3.125 \(\int \frac{1}{\sqrt{\frac{b^2+4 c}{4 c}+b x+c x^2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{\sinh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]

[Out]

ArcSinh[(b + 2*c*x)/(2*Sqrt[c])]/Sqrt[c]

________________________________________________________________________________________

Rubi [A]  time = 0.0131654, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.074, Rules used = {619, 215} \[ \frac{\sinh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[(b^2 + 4*c)/(4*c) + b*x + c*x^2],x]

[Out]

ArcSinh[(b + 2*c*x)/(2*Sqrt[c])]/Sqrt[c]

Rule 619

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dist[1/(2*c*((-4*c)/(b^2 - 4*a*c))^p), Subst[Int[Si
mp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 0]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

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

Mathematica [A]  time = 0.0274324, size = 22, normalized size = 1. \[ \frac{\sinh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{\sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[(b^2 + 4*c)/(4*c) + b*x + c*x^2],x]

[Out]

ArcSinh[(b + 2*c*x)/(2*Sqrt[c])]/Sqrt[c]

________________________________________________________________________________________

Maple [B]  time = 0.264, size = 51, normalized size = 2.3 \begin{align*}{\frac{\sqrt{4}}{2}\ln \left ({\frac{ \left ( 4\,cx+2\,b \right ) \sqrt{4}}{4}{\frac{1}{\sqrt{c}}}}+\sqrt{{\frac{{b}^{2}+4\,c}{c}}+4\,bx+4\,c{x}^{2}} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/((b^2+4*c)/c+4*b*x+4*c*x^2)^(1/2),x)

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.07685, size = 320, normalized size = 14.55 \begin{align*} \left [\frac{\log \left (-4 \, c^{2} x^{2} - 4 \, b c x - b^{2} -{\left (2 \, c x + b\right )} \sqrt{c} \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2} + 4 \, c}{c}} - 2 \, c\right )}{2 \, \sqrt{c}}, -\frac{\sqrt{-c} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c} \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2} + 4 \, c}{c}}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2} + 4 \, c}\right )}{c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/((b^2+4*c)/c+4*b*x+4*c*x^2)^(1/2),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 2 \int \frac{1}{\sqrt{\frac{b^{2}}{c} + 4 b x + 4 c x^{2} + 4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/((b**2+4*c)/c+4*b*x+4*c*x**2)**(1/2),x)

[Out]

2*Integral(1/sqrt(b**2/c + 4*b*x + 4*c*x**2 + 4), x)

________________________________________________________________________________________

Giac [B]  time = 1.43829, size = 66, normalized size = 3. \begin{align*} -\frac{\log \left ({\left | -{\left (2 \, \sqrt{c} x - \sqrt{4 \, c x^{2} + 4 \, b x + \frac{b^{2} + 4 \, c}{c}}\right )} \sqrt{c} - b \right |}\right )}{\sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/((b^2+4*c)/c+4*b*x+4*c*x^2)^(1/2),x, algorithm="giac")

[Out]

-log(abs(-(2*sqrt(c)*x - sqrt(4*c*x^2 + 4*b*x + (b^2 + 4*c)/c))*sqrt(c) - b))/sqrt(c)