3.2990 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x} \, dx\)

Optimal. Leaf size=31 \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

(4*ArcTanh[Sqrt[a + b*Sqrt[c/x]]/Sqrt[a]])/Sqrt[a]

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Rubi [A]  time = 0.0326503, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {369, 266, 63, 208} \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*Sqrt[c/x]]*x),x]

[Out]

(4*ArcTanh[Sqrt[a + b*Sqrt[c/x]]/Sqrt[a]])/Sqrt[a]

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

Rule 266

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

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{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}} x} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (\frac{4 \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b \sqrt{c}}+\frac{x^2}{b \sqrt{c}}} \, dx,x,\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}}\right )}{b \sqrt{c}},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{\sqrt{a}}\\ \end{align*}

Mathematica [A]  time = 0.0430557, size = 31, normalized size = 1. \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*Sqrt[c/x]]*x),x]

[Out]

(4*ArcTanh[Sqrt[a + b*Sqrt[c/x]]/Sqrt[a]])/Sqrt[a]

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Maple [B]  time = 0.041, size = 203, normalized size = 6.6 \begin{align*} -{\frac{1}{b}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( -b\sqrt{{\frac{c}{x}}}\sqrt{x}\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) -b\sqrt{{\frac{c}{x}}}\sqrt{x}\ln \left ({\frac{1}{2} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ){\frac{1}{\sqrt{a}}}} \right ) +2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}-2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a} \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{\frac{1}{\sqrt{{\frac{c}{x}}}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.44002, size = 194, normalized size = 6.26 \begin{align*} \left [\frac{2 \, \log \left (2 \, \sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{a} x \sqrt{\frac{c}{x}} + 2 \, a x \sqrt{\frac{c}{x}} + b c\right )}{\sqrt{a}}, -\frac{4 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}{a}\right )}{a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[2*log(2*sqrt(b*sqrt(c/x) + a)*sqrt(a)*x*sqrt(c/x) + 2*a*x*sqrt(c/x) + b*c)/sqrt(a), -4*sqrt(-a)*arctan(sqrt(b
*sqrt(c/x) + a)*sqrt(-a)/a)/a]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{x \sqrt{a + b \sqrt{\frac{c}{x}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/(x*sqrt(a + b*sqrt(c/x))), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sqrt{\frac{c}{x}} + a} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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