\(\int \frac {3+2 x}{\sqrt {-3-4 x-x^2} (3+4 x+2 x^2)} \, dx\) [37]

Optimal result

Integrand size = 32, antiderivative size = 17 \[ \int \frac {3+2 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]

[Out]

arctanh(x/(-x^2-4*x-3)^(1/2))

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1041, 212} \[ \int \frac {3+2 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {-x^2-4 x-3}}\right ) \]

[In]

Int[(3 + 2*x)/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]

[Out]

ArcTanh[x/Sqrt[-3 - 4*x - x^2]]

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 1041

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

Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {1}{3-3 x^2} \, dx,x,\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \\ & = \tanh ^{-1}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {3+2 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\text {arctanh}\left (\frac {x}{\sqrt {-3-4 x-x^2}}\right ) \]

[In]

Integrate[(3 + 2*x)/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]

[Out]

ArcTanh[x/Sqrt[-3 - 4*x - x^2]]

Maple [A] (verified)

Time = 0.82 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.06

method	result	size
pseudoelliptic	\(\operatorname {arctanh}\left (\frac {\sqrt {-x^{2}-4 x -3}}{x}\right )\)	\(18\)
trager	\(\frac {\ln \left (\frac {2 x \sqrt {-x^{2}-4 x -3}-4 x -3}{2 x^{2}+4 x +3}\right )}{2}\)	\(37\)
default	\(-\frac {\sqrt {3}\, \sqrt {4}\, \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}\, \operatorname {arctanh}\left (\frac {3 x}{\left (-\frac {3}{2}-x \right ) \sqrt {\frac {3 x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-12}}\right )}{6 \sqrt {\frac {\frac {x^{2}}{\left (-\frac {3}{2}-x \right )^{2}}-4}{\left (1+\frac {x}{-\frac {3}{2}-x}\right )^{2}}}\, \left (1+\frac {x}{-\frac {3}{2}-x}\right )}\)	\(94\)

[In]

int((3+2*x)/(2*x^2+4*x+3)/(-x^2-4*x-3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

arctanh((-x^2-4*x-3)^(1/2)/x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (15) = 30\).

Time = 0.31 (sec) , antiderivative size = 56, normalized size of antiderivative = 3.29 \[ \int \frac {3+2 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=-\frac {1}{4} \, \log \left (-\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) + \frac {1}{4} \, \log \left (\frac {2 \, \sqrt {-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) \]

[In]

integrate((3+2*x)/(2*x^2+4*x+3)/(-x^2-4*x-3)^(1/2),x, algorithm="fricas")

[Out]

-1/4*log(-(2*sqrt(-x^2 - 4*x - 3)*x + 4*x + 3)/x^2) + 1/4*log((2*sqrt(-x^2 - 4*x - 3)*x - 4*x - 3)/x^2)

Sympy [F]

\[ \int \frac {3+2 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {2 x + 3}{\sqrt {- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]

[In]

integrate((3+2*x)/(2*x**2+4*x+3)/(-x**2-4*x-3)**(1/2),x)

[Out]

Integral((2*x + 3)/(sqrt(-(x + 1)*(x + 3))*(2*x**2 + 4*x + 3)), x)

Maxima [F]

\[ \int \frac {3+2 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int { \frac {2 \, x + 3}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt {-x^{2} - 4 \, x - 3}} \,d x } \]

[In]

integrate((3+2*x)/(2*x^2+4*x+3)/(-x^2-4*x-3)^(1/2),x, algorithm="maxima")

[Out]

integrate((2*x + 3)/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 98 vs. \(2 (15) = 30\).

Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 5.76 \[ \int \frac {3+2 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\frac {1}{2} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {3 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) - \frac {1}{2} \, \log \left (\frac {2 \, {\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac {{\left (\sqrt {-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]

[In]

integrate((3+2*x)/(2*x^2+4*x+3)/(-x^2-4*x-3)^(1/2),x, algorithm="giac")

[Out]

1/2*log(2*(sqrt(-x^2 - 4*x - 3) - 1)/(x + 2) + 3*(sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 1) - 1/2*log(2*(sqrt
(-x^2 - 4*x - 3) - 1)/(x + 2) + (sqrt(-x^2 - 4*x - 3) - 1)^2/(x + 2)^2 + 3)

Mupad [F(-1)]

Timed out. \[ \int \frac {3+2 x}{\sqrt {-3-4 x-x^2} \left (3+4 x+2 x^2\right )} \, dx=\int \frac {2\,x+3}{\sqrt {-x^2-4\,x-3}\,\left (2\,x^2+4\,x+3\right )} \,d x \]

[In]

int((2*x + 3)/((- 4*x - x^2 - 3)^(1/2)*(4*x + 2*x^2 + 3)),x)

[Out]

int((2*x + 3)/((- 4*x - x^2 - 3)^(1/2)*(4*x + 2*x^2 + 3)), x)