3.370 \(\int \frac{x^{\frac{1-n}{2}+\frac{1}{2} (-3+n)}}{\sqrt{a+b x}} \, dx\)

Optimal. Leaf size=23 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0240003, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[x^((1 - n)/2 + (-3 + n)/2)/Sqrt[a + b*x],x]

[Out]

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

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Rubi in Sympy [A]  time = 3.24478, size = 22, normalized size = 0.96 \[ - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0118643, size = 23, normalized size = 1. \[ -\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^((1 - n)/2 + (-3 + n)/2)/Sqrt[a + b*x],x]

[Out]

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

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Maple [A]  time = 0., size = 18, normalized size = 0.8 \[ -2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x+a)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.220482, size = 1, normalized size = 0.04 \[ \left [\frac{\log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right )}{\sqrt{a}}, \frac{2 \, \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right )}{\sqrt{-a}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.58393, size = 24, normalized size = 1.04 \[ - \frac{2 \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x+a)**(1/2),x)

[Out]

-2*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/sqrt(a)

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GIAC/XCAS [A]  time = 0.201821, size = 28, normalized size = 1.22 \[ \frac{2 \, \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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