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

Optimal. Leaf size=149 \[ -\frac{1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}-\frac{873 \left (12 x^2+17 x+6\right )^{7/2}}{1792}+\frac{25091 (24 x+17) \left (12 x^2+17 x+6\right )^{5/2}}{24576}-\frac{125455 (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}}{4718592}+\frac{125455 (24 x+17) \sqrt{12 x^2+17 x+6}}{150994944}-\frac{125455 \tanh ^{-1}\left (\frac{24 x+17}{4 \sqrt{3} \sqrt{12 x^2+17 x+6}}\right )}{603979776 \sqrt{3}} \]

[Out]

(125455*(17 + 24*x)*Sqrt[6 + 17*x + 12*x^2])/150994944 - (125455*(17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2))/471859
2 + (25091*(17 + 24*x)*(6 + 17*x + 12*x^2)^(5/2))/24576 - (873*(6 + 17*x + 12*x^2)^(7/2))/1792 - ((10 - 3*x)*(
6 + 17*x + 12*x^2)^(7/2))/32 - (125455*ArcTanh[(17 + 24*x)/(4*Sqrt[3]*Sqrt[6 + 17*x + 12*x^2])])/(603979776*Sq
rt[3])

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Rubi [A]  time = 0.0933228, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1002, 742, 640, 612, 621, 206} \[ -\frac{1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}-\frac{873 \left (12 x^2+17 x+6\right )^{7/2}}{1792}+\frac{25091 (24 x+17) \left (12 x^2+17 x+6\right )^{5/2}}{24576}-\frac{125455 (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}}{4718592}+\frac{125455 (24 x+17) \sqrt{12 x^2+17 x+6}}{150994944}-\frac{125455 \tanh ^{-1}\left (\frac{24 x+17}{4 \sqrt{3} \sqrt{12 x^2+17 x+6}}\right )}{603979776 \sqrt{3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(125455*(17 + 24*x)*Sqrt[6 + 17*x + 12*x^2])/150994944 - (125455*(17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2))/471859
2 + (25091*(17 + 24*x)*(6 + 17*x + 12*x^2)^(5/2))/24576 - (873*(6 + 17*x + 12*x^2)^(7/2))/1792 - ((10 - 3*x)*(
6 + 17*x + 12*x^2)^(7/2))/32 - (125455*ArcTanh[(17 + 24*x)/(4*Sqrt[3]*Sqrt[6 + 17*x + 12*x^2])])/(603979776*Sq
rt[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 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

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)^2 \left (30+31 x-12 x^2\right )^2 \sqrt{6+17 x+12 x^2} \, dx &=\int (10-3 x)^2 \left (6+17 x+12 x^2\right )^{5/2} \, dx\\ &=-\frac{1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}+\frac{1}{96} \int \left (11331-\frac{7857 x}{2}\right ) \left (6+17 x+12 x^2\right )^{5/2} \, dx\\ &=-\frac{873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac{1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}+\frac{75273}{512} \int \left (6+17 x+12 x^2\right )^{5/2} \, dx\\ &=\frac{25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac{873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac{1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac{125455 \int \left (6+17 x+12 x^2\right )^{3/2} \, dx}{49152}\\ &=-\frac{125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac{25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac{873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac{1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}+\frac{125455 \int \sqrt{6+17 x+12 x^2} \, dx}{3145728}\\ &=\frac{125455 (17+24 x) \sqrt{6+17 x+12 x^2}}{150994944}-\frac{125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac{25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac{873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac{1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac{125455 \int \frac{1}{\sqrt{6+17 x+12 x^2}} \, dx}{301989888}\\ &=\frac{125455 (17+24 x) \sqrt{6+17 x+12 x^2}}{150994944}-\frac{125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac{25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac{873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac{1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac{125455 \operatorname{Subst}\left (\int \frac{1}{48-x^2} \, dx,x,\frac{17+24 x}{\sqrt{6+17 x+12 x^2}}\right )}{150994944}\\ &=\frac{125455 (17+24 x) \sqrt{6+17 x+12 x^2}}{150994944}-\frac{125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac{25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac{873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac{1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac{125455 \tanh ^{-1}\left (\frac{17+24 x}{4 \sqrt{3} \sqrt{6+17 x+12 x^2}}\right )}{603979776 \sqrt{3}}\\ \end{align*}

Mathematica [A]  time = 0.225207, size = 87, normalized size = 0.58 \[ \frac{12 \sqrt{12 x^2+17 x+6} \left (171228266496 x^7-732816211968 x^6-1190083166208 x^5+3438453030912 x^4+8974844476416 x^3+7899203409792 x^2+3132157281976 x+474999091769\right )-878185 \sqrt{3} \tanh ^{-1}\left (\frac{24 x+17}{4 \sqrt{36 x^2+51 x+18}}\right )}{12683575296} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*Sqrt[6 + 17*x + 12*x^2]*(474999091769 + 3132157281976*x + 7899203409792*x^2 + 8974844476416*x^3 + 34384530
30912*x^4 - 1190083166208*x^5 - 732816211968*x^6 + 171228266496*x^7) - 878185*Sqrt[3]*ArcTanh[(17 + 24*x)/(4*S
qrt[18 + 51*x + 36*x^2])])/12683575296

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Maple [A]  time = 0.057, size = 147, normalized size = 1. \begin{align*}{\frac{129220757\,x}{458752} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}}-{\frac{125455\,\sqrt{12}}{3623878656}\ln \left ({\frac{\sqrt{12}}{12} \left ({\frac{17}{2}}+12\,x \right ) }+\sqrt{12\,{x}^{2}+17\,x+6} \right ) }+{\frac{2132735+3010920\,x}{150994944}\sqrt{12\,{x}^{2}+17\,x+6}}+{\frac{27\,{x}^{5}}{2} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}}-{\frac{8613\,{x}^{4}}{112} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}}+{\frac{14991\,{x}^{3}}{1792} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}}+{\frac{4267751\,{x}^{2}}{14336} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}}+{\frac{2473875847}{33030144} \left ( 12\,{x}^{2}+17\,x+6 \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

129220757/458752*x*(12*x^2+17*x+6)^(3/2)-125455/3623878656*ln(1/12*(17/2+12*x)*12^(1/2)+(12*x^2+17*x+6)^(1/2))
*12^(1/2)+125455/150994944*(17+24*x)*(12*x^2+17*x+6)^(1/2)+27/2*x^5*(12*x^2+17*x+6)^(3/2)-8613/112*x^4*(12*x^2
+17*x+6)^(3/2)+14991/1792*x^3*(12*x^2+17*x+6)^(3/2)+4267751/14336*x^2*(12*x^2+17*x+6)^(3/2)[phone]/3303014
4*(12*x^2+17*x+6)^(3/2)

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Maxima [A]  time = 1.5339, size = 209, normalized size = 1.4 \begin{align*} \frac{27}{2} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} x^{5} - \frac{8613}{112} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} x^{4} + \frac{14991}{1792} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} x^{3} + \frac{4267751}{14336} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} x^{2} + \frac{129220757}{458752} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} x + \frac{2473875847}{33030144} \,{\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac{3}{2}} + \frac{125455}{6291456} \, \sqrt{12 \, x^{2} + 17 \, x + 6} x - \frac{125455}{1811939328} \, \sqrt{3} \log \left (4 \, \sqrt{3} \sqrt{12 \, x^{2} + 17 \, x + 6} + 24 \, x + 17\right ) + \frac{2132735}{150994944} \, \sqrt{12 \, x^{2} + 17 \, x + 6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/2*(12*x^2 + 17*x + 6)^(3/2)*x^5 - 8613/112*(12*x^2 + 17*x + 6)^(3/2)*x^4 + 14991/1792*(12*x^2 + 17*x + 6)^(
3/2)*x^3 + 4267751/14336*(12*x^2 + 17*x + 6)^(3/2)*x^2 + 129220757/458752*(12*x^2 + 17*x + 6)^(3/2)*x + 247387
5847/33030144*(12*x^2 + 17*x + 6)^(3/2) + 125455/6291456*sqrt(12*x^2 + 17*x + 6)*x - 125455/1811939328*sqrt(3)
*log(4*sqrt(3)*sqrt(12*x^2 + 17*x + 6) + 24*x + 17) + 2132735/150994944*sqrt(12*x^2 + 17*x + 6)

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Fricas [A]  time = 1.57645, size = 398, normalized size = 2.67 \begin{align*} \frac{1}{1056964608} \,{\left (171228266496 \, x^{7} - 732816211968 \, x^{6} - 1190083166208 \, x^{5} + 3438453030912 \, x^{4} + 8974844476416 \, x^{3} + 7899203409792 \, x^{2} + 3132157281976 \, x + 474999091769\right )} \sqrt{12 \, x^{2} + 17 \, x + 6} + \frac{125455}{3623878656} \, \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)^2*(-12*x^2+31*x+30)^2*(12*x^2+17*x+6)^(1/2),x, algorithm="fricas")

[Out]

1/1056964608*(171228266496*x^7 - 732816211968*x^6 - 1190083166208*x^5 + 3438453030912*x^4 + 8974844476416*x^3
+ 7899203409792*x^2 + 3132157281976*x + 474999091769)*sqrt(12*x^2 + 17*x + 6) + 125455/3623878656*sqrt(3)*log(
-8*sqrt(3)*sqrt(12*x^2 + 17*x + 6)*(24*x + 17) + 1152*x^2 + 1632*x + 577)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\left (3 x + 2\right ) \left (4 x + 3\right )} \left (3 x - 10\right )^{2} \left (3 x + 2\right )^{2} \left (4 x + 3\right )^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt((3*x + 2)*(4*x + 3))*(3*x - 10)**2*(3*x + 2)**2*(4*x + 3)**2, x)

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Giac [A]  time = 1.16059, size = 115, normalized size = 0.77 \begin{align*} \frac{1}{1056964608} \,{\left (8 \,{\left (48 \,{\left (24 \,{\left (96 \,{\left (24 \,{\left (48 \,{\left (168 \, x - 719\right )} x - 56047\right )} x + 3886417\right )} x + 973832951\right )} x + 20570842213\right )} x + 391519660247\right )} x + 474999091769\right )} \sqrt{12 \, x^{2} + 17 \, x + 6} + \frac{125455}{1811939328} \, \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)^2*(-12*x^2+31*x+30)^2*(12*x^2+17*x+6)^(1/2),x, algorithm="giac")

[Out]

1/1056964608*(8*(48*(24*(96*(24*(48*(168*x - 719)*x - 56047)*x + 3886417)*x + 973832951)*x + 20570842213)*x +
391519660247)*x + 474999091769)*sqrt(12*x^2 + 17*x + 6) + 125455/1811939328*sqrt(3)*log(abs(-4*sqrt(3)*(2*sqrt
(3)*x - sqrt(12*x^2 + 17*x + 6)) - 17))