3.108 \(\int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{(e+f x^2)^{3/2}} \, dx\)

Optimal. Leaf size=484 \[ -\frac{c^{3/2} \sqrt{a+b x^2} (b e-a f) \text{EllipticF}\left (\tan ^{-1}\left (\frac{x \sqrt{d e-c f}}{\sqrt{c} \sqrt{e+f x^2}}\right ),-\frac{e (b c-a d)}{a (d e-c f)}\right )}{a e f \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{x \sqrt{a+b x^2} (d e-c f)}{e f \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e f \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

-(((d*e - c*f)*x*Sqrt[a + b*x^2])/(e*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) + (Sqrt[c]*Sqrt[d*e - c*f]*Sqrt[a + b
*x^2]*EllipticE[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(e
*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(b*e - a*f)*Sqrt[a + b*x^2]*EllipticF[Arc
Tan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(a*e*f*Sqrt[d*e - c*f
]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*c*Sqrt[e]*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(
e*(c + d*x^2))]*EllipticPi[(d*e)/(d*e - c*f), ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])], -(((b*c -
a*d)*e)/(a*(d*e - c*f)))])/(a*f*Sqrt[d*e - c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2])

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Rubi [A]  time = 0.753475, antiderivative size = 484, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {557, 553, 537, 554, 422, 418, 492, 411} \[ -\frac{c^{3/2} \sqrt{a+b x^2} (b e-a f) F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e f \sqrt{c+d x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{x \sqrt{a+b x^2} (d e-c f)}{e f \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} \sqrt{d e-c f} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{f x^2+e}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e f \sqrt{c+d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}+\frac{b c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{d x^2+c}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a f \sqrt{e+f x^2} \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2),x]

[Out]

-(((d*e - c*f)*x*Sqrt[a + b*x^2])/(e*f*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])) + (Sqrt[c]*Sqrt[d*e - c*f]*Sqrt[a + b
*x^2]*EllipticE[ArcTan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(e
*f*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*(b*e - a*f)*Sqrt[a + b*x^2]*EllipticF[Arc
Tan[(Sqrt[d*e - c*f]*x)/(Sqrt[c]*Sqrt[e + f*x^2])], -(((b*c - a*d)*e)/(a*(d*e - c*f)))])/(a*e*f*Sqrt[d*e - c*f
]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) + (b*c*Sqrt[e]*Sqrt[a + b*x^2]*Sqrt[(c*(e + f*x^2))/(
e*(c + d*x^2))]*EllipticPi[(d*e)/(d*e - c*f), ArcSin[(Sqrt[d*e - c*f]*x)/(Sqrt[e]*Sqrt[c + d*x^2])], -(((b*c -
a*d)*e)/(a*(d*e - c*f)))])/(a*f*Sqrt[d*e - c*f]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]*Sqrt[e + f*x^2])

Rule 557

Int[(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2])/((e_) + (f_.)*(x_)^2)^(3/2), x_Symbol] :> Dist[b/f,
Int[Sqrt[c + d*x^2]/(Sqrt[a + b*x^2]*Sqrt[e + f*x^2]), x], x] - Dist[(b*e - a*f)/f, Int[Sqrt[c + d*x^2]/(Sqrt[
a + b*x^2]*(e + f*x^2)^(3/2)), x], x] /; FreeQ[{a, b, c, d, e, f}, x]

Rule 553

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

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 554

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)^(3/2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[(Sqrt
[c + d*x^2]*Sqrt[(a*(e + f*x^2))/(e*(a + b*x^2))])/(a*Sqrt[e + f*x^2]*Sqrt[(a*(c + d*x^2))/(c*(a + b*x^2))]),
Subst[Int[Sqrt[1 - ((b*c - a*d)*x^2)/c]/Sqrt[1 - ((b*e - a*f)*x^2)/e], x], x, x/Sqrt[a + b*x^2]], x] /; FreeQ[
{a, b, c, d, e, f}, x]

Rule 422

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

Rule 418

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticF[ArcT
an[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(a*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 492

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(x*Sqrt[a + b*x^2])/(b*Sqr
t[c + d*x^2]), x] - Dist[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]

Rule 411

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]*EllipticE[ArcTan
[Rt[d/c, 2]*x], 1 - (b*c)/(a*d)])/(c*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[(c*(a + b*x^2))/(a*(c + d*x^2))]), x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx &=\frac{b \int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \sqrt{e+f x^2}} \, dx}{f}-\frac{(b e-a f) \int \frac{\sqrt{c+d x^2}}{\sqrt{a+b x^2} \left (e+f x^2\right )^{3/2}} \, dx}{f}\\ &=-\frac{\left ((b e-a f) \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{(-d e+c f) x^2}{c}}}{\sqrt{1-\frac{(-b e+a f) x^2}{a}}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{e f \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\left (b c \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{1}{\left (1-d x^2\right ) \sqrt{1-\frac{(-b c+a d) x^2}{a}} \sqrt{1-\frac{(d e-c f) x^2}{e}}} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{a f \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{e+f x^2}}\\ &=\frac{b c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{c+d x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a f \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{e+f x^2}}-\frac{\left ((b e-a f) \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{(-b e+a f) x^2}{a}} \sqrt{1-\frac{(-d e+c f) x^2}{c}}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{e f \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\left ((b e-a f) (-d e+c f) \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-\frac{(-b e+a f) x^2}{a}} \sqrt{1-\frac{(-d e+c f) x^2}{c}}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{c e f \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=-\frac{(d e-c f) x \sqrt{a+b x^2}}{e f \sqrt{c+d x^2} \sqrt{e+f x^2}}-\frac{c^{3/2} (b e-a f) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e f \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{b c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{c+d x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a f \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{e+f x^2}}-\frac{\left (a (-d e+c f) \sqrt{c+d x^2} \sqrt{\frac{e \left (a+b x^2\right )}{a \left (e+f x^2\right )}}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{1-\frac{(-b e+a f) x^2}{a}}}{\left (1-\frac{(-d e+c f) x^2}{c}\right )^{3/2}} \, dx,x,\frac{x}{\sqrt{e+f x^2}}\right )}{c e f \sqrt{a+b x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}\\ &=-\frac{(d e-c f) x \sqrt{a+b x^2}}{e f \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{\sqrt{c} \sqrt{d e-c f} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{e f \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}-\frac{c^{3/2} (b e-a f) \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{c} \sqrt{e+f x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a e f \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{c+d x^2}}+\frac{b c \sqrt{e} \sqrt{a+b x^2} \sqrt{\frac{c \left (e+f x^2\right )}{e \left (c+d x^2\right )}} \Pi \left (\frac{d e}{d e-c f};\sin ^{-1}\left (\frac{\sqrt{d e-c f} x}{\sqrt{e} \sqrt{c+d x^2}}\right )|-\frac{(b c-a d) e}{a (d e-c f)}\right )}{a f \sqrt{d e-c f} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \sqrt{e+f x^2}}\\ \end{align*}

Mathematica [F]  time = 0.66681, size = 0, normalized size = 0. \[ \int \frac{\sqrt{a+b x^2} \sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2),x]

[Out]

Integrate[(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])/(e + f*x^2)^(3/2), x]

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Maple [F]  time = 0.06, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{b{x}^{2}+a}\sqrt{d{x}^{2}+c} \left ( f{x}^{2}+e \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)

[Out]

int((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(f*x^2 + e)^(3/2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}}}{\left (e + f x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**(1/2)*(d*x**2+c)**(1/2)/(f*x**2+e)**(3/2),x)

[Out]

Integral(sqrt(a + b*x**2)*sqrt(c + d*x**2)/(e + f*x**2)**(3/2), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b x^{2} + a} \sqrt{d x^{2} + c}}{{\left (f x^{2} + e\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)/(f*x^2+e)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*x^2 + a)*sqrt(d*x^2 + c)/(f*x^2 + e)^(3/2), x)