3.25.10 \(\int \frac {1}{(\frac {b e}{2 c}+e x) \sqrt {\frac {b^2}{4 c}+b x+c x^2}} \, dx\) [2410]

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

[Out]

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

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Rubi [A]
time = 0.02, 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 = {657, 643} \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 643

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 657

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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Maple [A]
time = 0.71, size = 29, normalized size = 1.07

method	result	size
risch	\(-\frac {2}{e \sqrt {\frac {\left (2 c x +b \right )^{2}}{c}}}\)	\(20\)
gosper	\(-\frac {2}{\sqrt {\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{c}}\, e}\)	\(29\)
default	\(-\frac {2}{\sqrt {\frac {4 c^{2} x^{2}+4 b c x +b^{2}}{c}}\, e}\)	\(29\)
trager	\(\frac {4 c^{2} x \sqrt {-\frac {-4 c^{2} x^{2}-4 b c x -b^{2}}{c}}}{b e \left (2 c x +b \right )^{2}}\)	\(47\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.28, size = 20, normalized size = 0.74 \begin {gather*} -\frac {2}{2 \, \sqrt {c} x e + \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)/(b^2/c+4*b*x+4*c*x^2)^(1/2),x, algorithm="maxima")

[Out]

-2/(2*sqrt(c)*x*e + b*e/sqrt(c))

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Fricas [A]
time = 2.55, size = 47, normalized size = 1.74 \begin {gather*} -\frac {2 \, c \sqrt {\frac {4 \, c^{2} x^{2} + 4 \, b c x + b^{2}}{c}} e^{\left (-1\right )}}{4 \, c^{2} x^{2} + 4 \, b c x + b^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(2/(1/2*b*e/c+e*x)/(b^2/c+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)*e^(-1)/(4*c^2*x^2 + 4*b*c*x + b^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)/(b**2/c+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 = 1.20, size = 28, normalized size = 1.04 \begin {gather*} -\frac {2 \, c^{\frac {3}{2}} e^{\left (-1\right )}}{{\left (2 \, c x + b\right )} {\left | c \right |} \mathrm {sgn}\left (2 \, c x + b\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*c^(3/2)*e^(-1)/((2*c*x + b)*abs(c)*sgn(2*c*x + b))

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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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