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

Optimal. Leaf size=43 \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.03, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {446, 80, 63, 208} \[ \frac {a \tanh ^{-1}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b \sqrt {c+\frac {d}{x^2}}}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*Sqrt[c + d/x^2])/d) + (a*ArcTanh[Sqrt[c + d/x^2]/Sqrt[c]])/Sqrt[c]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 80

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

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 446

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 73, normalized size = 1.70 \[ \frac {a d x \sqrt {c x^2+d} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {c x^2+d}}\right )-b \sqrt {c} \left (c x^2+d\right )}{\sqrt {c} d x^2 \sqrt {c+\frac {d}{x^2}}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(b*Sqrt[c]*(d + c*x^2)) + a*d*x*Sqrt[d + c*x^2]*ArcTanh[(Sqrt[c]*x)/Sqrt[d + c*x^2]])/(Sqrt[c]*d*Sqrt[c + d/
x^2]*x^2)

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 130, normalized size = 3.02 \[ \left [\frac {a \sqrt {c} d \log \left (-2 \, c x^{2} - 2 \, \sqrt {c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}} - d\right ) - 2 \, b c \sqrt {\frac {c x^{2} + d}{x^{2}}}}{2 \, c d}, -\frac {a \sqrt {-c} d \arctan \left (\frac {\sqrt {-c} x^{2} \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c x^{2} + d}\right ) + b c \sqrt {\frac {c x^{2} + d}{x^{2}}}}{c d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(a*sqrt(c)*d*log(-2*c*x^2 - 2*sqrt(c)*x^2*sqrt((c*x^2 + d)/x^2) - d) - 2*b*c*sqrt((c*x^2 + d)/x^2))/(c*d)
, -(a*sqrt(-c)*d*arctan(sqrt(-c)*x^2*sqrt((c*x^2 + d)/x^2)/(c*x^2 + d)) + b*c*sqrt((c*x^2 + d)/x^2))/(c*d)]

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(t_nostep)]Warning, choosing root of [1,0,%%%{-2,[1,2,0]%%%}+%%%{-2,[1,0,0]%%%}+%%%{-2,[0,1,1]%%%},0,%%%{1,[2
,4,0]%%%}+%%%{-2,[2,2,0]%%%}+%%%{1,[2,0,0]%%%}+%%%{2,[1,3,1]%%%}+%%%{-2,[1,1,1]%%%}+%%%{1,[0,2,2]%%%}] at para
meters values [86,-97,-82]Warning, choosing root of [1,0,%%%{-2,[1,0,1]%%%}+%%%{-4,[0,1,0]%%%},0,%%%{1,[2,0,2]
%%%}] at parameters values [-64,61.7937478349,70]Sign error (%%%{d,0%%%}+%%%{-2*sqrt(c)*sqrt(d),1/2%%%}+%%%{2*
c,1%%%}+%%%{-c*sqrt(c)*sqrt(d)/d,3/2%%%}+%%%{c^2*sqrt(c)*sqrt(d)/(4*d^2),5/2%%%}+%%%{undef,7/2%%%})Limit: Max
order reached or unable to make series expansion Error: Bad Argument Value

________________________________________________________________________________________

maple [A]  time = 0.05, size = 70, normalized size = 1.63 \[ -\frac {\sqrt {c \,x^{2}+d}\, \left (-a d x \ln \left (\sqrt {c}\, x +\sqrt {c \,x^{2}+d}\right )+\sqrt {c \,x^{2}+d}\, b \sqrt {c}\right )}{\sqrt {\frac {c \,x^{2}+d}{x^{2}}}\, \sqrt {c}\, d \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.10, size = 54, normalized size = 1.26 \[ -\frac {a \log \left (\frac {\sqrt {c + \frac {d}{x^{2}}} - \sqrt {c}}{\sqrt {c + \frac {d}{x^{2}}} + \sqrt {c}}\right )}{2 \, \sqrt {c}} - \frac {b \sqrt {c + \frac {d}{x^{2}}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 4.85, size = 35, normalized size = 0.81 \[ \frac {a\,\mathrm {atanh}\left (\frac {\sqrt {c+\frac {d}{x^2}}}{\sqrt {c}}\right )}{\sqrt {c}}-\frac {b\,\sqrt {c+\frac {d}{x^2}}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 22.34, size = 63, normalized size = 1.47 \[ - \frac {a \operatorname {atan}{\left (\frac {1}{\sqrt {- \frac {1}{c}} \sqrt {c + \frac {d}{x^{2}}}} \right )}}{c \sqrt {- \frac {1}{c}}} + \frac {b \left (\begin {cases} - \frac {1}{\sqrt {c} x^{2}} & \text {for}\: d = 0 \\- \frac {2 \sqrt {c + \frac {d}{x^{2}}}}{d} & \text {otherwise} \end {cases}\right )}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*atan(1/(sqrt(-1/c)*sqrt(c + d/x**2)))/(c*sqrt(-1/c)) + b*Piecewise((-1/(sqrt(c)*x**2), Eq(d, 0)), (-2*sqrt(
c + d/x**2)/d, True))/2

________________________________________________________________________________________