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

Optimal. Leaf size=51 \[ -\frac{(a+b x)^{3/2}}{x}+3 b \sqrt{a+b x}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0499279, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{(a+b x)^{3/2}}{x}+3 b \sqrt{a+b x}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^(3/2)/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.72651, size = 44, normalized size = 0.86 \[ - 3 \sqrt{a} b \operatorname{atanh}{\left (\frac{\sqrt{a + b x}}{\sqrt{a}} \right )} + 3 b \sqrt{a + b x} - \frac{\left (a + b x\right )^{\frac{3}{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0376969, size = 45, normalized size = 0.88 \[ \left (2 b-\frac{a}{x}\right ) \sqrt{a+b x}-3 \sqrt{a} b \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.014, size = 47, normalized size = 0.9 \[ 2\,b \left ( \sqrt{bx+a}+a \left ( -1/2\,{\frac{\sqrt{bx+a}}{bx}}-3/2\,{\frac{1}{\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{bx+a}}{\sqrt{a}}} \right ) } \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^(3/2)/x^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236167, size = 1, normalized size = 0.02 \[ \left [\frac{3 \, \sqrt{a} b x \log \left (\frac{b x - 2 \, \sqrt{b x + a} \sqrt{a} + 2 \, a}{x}\right ) + 2 \,{\left (2 \, b x - a\right )} \sqrt{b x + a}}{2 \, x}, -\frac{3 \, \sqrt{-a} b x \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right ) -{\left (2 \, b x - a\right )} \sqrt{b x + a}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 8.54782, size = 92, normalized size = 1.8 \[ - 3 \sqrt{a} b \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )} - \frac{a^{2}}{\sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} + 1}} + \frac{a \sqrt{b}}{\sqrt{x} \sqrt{\frac{a}{b x} + 1}} + \frac{2 b^{\frac{3}{2}} \sqrt{x}}{\sqrt{\frac{a}{b x} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.207331, size = 76, normalized size = 1.49 \[ \frac{\frac{3 \, a b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} + 2 \, \sqrt{b x + a} b^{2} - \frac{\sqrt{b x + a} a b}{x}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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