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

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

[Out]

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

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Rubi [A]  time = 0.0482939, antiderivative size = 28, normalized size of antiderivative = 1., 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} \[ \frac{1}{42} \tanh ^{-1}\left (\frac{291 x+206}{84 \sqrt{12 x^2+17 x+6}}\right ) \]

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 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]

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 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])

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*}

Mathematica [A]  time = 0.109895, size = 37, normalized size = 1.32 \[ \frac{1}{42} \log \left (84 \sqrt{12 x^2+17 x+6}+291 x+206\right )-\frac{1}{42} \log (10-3 x) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.058, size = 163, normalized size = 5.8 \begin{align*} -{\frac{4}{49}\sqrt{12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}}}}+{\frac{\sqrt{12}}{294}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\, \left ( x+3/4 \right ) ^{2}-x-{\frac{3}{4}}} \right ) }-{\frac{1}{588}\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}}}-{\frac{97\,\sqrt{12}}{14112}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}} \right ) }+{\frac{1}{42}{\it Artanh} \left ({\frac{1}{28} \left ({\frac{206}{3}}+97\,x \right ){\frac{1}{\sqrt{12\, \left ( x-10/3 \right ) ^{2}+97\,x-{\frac{382}{3}}}}}} \right ) }+{\frac{1}{12}\sqrt{12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}}}}+{\frac{\sqrt{12}}{288}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\, \left ( x+2/3 \right ) ^{2}+x+{\frac{2}{3}}} \right ) } \end{align*}

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*(17/2+12*x)*12^(1/2)+(12*(x+3/4)^2-x-3/4)^(1/2))*12^(1/2)-1/588
*(12*(x-10/3)^2+97*x-382/3)^(1/2)-97/14112*ln(1/12*(17/2+12*x)*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))+1/12*(12*(x+2/3)^2+x+2/3)^(1/2)+1/288*ln
(1/12*(17/2+12*x)*12^(1/2)+(12*(x+2/3)^2+x+2/3)^(1/2))*12^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\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{align*}

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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Fricas [B]  time = 1.54378, size = 153, normalized size = 5.46 \begin{align*} \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{align*}

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

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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Giac [B]  time = 1.1727, size = 85, normalized size = 3.04 \begin{align*} \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{align*}

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))