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 = 0.0278372, 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 \[ \sqrt{x} \sqrt{a x+1}+\frac{\sinh ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{\sqrt{a}} \]

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]

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Rubi in Sympy [A]  time = 5.08801, 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]  rubi_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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Mathematica [A]  time = 0.0222779, 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 = 0.008, size = 57, normalized size = 1.7 \[ \sqrt{x}\sqrt{ax+1}+{\frac{1}{2}\sqrt{ \left ( ax+1 \right ) x}\ln \left ({1 \left ({\frac{1}{2}}+ax \right ){\frac{1}{\sqrt{a}}}}+\sqrt{a{x}^{2}+x} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{a}}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x + 1)/sqrt(x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.287526, size = 1, normalized size = 0.03 \[ \left [\frac{2 \, \sqrt{a x + 1} \sqrt{a} \sqrt{x} + \log \left (2 \, \sqrt{a x + 1} a \sqrt{x} +{\left (2 \, a x + 1\right )} \sqrt{a}\right )}{2 \, \sqrt{a}}, \frac{\sqrt{a x + 1} \sqrt{-a} \sqrt{x} + \arctan \left (\frac{\sqrt{a x + 1} \sqrt{-a}}{a \sqrt{x}}\right )}{\sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x + 1)/sqrt(x),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.78381, 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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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(a*x + 1)/sqrt(x),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError