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

Optimal. Leaf size=44 \[ \sqrt{x} \sqrt{a+b x}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{\sqrt{b}} \]

[Out]

Sqrt[x]*Sqrt[a + b*x] + (a*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/Sqrt[b]

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Rubi [A]  time = 0.0366256, antiderivative size = 44, 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 \[ \sqrt{x} \sqrt{a+b x}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x]/Sqrt[x],x]

[Out]

Sqrt[x]*Sqrt[a + b*x] + (a*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a + b*x]])/Sqrt[b]

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Rubi in Sympy [A]  time = 4.88176, size = 39, normalized size = 0.89 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{b}} + \sqrt{x} \sqrt{a + b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**(1/2)/x**(1/2),x)

[Out]

a*atanh(sqrt(a + b*x)/(sqrt(b)*sqrt(x)))/sqrt(b) + sqrt(x)*sqrt(a + b*x)

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Mathematica [A]  time = 0.0210274, size = 47, normalized size = 1.07 \[ \sqrt{x} \sqrt{a+b x}+\frac{a \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*x]/Sqrt[x],x]

[Out]

Sqrt[x]*Sqrt[a + b*x] + (a*Log[b*Sqrt[x] + Sqrt[b]*Sqrt[a + b*x]])/Sqrt[b]

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Maple [A]  time = 0.01, size = 62, normalized size = 1.4 \[ \sqrt{x}\sqrt{bx+a}+{\frac{a}{2}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(1/2)/x^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219783, size = 1, normalized size = 0.02 \[ \left [\frac{a \log \left (2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x}}{2 \, \sqrt{b}}, \frac{a \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) + \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{\sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 6.49882, size = 42, normalized size = 0.95 \[ \sqrt{a} \sqrt{x} \sqrt{1 + \frac{b x}{a}} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{\sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 12.3072, size = 4, normalized size = 0.09 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x