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

Optimal. Leaf size=60 \[ \frac{(b+2 c x) \sqrt{b x+c x^2}}{4 c}-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{3/2}} \]

[Out]

((b + 2*c*x)*Sqrt[b*x + c*x^2])/(4*c) - (b^2*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^
2]])/(4*c^(3/2))

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Rubi [A]  time = 0.0440396, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{(b+2 c x) \sqrt{b x+c x^2}}{4 c}-\frac{b^2 \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{b x+c x^2}}\right )}{4 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[b*x + c*x^2],x]

[Out]

((b + 2*c*x)*Sqrt[b*x + c*x^2])/(4*c) - (b^2*ArcTanh[(Sqrt[c]*x)/Sqrt[b*x + c*x^
2]])/(4*c^(3/2))

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Rubi in Sympy [A]  time = 4.08321, size = 51, normalized size = 0.85 \[ - \frac{b^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x}{\sqrt{b x + c x^{2}}} \right )}}{4 c^{\frac{3}{2}}} + \frac{\left (b + 2 c x\right ) \sqrt{b x + c x^{2}}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**2*atanh(sqrt(c)*x/sqrt(b*x + c*x**2))/(4*c**(3/2)) + (b + 2*c*x)*sqrt(b*x +
c*x**2)/(4*c)

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Mathematica [A]  time = 0.0730019, size = 76, normalized size = 1.27 \[ \frac{\sqrt{x (b+c x)} \left (\sqrt{c} (b+2 c x)-\frac{b^2 \log \left (\sqrt{c} \sqrt{b+c x}+c \sqrt{x}\right )}{\sqrt{x} \sqrt{b+c x}}\right )}{4 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[b*x + c*x^2],x]

[Out]

(Sqrt[x*(b + c*x)]*(Sqrt[c]*(b + 2*c*x) - (b^2*Log[c*Sqrt[x] + Sqrt[c]*Sqrt[b +
c*x]])/(Sqrt[x]*Sqrt[b + c*x])))/(4*c^(3/2))

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \[{\frac{2\,cx+b}{4\,c}\sqrt{c{x}^{2}+bx}}-{\frac{{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x)^(1/2),x)

[Out]

1/4*(2*c*x+b)*(c*x^2+b*x)^(1/2)/c-1/8*b^2/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(sqrt(c*x^2 + b*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225271, size = 1, normalized size = 0.02 \[ \left [\frac{b^{2} \log \left ({\left (2 \, c x + b\right )} \sqrt{c} - 2 \, \sqrt{c x^{2} + b x} c\right ) + 2 \, \sqrt{c x^{2} + b x}{\left (2 \, c x + b\right )} \sqrt{c}}{8 \, c^{\frac{3}{2}}}, -\frac{b^{2} \arctan \left (\frac{\sqrt{c x^{2} + b x} \sqrt{-c}}{c x}\right ) - \sqrt{c x^{2} + b x}{\left (2 \, c x + b\right )} \sqrt{-c}}{4 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(b^2*log((2*c*x + b)*sqrt(c) - 2*sqrt(c*x^2 + b*x)*c) + 2*sqrt(c*x^2 + b*x)
*(2*c*x + b)*sqrt(c))/c^(3/2), -1/4*(b^2*arctan(sqrt(c*x^2 + b*x)*sqrt(-c)/(c*x)
) - sqrt(c*x^2 + b*x)*(2*c*x + b)*sqrt(-c))/(sqrt(-c)*c)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 0.216791, size = 82, normalized size = 1.37 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x}{\left (2 \, x + \frac{b}{c}\right )} + \frac{b^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(c*x^2 + b*x)*(2*x + b/c) + 1/8*b^2*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 +
b*x))*sqrt(c) - b))/c^(3/2)