3.222 \(\int \frac{\sqrt{1+a x}}{\sqrt{x}} \, dx\)

Optimal. Leaf size=34 \[ \sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]

[Out]

Sqrt[x]*Sqrt[1 + a*x] + ArcSinh[Sqrt[a]*Sqrt[x]]/Sqrt[a]

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Rubi [A]  time = 0.0087351, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {50, 54, 215} \[ \sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 + a*x]/Sqrt[x],x]

[Out]

Sqrt[x]*Sqrt[1 + a*x] + ArcSinh[Sqrt[a]*Sqrt[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 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.013254, size = 34, normalized size = 1. \[ \sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 + a*x]/Sqrt[x],x]

[Out]

Sqrt[x]*Sqrt[1 + a*x] + ArcSinh[Sqrt[a]*Sqrt[x]]/Sqrt[a]

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Maple [B]  time = 0.167, size = 57, normalized size = 1.7 \begin{align*} \sqrt{x}\sqrt{ax+1}+{\frac{1}{2}\sqrt{ \left ( ax+1 \right ) x}\ln \left ({ \left ({\frac{1}{2}}+ax \right ){\frac{1}{\sqrt{a}}}}+\sqrt{a{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.59336, size = 244, normalized size = 7.18 \begin{align*} \left [\frac{2 \, \sqrt{a x + 1} a \sqrt{x} + \sqrt{a} \log \left (2 \, a x + 2 \, \sqrt{a x + 1} \sqrt{a} \sqrt{x} + 1\right )}{2 \, a}, \frac{\sqrt{a x + 1} a \sqrt{x} - \sqrt{-a} \arctan \left (\frac{\sqrt{a x + 1} \sqrt{-a}}{a \sqrt{x}}\right )}{a}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*(2*sqrt(a*x + 1)*a*sqrt(x) + sqrt(a)*log(2*a*x + 2*sqrt(a*x + 1)*sqrt(a)*sqrt(x) + 1))/a, (sqrt(a*x + 1)*
a*sqrt(x) - sqrt(-a)*arctan(sqrt(a*x + 1)*sqrt(-a)/(a*sqrt(x))))/a]

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Sympy [A]  time = 1.92732, size = 29, normalized size = 0.85 \begin{align*} \sqrt{x} \sqrt{a x + 1} + \frac{\operatorname{asinh}{\left (\sqrt{a} \sqrt{x} \right )}}{\sqrt{a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(x)*sqrt(a*x + 1) + asinh(sqrt(a)*sqrt(x))/sqrt(a)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError