Integrand size = 34, antiderivative size = 149 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {125455 (17+24 x) \sqrt {6+17 x+12 x^2}}{150994944}-\frac {125455 (17+24 x) \left (6+17 x+12 x^2\right )^{3/2}}{4718592}+\frac {25091 (17+24 x) \left (6+17 x+12 x^2\right )^{5/2}}{24576}-\frac {873 \left (6+17 x+12 x^2\right )^{7/2}}{1792}-\frac {1}{32} (10-3 x) \left (6+17 x+12 x^2\right )^{7/2}-\frac {125455 \text {arctanh}\left (\frac {17+24 x}{4 \sqrt {3} \sqrt {6+17 x+12 x^2}}\right )}{603979776 \sqrt {3}} \]
-125455/4718592*(17+24*x)*(12*x^2+17*x+6)^(3/2)+25091/24576*(17+24*x)*(12* x^2+17*x+6)^(5/2)-873/1792*(12*x^2+17*x+6)^(7/2)-1/32*(10-3*x)*(12*x^2+17* x+6)^(7/2)-125455/1811939328*arctanh(1/12*(17+24*x)*3^(1/2)/(12*x^2+17*x+6 )^(1/2))*3^(1/2)+125455/150994944*(17+24*x)*(12*x^2+17*x+6)^(1/2)
Time = 0.50 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.60 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {6 \sqrt {6+17 x+12 x^2} \left (474999091769+3132157281976 x+7899203409792 x^2+8974844476416 x^3+3438453030912 x^4-1190083166208 x^5-732816211968 x^6+171228266496 x^7\right )-878185 \sqrt {3} \text {arctanh}\left (\frac {2 \sqrt {2+\frac {17 x}{3}+4 x^2}}{3+4 x}\right )}{6341787648} \]
(6*Sqrt[6 + 17*x + 12*x^2]*(474999091769 + 3132157281976*x + 7899203409792 *x^2 + 8974844476416*x^3 + 3438453030912*x^4 - 1190083166208*x^5 - 7328162 11968*x^6 + 171228266496*x^7) - 878185*Sqrt[3]*ArcTanh[(2*Sqrt[2 + (17*x)/ 3 + 4*x^2])/(3 + 4*x)])/6341787648
Time = 0.32 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {1335, 1166, 27, 1160, 1087, 1087, 1087, 1092, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (3 x+2)^2 \left (-12 x^2+31 x+30\right )^2 \sqrt {12 x^2+17 x+6} \, dx\) |
| \(\Big \downarrow \) 1335 |
| \(\displaystyle \int (10-3 x)^2 \left (12 x^2+17 x+6\right )^{5/2}dx\) |
| \(\Big \downarrow \) 1166 |
| \(\displaystyle \frac {1}{96} \int \frac {9}{2} (2518-873 x) \left (12 x^2+17 x+6\right )^{5/2}dx-\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{64} \int (2518-873 x) \left (12 x^2+17 x+6\right )^{5/2}dx-\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle \frac {3}{64} \left (\frac {25091}{8} \int \left (12 x^2+17 x+6\right )^{5/2}dx-\frac {291}{28} \left (12 x^2+17 x+6\right )^{7/2}\right )-\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {3}{64} \left (\frac {25091}{8} \left (\frac {1}{144} (24 x+17) \left (12 x^2+17 x+6\right )^{5/2}-\frac {5}{288} \int \left (12 x^2+17 x+6\right )^{3/2}dx\right )-\frac {291}{28} \left (12 x^2+17 x+6\right )^{7/2}\right )-\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {3}{64} \left (\frac {25091}{8} \left (\frac {1}{144} (24 x+17) \left (12 x^2+17 x+6\right )^{5/2}-\frac {5}{288} \left (\frac {1}{96} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}-\frac {1}{64} \int \sqrt {12 x^2+17 x+6}dx\right )\right )-\frac {291}{28} \left (12 x^2+17 x+6\right )^{7/2}\right )-\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}\) |
| \(\Big \downarrow \) 1087 |
| \(\displaystyle \frac {3}{64} \left (\frac {25091}{8} \left (\frac {1}{144} (24 x+17) \left (12 x^2+17 x+6\right )^{5/2}-\frac {5}{288} \left (\frac {1}{64} \left (\frac {1}{96} \int \frac {1}{\sqrt {12 x^2+17 x+6}}dx-\frac {1}{48} (24 x+17) \sqrt {12 x^2+17 x+6}\right )+\frac {1}{96} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}\right )\right )-\frac {291}{28} \left (12 x^2+17 x+6\right )^{7/2}\right )-\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {3}{64} \left (\frac {25091}{8} \left (\frac {1}{144} (24 x+17) \left (12 x^2+17 x+6\right )^{5/2}-\frac {5}{288} \left (\frac {1}{64} \left (\frac {1}{48} \int \frac {1}{48-\frac {(24 x+17)^2}{12 x^2+17 x+6}}d\frac {24 x+17}{\sqrt {12 x^2+17 x+6}}-\frac {1}{48} (24 x+17) \sqrt {12 x^2+17 x+6}\right )+\frac {1}{96} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}\right )\right )-\frac {291}{28} \left (12 x^2+17 x+6\right )^{7/2}\right )-\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{64} \left (\frac {25091}{8} \left (\frac {1}{144} (24 x+17) \left (12 x^2+17 x+6\right )^{5/2}-\frac {5}{288} \left (\frac {1}{64} \left (\frac {\text {arctanh}\left (\frac {24 x+17}{4 \sqrt {3} \sqrt {12 x^2+17 x+6}}\right )}{192 \sqrt {3}}-\frac {1}{48} (24 x+17) \sqrt {12 x^2+17 x+6}\right )+\frac {1}{96} (24 x+17) \left (12 x^2+17 x+6\right )^{3/2}\right )\right )-\frac {291}{28} \left (12 x^2+17 x+6\right )^{7/2}\right )-\frac {1}{32} (10-3 x) \left (12 x^2+17 x+6\right )^{7/2}\) |
-1/32*((10 - 3*x)*(6 + 17*x + 12*x^2)^(7/2)) + (3*((-291*(6 + 17*x + 12*x^ 2)^(7/2))/28 + (25091*(((17 + 24*x)*(6 + 17*x + 12*x^2)^(5/2))/144 - (5*(( (17 + 24*x)*(6 + 17*x + 12*x^2)^(3/2))/96 + (-1/48*((17 + 24*x)*Sqrt[6 + 1 7*x + 12*x^2]) + ArcTanh[(17 + 24*x)/(4*Sqrt[3]*Sqrt[6 + 17*x + 12*x^2])]/ (192*Sqrt[3]))/64))/288))/8))/64
3.2.33.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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] - Simp[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] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[3*p])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
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] + Simp[(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[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[e*(d + e*x)^(m - 1)*((a + b*x + c*x^2)^(p + 1)/(c*(m + 2*p + 1))), x] + Simp[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] && If[Ration alQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadrat icQ[a, b, c, d, e, m, p, x]
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]
Time = 0.66 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.54
| method | result | size |
| risch | \(\frac {\left (171228266496 x^{7}-732816211968 x^{6}-1190083166208 x^{5}+3438453030912 x^{4}+8974844476416 x^{3}+7899203409792 x^{2}+3132157281976 x +474999091769\right ) \sqrt {12 x^{2}+17 x +6}}{1056964608}-\frac {125455 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right ) \sqrt {12}}{3623878656}\) | \(80\) |
| trager | \(\left (162 x^{7}-\frac {19413}{28} x^{6}-\frac {504423}{448} x^{5}+\frac {11659251}{3584} x^{4}+\frac {139118993}{16384} x^{3}+\frac {20570842213}{2752512} x^{2}+\frac {391519660247}{132120576} x +\frac {474999091769}{1056964608}\right ) \sqrt {12 x^{2}+17 x +6}+\frac {125455 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-24 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x -17 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+12 \sqrt {12 x^{2}+17 x +6}\right )}{1811939328}\) | \(91\) |
| default | \(\frac {125455 \left (17+24 x \right ) \sqrt {12 x^{2}+17 x +6}}{150994944}-\frac {125455 \ln \left (\frac {\left (\frac {17}{2}+12 x \right ) \sqrt {12}}{12}+\sqrt {12 x^{2}+17 x +6}\right ) \sqrt {12}}{3623878656}+\frac {2473875847 \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{33030144}+\frac {27 x^{5} \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{2}-\frac {8613 x^{4} \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{112}+\frac {14991 x^{3} \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{1792}+\frac {4267751 x^{2} \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{14336}+\frac {129220757 x \left (12 x^{2}+17 x +6\right )^{\frac {3}{2}}}{458752}\) | \(147\) |
1/1056964608*(171228266496*x^7-732816211968*x^6-1190083166208*x^5+34384530 30912*x^4+8974844476416*x^3+7899203409792*x^2+3132157281976*x+474999091769 )*(12*x^2+17*x+6)^(1/2)-125455/3623878656*ln(1/12*(17/2+12*x)*12^(1/2)+(12 *x^2+17*x+6)^(1/2))*12^(1/2)
Time = 0.28 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.59 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {1}{1056964608} \, {\left (171228266496 \, x^{7} - 732816211968 \, x^{6} - 1190083166208 \, x^{5} + 3438453030912 \, x^{4} + 8974844476416 \, x^{3} + 7899203409792 \, x^{2} + 3132157281976 \, x + 474999091769\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} + \frac {125455}{3623878656} \, \sqrt {3} \log \left (-8 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} {\left (24 \, x + 17\right )} + 1152 \, x^{2} + 1632 \, x + 577\right ) \]
1/1056964608*(171228266496*x^7 - 732816211968*x^6 - 1190083166208*x^5 + 34 38453030912*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)
Time = 0.53 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.64 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\sqrt {12 x^{2} + 17 x + 6} \cdot \left (162 x^{7} - \frac {19413 x^{6}}{28} - \frac {504423 x^{5}}{448} + \frac {11659251 x^{4}}{3584} + \frac {139118993 x^{3}}{16384} + \frac {20570842213 x^{2}}{2752512} + \frac {391519660247 x}{132120576} + \frac {474999091769}{1056964608}\right ) - \frac {125455 \sqrt {3} \log {\left (24 x + 4 \sqrt {3} \sqrt {12 x^{2} + 17 x + 6} + 17 \right )}}{1811939328} \]
sqrt(12*x**2 + 17*x + 6)*(162*x**7 - 19413*x**6/28 - 504423*x**5/448 + 116 59251*x**4/3584 + 139118993*x**3/16384 + 20570842213*x**2/2752512 + 391519 660247*x/132120576 + 474999091769/1056964608) - 125455*sqrt(3)*log(24*x + 4*sqrt(3)*sqrt(12*x**2 + 17*x + 6) + 17)/1811939328
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.04 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {27}{2} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{5} - \frac {8613}{112} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{4} + \frac {14991}{1792} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{3} + \frac {4267751}{14336} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x^{2} + \frac {129220757}{458752} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} x + \frac {2473875847}{33030144} \, {\left (12 \, x^{2} + 17 \, x + 6\right )}^{\frac {3}{2}} + \frac {125455}{6291456} \, \sqrt {12 \, x^{2} + 17 \, x + 6} x - \frac {125455}{1811939328} \, \sqrt {3} \log \left (4 \, \sqrt {3} \sqrt {12 \, x^{2} + 17 \, x + 6} + 24 \, x + 17\right ) + \frac {2132735}{150994944} \, \sqrt {12 \, x^{2} + 17 \, x + 6} \]
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 + 24738758 47/33030144*(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)
Time = 0.30 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.57 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {1}{1056964608} \, {\left (8 \, {\left (48 \, {\left (24 \, {\left (96 \, {\left (24 \, {\left (48 \, {\left (168 \, x - 719\right )} x - 56047\right )} x + 3886417\right )} x + 973832951\right )} x + 20570842213\right )} x + 391519660247\right )} x + 474999091769\right )} \sqrt {12 \, x^{2} + 17 \, x + 6} + \frac {125455}{1811939328} \, \sqrt {3} \log \left ({\left | -4 \, \sqrt {3} {\left (2 \, \sqrt {3} x - \sqrt {12 \, x^{2} + 17 \, x + 6}\right )} - 17 \right |}\right ) \]
1/1056964608*(8*(48*(24*(96*(24*(48*(168*x - 719)*x - 56047)*x + 3886417)* x + 973832951)*x + 20570842213)*x + 391519660247)*x + 474999091769)*sqrt(1 2*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))
Time = 13.92 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.26 \[ \int (2+3 x)^2 \left (30+31 x-12 x^2\right )^2 \sqrt {6+17 x+12 x^2} \, dx=\frac {4267751\,x^2\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{14336}+\frac {14991\,x^3\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{1792}-\frac {8613\,x^4\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{112}+\frac {27\,x^5\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{2}-\frac {146030443\,\sqrt {12}\,\ln \left (\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (12\,x+\frac {17}{2}\right )}{12}\right )}{88080384}+\frac {438091329\,\left (\frac {x}{2}+\frac {17}{48}\right )\,\sqrt {12\,x^2+17\,x+6}}{229376}+\frac {2473875847\,\sqrt {12\,x^2+17\,x+6}\,\left (1152\,x^2+408\,x-291\right )}{3170893824}+\frac {129220757\,x\,{\left (12\,x^2+17\,x+6\right )}^{3/2}}{458752}+\frac {42055889399\,\sqrt {12}\,\ln \left (2\,\sqrt {12\,x^2+17\,x+6}+\frac {\sqrt {12}\,\left (24\,x+17\right )}{12}\right )}{25367150592} \]
(4267751*x^2*(17*x + 12*x^2 + 6)^(3/2))/14336 + (14991*x^3*(17*x + 12*x^2 + 6)^(3/2))/1792 - (8613*x^4*(17*x + 12*x^2 + 6)^(3/2))/112 + (27*x^5*(17* x + 12*x^2 + 6)^(3/2))/2 - (146030443*12^(1/2)*log((17*x + 12*x^2 + 6)^(1/ 2) + (12^(1/2)*(12*x + 17/2))/12))/88080384 + (438091329*(x/2 + 17/48)*(17 *x + 12*x^2 + 6)^(1/2))/229376 + (2473875847*(17*x + 12*x^2 + 6)^(1/2)*(40 8*x + 1152*x^2 - 291))/3170893824 + (129220757*x*(17*x + 12*x^2 + 6)^(3/2) )/458752 + (42055889399*12^(1/2)*log(2*(17*x + 12*x^2 + 6)^(1/2) + (12^(1/ 2)*(24*x + 17))/12))/25367150592