3.2985 \(\int \frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{x^2} \, dx\)

Optimal. Leaf size=56 \[ \frac{4 a \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^2 c} \]

[Out]

(4*a*(a + b*Sqrt[c/x])^(3/2))/(3*b^2*c) - (4*(a + b*Sqrt[c/x])^(5/2))/(5*b^2*c)

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Rubi [A]  time = 0.0380851, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {369, 266, 43} \[ \frac{4 a \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^2 c}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^2 c} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c/x]]/x^2,x]

[Out]

(4*a*(a + b*Sqrt[c/x])^(3/2))/(3*b^2*c) - (4*(a + b*Sqrt[c/x])^(5/2))/(5*b^2*c)

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0335908, size = 43, normalized size = 0.77 \[ \frac{4 \left (2 a-3 b \sqrt{\frac{c}{x}}\right ) \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{15 b^2 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(2*a - 3*b*Sqrt[c/x])*(a + b*Sqrt[c/x])^(3/2))/(15*b^2*c)

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Maple [A]  time = 0.022, size = 70, normalized size = 1.3 \begin{align*} -{\frac{4}{15\,cx{b}^{2}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{{\frac{3}{2}}} \left ( 3\,b\sqrt{{\frac{c}{x}}}-2\,a \right ){\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*(a+b*(c/x)^(1/2))^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)/c/x*(3*b*(c/x)^(1/2)-2*a)/(x*(a+b*(c/x)^(1/2)))^(1/2
)/b^2

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Maxima [A]  time = 0.932362, size = 58, normalized size = 1.04 \begin{align*} -\frac{4 \,{\left (\frac{3 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}}}{b^{2}} - \frac{5 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a}{b^{2}}\right )}}{15 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*(3*(b*sqrt(c/x) + a)^(5/2)/b^2 - 5*(b*sqrt(c/x) + a)^(3/2)*a/b^2)/c

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Fricas [A]  time = 1.43465, size = 104, normalized size = 1.86 \begin{align*} -\frac{4 \,{\left (a b x \sqrt{\frac{c}{x}} + 3 \, b^{2} c - 2 \, a^{2} x\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{15 \, b^{2} c x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/15*(a*b*x*sqrt(c/x) + 3*b^2*c - 2*a^2*x)*sqrt(b*sqrt(c/x) + a)/(b^2*c*x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sqrt(c/x))/x**2, x)

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Giac [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out