3.2397 \(\int \frac{1}{\left (\frac{b e}{2 c}+e x\right ) \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=56 \[ \frac{2 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]

[Out]

(2*Sqrt[c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^
2 - 4*a*c]*e)

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Rubi [A]  time = 0.123101, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{2 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]  Int[1/(((b*e)/(2*c) + e*x)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(2*Sqrt[c]*ArcTan[(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])/Sqrt[b^2 - 4*a*c]])/(Sqrt[b^
2 - 4*a*c]*e)

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Rubi in Sympy [A]  time = 22.9427, size = 53, normalized size = 0.95 \[ \frac{2 \sqrt{c} \operatorname{atan}{\left (\frac{2 \sqrt{c} \sqrt{a + b x + c x^{2}}}{\sqrt{- 4 a c + b^{2}}} \right )}}{e \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(c)*atan(2*sqrt(c)*sqrt(a + b*x + c*x**2)/sqrt(-4*a*c + b**2))/(e*sqrt(-4*
a*c + b**2))

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Mathematica [A]  time = 0.137636, size = 84, normalized size = 1.5 \[ \frac{2 \sqrt{c} \left (\log (b+2 c x)-\log \left (2 c \sqrt{4 a c-b^2} \sqrt{a+x (b+c x)}+4 a c^{3/2}+b^2 \left (-\sqrt{c}\right )\right )\right )}{e \sqrt{4 a c-b^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(((b*e)/(2*c) + e*x)*Sqrt[a + b*x + c*x^2]),x]

[Out]

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

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Maple [B]  time = 0.013, size = 98, normalized size = 1.8 \[ -2\,{\frac{1}{e}\ln \left ({1 \left ( 1/2\,{\frac{4\,ac-{b}^{2}}{c}}+1/2\,\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}\sqrt{4\, \left ( x+1/2\,{\frac{b}{c}} \right ) ^{2}c+{\frac{4\,ac-{b}^{2}}{c}}} \right ) \left ( x+1/2\,{\frac{b}{c}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{4\,ac-{b}^{2}}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(1/2*b*e/c+e*x)/(c*x^2+b*x+a)^(1/2),x)

[Out]

-2/e/((4*a*c-b^2)/c)^(1/2)*ln((1/2*(4*a*c-b^2)/c+1/2*((4*a*c-b^2)/c)^(1/2)*(4*(x
+1/2*b/c)^2*c+(4*a*c-b^2)/c)^(1/2))/(x+1/2*b/c))

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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(2/(sqrt(c*x^2 + b*x + a)*(2*e*x + b*e/c)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253295, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{-\frac{c}{b^{2} - 4 \, a c}} \log \left (-\frac{4 \, c^{2} x^{2} + 4 \, b c x - b^{2} + 8 \, a c + 4 \, \sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{-\frac{c}{b^{2} - 4 \, a c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}\right )}{e}, -\frac{2 \, \sqrt{\frac{c}{b^{2} - 4 \, a c}} \arctan \left (\frac{1}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{\frac{c}{b^{2} - 4 \, a c}}}\right )}{e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 c \int \frac{1}{b \sqrt{a + b x + c x^{2}} + 2 c x \sqrt{a + b x + c x^{2}}}\, dx}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*Integral(1/(b*sqrt(a + b*x + c*x**2) + 2*c*x*sqrt(a + b*x + c*x**2)), x)/e

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GIAC/XCAS [A]  time = 0.218404, size = 88, normalized size = 1.57 \[ \frac{4 \, c \arctan \left (-\frac{2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} c + b \sqrt{c}}{\sqrt{b^{2} c - 4 \, a c^{2}}}\right ) e^{\left (-1\right )}}{\sqrt{b^{2} c - 4 \, a c^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*c*arctan(-(2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*c + b*sqrt(c))/sqrt(b^2*c - 4
*a*c^2))*e^(-1)/sqrt(b^2*c - 4*a*c^2)