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

Optimal. Leaf size=52 \[ -\frac{\sqrt{a x^2+b x^3}}{x^2}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{\sqrt{a}} \]

[Out]

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

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Rubi [A]  time = 0.0930168, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{\sqrt{a x^2+b x^3}}{x^2}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{a x^2+b x^3}}\right )}{\sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a*x^2 + b*x^3]/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.35665, size = 46, normalized size = 0.88 \[ - \frac{\sqrt{a x^{2} + b x^{3}}}{x^{2}} - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a} x}{\sqrt{a x^{2} + b x^{3}}} \right )}}{\sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0440057, size = 64, normalized size = 1.23 \[ -\frac{\sqrt{a+b x} \left (\sqrt{a} \sqrt{a+b x}+b x \tanh ^{-1}\left (\frac{\sqrt{a+b x}}{\sqrt{a}}\right )\right )}{\sqrt{a} \sqrt{x^2 (a+b x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 56, normalized size = 1.1 \[ -{\frac{1}{{x}^{2}}\sqrt{b{x}^{3}+a{x}^{2}} \left ({\it Artanh} \left ({1\sqrt{bx+a}{\frac{1}{\sqrt{a}}}} \right ) xb+\sqrt{bx+a}\sqrt{a} \right ){\frac{1}{\sqrt{bx+a}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a*x^2)^(1/2)/x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229019, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{a} b x^{2} \log \left (\frac{{\left (b x^{2} + 2 \, a x\right )} \sqrt{a} - 2 \, \sqrt{b x^{3} + a x^{2}} a}{x^{2}}\right ) - 2 \, \sqrt{b x^{3} + a x^{2}} a}{2 \, a x^{2}}, -\frac{\sqrt{-a} b x^{2} \arctan \left (\frac{a x}{\sqrt{b x^{3} + a x^{2}} \sqrt{-a}}\right ) + \sqrt{b x^{3} + a x^{2}} a}{a x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{x^{2} \left (a + b x\right )}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x**2*(a + b*x))/x**3, x)

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GIAC/XCAS [A]  time = 0.23909, size = 58, normalized size = 1.12 \[ \frac{{\left (\frac{b^{2} \arctan \left (\frac{\sqrt{b x + a}}{\sqrt{-a}}\right )}{\sqrt{-a}} - \frac{\sqrt{b x + a} b}{x}\right )}{\rm sign}\left (x\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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