3.2236 \(\int \frac{\sqrt{a+b \sqrt{x}}}{x} \, dx\)

Optimal. Leaf size=43 \[ 4 \sqrt{a+b \sqrt{x}}-4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0217346, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {266, 50, 63, 208} \[ 4 \sqrt{a+b \sqrt{x}}-4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[x]]/x,x]

[Out]

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

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 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]

Rubi steps

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

Mathematica [A]  time = 0.0106248, size = 43, normalized size = 1. \[ 4 \sqrt{a+b \sqrt{x}}-4 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{x}}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sqrt[x]]/x,x]

[Out]

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

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Maple [A]  time = 0.003, size = 32, normalized size = 0.7 \begin{align*} -4\,{\it Artanh} \left ({\frac{\sqrt{a+b\sqrt{x}}}{\sqrt{a}}} \right ) \sqrt{a}+4\,\sqrt{a+b\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.3125, size = 244, normalized size = 5.67 \begin{align*} \left [2 \, \sqrt{a} \log \left (\frac{b x - 2 \, \sqrt{b \sqrt{x} + a} \sqrt{a} \sqrt{x} + 2 \, a \sqrt{x}}{x}\right ) + 4 \, \sqrt{b \sqrt{x} + a}, 4 \, \sqrt{-a} \arctan \left (\frac{\sqrt{b \sqrt{x} + a} \sqrt{-a}}{a}\right ) + 4 \, \sqrt{b \sqrt{x} + a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 1.94152, size = 75, normalized size = 1.74 \begin{align*} - 4 \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt [4]{x}} \right )} + \frac{4 a}{\sqrt{b} \sqrt [4]{x} \sqrt{\frac{a}{b \sqrt{x}} + 1}} + \frac{4 \sqrt{b} \sqrt [4]{x}}{\sqrt{\frac{a}{b \sqrt{x}} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sqrt(a)*asinh(sqrt(a)/(sqrt(b)*x**(1/4))) + 4*a/(sqrt(b)*x**(1/4)*sqrt(a/(b*sqrt(x)) + 1)) + 4*sqrt(b)*x**(
1/4)/sqrt(a/(b*sqrt(x)) + 1)

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Giac [A]  time = 1.11227, size = 49, normalized size = 1.14 \begin{align*} \frac{4 \, a \arctan \left (\frac{\sqrt{b \sqrt{x} + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 4 \, \sqrt{b \sqrt{x} + a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*a*arctan(sqrt(b*sqrt(x) + a)/sqrt(-a))/sqrt(-a) + 4*sqrt(b*sqrt(x) + a)