\(\int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx\) [271]

Optimal result

Integrand size = 23, antiderivative size = 233 \[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=-\frac {2 d \sqrt {a+\frac {b}{x^2}}}{\sqrt {c+\frac {d}{x^2}} x}+\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x+\frac {2 \sqrt {c} \sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}}-\frac {\sqrt {c} (b c+a d) \sqrt {a+\frac {b}{x^2}} \operatorname {EllipticF}\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}} \sqrt {c+\frac {d}{x^2}}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {382, 484, 545, 429, 506, 422} \[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=-\frac {2 d \sqrt {a+\frac {b}{x^2}}}{x \sqrt {c+\frac {d}{x^2}}}+x \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}}-\frac {\sqrt {c} \sqrt {a+\frac {b}{x^2}} (a d+b c) \operatorname {EllipticF}\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right ),1-\frac {b c}{a d}\right )}{a \sqrt {d} \sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}}+\frac {2 \sqrt {c} \sqrt {d} \sqrt {a+\frac {b}{x^2}} E\left (\cot ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {d}}\right )|1-\frac {b c}{a d}\right )}{\sqrt {c+\frac {d}{x^2}} \sqrt {\frac {c \left (a+\frac {b}{x^2}\right )}{a \left (c+\frac {d}{x^2}\right )}}} \]

[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 382

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 422

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

Rule 429

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

Rule 484

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 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[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 545

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]

Rubi steps \begin{align*} \text {integral}& = -\text {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 \text {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) \text {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) \text {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) \text {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] (verified)

Result contains complex when optimal does not.

Time = 2.47 (sec) , antiderivative size = 205, normalized size of antiderivative = 0.88 \[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=-\frac {\sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} x \left (\sqrt {\frac {a}{b}} \left (b+a x^2\right ) \left (d+c x^2\right )+2 i a d x \sqrt {1+\frac {a x^2}{b}} \sqrt {1+\frac {c x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {a}{b}} x\right )|\frac {b c}{a d}\right )+i (b c-a d) x \sqrt {1+\frac {a x^2}{b}} \sqrt {1+\frac {c x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {a}{b}} x\right ),\frac {b c}{a d}\right )\right )}{\sqrt {\frac {a}{b}} \left (b+a x^2\right ) \left (d+c x^2\right )} \]

[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)))

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.19

method	result	size
default	\(\frac {\sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \left (-\sqrt {-\frac {c}{d}}\, a c \,x^{4}+\sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, F\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right ) a d x -c b \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, x F\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right )+2 c b \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, x E\left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right )-\sqrt {-\frac {c}{d}}\, a d \,x^{2}-\sqrt {-\frac {c}{d}}\, b c \,x^{2}-\sqrt {-\frac {c}{d}}\, b d \right )}{\left (a \,x^{4} c +a d \,x^{2}+c b \,x^{2}+b d \right ) \sqrt {-\frac {c}{d}}}\)	\(277\)
risch	\(-x \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \sqrt {\frac {c \,x^{2}+d}{x^{2}}}+\frac {\left (\frac {a d \sqrt {1+\frac {c \,x^{2}}{d}}\, \sqrt {1+\frac {a \,x^{2}}{b}}\, F\left (x \sqrt {-\frac {c}{d}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {c}{d}}\, \sqrt {a \,x^{4} c +a d \,x^{2}+c b \,x^{2}+b d}}+\frac {b c \sqrt {1+\frac {c \,x^{2}}{d}}\, \sqrt {1+\frac {a \,x^{2}}{b}}\, F\left (x \sqrt {-\frac {c}{d}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {c}{d}}\, \sqrt {a \,x^{4} c +a d \,x^{2}+c b \,x^{2}+b d}}-\frac {2 c b \sqrt {1+\frac {c \,x^{2}}{d}}\, \sqrt {1+\frac {a \,x^{2}}{b}}\, \left (F\left (x \sqrt {-\frac {c}{d}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-E\left (x \sqrt {-\frac {c}{d}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {c}{d}}\, \sqrt {a \,x^{4} c +a d \,x^{2}+c b \,x^{2}+b d}}\right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, x^{2} \sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \sqrt {\left (a \,x^{2}+b \right ) \left (c \,x^{2}+d \right )}}{\left (a \,x^{2}+b \right ) \left (c \,x^{2}+d \right )}\)	\(394\)

[In]

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

[Out]

((a*x^2+b)/x^2)^(1/2)*x*((c*x^2+d)/x^2)^(1/2)*(-(-c/d)^(1/2)*a*c*x^4+((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))*a*d*x-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)*a*d*x^2-(-c/d)^(1/2)*b*c*x^2-(-c/d)^(1/2)*b*d)/(a*c*x^4+a*d*x^2+b*c*x^2+b*d)/(-c/d)^(1/2)

Fricas [F]

\[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=\int { \sqrt {a + \frac {b}{x^{2}}} \sqrt {c + \frac {d}{x^{2}}} \,d x } \]

[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)

Sympy [F]

\[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=\int \sqrt {a + \frac {b}{x^{2}}} \sqrt {c + \frac {d}{x^{2}}}\, dx \]

[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)

Maxima [F]

\[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=\int { \sqrt {a + \frac {b}{x^{2}}} \sqrt {c + \frac {d}{x^{2}}} \,d x } \]

[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)

Giac [F]

\[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=\int { \sqrt {a + \frac {b}{x^{2}}} \sqrt {c + \frac {d}{x^{2}}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+\frac {b}{x^2}} \sqrt {c+\frac {d}{x^2}} \, dx=\int \sqrt {a+\frac {b}{x^2}}\,\sqrt {c+\frac {d}{x^2}} \,d x \]

[In]

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

[Out]

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