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

Optimal. Leaf size=48 \[ \frac{\sqrt{x} \sqrt{a+b x}}{b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0353076, antiderivative size = 48, 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 \[ \frac{\sqrt{x} \sqrt{a+b x}}{b}-\frac{a \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a+b x}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[x]/Sqrt[a + b*x],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 5.1483, size = 41, normalized size = 0.85 \[ - \frac{a \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{b} \sqrt{x}} \right )}}{b^{\frac{3}{2}}} + \frac{\sqrt{x} \sqrt{a + b x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0303264, size = 51, normalized size = 1.06 \[ \frac{\sqrt{x} \sqrt{a+b x}}{b}-\frac{a \log \left (\sqrt{b} \sqrt{a+b x}+b \sqrt{x}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 65, normalized size = 1.4 \[{\frac{1}{b}\sqrt{x}\sqrt{bx+a}}-{\frac{a}{2}\sqrt{x \left ( bx+a \right ) }\ln \left ({1 \left ({\frac{a}{2}}+bx \right ){\frac{1}{\sqrt{b}}}}+\sqrt{b{x}^{2}+ax} \right ){b}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}{\frac{1}{\sqrt{bx+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(1/2)/(b*x+a)^(1/2),x)

[Out]

x^(1/2)*(b*x+a)^(1/2)/b-1/2*a/b^(3/2)*(x*(b*x+a))^(1/2)/x^(1/2)/(b*x+a)^(1/2)*ln
((1/2*a+b*x)/b^(1/2)+(b*x^2+a*x)^(1/2))

_______________________________________________________________________________________

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)/sqrt(b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.224826, size = 1, normalized size = 0.02 \[ \left [\frac{a \log \left (-2 \, \sqrt{b x + a} b \sqrt{x} +{\left (2 \, b x + a\right )} \sqrt{b}\right ) + 2 \, \sqrt{b x + a} \sqrt{b} \sqrt{x}}{2 \, b^{\frac{3}{2}}}, -\frac{a \arctan \left (\frac{\sqrt{b x + a} \sqrt{-b}}{b \sqrt{x}}\right ) - \sqrt{b x + a} \sqrt{-b} \sqrt{x}}{\sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(x)/sqrt(b*x + a),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)*
sqrt(b)*sqrt(x))/b^(3/2), -(a*arctan(sqrt(b*x + a)*sqrt(-b)/(b*sqrt(x))) - sqrt(
b*x + a)*sqrt(-b)*sqrt(x))/(sqrt(-b)*b)]

_______________________________________________________________________________________

Sympy [A]  time = 8.22372, size = 44, normalized size = 0.92 \[ \frac{\sqrt{a} \sqrt{x} \sqrt{1 + \frac{b x}{a}}}{b} - \frac{a \operatorname{asinh}{\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}} \right )}}{b^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*sqrt(x)*sqrt(1 + b*x/a)/b - a*asinh(sqrt(b)*sqrt(x)/sqrt(a))/b**(3/2)

_______________________________________________________________________________________

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(x)/sqrt(b*x + a),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError