c]*Sqrt[1 + (f*x^2)/e]*EllipticF[I*ArcSinh[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]*Ellipt
icPi[-((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])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(772\) vs. \(2(302)=604\).
time = 0.16, size = 773, normalized size = 2.15

method	result	size
elliptic	\(\frac {\sqrt {\left (-d \,x^{2}+c \right ) \left (f \,x^{2}+e \right )}\, \left (\frac {x \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}{2 a \left (b \,x^{2}+a \right )}-\frac {d f \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {c f -d e}{e d}}\right )}{2 b^{2} \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}-\frac {d e \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticF \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {c f -d e}{e d}}\right )}{2 a b \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}+\frac {d e \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticE \left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {c f -d e}{e d}}\right )}{2 a b \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}+\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {\frac {d}{c}}, -\frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {\frac {d}{c}}}\right ) d f}{2 b^{2} \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}+\frac {\sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {f \,x^{2}}{e}}\, \EllipticPi \left (x \sqrt {\frac {d}{c}}, -\frac {b c}{a d}, \frac {\sqrt {-\frac {f}{e}}}{\sqrt {\frac {d}{c}}}\right ) c e}{2 a^{2} \sqrt {\frac {d}{c}}\, \sqrt {-d f \,x^{4}+c f \,x^{2}-d e \,x^{2}+c e}}\right )}{\sqrt {-d \,x^{2}+c}\, \sqrt {f \,x^{2}+e}}\)	\(574\)
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}}\, \EllipticF \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}}\, \EllipticF \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}}\, \EllipticE \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}}\, \EllipticPi \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}}\, \EllipticPi \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}}\, \EllipticF \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}}\, \EllipticF \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}}\, \EllipticE \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}}\, \EllipticPi \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}}\, \EllipticPi \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}}}\)	\(773\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(-d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*((d/c)^(1/2)*a*b^2*d*f*x^5+((-d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Elli
pticF(x*(d/c)^(1/2),(-c*f/d/e)^(1/2))*a^2*b*d*f*x^2+((-d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticF(x*(d/c)
^(1/2),(-c*f/d/e)^(1/2))*a*b^2*d*e*x^2-((-d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(d/c)^(1/2),(-c*f/
d/e)^(1/2))*a*b^2*d*e*x^2-((-d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/e)^(1
/2)/(d/c)^(1/2))*a^2*b*d*f*x^2-((-d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f/
e)^(1/2)/(d/c)^(1/2))*b^3*c*e*x^2-(d/c)^(1/2)*a*b^2*c*f*x^3+(d/c)^(1/2)*a*b^2*d*e*x^3+((-d*x^2+c)/c)^(1/2)*((f
*x^2+e)/e)^(1/2)*EllipticF(x*(d/c)^(1/2),(-c*f/d/e)^(1/2))*a^3*d*f+((-d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*El
lipticF(x*(d/c)^(1/2),(-c*f/d/e)^(1/2))*a^2*b*d*e-((-d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticE(x*(d/c)^(
1/2),(-c*f/d/e)^(1/2))*a^2*b*d*e-((-d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-
f/e)^(1/2)/(d/c)^(1/2))*a^3*d*f-((-d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*EllipticPi(x*(d/c)^(1/2),-b*c/a/d,(-f
/e)^(1/2)/(d/c)^(1/2))*a*b^2*c*e-(d/c)^(1/2)*a*b^2*c*e*x)/(-d*f*x^4+c*f*x^2-d*e*x^2+c*e)/a^2/(b*x^2+a)/b^2/(d/
c)^(1/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-d*x^2 + c)*sqrt(f*x^2 + e)/(b*x^2 + a)^2, x)

________________________________________________________________________________________

Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {c - d x^{2}} \sqrt {e + f x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-d*x**2+c)**(1/2)*(f*x**2+e)**(1/2)/(b*x**2+a)**2,x)

[Out]

Integral(sqrt(c - d*x**2)*sqrt(e + f*x**2)/(a + b*x**2)**2, x)

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(-d*x^2 + c)*sqrt(f*x^2 + e)/(b*x^2 + a)^2, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c-d\,x^2}\,\sqrt {f\,x^2+e}}{{\left (b\,x^2+a\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c - d*x^2)^(1/2)*(e + f*x^2)^(1/2))/(a + b*x^2)^2,x)

[Out]

int(((c - d*x^2)^(1/2)*(e + f*x^2)^(1/2))/(a + b*x^2)^2, x)

________________________________________________________________________________________