3.6 \(\int \frac{\sqrt{b x+c x^2}}{x^2} \, dx\)

Optimal. Leaf size=47 \[ 2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 \sqrt{b x+c x^2}}{x} \]

[Out]

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

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Rubi [A]  time = 0.0549545, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ 2 \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )-\frac{2 \sqrt{b x+c x^2}}{x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.71895, size = 41, normalized size = 0.87 \[ 2 \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )} - \frac{2 \sqrt{b x + c x^{2}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.053617, size = 63, normalized size = 1.34 \[ -\frac{2 \left (-\sqrt{c} \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )+b+c x\right )}{\sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(b + c*x - Sqrt[c]*Sqrt[x]*Sqrt[b + c*x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*
x]]))/Sqrt[x*(b + c*x)]

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Maple [A]  time = 0.007, size = 66, normalized size = 1.4 \[ -2\,{\frac{ \left ( c{x}^{2}+bx \right ) ^{3/2}}{b{x}^{2}}}+2\,{\frac{c\sqrt{c{x}^{2}+bx}}{b}}+\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.239393, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{c} x \log \left (2 \, c x + b + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}\right ) - 2 \, \sqrt{c x^{2} + b x}}{x}, \frac{2 \,{\left (\sqrt{-c} x \arctan \left (\frac{\sqrt{c x^{2} + b x}}{\sqrt{-c} x}\right ) - \sqrt{c x^{2} + b x}\right )}}{x}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x*(b + c*x))/x**2, x)

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GIAC/XCAS [A]  time = 0.223876, size = 81, normalized size = 1.72 \[ -\sqrt{c}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right ) + \frac{2 \, b}{\sqrt{c} x - \sqrt{c x^{2} + b x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b)) + 2*b/(sqrt(c)*
x - sqrt(c*x^2 + b*x))