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

Optimal. Leaf size=87 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{2 a^{5/2}}-\frac{3 b \sqrt{a x+b \sqrt{x}}}{2 a^2}+\frac{\sqrt{x} \sqrt{a x+b \sqrt{x}}}{a} \]

[Out]

(-3*b*Sqrt[b*Sqrt[x] + a*x])/(2*a^2) + (Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/a + (3*b^
2*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/(2*a^(5/2))

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Rubi [A]  time = 0.162198, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{3 b^2 \tanh ^{-1}\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x+b \sqrt{x}}}\right )}{2 a^{5/2}}-\frac{3 b \sqrt{a x+b \sqrt{x}}}{2 a^2}+\frac{\sqrt{x} \sqrt{a x+b \sqrt{x}}}{a} \]

Antiderivative was successfully verified.

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

[Out]

(-3*b*Sqrt[b*Sqrt[x] + a*x])/(2*a^2) + (Sqrt[x]*Sqrt[b*Sqrt[x] + a*x])/a + (3*b^
2*ArcTanh[(Sqrt[a]*Sqrt[x])/Sqrt[b*Sqrt[x] + a*x]])/(2*a^(5/2))

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Rubi in Sympy [A]  time = 16.2908, size = 78, normalized size = 0.9 \[ \frac{\sqrt{x} \sqrt{a x + b \sqrt{x}}}{a} - \frac{3 b \sqrt{a x + b \sqrt{x}}}{2 a^{2}} + \frac{3 b^{2} \operatorname{atanh}{\left (\frac{\sqrt{a} \sqrt{x}}{\sqrt{a x + b \sqrt{x}}} \right )}}{2 a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(x)*sqrt(a*x + b*sqrt(x))/a - 3*b*sqrt(a*x + b*sqrt(x))/(2*a**2) + 3*b**2*at
anh(sqrt(a)*sqrt(x)/sqrt(a*x + b*sqrt(x)))/(2*a**(5/2))

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Mathematica [A]  time = 0.0844058, size = 80, normalized size = 0.92 \[ \frac{3 b^2 \log \left (2 \sqrt{a} \sqrt{a x+b \sqrt{x}}+2 a \sqrt{x}+b\right )}{4 a^{5/2}}+\frac{\left (2 a \sqrt{x}-3 b\right ) \sqrt{a x+b \sqrt{x}}}{2 a^2} \]

Antiderivative was successfully verified.

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

[Out]

((-3*b + 2*a*Sqrt[x])*Sqrt[b*Sqrt[x] + a*x])/(2*a^2) + (3*b^2*Log[b + 2*a*Sqrt[x
] + 2*Sqrt[a]*Sqrt[b*Sqrt[x] + a*x]])/(4*a^(5/2))

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Maple [B]  time = 0.009, size = 163, normalized size = 1.9 \[ -{\frac{1}{4}\sqrt{b\sqrt{x}+ax} \left ( -4\,\sqrt{b\sqrt{x}+ax}\sqrt{x}{a}^{7/2}+8\,b\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }{a}^{5/2}-2\,\sqrt{b\sqrt{x}+ax}b{a}^{5/2}+{b}^{2}\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{b\sqrt{x}+ax}\sqrt{a}+2\,\sqrt{x}a+b \right ){\frac{1}{\sqrt{a}}}} \right ){a}^{2}-4\,{b}^{2}\ln \left ( 1/2\,{\frac{2\,\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }\sqrt{a}+2\,\sqrt{x}a+b}{\sqrt{a}}} \right ){a}^{2} \right ){\frac{1}{\sqrt{\sqrt{x} \left ( b+\sqrt{x}a \right ) }}}{a}^{-{\frac{9}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(b*x^(1/2)+a*x)^(1/2)*(-4*(b*x^(1/2)+a*x)^(1/2)*x^(1/2)*a^(7/2)+8*b*(x^(1/2
)*(b+x^(1/2)*a))^(1/2)*a^(5/2)-2*(b*x^(1/2)+a*x)^(1/2)*b*a^(5/2)+b^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))*a^2-4*b^2*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))*a^2)/(x^(1/2)*(b+x^(1/2)*
a))^(1/2)/a^(9/2)

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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)/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(sqrt(x)/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. \[ \int \frac{\sqrt{x}}{\sqrt{a x + b \sqrt{x}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.277376, size = 93, normalized size = 1.07 \[ \frac{1}{2} \, \sqrt{a x + b \sqrt{x}}{\left (\frac{2 \, \sqrt{x}}{a} - \frac{3 \, b}{a^{2}}\right )} - \frac{3 \, b^{2}{\rm ln}\left ({\left | -2 \, \sqrt{a}{\left (\sqrt{a} \sqrt{x} - \sqrt{a x + b \sqrt{x}}\right )} - b \right |}\right )}{4 \, a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(a*x + b*sqrt(x))*(2*sqrt(x)/a - 3*b/a^2) - 3/4*b^2*ln(abs(-2*sqrt(a)*(s
qrt(a)*sqrt(x) - sqrt(a*x + b*sqrt(x))) - b))/a^(5/2)