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

Optimal. Leaf size=47 \[ \frac{\sqrt{b x+c x^2}}{c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0513003, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\sqrt{b x+c x^2}}{c}-\frac{b \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[b*x + c*x^2],x]

[Out]

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

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Rubi in Sympy [A]  time = 5.78093, size = 39, normalized size = 0.83 \[ - \frac{b \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} + \frac{\sqrt{b x + c x^{2}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0410577, size = 71, normalized size = 1.51 \[ \frac{\sqrt{c} x (b+c x)-b \sqrt{x} \sqrt{b+c x} \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{c^{3/2} \sqrt{x (b+c x)}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*x*(b + c*x) - b*Sqrt[x]*Sqrt[b + c*x]*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b +
c*x]])/(c^(3/2)*Sqrt[x*(b + c*x)])

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Maple [A]  time = 0.006, size = 47, normalized size = 1. \[{\frac{1}{c}\sqrt{c{x}^{2}+bx}}-{\frac{b}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c*x^2+b*x)^(1/2),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229014, size = 1, normalized size = 0.02 \[ \left [\frac{b \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \, \sqrt{c x^{2} + b x} \sqrt{c}}{2 \, c^{\frac{3}{2}}}, -\frac{b \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) - \sqrt{c x^{2} + b x} \sqrt{-c}}{\sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/sqrt(c*x^2 + b*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))/c^(3/2), -(b*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)) - sqrt(c*x^2 + b*x
)*sqrt(-c))/(sqrt(-c)*c)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/sqrt(x*(b + c*x)), x)

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GIAC/XCAS [A]  time = 0.227401, size = 70, normalized size = 1.49 \[ \frac{b{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{2 \, c^{\frac{3}{2}}} + \frac{\sqrt{c x^{2} + b x}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x))*sqrt(c) - b))/c^(3/2) + sqrt(c*x
^2 + b*x)/c