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

Optimal. Leaf size=28 \[ \frac {1}{42} \tanh ^{-1}\left (\frac {291 x+206}{84 \sqrt {12 x^2+17 x+6}}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {1002, 724, 206} \begin {gather*} \frac {1}{42} \tanh ^{-1}\left (\frac {291 x+206}{84 \sqrt {12 x^2+17 x+6}}\right ) \end {gather*}

Antiderivative was successfully verified.

[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 206

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

Rule 724

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 1002

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*} \int \frac {\sqrt {6+17 x+12 x^2}}{(2+3 x) \left (30+31 x-12 x^2\right )} \, dx &=\int \frac {1}{(10-3 x) \sqrt {6+17 x+12 x^2}} \, dx\\ &=-\left (2 \operatorname {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*}

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Mathematica [A]  time = 0.10, size = 37, normalized size = 1.32 \begin {gather*} \frac {1}{42} \log \left (84 \sqrt {12 x^2+17 x+6}+291 x+206\right )-\frac {1}{42} \log (10-3 x) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/42*Log[10 - 3*x] + Log[206 + 291*x + 84*Sqrt[6 + 17*x + 12*x^2]]/42

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IntegrateAlgebraic [A]  time = 0.30, size = 30, normalized size = 1.07 \begin {gather*} \frac {1}{21} \tanh ^{-1}\left (\frac {6 \sqrt {12 x^2+17 x+6}}{7 (3 x+2)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[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

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fricas [B]  time = 0.63, size = 53, normalized size = 1.89 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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
)

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giac [B]  time = 0.27, size = 63, normalized size = 2.25 \begin {gather*} \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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [B]  time = 0.02, size = 163, normalized size = 5.82 \begin {gather*} \frac {\arctanh \left (\frac {97 x +\frac {206}{3}}{28 \sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}}\right )}{42}-\frac {97 \sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}\right )}{14112}+\frac {\sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {x +12 \left (x +\frac {2}{3}\right )^{2}+\frac {2}{3}}\right )}{288}+\frac {\sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {-x +12 \left (x +\frac {3}{4}\right )^{2}-\frac {3}{4}}\right )}{294}-\frac {4 \sqrt {-x +12 \left (x +\frac {3}{4}\right )^{2}-\frac {3}{4}}}{49}+\frac {\sqrt {x +12 \left (x +\frac {2}{3}\right )^{2}+\frac {2}{3}}}{12}-\frac {\sqrt {97 x +12 \left (x -\frac {10}{3}\right )^{2}-\frac {382}{3}}}{588} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/49*(12*(x+3/4)^2-x-3/4)^(1/2)+1/294*ln(1/12*(12*x+17/2)*12^(1/2)+(12*(x+3/4)^2-x-3/4)^(1/2))*12^(1/2)+1/12*
(12*(x+2/3)^2+x+2/3)^(1/2)+1/288*ln(1/12*(12*x+17/2)*12^(1/2)+(12*(x+2/3)^2+x+2/3)^(1/2))*12^(1/2)-1/588*(12*(
x-10/3)^2+97*x-382/3)^(1/2)-97/14112*ln(1/12*(12*x+17/2)*12^(1/2)+(12*(x-10/3)^2+97*x-382/3)^(1/2))*12^(1/2)+1
/42*arctanh(1/28*(206/3+97*x)/(12*(x-10/3)^2+97*x-382/3)^(1/2))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} -\int \frac {\sqrt {12 \, x^{2} + 17 \, x + 6}}{{\left (12 \, x^{2} - 31 \, x - 30\right )} {\left (3 \, x + 2\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\sqrt {12\,x^2+17\,x+6}}{\left (3\,x+2\right )\,\left (-12\,x^2+31\,x+30\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \int \frac {\sqrt {12 x^{2} + 17 x + 6}}{36 x^{3} - 69 x^{2} - 152 x - 60}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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