3.3.73 \(\int \frac {\sqrt {a+\frac {b}{x^2}}}{(c+\frac {d}{x^2})^{3/2}} \, dx\) [273]

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

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 262, normalized size of antiderivative = 1.00, 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 = {382, 480, 597, 545, 429, 506, 422} \begin {gather*} \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 {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 {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 )}}} \end {gather*}

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 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 480

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 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]

Rule 597

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]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+\frac {b}{x^2}}}{\left (c+\frac {d}{x^2}\right )^{3/2}} \, dx &=-\text {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 {\text {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 {\text {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 \text {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) \text {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) \text {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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 2.06, size = 191, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {a+\frac {b}{x^2}} \left (\sqrt {\frac {a}{b}} c x \left (b+a x^2\right )+2 i a d \sqrt {1+\frac {a x^2}{b}} \sqrt {1+\frac {c x^2}{d}} E\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{b}} x\right )|\frac {b c}{a d}\right )+i (b c-2 a d) \sqrt {1+\frac {a x^2}{b}} \sqrt {1+\frac {c x^2}{d}} F\left (i \sinh ^{-1}\left (\sqrt {\frac {a}{b}} x\right )|\frac {b c}{a d}\right )\right )}{\sqrt {\frac {a}{b}} c^2 \sqrt {c+\frac {d}{x^2}} \left (b+a x^2\right )} \end {gather*}

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 = 0.06, size = 185, normalized size = 0.71

method	result	size
default	\(-\frac {\left (\sqrt {-\frac {c}{d}}\, a \,x^{3}+b \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, \EllipticF \left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right )-2 b \sqrt {\frac {c \,x^{2}+d}{d}}\, \sqrt {\frac {a \,x^{2}+b}{b}}\, \EllipticE \left (x \sqrt {-\frac {c}{d}}, \sqrt {\frac {a d}{b c}}\right )+\sqrt {-\frac {c}{d}}\, b x \right ) \sqrt {\frac {a \,x^{2}+b}{x^{2}}}\, \left (c \,x^{2}+d \right )}{\sqrt {-\frac {c}{d}}\, \left (a \,x^{2}+b \right ) c \,x^{2} \left (\frac {c \,x^{2}+d}{x^{2}}\right )^{\frac {3}{2}}}\)	\(185\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b/x^2+a)^(1/2)/(c+d/x^2)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

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]

Exception raised: TypeError >> Symbolic function elliptic_ec takes exactly 1 arguments (2 given)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + \frac {b}{x^{2}}}}{\left (c + \frac {d}{x^{2}}\right )^{\frac {3}{2}}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a+\frac {b}{x^2}}}{{\left (c+\frac {d}{x^2}\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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