Integrand size = 26, antiderivative size = 56 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right )^2 \, dx=\frac {1}{21} (2+3 x)^7+\frac {1}{21} (2+3 x)^{14}+\frac {1}{21} (2+3 x)^{21}+\frac {1}{42} (2+3 x)^{28}+\frac {1}{105} (2+3 x)^{35} \]
Leaf count is larger than twice the leaf count of optimal. \(188\) vs. \(2(56)=112\).
Time = 0.01 (sec) , antiderivative size = 188, normalized size of antiderivative = 3.36 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right )^2 \, dx=17451466816 x+443569828128 x^2+7299544818384 x^3+87406679578680 x^4+\frac {4057390785756924 x^5}{5}+6077684727888102 x^6+37727143432895007 x^7+197897276851452864 x^8+889942562270387136 x^9+\frac {17344958593049772048 x^{10}}{5}+11821487501620716192 x^{11}+35454069480572048124 x^{12}+94069263918929616324 x^{13}+221699757548270194389 x^{14}+465517091041681015296 x^{15}+872775774067455498528 x^{16}+1463104032160519033200 x^{17}+2194577166014752240080 x^{18}+2945285062308448290360 x^{19}+3534290697929473864098 x^{20}+\frac {26506949038858918036881 x^{21}}{7}+3614565944605222108800 x^{22}+3064515076512846852480 x^{23}+2298383223254096766840 x^{24}+\frac {7584660010542711771792 x^{25}}{5}+875152864622814086340 x^{26}+437576396725285446564 x^{27}+\frac {2625458326972530284475 x^{28}}{14}+67899784121041365504 x^{29}+\frac {101849676181562048256 x^{30}}{5}+4928210137817518464 x^{31}+924039400840784712 x^{32}+126005372841925188 x^{33}+11118121133111046 x^{34}+\frac {16677181699666569 x^{35}}{35} \]
17451466816*x + 443569828128*x^2 + 7299544818384*x^3 + 87406679578680*x^4 + (4057390785756924*x^5)/5 + 6077684727888102*x^6 + 37727143432895007*x^7 + 197897276851452864*x^8 + 889942562270387136*x^9 + (17344958593049772048* x^10)/5 + 11821487501620716192*x^11 + 35454069480572048124*x^12 + 94069263 918929616324*x^13 + 221699757548270194389*x^14 + 465517091041681015296*x^1 5 + 872775774067455498528*x^16 + 1463104032160519033200*x^17 + 21945771660 14752240080*x^18 + 2945285062308448290360*x^19 + 3534290697929473864098*x^ 20 + (26506949038858918036881*x^21)/7 + 3614565944605222108800*x^22 + 3064 515076512846852480*x^23 + 2298383223254096766840*x^24 + (75846600105427117 71792*x^25)/5 + 875152864622814086340*x^26 + 437576396725285446564*x^27 + (2625458326972530284475*x^28)/14 + 67899784121041365504*x^29 + (1018496761 81562048256*x^30)/5 + 4928210137817518464*x^31 + 924039400840784712*x^32 + 126005372841925188*x^33 + 11118121133111046*x^34 + (16677181699666569*x^3 5)/35
Time = 0.22 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.86, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {1725, 1690, 1085, 2009}
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.
| \(\displaystyle \int (3 x+2)^6 \left ((3 x+2)^{14}+(3 x+2)^7+1\right )^2 \, dx\) |
| \(\Big \downarrow \) 1725 |
| \(\displaystyle \frac {1}{3} \int (3 x+2)^6 \left ((3 x+2)^{14}+(3 x+2)^7+1\right )^2d(3 x+2)\) |
| \(\Big \downarrow \) 1690 |
| \(\displaystyle \frac {1}{21} \int \left ((3 x+2)^{14}+3 x+3\right )^2d(3 x+2)^7\) |
| \(\Big \downarrow \) 1085 |
| \(\displaystyle \frac {1}{21} \int \left ((3 x+2)^{28}+2 (3 x+2)^{21}+3 (3 x+2)^{14}+2 (3 x+2)^7+1\right )d(3 x+2)^7\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{21} \left ((3 x+2)^7+\frac {1}{5} (3 x+2)^5+\frac {1}{2} (3 x+2)^4+(3 x+2)^3+(3 x+2)^2\right )\) |
3.7.64.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegr and[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && IntegerQ[p] && (G tQ[p, 0] || EqQ[a, 0])
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol ] :> Simp[1/n Subst[Int[(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
Int[(u_)^(m_.)*((a_.) + (c_.)*(v_)^(n2_.) + (b_.)*(v_)^(n_))^(p_.), x_Symbo l] :> Simp[u^m/(Coefficient[v, x, 1]*v^m) Subst[Int[x^m*(a + b*x^n + c*x^ (2*n))^p, x], x, v], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && LinearPairQ[u, v, x]
Leaf count of result is larger than twice the leaf count of optimal. \(173\) vs. \(2(46)=92\).
Time = 0.59 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.11
| method | result | size |
| gosper | \(\frac {x \left (33354363399333138 x^{34}+778268479317773220 x^{33}+8820376098934763160 x^{32}+64682758058854929840 x^{31}+344974709647226292480 x^{30}+1425895466541868675584 x^{29}+4752984888472895585280 x^{28}+13127291634862651422375 x^{27}+30630347770769981259480 x^{26}+61260700523596986043800 x^{25}+106185240147597964805088 x^{24}+160886825627786773678800 x^{23}+214516055355899279673600 x^{22}+253019616122365547616000 x^{21}+265069490388589180368810 x^{20}+247400348855063170486860 x^{19}+206169954361591380325200 x^{18}+153620401621032656805600 x^{17}+102417282251236332324000 x^{16}+61094304184721884896960 x^{15}+32586196372917671070720 x^{14}+15518983028378913607230 x^{13}+6584848474325073142680 x^{12}+2481784863640043368680 x^{11}+827504125113450133440 x^{10}+242829420302696808672 x^{9}+62295979358927099520 x^{8}+13852809379601700480 x^{7}+2640900040302650490 x^{6}+425437930952167140 x^{5}+56803471000596936 x^{4}+6118467570507600 x^{3}+510968137286880 x^{2}+31049887968960 x +1221602677120\right )}{70}\) | \(174\) |
| default | \(\frac {17344958593049772048}{5} x^{10}+11821487501620716192 x^{11}+35454069480572048124 x^{12}+94069263918929616324 x^{13}+17451466816 x +7299544818384 x^{3}+\frac {4057390785756924}{5} x^{5}+889942562270387136 x^{9}+37727143432895007 x^{7}+87406679578680 x^{4}+221699757548270194389 x^{14}+197897276851452864 x^{8}+6077684727888102 x^{6}+443569828128 x^{2}+437576396725285446564 x^{27}+\frac {7584660010542711771792}{5} x^{25}+3534290697929473864098 x^{20}+2945285062308448290360 x^{19}+2194577166014752240080 x^{18}+1463104032160519033200 x^{17}+872775774067455498528 x^{16}+465517091041681015296 x^{15}+3064515076512846852480 x^{23}+3614565944605222108800 x^{22}+\frac {26506949038858918036881}{7} x^{21}+2298383223254096766840 x^{24}+875152864622814086340 x^{26}+\frac {2625458326972530284475}{14} x^{28}+67899784121041365504 x^{29}+\frac {101849676181562048256}{5} x^{30}+4928210137817518464 x^{31}+924039400840784712 x^{32}+126005372841925188 x^{33}+11118121133111046 x^{34}+\frac {16677181699666569}{35} x^{35}\) | \(175\) |
| norman | \(\frac {17344958593049772048}{5} x^{10}+11821487501620716192 x^{11}+35454069480572048124 x^{12}+94069263918929616324 x^{13}+17451466816 x +7299544818384 x^{3}+\frac {4057390785756924}{5} x^{5}+889942562270387136 x^{9}+37727143432895007 x^{7}+87406679578680 x^{4}+221699757548270194389 x^{14}+197897276851452864 x^{8}+6077684727888102 x^{6}+443569828128 x^{2}+437576396725285446564 x^{27}+\frac {7584660010542711771792}{5} x^{25}+3534290697929473864098 x^{20}+2945285062308448290360 x^{19}+2194577166014752240080 x^{18}+1463104032160519033200 x^{17}+872775774067455498528 x^{16}+465517091041681015296 x^{15}+3064515076512846852480 x^{23}+3614565944605222108800 x^{22}+\frac {26506949038858918036881}{7} x^{21}+2298383223254096766840 x^{24}+875152864622814086340 x^{26}+\frac {2625458326972530284475}{14} x^{28}+67899784121041365504 x^{29}+\frac {101849676181562048256}{5} x^{30}+4928210137817518464 x^{31}+924039400840784712 x^{32}+126005372841925188 x^{33}+11118121133111046 x^{34}+\frac {16677181699666569}{35} x^{35}\) | \(175\) |
| risch | \(\frac {17344958593049772048}{5} x^{10}+11821487501620716192 x^{11}+35454069480572048124 x^{12}+94069263918929616324 x^{13}+17451466816 x +7299544818384 x^{3}+\frac {4057390785756924}{5} x^{5}+889942562270387136 x^{9}+37727143432895007 x^{7}+87406679578680 x^{4}+221699757548270194389 x^{14}+197897276851452864 x^{8}+6077684727888102 x^{6}+443569828128 x^{2}+437576396725285446564 x^{27}+\frac {7584660010542711771792}{5} x^{25}+3534290697929473864098 x^{20}+2945285062308448290360 x^{19}+2194577166014752240080 x^{18}+1463104032160519033200 x^{17}+872775774067455498528 x^{16}+465517091041681015296 x^{15}+3064515076512846852480 x^{23}+3614565944605222108800 x^{22}+\frac {26506949038858918036881}{7} x^{21}+2298383223254096766840 x^{24}+875152864622814086340 x^{26}+\frac {2625458326972530284475}{14} x^{28}+67899784121041365504 x^{29}+\frac {101849676181562048256}{5} x^{30}+4928210137817518464 x^{31}+924039400840784712 x^{32}+126005372841925188 x^{33}+11118121133111046 x^{34}+\frac {16677181699666569}{35} x^{35}\) | \(175\) |
| parallelrisch | \(\frac {17344958593049772048}{5} x^{10}+11821487501620716192 x^{11}+35454069480572048124 x^{12}+94069263918929616324 x^{13}+17451466816 x +7299544818384 x^{3}+\frac {4057390785756924}{5} x^{5}+889942562270387136 x^{9}+37727143432895007 x^{7}+87406679578680 x^{4}+221699757548270194389 x^{14}+197897276851452864 x^{8}+6077684727888102 x^{6}+443569828128 x^{2}+437576396725285446564 x^{27}+\frac {7584660010542711771792}{5} x^{25}+3534290697929473864098 x^{20}+2945285062308448290360 x^{19}+2194577166014752240080 x^{18}+1463104032160519033200 x^{17}+872775774067455498528 x^{16}+465517091041681015296 x^{15}+3064515076512846852480 x^{23}+3614565944605222108800 x^{22}+\frac {26506949038858918036881}{7} x^{21}+2298383223254096766840 x^{24}+875152864622814086340 x^{26}+\frac {2625458326972530284475}{14} x^{28}+67899784121041365504 x^{29}+\frac {101849676181562048256}{5} x^{30}+4928210137817518464 x^{31}+924039400840784712 x^{32}+126005372841925188 x^{33}+11118121133111046 x^{34}+\frac {16677181699666569}{35} x^{35}\) | \(175\) |
1/70*x*(33354363399333138*x^34+778268479317773220*x^33+8820376098934763160 *x^32+64682758058854929840*x^31+344974709647226292480*x^30+142589546654186 8675584*x^29+4752984888472895585280*x^28+13127291634862651422375*x^27+3063 0347770769981259480*x^26+61260700523596986043800*x^25+10618524014759796480 5088*x^24+160886825627786773678800*x^23+214516055355899279673600*x^22+2530 19616122365547616000*x^21+265069490388589180368810*x^20+247400348855063170 486860*x^19+206169954361591380325200*x^18+153620401621032656805600*x^17+10 2417282251236332324000*x^16+61094304184721884896960*x^15+32586196372917671 070720*x^14+15518983028378913607230*x^13+6584848474325073142680*x^12+24817 84863640043368680*x^11+827504125113450133440*x^10+242829420302696808672*x^ 9+62295979358927099520*x^8+13852809379601700480*x^7+2640900040302650490*x^ 6+425437930952167140*x^5+56803471000596936*x^4+6118467570507600*x^3+510968 137286880*x^2+31049887968960*x+1221602677120)
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (46) = 92\).
Time = 0.24 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.11 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right )^2 \, dx=\frac {16677181699666569}{35} \, x^{35} + 11118121133111046 \, x^{34} + 126005372841925188 \, x^{33} + 924039400840784712 \, x^{32} + 4928210137817518464 \, x^{31} + \frac {101849676181562048256}{5} \, x^{30} + 67899784121041365504 \, x^{29} + \frac {2625458326972530284475}{14} \, x^{28} + 437576396725285446564 \, x^{27} + 875152864622814086340 \, x^{26} + \frac {7584660010542711771792}{5} \, x^{25} + 2298383223254096766840 \, x^{24} + 3064515076512846852480 \, x^{23} + 3614565944605222108800 \, x^{22} + \frac {26506949038858918036881}{7} \, x^{21} + 3534290697929473864098 \, x^{20} + 2945285062308448290360 \, x^{19} + 2194577166014752240080 \, x^{18} + 1463104032160519033200 \, x^{17} + 872775774067455498528 \, x^{16} + 465517091041681015296 \, x^{15} + 221699757548270194389 \, x^{14} + 94069263918929616324 \, x^{13} + 35454069480572048124 \, x^{12} + 11821487501620716192 \, x^{11} + \frac {17344958593049772048}{5} \, x^{10} + 889942562270387136 \, x^{9} + 197897276851452864 \, x^{8} + 37727143432895007 \, x^{7} + 6077684727888102 \, x^{6} + \frac {4057390785756924}{5} \, x^{5} + 87406679578680 \, x^{4} + 7299544818384 \, x^{3} + 443569828128 \, x^{2} + 17451466816 \, x \]
16677181699666569/35*x^35 + 11118121133111046*x^34 + 126005372841925188*x^ 33 + 924039400840784712*x^32 + 4928210137817518464*x^31 + 1018496761815620 48256/5*x^30 + 67899784121041365504*x^29 + 2625458326972530284475/14*x^28 + 437576396725285446564*x^27 + 875152864622814086340*x^26 + 75846600105427 11771792/5*x^25 + 2298383223254096766840*x^24 + 3064515076512846852480*x^2 3 + 3614565944605222108800*x^22 + 26506949038858918036881/7*x^21 + 3534290 697929473864098*x^20 + 2945285062308448290360*x^19 + 219457716601475224008 0*x^18 + 1463104032160519033200*x^17 + 872775774067455498528*x^16 + 465517 091041681015296*x^15 + 221699757548270194389*x^14 + 94069263918929616324*x ^13 + 35454069480572048124*x^12 + 11821487501620716192*x^11 + 173449585930 49772048/5*x^10 + 889942562270387136*x^9 + 197897276851452864*x^8 + 377271 43432895007*x^7 + 6077684727888102*x^6 + 4057390785756924/5*x^5 + 87406679 578680*x^4 + 7299544818384*x^3 + 443569828128*x^2 + 17451466816*x
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (41) = 82\).
Time = 0.07 (sec) , antiderivative size = 187, normalized size of antiderivative = 3.34 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right )^2 \, dx=\frac {16677181699666569 x^{35}}{35} + 11118121133111046 x^{34} + 126005372841925188 x^{33} + 924039400840784712 x^{32} + 4928210137817518464 x^{31} + \frac {101849676181562048256 x^{30}}{5} + 67899784121041365504 x^{29} + \frac {2625458326972530284475 x^{28}}{14} + 437576396725285446564 x^{27} + 875152864622814086340 x^{26} + \frac {7584660010542711771792 x^{25}}{5} + 2298383223254096766840 x^{24} + 3064515076512846852480 x^{23} + 3614565944605222108800 x^{22} + \frac {26506949038858918036881 x^{21}}{7} + 3534290697929473864098 x^{20} + 2945285062308448290360 x^{19} + 2194577166014752240080 x^{18} + 1463104032160519033200 x^{17} + 872775774067455498528 x^{16} + 465517091041681015296 x^{15} + 221699757548270194389 x^{14} + 94069263918929616324 x^{13} + 35454069480572048124 x^{12} + 11821487501620716192 x^{11} + \frac {17344958593049772048 x^{10}}{5} + 889942562270387136 x^{9} + 197897276851452864 x^{8} + 37727143432895007 x^{7} + 6077684727888102 x^{6} + \frac {4057390785756924 x^{5}}{5} + 87406679578680 x^{4} + 7299544818384 x^{3} + 443569828128 x^{2} + 17451466816 x \]
16677181699666569*x**35/35 + 11118121133111046*x**34 + 126005372841925188* x**33 + 924039400840784712*x**32 + 4928210137817518464*x**31 + 10184967618 1562048256*x**30/5 + 67899784121041365504*x**29 + 2625458326972530284475*x **28/14 + 437576396725285446564*x**27 + 875152864622814086340*x**26 + 7584 660010542711771792*x**25/5 + 2298383223254096766840*x**24 + 30645150765128 46852480*x**23 + 3614565944605222108800*x**22 + 26506949038858918036881*x* *21/7 + 3534290697929473864098*x**20 + 2945285062308448290360*x**19 + 2194 577166014752240080*x**18 + 1463104032160519033200*x**17 + 8727757740674554 98528*x**16 + 465517091041681015296*x**15 + 221699757548270194389*x**14 + 94069263918929616324*x**13 + 35454069480572048124*x**12 + 1182148750162071 6192*x**11 + 17344958593049772048*x**10/5 + 889942562270387136*x**9 + 1978 97276851452864*x**8 + 37727143432895007*x**7 + 6077684727888102*x**6 + 405 7390785756924*x**5/5 + 87406679578680*x**4 + 7299544818384*x**3 + 44356982 8128*x**2 + 17451466816*x
Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (46) = 92\).
Time = 0.23 (sec) , antiderivative size = 174, normalized size of antiderivative = 3.11 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right )^2 \, dx=\frac {16677181699666569}{35} \, x^{35} + 11118121133111046 \, x^{34} + 126005372841925188 \, x^{33} + 924039400840784712 \, x^{32} + 4928210137817518464 \, x^{31} + \frac {101849676181562048256}{5} \, x^{30} + 67899784121041365504 \, x^{29} + \frac {2625458326972530284475}{14} \, x^{28} + 437576396725285446564 \, x^{27} + 875152864622814086340 \, x^{26} + \frac {7584660010542711771792}{5} \, x^{25} + 2298383223254096766840 \, x^{24} + 3064515076512846852480 \, x^{23} + 3614565944605222108800 \, x^{22} + \frac {26506949038858918036881}{7} \, x^{21} + 3534290697929473864098 \, x^{20} + 2945285062308448290360 \, x^{19} + 2194577166014752240080 \, x^{18} + 1463104032160519033200 \, x^{17} + 872775774067455498528 \, x^{16} + 465517091041681015296 \, x^{15} + 221699757548270194389 \, x^{14} + 94069263918929616324 \, x^{13} + 35454069480572048124 \, x^{12} + 11821487501620716192 \, x^{11} + \frac {17344958593049772048}{5} \, x^{10} + 889942562270387136 \, x^{9} + 197897276851452864 \, x^{8} + 37727143432895007 \, x^{7} + 6077684727888102 \, x^{6} + \frac {4057390785756924}{5} \, x^{5} + 87406679578680 \, x^{4} + 7299544818384 \, x^{3} + 443569828128 \, x^{2} + 17451466816 \, x \]
16677181699666569/35*x^35 + 11118121133111046*x^34 + 126005372841925188*x^ 33 + 924039400840784712*x^32 + 4928210137817518464*x^31 + 1018496761815620 48256/5*x^30 + 67899784121041365504*x^29 + 2625458326972530284475/14*x^28 + 437576396725285446564*x^27 + 875152864622814086340*x^26 + 75846600105427 11771792/5*x^25 + 2298383223254096766840*x^24 + 3064515076512846852480*x^2 3 + 3614565944605222108800*x^22 + 26506949038858918036881/7*x^21 + 3534290 697929473864098*x^20 + 2945285062308448290360*x^19 + 219457716601475224008 0*x^18 + 1463104032160519033200*x^17 + 872775774067455498528*x^16 + 465517 091041681015296*x^15 + 221699757548270194389*x^14 + 94069263918929616324*x ^13 + 35454069480572048124*x^12 + 11821487501620716192*x^11 + 173449585930 49772048/5*x^10 + 889942562270387136*x^9 + 197897276851452864*x^8 + 377271 43432895007*x^7 + 6077684727888102*x^6 + 4057390785756924/5*x^5 + 87406679 578680*x^4 + 7299544818384*x^3 + 443569828128*x^2 + 17451466816*x
Time = 0.33 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right )^2 \, dx=\frac {1}{105} \, {\left (3 \, x + 2\right )}^{35} + \frac {1}{42} \, {\left (3 \, x + 2\right )}^{28} + \frac {1}{21} \, {\left (3 \, x + 2\right )}^{21} + \frac {1}{21} \, {\left (3 \, x + 2\right )}^{14} + \frac {1}{21} \, {\left (3 \, x + 2\right )}^{7} \]
1/105*(3*x + 2)^35 + 1/42*(3*x + 2)^28 + 1/21*(3*x + 2)^21 + 1/21*(3*x + 2 )^14 + 1/21*(3*x + 2)^7
Time = 0.08 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.82 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right )^2 \, dx=\frac {{\left (3\,x+2\right )}^7}{21}+\frac {{\left (3\,x+2\right )}^{14}}{21}+\frac {{\left (3\,x+2\right )}^{21}}{21}+\frac {{\left (3\,x+2\right )}^{28}}{42}+\frac {{\left (3\,x+2\right )}^{35}}{105} \]