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

Optimal. Leaf size=233 \[ -\frac{\sqrt{c} \sqrt{a+\frac{b}{x^2}} (a d+b c) \text{EllipticF}\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ),1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}-\frac{2 d \sqrt{a+\frac{b}{x^2}}}{x \sqrt{c+\frac{d}{x^2}}}+x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}+\frac{2 \sqrt{c} \sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]

[Out]

(-2*d*Sqrt[a + b/x^2])/(Sqrt[c + d/x^2]*x) + Sqrt[a + b/x^2]*Sqrt[c + d/x^2]*x + (2*Sqrt[c]*Sqrt[d]*Sqrt[a + b
/x^2]*EllipticE[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c +
d/x^2]) - (Sqrt[c]*(b*c + a*d)*Sqrt[a + b/x^2]*EllipticF[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(a*Sq
rt[d]*Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2])

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Rubi [A]  time = 0.217579, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {375, 473, 531, 418, 492, 411} \[ -\frac{2 d \sqrt{a+\frac{b}{x^2}}}{x \sqrt{c+\frac{d}{x^2}}}+x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}-\frac{\sqrt{c} \sqrt{a+\frac{b}{x^2}} (a d+b c) F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}+\frac{2 \sqrt{c} \sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b/x^2]*Sqrt[c + d/x^2],x]

[Out]

(-2*d*Sqrt[a + b/x^2])/(Sqrt[c + d/x^2]*x) + Sqrt[a + b/x^2]*Sqrt[c + d/x^2]*x + (2*Sqrt[c]*Sqrt[d]*Sqrt[a + b
/x^2]*EllipticE[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c +
d/x^2]) - (Sqrt[c]*(b*c + a*d)*Sqrt[a + b/x^2]*EllipticF[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(a*Sq
rt[d]*Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2])

Rule 375

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> -Subst[Int[((a + b/x^n)^p*(c +
d/x^n)^q)/x^2, x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]

Rule 473

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((e*x)^(m
+ 1)*(a + b*x^n)^p*(c + d*x^n)^q)/(e*(m + 1)), x] - Dist[n/(e^n*(m + 1)), Int[(e*x)^(m + n)*(a + b*x^n)^(p -
1)*(c + d*x^n)^(q - 1)*Simp[b*c*p + a*d*q + b*d*(p + q)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b*
c - a*d, 0] && IGtQ[n, 0] && GtQ[q, 0] && LtQ[m, -1] && GtQ[p, 0] && IntBinomialQ[a, b, c, d, e, m, n, p, q, 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]

Rubi steps

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

Mathematica [C]  time = 0.296365, size = 205, normalized size = 0.88 \[ -\frac{x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} \left (i x \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} (b c-a d) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{a}{b}}\right ),\frac{b c}{a d}\right )+\sqrt{\frac{a}{b}} \left (a x^2+b\right ) \left (c x^2+d\right )+2 i a d x \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a}{b}} x\right )|\frac{b c}{a d}\right )\right )}{\sqrt{\frac{a}{b}} \left (a x^2+b\right ) \left (c x^2+d\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b/x^2]*Sqrt[c + d/x^2],x]

[Out]

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

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Maple [A]  time = 0.038, size = 277, normalized size = 1.2 \begin{align*}{\frac{x}{ac{x}^{4}+ad{x}^{2}+bc{x}^{2}+bd}\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( -\sqrt{-{\frac{c}{d}}}{x}^{4}ac+\sqrt{{\frac{c{x}^{2}+d}{d}}}\sqrt{{\frac{a{x}^{2}+b}{b}}}{\it EllipticF} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) xad-cb\sqrt{{\frac{c{x}^{2}+d}{d}}}\sqrt{{\frac{a{x}^{2}+b}{b}}}x{\it EllipticF} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) +2\,cb\sqrt{{\frac{c{x}^{2}+d}{d}}}\sqrt{{\frac{a{x}^{2}+b}{b}}}x{\it EllipticE} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) -\sqrt{-{\frac{c}{d}}}{x}^{2}ad-\sqrt{-{\frac{c}{d}}}{x}^{2}bc-\sqrt{-{\frac{c}{d}}}bd \right ){\frac{1}{\sqrt{-{\frac{c}{d}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*x^2+b)/x^2)^(1/2)*x*((c*x^2+d)/x^2)^(1/2)*(-(-c/d)^(1/2)*x^4*a*c+((c*x^2+d)/d)^(1/2)*((a*x^2+b)/b)^(1/2)*E
llipticF(x*(-c/d)^(1/2),(a*d/b/c)^(1/2))*x*a*d-c*b*((c*x^2+d)/d)^(1/2)*((a*x^2+b)/b)^(1/2)*x*EllipticF(x*(-c/d
)^(1/2),(a*d/b/c)^(1/2))+2*c*b*((c*x^2+d)/d)^(1/2)*((a*x^2+b)/b)^(1/2)*x*EllipticE(x*(-c/d)^(1/2),(a*d/b/c)^(1
/2))-(-c/d)^(1/2)*x^2*a*d-(-c/d)^(1/2)*x^2*b*c-(-c/d)^(1/2)*b*d)/(a*c*x^4+a*d*x^2+b*c*x^2+b*d)/(-c/d)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt((a*x^2 + b)/x^2)*sqrt((c*x^2 + d)/x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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