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]

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

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Rubi [A]  time = 0.033241, antiderivative size = 43, normalized size of antiderivative = 1., 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 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]]

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 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 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]

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 = 0.0285701, size = 73, normalized size = 1.7 \[ \frac{a d x \sqrt{c x^2+d} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{c x^2+d}}\right )-b \sqrt{c} \left (c x^2+d\right )}{\sqrt{c} d x^2 \sqrt{c+\frac{d}{x^2}}} \]

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)

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Maple [A]  time = 0.009, size = 69, normalized size = 1.6 \begin{align*}{\frac{1}{{x}^{2}d}\sqrt{c{x}^{2}+d} \left ( a\ln \left ( \sqrt{c}x+\sqrt{c{x}^{2}+d} \right ) xd-b\sqrt{c{x}^{2}+d}\sqrt{c} \right ){\frac{1}{\sqrt{{\frac{c{x}^{2}+d}{{x}^{2}}}}}}{\frac{1}{\sqrt{c}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.37018, size = 298, normalized size = 6.93 \begin{align*} \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 ] \end{align*}

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)]

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Sympy [A]  time = 14.1264, size = 63, normalized size = 1.47 \begin{align*} - \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} \end{align*}

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

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

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]

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