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

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

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Rubi [A]  time = 0.03, antiderivative size = 27, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2}{e \sqrt {\frac {b^2}{c}+4 b x+4 c x^2}} \end {gather*}

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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 21, normalized size = 0.78 \begin {gather*} -\frac {2}{e \sqrt {\frac {(b+2 c x)^2}{c}}} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.05, size = 30, normalized size = 1.11 \begin {gather*} -\frac {2 c \sqrt {\frac {(b+2 c x)^2}{c}}}{e (b+2 c x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 49, normalized size = 1.81 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.20, size = 44, normalized size = 1.63 \begin {gather*} \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 {gather*}

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)

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maple [A]  time = 0.05, size = 29, normalized size = 1.07 \begin {gather*} -\frac {2}{\sqrt {\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{c}}\, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(2/(1/2*b/c*e+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 = 0.91, size = 18, normalized size = 0.67 \begin {gather*} -\frac {2}{2 \, \sqrt {c} e x + \frac {b e}{\sqrt {c}}} \end {gather*}

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*sqrt(c)*e*x + b*e/sqrt(c))

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mupad [B]  time = 1.12, size = 25, normalized size = 0.93 \begin {gather*} -\frac {2}{e\,\sqrt {4\,b\,x+4\,c\,x^2+\frac {b^2}{c}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 {gather*}

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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