3.2986 \(\int \frac{\sqrt{a+b \sqrt{\frac{c}{x}}}}{x^3} \, dx\)

Optimal. Leaf size=116 \[ -\frac{12 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2}+\frac{4 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2} \]

[Out]

(4*a^3*(a + b*Sqrt[c/x])^(3/2))/(3*b^4*c^2) - (12*a^2*(a + b*Sqrt[c/x])^(5/2))/(5*b^4*c^2) + (12*a*(a + b*Sqrt
[c/x])^(7/2))/(7*b^4*c^2) - (4*(a + b*Sqrt[c/x])^(9/2))/(9*b^4*c^2)

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Rubi [A]  time = 0.0686195, antiderivative size = 116, 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} \[ -\frac{12 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2}+\frac{4 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c/x]]/x^3,x]

[Out]

(4*a^3*(a + b*Sqrt[c/x])^(3/2))/(3*b^4*c^2) - (12*a^2*(a + b*Sqrt[c/x])^(5/2))/(5*b^4*c^2) + (12*a*(a + b*Sqrt
[c/x])^(7/2))/(7*b^4*c^2) - (4*(a + b*Sqrt[c/x])^(9/2))/(9*b^4*c^2)

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^3} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}}}{x^3} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int 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 )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \left (-\frac{a^3 \sqrt{a+b \sqrt{c} x}}{b^3 c^{3/2}}+\frac{3 a^2 \left (a+b \sqrt{c} x\right )^{3/2}}{b^3 c^{3/2}}-\frac{3 a \left (a+b \sqrt{c} x\right )^{5/2}}{b^3 c^{3/2}}+\frac{\left (a+b \sqrt{c} x\right )^{7/2}}{b^3 c^{3/2}}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\frac{4 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^4 c^2}-\frac{12 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{5 b^4 c^2}+\frac{12 a \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^4 c^2}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^4 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0500101, size = 75, normalized size = 0.65 \[ \frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2} \left (-24 a^2 b x \sqrt{\frac{c}{x}}+16 a^3 x+30 a b^2 c-35 b^3 c \sqrt{\frac{c}{x}}\right )}{315 b^4 c^2 x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sqrt[c/x]]/x^3,x]

[Out]

(4*(a + b*Sqrt[c/x])^(3/2)*(30*a*b^2*c - 35*b^3*c*Sqrt[c/x] + 16*a^3*x - 24*a^2*b*Sqrt[c/x]*x))/(315*b^4*c^2*x
)

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Maple [A]  time = 0.022, size = 97, normalized size = 0.8 \begin{align*} -{\frac{4}{315\,{c}^{2}{x}^{2}{b}^{4}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{{\frac{3}{2}}} \left ( 35\, \left ({\frac{c}{x}} \right ) ^{3/2}x{b}^{3}+24\,\sqrt{{\frac{c}{x}}}x{a}^{2}b-30\,ac{b}^{2}-16\,{a}^{3}x \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^3,x)

[Out]

-4/315*(a+b*(c/x)^(1/2))^(1/2)*(a*x+b*(c/x)^(1/2)*x)^(3/2)/c^2/x^2*(35*(c/x)^(3/2)*x*b^3+24*(c/x)^(1/2)*x*a^2*
b-30*a*c*b^2-16*a^3*x)/(x*(a+b*(c/x)^(1/2)))^(1/2)/b^4

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Maxima [A]  time = 0.931915, size = 115, normalized size = 0.99 \begin{align*} -\frac{4 \,{\left (\frac{35 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{9}{2}}}{b^{4}} - \frac{135 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{7}{2}} a}{b^{4}} + \frac{189 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}} a^{2}}{b^{4}} - \frac{105 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a^{3}}{b^{4}}\right )}}{315 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c/x)^(1/2))^(1/2)/x^3,x, algorithm="maxima")

[Out]

-4/315*(35*(b*sqrt(c/x) + a)^(9/2)/b^4 - 135*(b*sqrt(c/x) + a)^(7/2)*a/b^4 + 189*(b*sqrt(c/x) + a)^(5/2)*a^2/b
^4 - 105*(b*sqrt(c/x) + a)^(3/2)*a^3/b^4)/c^2

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Fricas [A]  time = 1.48611, size = 170, normalized size = 1.47 \begin{align*} -\frac{4 \,{\left (35 \, b^{4} c^{2} - 6 \, a^{2} b^{2} c x - 16 \, a^{4} x^{2} +{\left (5 \, a b^{3} c x + 8 \, a^{3} b x^{2}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{315 \, b^{4} c^{2} x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c/x)^(1/2))^(1/2)/x^3,x, algorithm="fricas")

[Out]

-4/315*(35*b^4*c^2 - 6*a^2*b^2*c*x - 16*a^4*x^2 + (5*a*b^3*c*x + 8*a^3*b*x^2)*sqrt(c/x))*sqrt(b*sqrt(c/x) + a)
/(b^4*c^2*x^2)

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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^{3}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*(c/x)**(1/2))**(1/2)/x**3,x)

[Out]

Integral(sqrt(a + b*sqrt(c/x))/x**3, 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^3,x, algorithm="giac")

[Out]

Timed out