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

Optimal. Leaf size=40 \[ \frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2}} \]

[Out]

(2*Sqrt[x])/b - (2*Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(3/2)

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Rubi [A]  time = 0.0319913, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/(a + b*x),x]

[Out]

(2*Sqrt[x])/b - (2*Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(3/2)

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Rubi in Sympy [A]  time = 6.25855, size = 36, normalized size = 0.9 \[ - \frac{2 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} + \frac{2 \sqrt{x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*sqrt(a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/b**(3/2) + 2*sqrt(x)/b

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Mathematica [A]  time = 0.0164958, size = 40, normalized size = 1. \[ \frac{2 \sqrt{x}}{b}-\frac{2 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[x]/(a + b*x),x]

[Out]

(2*Sqrt[x])/b - (2*Sqrt[a]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/b^(3/2)

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Maple [A]  time = 0.009, size = 32, normalized size = 0.8 \[ 2\,{\frac{\sqrt{x}}{b}}-2\,{\frac{a}{b\sqrt{ab}}\arctan \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(b*x+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220579, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-\frac{a}{b}} \log \left (\frac{b x - 2 \, b \sqrt{x} \sqrt{-\frac{a}{b}} - a}{b x + a}\right ) + 2 \, \sqrt{x}}{b}, -\frac{2 \,{\left (\sqrt{\frac{a}{b}} \arctan \left (\frac{\sqrt{x}}{\sqrt{\frac{a}{b}}}\right ) - \sqrt{x}\right )}}{b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 2.85297, size = 36, normalized size = 0.9 \[ - \frac{2 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} + \frac{2 \sqrt{x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(1/2)/(b*x+a),x)

[Out]

-2*sqrt(a)*atan(sqrt(b)*sqrt(x)/sqrt(a))/b**(3/2) + 2*sqrt(x)/b

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GIAC/XCAS [A]  time = 0.207195, size = 42, normalized size = 1.05 \[ -\frac{2 \, a \arctan \left (\frac{b \sqrt{x}}{\sqrt{a b}}\right )}{\sqrt{a b} b} + \frac{2 \, \sqrt{x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*a*arctan(b*sqrt(x)/sqrt(a*b))/(sqrt(a*b)*b) + 2*sqrt(x)/b