3.2984 \(\int \frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{x} \, dx\)

Optimal. Leaf size=51 \[ 4 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )-4 \sqrt {a+b \sqrt {\frac {c}{x}}} \]

[Out]

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

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Rubi [A]  time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {369, 266, 50, 63, 208} \[ 4 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )-4 \sqrt {a+b \sqrt {\frac {c}{x}}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c/x]]/x,x]

[Out]

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

Rule 50

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

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

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Mathematica [A]  time = 0.05, size = 51, normalized size = 1.00 \[ 4 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {\frac {c}{x}}}}{\sqrt {a}}\right )-4 \sqrt {a+b \sqrt {\frac {c}{x}}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sqrt[c/x]]/x,x]

[Out]

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

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fricas [A]  time = 0.94, size = 110, normalized size = 2.16 \[ \left [2 \, \sqrt {a} \log \left (2 \, \sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {a} x \sqrt {\frac {c}{x}} + 2 \, a x \sqrt {\frac {c}{x}} + b c\right ) - 4 \, \sqrt {b \sqrt {\frac {c}{x}} + a}, -4 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} \sqrt {-a}}{a}\right ) - 4 \, \sqrt {b \sqrt {\frac {c}{x}} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x)]Warning, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):
Check [abs(t_nostep)]Warning, choosing root of [1,0,%%%{-2,[1,1,2,0]%%%}+%%%{-2,[1,1,0,0]%%%}+%%%{-2,[0,2,1,1]
%%%},0,%%%{1,[2,2,4,0]%%%}+%%%{-2,[2,2,2,0]%%%}+%%%{1,[2,2,0,0]%%%}+%%%{2,[1,3,3,1]%%%}+%%%{-2,[1,3,1,1]%%%}+%
%%{1,[0,4,2,2]%%%}] at parameters values [-97,-82,7,-27]Sign error (%%%{(-c*sqrt(a*c))*b,0%%%}+%%%{2*a*sqrt(b)
*abs(c),1/2%%%}+%%%{-2*a*sqrt(a*c),1%%%}+%%%{a^2*sqrt(b)*abs(c)/(b*c),3/2%%%}+%%%{-a^3*sqrt(b)*abs(c)/(4*b^2*c
^2),5/2%%%}+%%%{undef,7/2%%%})Limit: Max order reached or unable to make series expansion Error: Bad Argument
Value

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maple [B]  time = 0.02, size = 150, normalized size = 2.94 \[ \frac {2 \sqrt {a +\sqrt {\frac {c}{x}}\, b}\, \left (\sqrt {\frac {c}{x}}\, a b \,x^{\frac {3}{2}} \ln \left (\frac {2 a \sqrt {x}+\sqrt {\frac {c}{x}}\, b \sqrt {x}+2 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, \sqrt {a}}{2 \sqrt {a}}\right )+2 \sqrt {a x +\sqrt {\frac {c}{x}}\, b x}\, a^{\frac {3}{2}} x -2 \left (a x +\sqrt {\frac {c}{x}}\, b x \right )^{\frac {3}{2}} \sqrt {a}\right )}{\sqrt {\left (a +\sqrt {\frac {c}{x}}\, b \right ) x}\, \sqrt {\frac {c}{x}}\, \sqrt {a}\, b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.29, size = 61, normalized size = 1.20 \[ -2 \, \sqrt {a} \log \left (\frac {\sqrt {b \sqrt {\frac {c}{x}} + a} - \sqrt {a}}{\sqrt {b \sqrt {\frac {c}{x}} + a} + \sqrt {a}}\right ) - 4 \, \sqrt {b \sqrt {\frac {c}{x}} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+b\,\sqrt {\frac {c}{x}}}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c/x)^(1/2))^(1/2)/x,x)

[Out]

int((a + b*(c/x)^(1/2))^(1/2)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sqrt(c/x))/x, x)

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