3.2989 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}}} \, dx\)

Optimal. Leaf size=95 \[ \frac{3 b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a^2 \sqrt{\frac{c}{x}}}+\frac{x \sqrt{a+b \sqrt{\frac{c}{x}}}}{a} \]

[Out]

(-3*b*c*Sqrt[a + b*Sqrt[c/x]])/(2*a^2*Sqrt[c/x]) + (Sqrt[a + b*Sqrt[c/x]]*x)/a + (3*b^2*c*ArcTanh[Sqrt[a + b*S
qrt[c/x]]/Sqrt[a]])/(2*a^(5/2))

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Rubi [A]  time = 0.0516822, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {255, 190, 51, 63, 208} \[ \frac{3 b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{3 b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a^2 \sqrt{\frac{c}{x}}}+\frac{x \sqrt{a+b \sqrt{\frac{c}{x}}}}{a} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a + b*Sqrt[c/x]],x]

[Out]

(-3*b*c*Sqrt[a + b*Sqrt[c/x]])/(2*a^2*Sqrt[c/x]) + (Sqrt[a + b*Sqrt[c/x]]*x)/a + (3*b^2*c*ArcTanh[Sqrt[a + b*S
qrt[c/x]]/Sqrt[a]])/(2*a^(5/2))

Rule 255

Int[((a_) + (b_.)*((c_.)*(x_)^(q_.))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Subst[Int[(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, p, q}, x] && Fraction
Q[n]

Rule 190

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

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

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}}}} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}}} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{1}{x^3 \sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{\sqrt{a+b \sqrt{\frac{c}{x}}} x}{a}+\operatorname{Subst}\left (\frac{\left (3 b \sqrt{c}\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{2 a},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\frac{3 b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a^2 \sqrt{\frac{c}{x}}}+\frac{\sqrt{a+b \sqrt{\frac{c}{x}}} x}{a}-\operatorname{Subst}\left (\frac{\left (3 b^2 c\right ) \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right )}{4 a^2},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\frac{3 b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a^2 \sqrt{\frac{c}{x}}}+\frac{\sqrt{a+b \sqrt{\frac{c}{x}}} x}{a}-\operatorname{Subst}\left (\frac{\left (3 b \sqrt{c}\right ) \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 )}{2 a^2},\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\frac{3 b c \sqrt{a+b \sqrt{\frac{c}{x}}}}{2 a^2 \sqrt{\frac{c}{x}}}+\frac{\sqrt{a+b \sqrt{\frac{c}{x}}} x}{a}+\frac{3 b^2 c \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{\sqrt{a}}\right )}{2 a^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.0930115, size = 89, normalized size = 0.94 \[ \frac{2 a^2 x-a b x \sqrt{\frac{c}{x}}-3 b^2 c}{2 a^2 \sqrt{a+b \sqrt{\frac{c}{x}}}}+\frac{3 b^2 c \tanh ^{-1}\left (\frac{\sqrt{a}}{\sqrt{a+b \sqrt{\frac{c}{x}}}}\right )}{2 a^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a + b*Sqrt[c/x]],x]

[Out]

(-3*b^2*c + 2*a^2*x - a*b*Sqrt[c/x]*x)/(2*a^2*Sqrt[a + b*Sqrt[c/x]]) + (3*b^2*c*ArcTanh[Sqrt[a]/Sqrt[a + b*Sqr
t[c/x]]])/(2*a^(5/2))

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Maple [B]  time = 0.037, size = 229, normalized size = 2.4 \begin{align*} -{\frac{1}{4}\sqrt{a+b\sqrt{{\frac{c}{x}}}}\sqrt{x} \left ( 8\,{a}^{3/2}\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{{\frac{c}{x}}}\sqrt{x}b-2\,{a}^{3/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{{\frac{c}{x}}}\sqrt{x}b-4\,{a}^{5/2}\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{x}-4\,a\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \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 ) } \right ) c{b}^{2}+{b}^{2}c\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 ) a \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}{a}^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(a+b*(c/x)^(1/2))^(1/2)*x^(1/2)*(8*a^(3/2)*(x*(a+b*(c/x)^(1/2)))^(1/2)*(c/x)^(1/2)*x^(1/2)*b-2*a^(3/2)*(a
*x+b*(c/x)^(1/2)*x)^(1/2)*(c/x)^(1/2)*x^(1/2)*b-4*a^(5/2)*(a*x+b*(c/x)^(1/2)*x)^(1/2)*x^(1/2)-4*a*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))*c*b^2+b^2*c*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))*a)/(x*(a+b*(c/x)^(1/2)))^(1/2)/a^(7/2)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.47758, size = 383, normalized size = 4.03 \begin{align*} \left [\frac{3 \, \sqrt{a} b^{2} c \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 ) - 2 \,{\left (3 \, a b x \sqrt{\frac{c}{x}} - 2 \, a^{2} x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{4 \, a^{3}}, -\frac{3 \, \sqrt{-a} b^{2} c \arctan \left (\frac{\sqrt{b \sqrt{\frac{c}{x}} + a} \sqrt{-a}}{a}\right ) +{\left (3 \, a b x \sqrt{\frac{c}{x}} - 2 \, a^{2} x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{2 \, a^{3}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(3*sqrt(a)*b^2*c*log(2*sqrt(b*sqrt(c/x) + a)*sqrt(a)*x*sqrt(c/x) + 2*a*x*sqrt(c/x) + b*c) - 2*(3*a*b*x*sq
rt(c/x) - 2*a^2*x)*sqrt(b*sqrt(c/x) + a))/a^3, -1/2*(3*sqrt(-a)*b^2*c*arctan(sqrt(b*sqrt(c/x) + a)*sqrt(-a)/a)
+ (3*a*b*x*sqrt(c/x) - 2*a^2*x)*sqrt(b*sqrt(c/x) + a))/a^3]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/sqrt(a + b*sqrt(c/x)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError