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

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

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 103, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\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}} \end {gather*}

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 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 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 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 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 (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 = 0.03, size = 72, normalized size = 0.70 \begin {gather*} \frac {485 \sqrt {3} \tanh ^{-1}\left (\frac {24 x+17}{4 \sqrt {36 x^2+51 x+18}}\right )+12 \sqrt {12 x^2+17 x+6} \left (-884736 x^4+1963008 x^3+6837888 x^2+5455144 x+1353611\right )}{1474560} \end {gather*}

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.44, size = 77, normalized size = 0.75 \begin {gather*} \frac {97 \tanh ^{-1}\left (\frac {2 \sqrt {12 x^2+17 x+6}}{\sqrt {3} (4 x+3)}\right )}{49152 \sqrt {3}}+\frac {\sqrt {12 x^2+17 x+6} \left (-884736 x^4+1963008 x^3+6837888 x^2+5455144 x+1353611\right )}{122880} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[6 + 17*x + 12*x^2]*(1353611 + 5455144*x + 6837888*x^2 + 1963008*x^3 - 884736*x^4))/122880 + (97*ArcTanh[
(2*Sqrt[6 + 17*x + 12*x^2])/(Sqrt[3]*(3 + 4*x))])/(49152*Sqrt[3])

________________________________________________________________________________________

fricas [A]  time = 0.87, size = 73, normalized size = 0.71 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

giac [A]  time = 0.20, size = 70, normalized size = 0.68 \begin {gather*} -\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 {gather*}

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))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 96, normalized size = 0.93 \begin {gather*} -\frac {3 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}} x^{2}}{5}+\frac {349 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}} x}{160}+\frac {97 \sqrt {12}\, \ln \left (\frac {\left (12 x +\frac {17}{2}\right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right )}{589824}+\frac {7093 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{3840}-\frac {97 \left (24 x +17\right ) \sqrt {12 x^{2}+17 x +6}}{24576} \end {gather*}

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*(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)-97/24576*(24*x+
17)*(12*x^2+17*x+6)^(1/2)+97/589824*12^(1/2)*ln(1/12*(12*x+17/2)*12^(1/2)+(12*x^2+17*x+6)^(1/2))

________________________________________________________________________________________

maxima [A]  time = 0.98, size = 104, normalized size = 1.01 \begin {gather*} -\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 {gather*}

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)

________________________________________________________________________________________

mupad [B]  time = 4.69, size = 136, normalized size = 1.32 \begin {gather*} \frac {3753\,\left (\frac {x}{2}+\frac {17}{48}\right )\,\sqrt {12\,x^2+17\,x+6}}{80}-\frac {417\,\sqrt {12}\,\ln \left (\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (12\,x+\frac {17}{2}\right )}{12}\right )}{10240}-\frac {3\,x^2\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{5}+\frac {7093\,\sqrt {12\,x^2+17\,x+6}\,\left (1152\,x^2+408\,x-291\right )}{368640}+\frac {349\,x\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{160}+\frac {120581\,\sqrt {12}\,\ln \left (2\,\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (24\,x+17\right )}{12}\right )}{2949120} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3753*(x/2 + 17/48)*(17*x + 12*x^2 + 6)^(1/2))/80 - (417*12^(1/2)*log((17*x + 12*x^2 + 6)^(1/2) + (12^(1/2)*(1
2*x + 17/2))/12))/10240 - (3*x^2*(17*x + 12*x^2 + 6)^(3/2))/5 + (7093*(17*x + 12*x^2 + 6)^(1/2)*(408*x + 1152*
x^2 - 291))/368640 + (349*x*(17*x + 12*x^2 + 6)^(3/2))/160 + (120581*12^(1/2)*log(2*(17*x + 12*x^2 + 6)^(1/2)
+ (12^(1/2)*(24*x + 17))/12))/2949120

________________________________________________________________________________________

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

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)

________________________________________________________________________________________