\(\int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) (30+31 x-12 x^2)} \, dx\) [135]

Optimal result

Integrand size = 34, antiderivative size = 28 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) \left (30+31 x-12 x^2\right )} \, dx=\frac {1}{42} \text {arctanh}\left (\frac {206+291 x}{84 \sqrt {6+17 x+12 x^2}}\right ) \]

[Out]

1/42*arctanh(1/84*(206+291*x)/(12*x^2+17*x+6)^(1/2))

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {1016, 738, 212} \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) \left (30+31 x-12 x^2\right )} \, dx=\frac {1}{42} \text {arctanh}\left (\frac {291 x+206}{84 \sqrt {12 x^2+17 x+6}}\right ) \]

[In]

Int[Sqrt[6 + 17*x + 12*x^2]/((2 + 3*x)*(30 + 31*x - 12*x^2)),x]

[Out]

ArcTanh[(206 + 291*x)/(84*Sqrt[6 + 17*x + 12*x^2])]/42

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 738

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

Rule 1016

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(10-3 x) \sqrt {6+17 x+12 x^2}} \, dx \\ & = -\left (2 \text {Subst}\left (\int \frac {1}{7056-x^2} \, dx,x,\frac {-206-291 x}{\sqrt {6+17 x+12 x^2}}\right )\right ) \\ & = \frac {1}{42} \tanh ^{-1}\left (\frac {206+291 x}{84 \sqrt {6+17 x+12 x^2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) \left (30+31 x-12 x^2\right )} \, dx=\frac {1}{21} \text {arctanh}\left (\frac {6 \sqrt {6+17 x+12 x^2}}{7 (2+3 x)}\right ) \]

[In]

Integrate[Sqrt[6 + 17*x + 12*x^2]/((2 + 3*x)*(30 + 31*x - 12*x^2)),x]

[Out]

ArcTanh[(6*Sqrt[6 + 17*x + 12*x^2])/(7*(2 + 3*x))]/21

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.14

method	result	size
trager	\(-\frac {\ln \left (-\frac {84 \sqrt {12 x^{2}+17 x +6}-206-291 x}{3 x -10}\right )}{42}\)	\(32\)
default	\(\frac {\sqrt {12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}}}{12}+\frac {\ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x +\frac {2}{3}\right )^{2}+x +\frac {2}{3}}\right ) \sqrt {12}}{288}-\frac {4 \sqrt {12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}}}{49}+\frac {\ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x +\frac {3}{4}\right )^{2}-x -\frac {3}{4}}\right ) \sqrt {12}}{294}-\frac {\sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}{588}-\frac {97 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}\right ) \sqrt {12}}{14112}+\frac {\operatorname {arctanh}\left (\frac {\frac {206}{3}+97 x}{28 \sqrt {12 \left (x -\frac {10}{3}\right )^{2}+97 x -\frac {382}{3}}}\right )}{42}\)	\(163\)

[In]

int((12*x^2+17*x+6)^(1/2)/(2+3*x)/(-12*x^2+31*x+30),x,method=_RETURNVERBOSE)

[Out]

-1/42*ln(-(84*(12*x^2+17*x+6)^(1/2)-206-291*x)/(3*x-10))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 53 vs. \(2 (22) = 44\).

Time = 0.32 (sec) , antiderivative size = 53, normalized size of antiderivative = 1.89 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) \left (30+31 x-12 x^2\right )} \, dx=\frac {1}{84} \, \log \left (\frac {291 \, x + 84 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) - \frac {1}{84} \, \log \left (\frac {291 \, x - 84 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 206}{x}\right ) \]

[In]

integrate((12*x^2+17*x+6)^(1/2)/(2+3*x)/(-12*x^2+31*x+30),x, algorithm="fricas")

[Out]

1/84*log((291*x + 84*sqrt(12*x^2 + 17*x + 6) + 206)/x) - 1/84*log((291*x - 84*sqrt(12*x^2 + 17*x + 6) + 206)/x
)

Sympy [F]

\[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) \left (30+31 x-12 x^2\right )} \, dx=- \int \frac {\sqrt {12 x^{2} + 17 x + 6}}{36 x^{3} - 69 x^{2} - 152 x - 60}\, dx \]

[In]

integrate((12*x**2+17*x+6)**(1/2)/(2+3*x)/(-12*x**2+31*x+30),x)

[Out]

-Integral(sqrt(12*x**2 + 17*x + 6)/(36*x**3 - 69*x**2 - 152*x - 60), x)

Maxima [F]

\[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) \left (30+31 x-12 x^2\right )} \, dx=\int { -\frac {\sqrt {12 \, x^{2} + 17 \, x + 6}}{{\left (12 \, x^{2} - 31 \, x - 30\right )} {\left (3 \, x + 2\right )}} \,d x } \]

[In]

integrate((12*x^2+17*x+6)^(1/2)/(2+3*x)/(-12*x^2+31*x+30),x, algorithm="maxima")

[Out]

-integrate(sqrt(12*x^2 + 17*x + 6)/((12*x^2 - 31*x - 30)*(3*x + 2)), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 63 vs. \(2 (22) = 44\).

Time = 0.31 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.25 \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) \left (30+31 x-12 x^2\right )} \, dx=\frac {1}{42} \, \log \left ({\left | -6 \, \sqrt {3} x + 20 \, \sqrt {3} + 3 \, \sqrt {12 \, x^{2} + 17 \, x + 6} + 42 \right |}\right ) - \frac {1}{42} \, \log \left ({\left | -6 \, \sqrt {3} x + 20 \, \sqrt {3} + 3 \, \sqrt {12 \, x^{2} + 17 \, x + 6} - 42 \right |}\right ) \]

[In]

integrate((12*x^2+17*x+6)^(1/2)/(2+3*x)/(-12*x^2+31*x+30),x, algorithm="giac")

[Out]

1/42*log(abs(-6*sqrt(3)*x + 20*sqrt(3) + 3*sqrt(12*x^2 + 17*x + 6) + 42)) - 1/42*log(abs(-6*sqrt(3)*x + 20*sqr
t(3) + 3*sqrt(12*x^2 + 17*x + 6) - 42))

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) \left (30+31 x-12 x^2\right )} \, dx=\int \frac {\sqrt {12\,x^2+17\,x+6}}{\left (3\,x+2\right )\,\left (-12\,x^2+31\,x+30\right )} \,d x \]

[In]

int((17*x + 12*x^2 + 6)^(1/2)/((3*x + 2)*(31*x - 12*x^2 + 30)),x)

[Out]

int((17*x + 12*x^2 + 6)^(1/2)/((3*x + 2)*(31*x - 12*x^2 + 30)), x)