3.2991 \(\int \frac {1}{\sqrt {a+b \sqrt {\frac {c}{x}}} x^2} \, dx\)

Optimal. Leaf size=54 \[ \frac {4 a \sqrt {a+b \sqrt {\frac {c}{x}}}}{b^2 c}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^2 c} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 54, normalized size of antiderivative = 1.00, 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} \[ \frac {4 a \sqrt {a+b \sqrt {\frac {c}{x}}}}{b^2 c}-\frac {4 \left (a+b \sqrt {\frac {c}{x}}\right )^{3/2}}{3 b^2 c} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[a + b*Sqrt[c/x]]*x^2),x]

[Out]

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

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])

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 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]

Rubi steps

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

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Mathematica [A]  time = 0.04, size = 42, normalized size = 0.78 \[ -\frac {4 \left (b \sqrt {\frac {c}{x}}-2 a\right ) \sqrt {a+b \sqrt {\frac {c}{x}}}}{3 b^2 c} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[a + b*Sqrt[c/x]]*x^2),x]

[Out]

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

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fricas [A]  time = 0.74, size = 34, normalized size = 0.63 \[ -\frac {4 \, \sqrt {b \sqrt {\frac {c}{x}} + a} {\left (b \sqrt {\frac {c}{x}} - 2 \, a\right )}}{3 \, b^{2} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.18, size = 60, normalized size = 1.11 \[ -\frac {4 \, {\left ({\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}} b - 3 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a b\right )} \mathrm {sgn}\left ({\left (b \sqrt {\frac {c}{x}} + a\right )} b - a b\right )}{3 \, b^{3} c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.04, size = 274, normalized size = 5.07 \[ -\frac {\sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (-3 \sqrt {\frac {c}{x}}\, a^{2} b \,x^{2} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \sqrt {a}}{2 \sqrt {a}}\right )+3 \sqrt {\frac {c}{x}}\, a^{2} b \,x^{2} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+6 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, a^{\frac {5}{2}} x^{\frac {3}{2}}+6 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, a^{\frac {5}{2}} x^{\frac {3}{2}}-12 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} a^{\frac {3}{2}} \sqrt {x}+4 \sqrt {\frac {c}{x}}\, \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \sqrt {a}\, b \sqrt {x}\right )}{3 \sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \left (\frac {c}{x}\right )^{\frac {3}{2}} \sqrt {a}\, b^{3} x^{\frac {5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(a+(c/x)^(1/2)*b)^(1/2)*(6*x^(3/2)*a^(5/2)*((a+(c/x)^(1/2)*b)*x)^(1/2)+6*x^(3/2)*a^(5/2)*(a*x+(c/x)^(1/2)
*b*x)^(1/2)-12*x^(1/2)*a^(3/2)*(a*x+(c/x)^(1/2)*b*x)^(3/2)+4*x^(1/2)*a^(1/2)*(c/x)^(1/2)*(a*x+(c/x)^(1/2)*b*x)
^(3/2)*b+3*(c/x)^(1/2)*ln(1/2*(2*a*x^(1/2)+(c/x)^(1/2)*b*x^(1/2)+2*(a*x+(c/x)^(1/2)*b*x)^(1/2)*a^(1/2))/a^(1/2
))*x^2*a^2*b-3*(c/x)^(1/2)*ln(1/2*(2*a*x^(1/2)+(c/x)^(1/2)*b*x^(1/2)+2*((a+(c/x)^(1/2)*b)*x)^(1/2)*a^(1/2))/a^
(1/2))*x^2*a^2*b)/x^(5/2)/((a+(c/x)^(1/2)*b)*x)^(1/2)/b^3/(c/x)^(3/2)/a^(1/2)

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maxima [A]  time = 0.56, size = 42, normalized size = 0.78 \[ -\frac {4 \, {\left (\frac {{\left (b \sqrt {\frac {c}{x}} + a\right )}^{\frac {3}{2}}}{b^{2}} - \frac {3 \, \sqrt {b \sqrt {\frac {c}{x}} + a} a}{b^{2}}\right )}}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.44, size = 52, normalized size = 0.96 \[ -\frac {\sqrt {\frac {b\,\sqrt {\frac {c}{x}}}{a}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},2;\ 3;\ -\frac {b\,\sqrt {\frac {c}{x}}}{a}\right )}{x\,\sqrt {a+b\,\sqrt {\frac {c}{x}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(((b*(c/x)^(1/2))/a + 1)^(1/2)*hypergeom([1/2, 2], 3, -(b*(c/x)^(1/2))/a))/(x*(a + b*(c/x)^(1/2))^(1/2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{2} \sqrt {a + b \sqrt {\frac {c}{x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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