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

Optimal. Leaf size=59 \[ \frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3 \sqrt{a+b x}}{a^2 x}+\frac{2}{a x \sqrt{a+b x}} \]

[Out]

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

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Rubi [A]  time = 0.0537651, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3 \sqrt{a+b x}}{a^2 x}+\frac{2}{a x \sqrt{a+b x}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x)^(3/2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.29956, size = 51, normalized size = 0.86 \[ \frac{2}{a x \sqrt{a + b x}} - \frac{3 \sqrt{a + b x}}{a^{2} x} + \frac{3 b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0791913, size = 48, normalized size = 0.81 \[ \frac{3 b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{a+3 b x}{a^2 x \sqrt{a+b x}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.02, size = 55, normalized size = 0.9 \[ 2\,b \left ( -{\frac{1}{{a}^{2}\sqrt{bx+a}}}-{\frac{1}{{a}^{2}} \left ( 1/2\,{\frac{\sqrt{bx+a}}{bx}}-3/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) } \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x+a)^(3/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.233278, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{b x + a} b x \log \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{a} + 2 \, \sqrt{b x + a} a}{x}\right ) - 2 \,{\left (3 \, b x + a\right )} \sqrt{a}}{2 \, \sqrt{b x + a} a^{\frac{5}{2}} x}, -\frac{3 \, \sqrt{b x + a} b x \arctan \left (\frac{a}{\sqrt{b x + a} \sqrt{-a}}\right ) +{\left (3 \, b x + a\right )} \sqrt{-a}}{\sqrt{b x + a} \sqrt{-a} a^{2} x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(3*sqrt(b*x + a)*b*x*log(((b*x + 2*a)*sqrt(a) + 2*sqrt(b*x + a)*a)/x) - 2*(
3*b*x + a)*sqrt(a))/(sqrt(b*x + a)*a^(5/2)*x), -(3*sqrt(b*x + a)*b*x*arctan(a/(s
qrt(b*x + a)*sqrt(-a))) + (3*b*x + a)*sqrt(-a))/(sqrt(b*x + a)*sqrt(-a)*a^2*x)]

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Sympy [A]  time = 12.6608, size = 73, normalized size = 1.24 \[ - \frac{1}{a \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} - \frac{3 \sqrt{b}}{a^{2} \sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{3 b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*sqrt(b)*x**(3/2)*sqrt(a/(b*x) + 1)) - 3*sqrt(b)/(a**2*sqrt(x)*sqrt(a/(b*x)
+ 1)) + 3*b*asinh(sqrt(a)/(sqrt(b)*sqrt(x)))/a**(5/2)

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GIAC/XCAS [A]  time = 0.205489, size = 86, normalized size = 1.46 \[ -\frac{3 \, b \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a} a^{2}} - \frac{3 \,{\left (b x + a\right )} b - 2 \, a b}{{\left ({\left (b x + a\right )}^{\frac{3}{2}} - \sqrt{b x + a} a\right )} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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