3.348 \(\int \frac{1}{x (a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=38 \[ \frac{2}{a \sqrt{a+b x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0357687, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{2}{a \sqrt{a+b x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*(a + b*x)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.04108, size = 32, normalized size = 0.84 \[ \frac{2}{a \sqrt{a + b x}} - \frac{2 \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x/(b*x+a)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.0331102, size = 38, normalized size = 1. \[ \frac{2}{a \sqrt{a+b x}}-\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 31, normalized size = 0.8 \[ -2\,{\frac{1}{{a}^{3/2}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) }+2\,{\frac{1}{a\sqrt{bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x/(b*x+a)^(3/2),x)

[Out]

-2*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(3/2)+2/a/(b*x+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/((b*x + a)^(3/2)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230624, size = 1, normalized size = 0.03 \[ \left [\frac{\sqrt{b x + a} \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} - 2 \, \sqrt{b x + a} a}{x}\right ) + 2 \, \sqrt{a}}{\sqrt{b x + a} a^{\frac{3}{2}}}, \frac{2 \,{\left (\sqrt{b x + a} \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) + \sqrt{-a}\right )}}{\sqrt{b x + a} \sqrt{-a} a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x),x, algorithm="fricas")

[Out]

[(sqrt(b*x + a)*log(((b*x + 2*a)*sqrt(a) - 2*sqrt(b*x + a)*a)/x) + 2*sqrt(a))/(s
qrt(b*x + a)*a^(3/2)), 2*(sqrt(b*x + a)*arctan(a/(sqrt(b*x + a)*sqrt(-a))) + sqr
t(-a))/(sqrt(b*x + a)*sqrt(-a)*a)]

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Sympy [A]  time = 5.84359, size = 146, normalized size = 3.84 \[ \frac{2 a^{3} \sqrt{1 + \frac{b x}{a}}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{a^{3} \log{\left (\frac{b x}{a} \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 a^{3} \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} + \frac{a^{2} b x \log{\left (\frac{b x}{a} \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} - \frac{2 a^{2} b x \log{\left (\sqrt{1 + \frac{b x}{a}} + 1 \right )}}{a^{\frac{9}{2}} + a^{\frac{7}{2}} b x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x/(b*x+a)**(3/2),x)

[Out]

2*a**3*sqrt(1 + b*x/a)/(a**(9/2) + a**(7/2)*b*x) + a**3*log(b*x/a)/(a**(9/2) + a
**(7/2)*b*x) - 2*a**3*log(sqrt(1 + b*x/a) + 1)/(a**(9/2) + a**(7/2)*b*x) + a**2*
b*x*log(b*x/a)/(a**(9/2) + a**(7/2)*b*x) - 2*a**2*b*x*log(sqrt(1 + b*x/a) + 1)/(
a**(9/2) + a**(7/2)*b*x)

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GIAC/XCAS [A]  time = 0.201086, size = 50, normalized size = 1.32 \[ \frac{2 \, \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a} + \frac{2}{\sqrt{b x + a} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x + a)^(3/2)*x),x, algorithm="giac")

[Out]

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