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

Optimal. Leaf size=34 \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0867439, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{4 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[x]*Sqrt[b*Sqrt[x] + a*x]),x]

[Out]

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

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Rubi in Sympy [A]  time = 8.3649, size = 31, normalized size = 0.91 \[ \frac{4 \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0330552, size = 40, normalized size = 1.18 \[ \frac{2 \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.012, size = 136, normalized size = 4. \[ -{\frac{1}{b}\sqrt{b\sqrt{x}+ax} \left ( 2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}-2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b \right ){\frac{1}{\sqrt{a}}}} \right ) b-b\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b \right ){\frac{1}{\sqrt{a}}}} \right ) \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(x)*sqrt(a*x + b*sqrt(x))), x)

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GIAC/XCAS [A]  time = 0.272275, size = 50, normalized size = 1.47 \[ -\frac{2 \,{\rm ln}\left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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