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

Optimal. Leaf size=103 \[ -\frac{1}{20} \left (12 x^2+17 x+6\right )^{5/2}+\frac{97}{768} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}-\frac{97 (24 x+17) \sqrt{12 x^2+17 x+6}}{24576}+\frac{97 \tanh ^{-1}\left (\frac{24 x+17}{4 \sqrt{3} \sqrt{12 x^2+17 x+6}}\right )}{98304 \sqrt{3}} \]

[Out]

(-97*(17 + 24*x)*Sqrt[6 + 17*x + 12*x^2])/24576 + (97*(17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2))/768 - (6 + 17*x +
12*x^2)^(5/2)/20 + (97*ArcTanh[(17 + 24*x)/(4*Sqrt[3]*Sqrt[6 + 17*x + 12*x^2])])/(98304*Sqrt[3])

________________________________________________________________________________________

Rubi [A]  time = 0.0417328, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1002, 640, 612, 621, 206} \[ -\frac{1}{20} \left (12 x^2+17 x+6\right )^{5/2}+\frac{97}{768} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}-\frac{97 (24 x+17) \sqrt{12 x^2+17 x+6}}{24576}+\frac{97 \tanh ^{-1}\left (\frac{24 x+17}{4 \sqrt{3} \sqrt{12 x^2+17 x+6}}\right )}{98304 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-97*(17 + 24*x)*Sqrt[6 + 17*x + 12*x^2])/24576 + (97*(17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2))/768 - (6 + 17*x +
12*x^2)^(5/2)/20 + (97*ArcTanh[(17 + 24*x)/(4*Sqrt[3]*Sqrt[6 + 17*x + 12*x^2])])/(98304*Sqrt[3])

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 640

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

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 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 (2+3 x) \left (30+31 x-12 x^2\right ) \sqrt{6+17 x+12 x^2} \, dx &=\int (10-3 x) \left (6+17 x+12 x^2\right )^{3/2} \, dx\\ &=-\frac{1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac{97}{8} \int \left (6+17 x+12 x^2\right )^{3/2} \, dx\\ &=\frac{97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac{1}{20} \left (6+17 x+12 x^2\right )^{5/2}-\frac{97}{512} \int \sqrt{6+17 x+12 x^2} \, dx\\ &=-\frac{97 (17+24 x) \sqrt{6+17 x+12 x^2}}{24576}+\frac{97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac{1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac{97 \int \frac{1}{\sqrt{6+17 x+12 x^2}} \, dx}{49152}\\ &=-\frac{97 (17+24 x) \sqrt{6+17 x+12 x^2}}{24576}+\frac{97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac{1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac{97 \operatorname{Subst}\left (\int \frac{1}{48-x^2} \, dx,x,\frac{17+24 x}{\sqrt{6+17 x+12 x^2}}\right )}{24576}\\ &=-\frac{97 (17+24 x) \sqrt{6+17 x+12 x^2}}{24576}+\frac{97}{768} (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}-\frac{1}{20} \left (6+17 x+12 x^2\right )^{5/2}+\frac{97 \tanh ^{-1}\left (\frac{17+24 x}{4 \sqrt{3} \sqrt{6+17 x+12 x^2}}\right )}{98304 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.0345904, size = 72, normalized size = 0.7 \[ \frac{12 \sqrt{12 x^2+17 x+6} \left (-884736 x^4+1963008 x^3+6837888 x^2+5455144 x+1353611\right )+485 \sqrt{3} \tanh ^{-1}\left (\frac{24 x+17}{4 \sqrt{36 x^2+51 x+18}}\right )}{1474560} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*Sqrt[6 + 17*x + 12*x^2]*(1353611 + 5455144*x + 6837888*x^2 + 1963008*x^3 - 884736*x^4) + 485*Sqrt[3]*ArcTa
nh[(17 + 24*x)/(4*Sqrt[18 + 51*x + 36*x^2])])/1474560

________________________________________________________________________________________

Maple [A]  time = 0.05, size = 96, normalized size = 0.9 \begin{align*} -{\frac{3\,{x}^{2}}{5} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}}+{\frac{349\,x}{160} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}}+{\frac{7093}{3840} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}}-{\frac{1649+2328\,x}{24576}\sqrt{12\,{x}^{2}+17\,x+6}}+{\frac{97\,\sqrt{12}}{589824}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\,{x}^{2}+17\,x+6} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*x^2*(12*x^2+17*x+6)^(3/2)+349/160*x*(12*x^2+17*x+6)^(3/2)+7093/3840*(12*x^2+17*x+6)^(3/2)-97/24576*(17+24
*x)*(12*x^2+17*x+6)^(1/2)+97/589824*ln(1/12*(17/2+12*x)*12^(1/2)+(12*x^2+17*x+6)^(1/2))*12^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 1.47097, size = 140, normalized size = 1.36 \begin{align*} -\frac{3}{5} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} x^{2} + \frac{349}{160} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} x + \frac{7093}{3840} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} - \frac{97}{1024} \, \sqrt{12 \, x^{2} + 17 \, x + 6} x + \frac{97}{294912} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{12 \, x^{2} + 17 \, x + 6} + 24 \, x + 17\right ) - \frac{1649}{24576} \, \sqrt{12 \, x^{2} + 17 \, x + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*(12*x^2 + 17*x + 6)^(3/2)*x^2 + 349/160*(12*x^2 + 17*x + 6)^(3/2)*x + 7093/3840*(12*x^2 + 17*x + 6)^(3/2)
- 97/1024*sqrt(12*x^2 + 17*x + 6)*x + 97/294912*sqrt(3)*log(4*sqrt(3)*sqrt(12*x^2 + 17*x + 6) + 24*x + 17) -
1649/24576*sqrt(12*x^2 + 17*x + 6)

________________________________________________________________________________________

Fricas [A]  time = 1.55974, size = 263, normalized size = 2.55 \begin{align*} -\frac{1}{122880} \,{\left (884736 \, x^{4} - 1963008 \, x^{3} - 6837888 \, x^{2} - 5455144 \, x - 1353611\right )} \sqrt{12 \, x^{2} + 17 \, x + 6} + \frac{97}{589824} \, \sqrt{3} \log \left (8 \, \sqrt{3} \sqrt{12 \, x^{2} + 17 \, x + 6}{\left (24 \, x + 17\right )} + 1152 \, x^{2} + 1632 \, x + 577\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/122880*(884736*x^4 - 1963008*x^3 - 6837888*x^2 - 5455144*x - 1353611)*sqrt(12*x^2 + 17*x + 6) + 97/589824*s
qrt(3)*log(8*sqrt(3)*sqrt(12*x^2 + 17*x + 6)*(24*x + 17) + 1152*x^2 + 1632*x + 577)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.15629, size = 95, normalized size = 0.92 \begin{align*} -\frac{1}{122880} \,{\left (8 \,{\left (48 \,{\left (72 \,{\left (32 \, x - 71\right )} x - 17807\right )} x - 681893\right )} x - 1353611\right )} \sqrt{12 \, x^{2} + 17 \, x + 6} - \frac{97}{294912} \, \sqrt{3} \log \left ({\left | -4 \, \sqrt{3}{\left (2 \, \sqrt{3} x - \sqrt{12 \, x^{2} + 17 \, x + 6}\right )} - 17 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/122880*(8*(48*(72*(32*x - 71)*x - 17807)*x - 681893)*x - 1353611)*sqrt(12*x^2 + 17*x + 6) - 97/294912*sqrt(
3)*log(abs(-4*sqrt(3)*(2*sqrt(3)*x - sqrt(12*x^2 + 17*x + 6)) - 17))