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

Optimal. Leaf size=42 \[ \sqrt{b x+c x^2}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0547881, antiderivative size = 42, 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 \[ \sqrt{b x+c x^2}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.71714, size = 37, normalized size = 0.88 \[ \frac{b \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{\sqrt{c}} + \sqrt{b x + c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0422125, size = 59, normalized size = 1.4 \[ \sqrt{x (b+c x)} \left (\frac{b \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{c} \sqrt{x} \sqrt{b+c x}}+1\right ) \]

Antiderivative was successfully verified.

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

[Out]

Sqrt[x*(b + c*x)]*(1 + (b*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b + c*x]])/(Sqrt[c]*Sqrt[
x]*Sqrt[b + c*x]))

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Maple [A]  time = 0.005, size = 43, normalized size = 1. \[ \sqrt{c{x}^{2}+bx}+{\frac{b}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){\frac{1}{\sqrt{c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2)/x,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24782, size = 1, normalized size = 0.02 \[ \left [\frac{b \log \left ({\left (2 \, c x + b\right )} \sqrt{c} + 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}}{2 \, \sqrt{c}}, \frac{b \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) + \sqrt{c x^{2} + b x} \sqrt{-c}}{\sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(x*(b + c*x))/x, x)

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GIAC/XCAS [A]  time = 0.220815, size = 65, normalized size = 1.55 \[ -\frac{b{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, \sqrt{c}} + \sqrt{c x^{2} + b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*b*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/sqrt(c) + sqrt(c*
x^2 + b*x)