3.2325 \(\int \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=75 \[ \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{4 c}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{3/2}} \]

[Out]

((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2))

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Rubi [A]  time = 0.059624, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{(b+2 c x) \sqrt{a+b x+c x^2}}{4 c}-\frac{\left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*x + c*x^2],x]

[Out]

((b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(4*c) - ((b^2 - 4*a*c)*ArcTanh[(b + 2*c*x)/(
2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(3/2))

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Rubi in Sympy [A]  time = 4.67208, size = 66, normalized size = 0.88 \[ \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}}}{4 c} - \frac{\left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(b + 2*c*x)*sqrt(a + b*x + c*x**2)/(4*c) - (-4*a*c + b**2)*atanh((b + 2*c*x)/(2*
sqrt(c)*sqrt(a + b*x + c*x**2)))/(8*c**(3/2))

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Mathematica [A]  time = 0.114529, size = 71, normalized size = 0.95 \[ \frac{(b+2 c x) \sqrt{a+x (b+c x)}}{4 c}-\frac{\left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{8 c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

((b + 2*c*x)*Sqrt[a + x*(b + c*x)])/(4*c) - ((b^2 - 4*a*c)*Log[b + 2*c*x + 2*Sqr
t[c]*Sqrt[a + x*(b + c*x)]])/(8*c^(3/2))

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Maple [A]  time = 0.005, size = 89, normalized size = 1.2 \[{\frac{2\,cx+b}{4\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{a}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{b}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2+b*x+a)^(1/2),x)

[Out]

1/4*(2*c*x+b)*(c*x^2+b*x+a)^(1/2)/c+1/2/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*
x+a)^(1/2))*a-1/8/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^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 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235197, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} -{\left (b^{2} - 4 \, a c\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{16 \, c^{\frac{3}{2}}}, \frac{2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{-c} -{\left (b^{2} - 4 \, a c\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{8 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(c) - (b^2 - 4*a*c)*log(-4*(2*c^2
*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)))/
c^(3/2), 1/8*(2*sqrt(c*x^2 + b*x + a)*(2*c*x + b)*sqrt(-c) - (b^2 - 4*a*c)*arcta
n(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 0.216594, size = 92, normalized size = 1.23 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x + a}{\left (2 \, x + \frac{b}{c}\right )} + \frac{{\left (b^{2} - 4 \, a c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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