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

Optimal. Leaf size=262 \[ -\frac{b \sqrt{a+\frac{b}{x^2}} \text{EllipticF}\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right ),1-\frac{b c}{a d}\right )}{a \sqrt{c} \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}-\frac{2 d \sqrt{a+\frac{b}{x^2}}}{c^2 x \sqrt{c+\frac{d}{x^2}}}+\frac{2 x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c^2}+\frac{2 \sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{c^{3/2} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}-\frac{x \sqrt{a+\frac{b}{x^2}}}{c \sqrt{c+\frac{d}{x^2}}} \]

[Out]

(-2*d*Sqrt[a + b/x^2])/(c^2*Sqrt[c + d/x^2]*x) - (Sqrt[a + b/x^2]*x)/(c*Sqrt[c + d/x^2]) + (2*Sqrt[a + b/x^2]*
Sqrt[c + d/x^2]*x)/c^2 + (2*Sqrt[d]*Sqrt[a + b/x^2]*EllipticE[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(
c^(3/2)*Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2]) - (b*Sqrt[a + b/x^2]*EllipticF[ArcCot[(Sqrt[c]*
x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(a*Sqrt[c]*Sqrt[d]*Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2])

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Rubi [A]  time = 0.280129, antiderivative size = 262, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {375, 469, 583, 531, 418, 492, 411} \[ -\frac{2 d \sqrt{a+\frac{b}{x^2}}}{c^2 x \sqrt{c+\frac{d}{x^2}}}+\frac{2 x \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}}}{c^2}+\frac{2 \sqrt{d} \sqrt{a+\frac{b}{x^2}} E\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{c^{3/2} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}}-\frac{x \sqrt{a+\frac{b}{x^2}}}{c \sqrt{c+\frac{d}{x^2}}}-\frac{b \sqrt{a+\frac{b}{x^2}} F\left (\cot ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{d}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{c} \sqrt{d} \sqrt{c+\frac{d}{x^2}} \sqrt{\frac{c \left (a+\frac{b}{x^2}\right )}{a \left (c+\frac{d}{x^2}\right )}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*d*Sqrt[a + b/x^2])/(c^2*Sqrt[c + d/x^2]*x) - (Sqrt[a + b/x^2]*x)/(c*Sqrt[c + d/x^2]) + (2*Sqrt[a + b/x^2]*
Sqrt[c + d/x^2]*x)/c^2 + (2*Sqrt[d]*Sqrt[a + b/x^2]*EllipticE[ArcCot[(Sqrt[c]*x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(
c^(3/2)*Sqrt[(c*(a + b/x^2))/(a*(c + d/x^2))]*Sqrt[c + d/x^2]) - (b*Sqrt[a + b/x^2]*EllipticF[ArcCot[(Sqrt[c]*
x)/Sqrt[d]], 1 - (b*c)/(a*d)])/(a*Sqrt[c]*Sqrt[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 469

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 + 1)*(c + d*x^n)^q)/(a*e*n*(p + 1)), x] + Dist[1/(a*n*(p + 1)), Int[(e*x)^m*(a + b*x^n)
^(p + 1)*(c + d*x^n)^(q - 1)*Simp[c*(m + n*(p + 1) + 1) + d*(m + n*(p + q + 1) + 1)*x^n, x], x], x] /; FreeQ[{
a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c
, d, e, m, n, p, q, x]

Rule 583

Int[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)),
x_Symbol] :> Simp[(e*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*c*g*(m + 1)), x] + Dist[1/(a*c*
g^n*(m + 1)), Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e
*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] &&
IGtQ[n, 0] && LtQ[m, -1]

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 \frac{\sqrt{a+\frac{b}{x^2}}}{\left (c+\frac{d}{x^2}\right )^{3/2}} \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{a+b x^2}}{x^2 \left (c+d x^2\right )^{3/2}} \, dx,x,\frac{1}{x}\right )\\ &=-\frac{\sqrt{a+\frac{b}{x^2}} x}{c \sqrt{c+\frac{d}{x^2}}}+\frac{\operatorname{Subst}\left (\int \frac{-2 a-b x^2}{x^2 \sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx,x,\frac{1}{x}\right )}{c}\\ &=-\frac{\sqrt{a+\frac{b}{x^2}} x}{c \sqrt{c+\frac{d}{x^2}}}+\frac{2 \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} x}{c^2}-\frac{\operatorname{Subst}\left (\int \frac{a b c+2 a b d x^2}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx,x,\frac{1}{x}\right )}{a c^2}\\ &=-\frac{\sqrt{a+\frac{b}{x^2}} x}{c \sqrt{c+\frac{d}{x^2}}}+\frac{2 \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} x}{c^2}-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+b x^2} \sqrt{c+d x^2}} \, dx,x,\frac{1}{x}\right )}{c}-\frac{(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 )}{c^2}\\ &=-\frac{2 d \sqrt{a+\frac{b}{x^2}}}{c^2 \sqrt{c+\frac{d}{x^2}} x}-\frac{\sqrt{a+\frac{b}{x^2}} x}{c \sqrt{c+\frac{d}{x^2}}}+\frac{2 \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} x}{c^2}-\frac{b \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{c} \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}}}+\frac{(2 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 )}{c}\\ &=-\frac{2 d \sqrt{a+\frac{b}{x^2}}}{c^2 \sqrt{c+\frac{d}{x^2}} x}-\frac{\sqrt{a+\frac{b}{x^2}} x}{c \sqrt{c+\frac{d}{x^2}}}+\frac{2 \sqrt{a+\frac{b}{x^2}} \sqrt{c+\frac{d}{x^2}} x}{c^2}+\frac{2 \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 )}{c^{3/2} \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{b \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{c} \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 = 0.224402, size = 191, normalized size = 0.73 \[ -\frac{\sqrt{a+\frac{b}{x^2}} \left (i \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} (b c-2 a d) \text{EllipticF}\left (i \sinh ^{-1}\left (x \sqrt{\frac{a}{b}}\right ),\frac{b c}{a d}\right )+2 i a d \sqrt{\frac{a x^2}{b}+1} \sqrt{\frac{c x^2}{d}+1} E\left (i \sinh ^{-1}\left (\sqrt{\frac{a}{b}} x\right )|\frac{b c}{a d}\right )+c x \sqrt{\frac{a}{b}} \left (a x^2+b\right )\right )}{c^2 \sqrt{\frac{a}{b}} \left (a x^2+b\right ) \sqrt{c+\frac{d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.071, size = 187, normalized size = 0.7 \begin{align*}{\frac{c{x}^{2}+d}{{x}^{2}c \left ( a{x}^{2}+b \right ) }\sqrt{{\frac{a{x}^{2}+b}{{x}^{2}}}} \left ( -{x}^{3}a\sqrt{-{\frac{c}{d}}}+2\,{\it EllipticE} \left ( x\sqrt{-{\frac{c}{d}}},\sqrt{{\frac{ad}{bc}}} \right ) b\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 ) b\sqrt{{\frac{c{x}^{2}+d}{d}}}\sqrt{{\frac{a{x}^{2}+b}{b}}}-xb\sqrt{-{\frac{c}{d}}} \right ){\frac{1}{\sqrt{-{\frac{c}{d}}}}} \left ({\frac{c{x}^{2}+d}{{x}^{2}}} \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((a*x^2+b)/x^2)^(1/2)/x^2/(a*x^2+b)*(-x^3*a*(-c/d)^(1/2)+2*EllipticE(x*(-c/d)^(1/2),(a*d/b/c)^(1/2))*b*((c*x^2
+d)/d)^(1/2)*((a*x^2+b)/b)^(1/2)-EllipticF(x*(-c/d)^(1/2),(a*d/b/c)^(1/2))*b*((c*x^2+d)/d)^(1/2)*((a*x^2+b)/b)
^(1/2)-x*b*(-c/d)^(1/2))*(c*x^2+d)/(-c/d)^(1/2)/c/((c*x^2+d)/x^2)^(3/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{4} \sqrt{\frac{a x^{2} + b}{x^{2}}} \sqrt{\frac{c x^{2} + d}{x^{2}}}}{c^{2} x^{4} + 2 \, c d x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(x^4*sqrt((a*x^2 + b)/x^2)*sqrt((c*x^2 + d)/x^2)/(c^2*x^4 + 2*c*d*x^2 + d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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