### 3.2410 $$\int \frac{1}{(\frac{b e}{2 c}+e x) \sqrt{\frac{b^2}{4 c}+b x+c x^2}} \, dx$$

Optimal. Leaf size=27 $-\frac{2}{e \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}}$

[Out]

-2/(e*Sqrt[b^2/c + 4*b*x + 4*c*x^2])

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Rubi [A]  time = 0.0284133, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.051, Rules used = {643, 629} $-\frac{2}{e \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-2/(e*Sqrt[b^2/c + 4*b*x + 4*c*x^2])

Rule 643

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

Rule 629

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

Rubi steps

\begin{align*} \int \frac{1}{\left (\frac{b e}{2 c}+e x\right ) \sqrt{\frac{b^2}{4 c}+b x+c x^2}} \, dx &=\frac{c \int \frac{\frac{b e}{2 c}+e x}{\left (\frac{b^2}{4 c}+b x+c x^2\right )^{3/2}} \, dx}{e^2}\\ &=-\frac{2}{e \sqrt{\frac{b^2}{c}+4 b x+4 c x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0128009, size = 21, normalized size = 0.78 $-\frac{2}{e \sqrt{\frac{(b+2 c x)^2}{c}}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-2/(e*Sqrt[(b + 2*c*x)^2/c])

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Maple [A]  time = 0.153, size = 29, normalized size = 1.1 \begin{align*} -2\,{\frac{1}{e}{\frac{1}{\sqrt{{\frac{4\,{c}^{2}{x}^{2}+4\,bcx+{b}^{2}}{c}}}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.984441, size = 45, normalized size = 1.67 \begin{align*} -\frac{2}{2 \, e^{2} x \sqrt{\frac{c}{e^{2}}} + \frac{b e^{2} \sqrt{\frac{c}{e^{2}}}}{c}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/(2*e^2*x*sqrt(c/e^2) + b*e^2*sqrt(c/e^2)/c)

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Fricas [A]  time = 1.96891, size = 103, normalized size = 3.81 \begin{align*} -\frac{2 \, c \sqrt{\frac{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}}}{4 \, c^{2} e x^{2} + 4 \, b c e x + b^{2} e} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.09872, size = 59, normalized size = 2.19 \begin{align*} \frac{4 \, \sqrt{c} e^{\left (-1\right )}}{{\left (2 \, \sqrt{c} x - \sqrt{4 \, c x^{2} + 4 \, b x + \frac{b^{2}}{c}}\right )} \sqrt{c} + b} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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