3.2408 \(\int \frac{1}{(\frac{b e}{2 c}+e x) \sqrt{a+b x+c x^2}} \, dx\)

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 = 0.0530509, 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, Rules used = {688, 205} \[ \frac{2 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+b x+c x^2}}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]

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)

Rule 688

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :> Dist[4*c, Subst[Int[1/(b^2*e
- 4*a*c*e + 4*c*e*x^2), x], x, Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0]
&& EqQ[2*c*d - b*e, 0]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{1}{\left (\frac{b e}{2 c}+e x\right ) \sqrt{a+b x+c x^2}} \, dx &=(4 c) \operatorname{Subst}\left (\int \frac{1}{b^2 e-4 a c e+4 c e x^2} \, dx,x,\sqrt{a+b x+c x^2}\right )\\ &=\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 )}{\sqrt{b^2-4 a c} e}\\ \end{align*}

Mathematica [A]  time = 0.040288, size = 55, normalized size = 0.98 \[ \frac{2 \sqrt{c} \tan ^{-1}\left (\frac{2 \sqrt{c} \sqrt{a+x (b+c x)}}{\sqrt{b^2-4 a c}}\right )}{e \sqrt{b^2-4 a c}} \]

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

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Maple [B]  time = 0.193, size = 98, normalized size = 1.8 \begin{align*} -2\,{\frac{1}{e}\ln \left ({ \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}}}}}} \end{align*}

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(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.29252, size = 385, normalized size = 6.88 \begin{align*} \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{\sqrt{c x^{2} + b x + a}{\left (b^{2} - 4 \, a c\right )} \sqrt{\frac{c}{b^{2} - 4 \, a c}}}{2 \,{\left (c^{2} x^{2} + b c x + a c\right )}}\right )}{e}\right ] \end{align*}

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, 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)*(b
^2 - 4*a*c)*sqrt(c/(b^2 - 4*a*c))/(c^2*x^2 + b*c*x + a*c))/e]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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 [A]  time = 1.14539, size = 88, normalized size = 1.57 \begin{align*} \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}}} \end{align*}

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, 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)