3.105 \(\int \frac{1}{\sqrt{b \sqrt{x}+a x}} \, dx\)

Optimal. Leaf size=56 \[ \frac{2 \sqrt{a x+b \sqrt{x}}}{a}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.106577, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{2 \sqrt{a x+b \sqrt{x}}}{a}-\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/Sqrt[b*Sqrt[x] + a*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 9.47211, size = 49, normalized size = 0.88 \[ \frac{2 \sqrt{a x + b \sqrt{x}}}{a} - \frac{2 b \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0471505, size = 62, normalized size = 1.11 \[ \frac{2 \sqrt{a x+b \sqrt{x}}}{a}-\frac{b \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/Sqrt[b*Sqrt[x] + a*x],x]

[Out]

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

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Maple [A]  time = 0.005, size = 83, normalized size = 1.5 \[ -{1\sqrt{b\sqrt{x}+ax} \left ( b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b \right ){\frac{1}{\sqrt{a}}}} \right ) -2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^(1/2)+a*x)^(1/2),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a x + b \sqrt{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/sqrt(a*x + b*sqrt(x)), x)

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GIAC/XCAS [A]  time = 0.274202, size = 73, normalized size = 1.3 \[ \frac{b{\rm ln}\left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{a^{\frac{3}{2}}} + \frac{2 \, \sqrt{a x + b \sqrt{x}}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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