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

Optimal. Leaf size=381 \[ \frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right ),1-\frac{d e}{c f}\right )}{2 b^2 c \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (b^2 c e-a^2 d f\right ) \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{2 a^2 b^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}-\frac{f x \sqrt{c+d x^2}}{2 a b \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 a b \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

[Out]

-(f*x*Sqrt[c + d*x^2])/(2*a*b*Sqrt[e + f*x^2]) + (x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)) + (Sqrt
[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(2*a*b*Sqrt[(e*(c + d*x^2
))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]
], 1 - (d*e)/(c*f)])/(2*b^2*c*Sqrt[(e*(c + d*x^2))/(c*(e + f*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])

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Rubi [A]  time = 0.291334, antiderivative size = 381, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.219, Rules used = {548, 531, 418, 492, 411, 538, 537} \[ \frac{\sqrt{-c} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \left (b^2 c e-a^2 d f\right ) \Pi \left (\frac{b c}{a d};\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{-c}}\right )|\frac{c f}{d e}\right )}{2 a^2 b^2 \sqrt{d} \sqrt{c+d x^2} \sqrt{e+f x^2}}+\frac{x \sqrt{c+d x^2} \sqrt{e+f x^2}}{2 a \left (a+b x^2\right )}-\frac{f x \sqrt{c+d x^2}}{2 a b \sqrt{e+f x^2}}+\frac{\sqrt{e} \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 a b \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}}+\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 b^2 c \sqrt{e+f x^2} \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(f*x*Sqrt[c + d*x^2])/(2*a*b*Sqrt[e + f*x^2]) + (x*Sqrt[c + d*x^2]*Sqrt[e + f*x^2])/(2*a*(a + b*x^2)) + (Sqrt
[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticE[ArcTan[(Sqrt[f]*x)/Sqrt[e]], 1 - (d*e)/(c*f)])/(2*a*b*Sqrt[(e*(c + d*x^2
))/(c*(e + f*x^2))]*Sqrt[e + f*x^2]) + (d*Sqrt[e]*Sqrt[f]*Sqrt[c + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt[e]
], 1 - (d*e)/(c*f)])/(2*b^2*c*Sqrt[(e*(c + d*x^2))/(c*(e + f*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 548

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

Rule 531

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Dist[
e, Int[(a + b*x^n)^p*(c + d*x^n)^q, x], x] + Dist[f, Int[x^n*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, n, p, q}, x]

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]

Rule 538

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

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)
])

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 f) \int \frac{1}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{2 b^2}-\frac{(d f) \int \frac{x^2}{\sqrt{c+d x^2} \sqrt{e+f x^2}} \, dx}{2 a b}+\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{f x \sqrt{c+d x^2}}{2 a b \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{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 b^2 c \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{(e f) \int \frac{\sqrt{c+d x^2}}{\left (e+f x^2\right )^{3/2}} \, dx}{2 a b}+\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{f x \sqrt{c+d x^2}}{2 a b \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{e} \sqrt{f} \sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 a b \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \sqrt{e+f x^2}}+\frac{d \sqrt{e} \sqrt{f} \sqrt{c+d x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{e}}\right )|1-\frac{d e}{c f}\right )}{2 b^2 c \sqrt{\frac{e \left (c+d x^2\right )}{c \left (e+f x^2\right )}} \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]  time = 1.99279, size = 401, normalized size = 1.05 \[ \frac{-\frac{i c \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} (a f+b e) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{d}{c}}\right ),\frac{c f}{d e}\right )}{b^2}+\frac{i a c f \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{b^2}-\frac{i c e \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} \Pi \left (\frac{b c}{a d};i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{a \sqrt{\frac{d}{c}}}+\frac{c e x}{a+b x^2}+\frac{c f x^3}{a+b x^2}+\frac{d e x^3}{a+b x^2}+\frac{d f x^5}{a+b x^2}+\frac{i c e \sqrt{\frac{d}{c}} \sqrt{\frac{d x^2}{c}+1} \sqrt{\frac{f x^2}{e}+1} E\left (i \sinh ^{-1}\left (\sqrt{\frac{d}{c}} x\right )|\frac{c f}{d e}\right )}{b}}{2 a \sqrt{c+d x^2} \sqrt{e+f x^2}} \]

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*Sqrt[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*c*
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*Sqr
t[d/c]) + (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])

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Maple [A]  time = 0.036, size = 765, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}

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)

[Out]

1/2*(d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)*((-d/c)^(1/2)*x^5*a*b^2*d*f+((d*x^2+c)/c)^(1/2)*((f*x^2+e)/e)^(1/2)*Ellipt
icF(x*(-d/c)^(1/2),(c*f/d/e)^(1/2))*x^2*a^2*b*d*f+((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))*x^2*a*b^2*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))*x^2*a*b^2*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))*x^2*a^2*b*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))*x^2*b^3*c*e+(-d/c)^(1/2)*x^3*a*b^2*c*f+(-d/c)^(1/2)*x^3*a*b^2*d*e+((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)*Ellipt
icF(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)*x*a*b^2*c*e)/(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)

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

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)

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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((d*x^2+c)^(1/2)*(f*x^2+e)^(1/2)/(b*x^2+a)^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{c + d x^{2}} \sqrt{e + f x^{2}}}{\left (a + b x^{2}\right )^{2}}\, dx \end{align*}

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)

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

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)