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]

arcsinh(a^(1/2)*x^(1/2))/a^(1/2)+x^(1/2)*(a*x+1)^(1/2)

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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*}

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Mathematica [A]  time = 0.01, size = 34, normalized size = 1.00 \[ \sqrt {x} \sqrt {a x+1}+\frac {\sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \]

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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fricas [A]  time = 0.89, size = 90, normalized size = 2.65 \[ \left [\frac {2 \, \sqrt {a x + 1} a \sqrt {x} + \sqrt {a} \log \left (2 \, a x + 2 \, \sqrt {a x + 1} \sqrt {a} \sqrt {x} + 1\right )}{2 \, a}, \frac {\sqrt {a x + 1} a \sqrt {x} - \sqrt {-a} \arctan \left (\frac {\sqrt {a x + 1} \sqrt {-a}}{a \sqrt {x}}\right )}{a}\right ] \]

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

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 >> Unable to parse Giac output: Warning, choosing root of [1,0,%%%{-4,[1
,1]%%%}+%%%{-4,[1,0]%%%}+%%%{-4,[0,1]%%%}+%%%{-4,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,[2,1]%%%}+%%%{6,[2,0]%%%}+%
%%{4,[1,2]%%%}+%%%{16,[1,1]%%%}+%%%{4,[1,0]%%%}+%%%{6,[0,2]%%%}+%%%{4,[0,1]%%%}+%%%{6,[0,0]%%%},0,%%%{-4,[3,3]
%%%}+%%%{4,[3,2]%%%}+%%%{4,[3,1]%%%}+%%%{-4,[3,0]%%%}+%%%{4,[2,3]%%%}+%%%{-52,[2,2]%%%}+%%%{12,[2,1]%%%}+%%%{4
,[2,0]%%%}+%%%{4,[1,3]%%%}+%%%{12,[1,2]%%%}+%%%{-52,[1,1]%%%}+%%%{4,[1,0]%%%}+%%%{-4,[0,3]%%%}+%%%{4,[0,2]%%%}
+%%%{4,[0,1]%%%}+%%%{-4,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{-4,[4,3]%%%}+%%%{6,[4,2]%%%}+%%%{-4,[4,1]%%%}+%%%{1,[4
,0]%%%}+%%%{-4,[3,4]%%%}+%%%{8,[3,3]%%%}+%%%{-8,[3,2]%%%}+%%%{8,[3,1]%%%}+%%%{-4,[3,0]%%%}+%%%{6,[2,4]%%%}+%%%
{-8,[2,3]%%%}+%%%{20,[2,2]%%%}+%%%{-8,[2,1]%%%}+%%%{6,[2,0]%%%}+%%%{-4,[1,4]%%%}+%%%{8,[1,3]%%%}+%%%{-8,[1,2]%
%%}+%%%{8,[1,1]%%%}+%%%{-4,[1,0]%%%}+%%%{1,[0,4]%%%}+%%%{-4,[0,3]%%%}+%%%{6,[0,2]%%%}+%%%{-4,[0,1]%%%}+%%%{1,[
0,0]%%%}] at parameters values [85.3561567818,61.7937478349]Warning, choosing root of [1,0,%%%{-4,[1,1]%%%}+%%
%{-4,[1,0]%%%}+%%%{-4,[0,1]%%%}+%%%{-4,[0,0]%%%},0,%%%{6,[2,2]%%%}+%%%{4,[2,1]%%%}+%%%{6,[2,0]%%%}+%%%{4,[1,2]
%%%}+%%%{16,[1,1]%%%}+%%%{4,[1,0]%%%}+%%%{6,[0,2]%%%}+%%%{4,[0,1]%%%}+%%%{6,[0,0]%%%},0,%%%{-4,[3,3]%%%}+%%%{4
,[3,2]%%%}+%%%{4,[3,1]%%%}+%%%{-4,[3,0]%%%}+%%%{4,[2,3]%%%}+%%%{-52,[2,2]%%%}+%%%{12,[2,1]%%%}+%%%{4,[2,0]%%%}
+%%%{4,[1,3]%%%}+%%%{12,[1,2]%%%}+%%%{-52,[1,1]%%%}+%%%{4,[1,0]%%%}+%%%{-4,[0,3]%%%}+%%%{4,[0,2]%%%}+%%%{4,[0,
1]%%%}+%%%{-4,[0,0]%%%},0,%%%{1,[4,4]%%%}+%%%{-4,[4,3]%%%}+%%%{6,[4,2]%%%}+%%%{-4,[4,1]%%%}+%%%{1,[4,0]%%%}+%%
%{-4,[3,4]%%%}+%%%{8,[3,3]%%%}+%%%{-8,[3,2]%%%}+%%%{8,[3,1]%%%}+%%%{-4,[3,0]%%%}+%%%{6,[2,4]%%%}+%%%{-8,[2,3]%
%%}+%%%{20,[2,2]%%%}+%%%{-8,[2,1]%%%}+%%%{6,[2,0]%%%}+%%%{-4,[1,4]%%%}+%%%{8,[1,3]%%%}+%%%{-8,[1,2]%%%}+%%%{8,
[1,1]%%%}+%%%{-4,[1,0]%%%}+%%%{1,[0,4]%%%}+%%%{-4,[0,3]%%%}+%%%{6,[0,2]%%%}+%%%{-4,[0,1]%%%}+%%%{1,[0,0]%%%}]
at parameters values [71.707969239,78.6493344628]1/abs(a)*a^2/a*(1/a*sqrt(a*x+1)*sqrt(a*(a*x+1)-a)-1/sqrt(a)*l
n(abs(sqrt(a*(a*x+1)-a)-sqrt(a)*sqrt(a*x+1))))

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maple [B]  time = 0.01, size = 57, normalized size = 1.68 \[ \sqrt {a x +1}\, \sqrt {x}+\frac {\sqrt {\left (a x +1\right ) x}\, \ln \left (\frac {a x +\frac {1}{2}}{\sqrt {a}}+\sqrt {a \,x^{2}+x}\right )}{2 \sqrt {a x +1}\, \sqrt {a}\, \sqrt {x}} \]

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 [B]  time = 0.96, size = 68, normalized size = 2.00 \[ -\frac {\log \left (-\frac {\sqrt {a} - \frac {\sqrt {a x + 1}}{\sqrt {x}}}{\sqrt {a} + \frac {\sqrt {a x + 1}}{\sqrt {x}}}\right )}{2 \, \sqrt {a}} - \frac {\sqrt {a x + 1}}{{\left (a - \frac {a x + 1}{x}\right )} \sqrt {x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*log(-(sqrt(a) - sqrt(a*x + 1)/sqrt(x))/(sqrt(a) + sqrt(a*x + 1)/sqrt(x)))/sqrt(a) - sqrt(a*x + 1)/((a - (
a*x + 1)/x)*sqrt(x))

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mupad [B]  time = 3.00, size = 36, normalized size = 1.06 \[ \sqrt {x}\,\sqrt {a\,x+1}+\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {a\,x+1}-1}\right )}{\sqrt {a}} \]

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) + (2*atanh((a^(1/2)*x^(1/2))/((a*x + 1)^(1/2) - 1)))/a^(1/2)

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sympy [A]  time = 1.97, size = 29, normalized size = 0.85 \[ \sqrt {x} \sqrt {a x + 1} + \frac {\operatorname {asinh}{\left (\sqrt {a} \sqrt {x} \right )}}{\sqrt {a}} \]

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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