3.7.64 \(\int (2+3 x)^6 (1+(2+3 x)^7+(2+3 x)^{14})^2 \, dx\) [664]

Optimal. Leaf size=56 \[ \frac {1}{21} (2+3 x)^7+\frac {1}{21} (2+3 x)^{14}+\frac {1}{21} (2+3 x)^{21}+\frac {1}{42} (2+3 x)^{28}+\frac {1}{105} (2+3 x)^{35} \]

[Out]

1/21*(2+3*x)^7+1/21*(2+3*x)^14+1/21*(2+3*x)^21+1/42*(2+3*x)^28+1/105*(2+3*x)^35

________________________________________________________________________________________

Rubi [A]
time = 0.07, antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {1404, 1366, 625} \begin {gather*} \frac {1}{105} (3 x+2)^{35}+\frac {1}{42} (3 x+2)^{28}+\frac {1}{21} (3 x+2)^{21}+\frac {1}{21} (3 x+2)^{14}+\frac {1}{21} (3 x+2)^7 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^6*(1 + (2 + 3*x)^7 + (2 + 3*x)^14)^2,x]

[Out]

(2 + 3*x)^7/21 + (2 + 3*x)^14/21 + (2 + 3*x)^21/21 + (2 + 3*x)^28/42 + (2 + 3*x)^35/105

Rule 625

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 0] && (EqQ[a, 0] ||  !PerfectSquareQ[b^2 - 4*a*c])

Rule 1366

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[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]

Rule 1404

Int[(u_)^(m_.)*((a_.) + (c_.)*(v_)^(n2_.) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[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]

Rubi steps

\begin {align*} \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right )^2 \, dx &=\frac {1}{3} \text {Subst}\left (\int x^6 \left (1+x^7+x^{14}\right )^2 \, dx,x,2+3 x\right )\\ &=\frac {1}{21} \text {Subst}\left (\int \left (1+x+x^2\right )^2 \, dx,x,(2+3 x)^7\right )\\ &=\frac {1}{21} \text {Subst}\left (\int \left (1+2 x+3 x^2+2 x^3+x^4\right ) \, dx,x,(2+3 x)^7\right )\\ &=\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}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(188\) vs. \(2(56)=112\).
time = 0.01, size = 188, normalized size = 3.36 \begin {gather*} 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} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^6*(1 + (2 + 3*x)^7 + (2 + 3*x)^14)^2,x]

[Out]

17451466816*x + 443569828128*x^2 + 7299544818384*x^3 + 87406679578680*x^4 + (4057390785756924*x^5)/5 + 6077684
727888102*x^6 + 37727143432895007*x^7 + 197897276851452864*x^8 + 889942562270387136*x^9 + (1734495859304977204
8*x^10)/5 + 11821487501620716192*x^11 + 35454069480572048124*x^12 + 94069263918929616324*x^13 + 22169975754827
0194389*x^14 + 465517091041681015296*x^15 + 872775774067455498528*x^16 + 1463104032160519033200*x^17 + 2194577
166014752240080*x^18 + 2945285062308448290360*x^19 + 3534290697929473864098*x^20 + (26506949038858918036881*x^
21)/7 + 3614565944605222108800*x^22 + 3064515076512846852480*x^23 + 2298383223254096766840*x^24 + (75846600105
42711771792*x^25)/5 + 875152864622814086340*x^26 + 437576396725285446564*x^27 + (2625458326972530284475*x^28)/
14 + 67899784121041365504*x^29 + (101849676181562048256*x^30)/5 + 4928210137817518464*x^31 + 92403940084078471
2*x^32 + 126005372841925188*x^33 + 11118121133111046*x^34 + (16677181699666569*x^35)/35

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(174\) vs. \(2(46)=92\).
time = 0.24, size = 175, normalized size = 3.12

method	result	size
gosper	\(\frac {x \left (33354363399333138 x^{34}+778268479317773220 x^{33}+8820376098934763160 x^{32}+64682758058854929840 x^{31}+344974709647226292480 x^{30}+1425895466541868675584 x^{29}+4752984888472895585280 x^{28}+13127291634862651422375 x^{27}+30630347770769981259480 x^{26}+61260700523596986043800 x^{25}+106185240147597964805088 x^{24}+160886825627786773678800 x^{23}+214516055355899279673600 x^{22}+253019616122365547616000 x^{21}+265069490388589180368810 x^{20}+247400348855063170486860 x^{19}+206169954361591380325200 x^{18}+153620401621032656805600 x^{17}+102417282251236332324000 x^{16}+61094304184721884896960 x^{15}+32586196372917671070720 x^{14}+15518983028378913607230 x^{13}+6584848474325073142680 x^{12}+2481784863640043368680 x^{11}+827504125113450133440 x^{10}+242829420302696808672 x^{9}+62295979358927099520 x^{8}+13852809379601700480 x^{7}+2640900040302650490 x^{6}+425437930952167140 x^{5}+56803471000596936 x^{4}+6118467570507600 x^{3}[phone] x^{2}[phone] x [phone]\right )}{70}\)	\(174\)
default	\(17451466816 x +924039400840784712 x^{32}+4928210137817518464 x^{31}+\frac {101849676181562048256}{5} x^{30}+\frac {16677181699666569}{35} x^{35}+11118121133111046 x^{34}+126005372841925188 x^{33}+3064515076512846852480 x^{23}+3614565944605222108800 x^{22}+875152864622814086340 x^{26}+\frac {7584660010542711771792}{5} x^{25}+2298383223254096766840 x^{24}+437576396725285446564 x^{27}+197897276851452864 x^{8}+889942562270387136 x^{9}+37727143432895007 x^{7}+6077684727888102 x^{6}[phone] x^{4}[phone] x^{2}[phone] x^{3}+\frac {4057390785756924}{5} x^{5}+\frac {17344958593049772048}{5} x^{10}+35454069480572048124 x^{12}+94069263918929616324 x^{13}+221699757548270194389 x^{14}+465517091041681015296 x^{15}+3534290697929473864098 x^{20}+2945285062308448290360 x^{19}+2194577166014752240080 x^{18}+1463104032160519033200 x^{17}+872775774067455498528 x^{16}+\frac {26506949038858918036881}{7} x^{21}+67899784121041365504 x^{29}+\frac {2625458326972530284475}{14} x^{28}+11821487501620716192 x^{11}\)	\(175\)
norman	\(17451466816 x +924039400840784712 x^{32}+4928210137817518464 x^{31}+\frac {101849676181562048256}{5} x^{30}+\frac {16677181699666569}{35} x^{35}+11118121133111046 x^{34}+126005372841925188 x^{33}+3064515076512846852480 x^{23}+3614565944605222108800 x^{22}+875152864622814086340 x^{26}+\frac {7584660010542711771792}{5} x^{25}+2298383223254096766840 x^{24}+437576396725285446564 x^{27}+197897276851452864 x^{8}+889942562270387136 x^{9}+37727143432895007 x^{7}+6077684727888102 x^{6}[phone] x^{4}[phone] x^{2}[phone] x^{3}+\frac {4057390785756924}{5} x^{5}+\frac {17344958593049772048}{5} x^{10}+35454069480572048124 x^{12}+94069263918929616324 x^{13}+221699757548270194389 x^{14}+465517091041681015296 x^{15}+3534290697929473864098 x^{20}+2945285062308448290360 x^{19}+2194577166014752240080 x^{18}+1463104032160519033200 x^{17}+872775774067455498528 x^{16}+\frac {26506949038858918036881}{7} x^{21}+67899784121041365504 x^{29}+\frac {2625458326972530284475}{14} x^{28}+11821487501620716192 x^{11}\)	\(175\)
risch	\(17451466816 x +924039400840784712 x^{32}+4928210137817518464 x^{31}+\frac {101849676181562048256}{5} x^{30}+\frac {16677181699666569}{35} x^{35}+11118121133111046 x^{34}+126005372841925188 x^{33}+3064515076512846852480 x^{23}+3614565944605222108800 x^{22}+875152864622814086340 x^{26}+\frac {7584660010542711771792}{5} x^{25}+2298383223254096766840 x^{24}+437576396725285446564 x^{27}+197897276851452864 x^{8}+889942562270387136 x^{9}+37727143432895007 x^{7}+6077684727888102 x^{6}[phone] x^{4}[phone] x^{2}[phone] x^{3}+\frac {4057390785756924}{5} x^{5}+\frac {17344958593049772048}{5} x^{10}+35454069480572048124 x^{12}+94069263918929616324 x^{13}+221699757548270194389 x^{14}+465517091041681015296 x^{15}+3534290697929473864098 x^{20}+2945285062308448290360 x^{19}+2194577166014752240080 x^{18}+1463104032160519033200 x^{17}+872775774067455498528 x^{16}+\frac {26506949038858918036881}{7} x^{21}+67899784121041365504 x^{29}+\frac {2625458326972530284475}{14} x^{28}+11821487501620716192 x^{11}\)	\(175\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6*(1+(2+3*x)^7+(2+3*x)^14)^2,x,method=_RETURNVERBOSE)

[Out]

17451466816*x+924039400840784712*x^32+4928210137817518464*x^31+101849676181562048256/5*x^30+16677181699666569/
35*x^35+11118121133111046*x^34+126005372841925188*x^33+3064515076512846852480*x^23+3614565944605222108800*x^22
+875152864622814086340*x^26+7584660010542711771792/5*x^25+2298383223254096766840*x^24+437576396725285446564*x^
27+197897276851452864*x^8+889942562270387136*x^9+37727143432895007*x^7+6077684727888102*x^6+87406679578680*x^4
[phone]*x^2+7299544818384*x^3+4057390785756924/5*x^5+17344958593049772048/5*x^10+35454069480572048124*x^
12+94069263918929616324*x^13+221699757548270194389*x^14+465517091041681015296*x^15+3534290697929473864098*x^20
+2945285062308448290360*x^19+2194577166014752240080*x^18+1463104032160519033200*x^17+872775774067455498528*x^1
6+26506949038858918036881/7*x^21+67899784121041365504*x^29+2625458326972530284475/14*x^28+11821487501620716192
*x^11

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (46) = 92\).
time = 0.27, size = 174, normalized size = 3.11 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(1+(2+3*x)^7+(2+3*x)^14)^2,x, algorithm="maxima")

[Out]

16677181699666569/35*x^35 + 11118121133111046*x^34 + 126005372841925188*x^33 + 924039400840784712*x^32 + 49282
10137817518464*x^31 + 101849676181562048256/5*x^30 + 67899784121041365504*x^29 + 2625458326972530284475/14*x^2
8 + 437576396725285446564*x^27 + 875152864622814086340*x^26 + 7584660010542711771792/5*x^25 + 2298383223254096
766840*x^24 + 3064515076512846852480*x^23 + 3614565944605222108800*x^22 + 26506949038858918036881/7*x^21 + 353
4290697929473864098*x^20 + 2945285062308448290360*x^19 + 2194577166014752240080*x^18 + 1463104032160519033200*
x^17 + 872775774067455498528*x^16 + 465517091041681015296*x^15 + 221699757548270194389*x^14 + 9406926391892961
6324*x^13 + 35454069480572048124*x^12 + 11821487501620716192*x^11 + 17344958593049772048/5*x^10 + 889942562270
387136*x^9 + 197897276851452864*x^8 + 37727143432895007*x^7 + 6077684727888102*x^6 + 4057390785756924/5*x^5 +
87406679578680*x^4 + 7299544818384*x^3 + 443569828128*x^2 + 17451466816*x

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (46) = 92\).
time = 0.40, size = 174, normalized size = 3.11 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(1+(2+3*x)^7+(2+3*x)^14)^2,x, algorithm="fricas")

[Out]

16677181699666569/35*x^35 + 11118121133111046*x^34 + 126005372841925188*x^33 + 924039400840784712*x^32 + 49282
10137817518464*x^31 + 101849676181562048256/5*x^30 + 67899784121041365504*x^29 + 2625458326972530284475/14*x^2
8 + 437576396725285446564*x^27 + 875152864622814086340*x^26 + 7584660010542711771792/5*x^25 + 2298383223254096
766840*x^24 + 3064515076512846852480*x^23 + 3614565944605222108800*x^22 + 26506949038858918036881/7*x^21 + 353
4290697929473864098*x^20 + 2945285062308448290360*x^19 + 2194577166014752240080*x^18 + 1463104032160519033200*
x^17 + 872775774067455498528*x^16 + 465517091041681015296*x^15 + 221699757548270194389*x^14 + 9406926391892961
6324*x^13 + 35454069480572048124*x^12 + 11821487501620716192*x^11 + 17344958593049772048/5*x^10 + 889942562270
387136*x^9 + 197897276851452864*x^8 + 37727143432895007*x^7 + 6077684727888102*x^6 + 4057390785756924/5*x^5 +
87406679578680*x^4 + 7299544818384*x^3 + 443569828128*x^2 + 17451466816*x

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (41) = 82\).
time = 0.05, size = 187, normalized size = 3.34 \begin {gather*} \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 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6*(1+(2+3*x)**7+(2+3*x)**14)**2,x)

[Out]

16677181699666569*x**35/35 + 11118121133111046*x**34 + 126005372841925188*x**33 + 924039400840784712*x**32 + 4
928210137817518464*x**31 + 101849676181562048256*x**30/5 + 67899784121041365504*x**29 + 2625458326972530284475
*x**28/14 + 437576396725285446564*x**27 + 875152864622814086340*x**26 + 7584660010542711771792*x**25/5 + 22983
83223254096766840*x**24 + 3064515076512846852480*x**23 + 3614565944605222108800*x**22 + 2650694903885891803688
1*x**21/7 + 3534290697929473864098*x**20 + 2945285062308448290360*x**19 + 2194577166014752240080*x**18 + 14631
04032160519033200*x**17 + 872775774067455498528*x**16 + 465517091041681015296*x**15 + 221699757548270194389*x*
*14 + 94069263918929616324*x**13 + 35454069480572048124*x**12 + 11821487501620716192*x**11 + 17344958593049772
048*x**10/5 + 889942562270387136*x**9 + 197897276851452864*x**8 + 37727143432895007*x**7 + 6077684727888102*x*
*6 + 4057390785756924*x**5/5 + 87406679578680*x**4 + 7299544818384*x**3 + 443569828128*x**2 + 17451466816*x

________________________________________________________________________________________

Giac [A]
time = 3.30, size = 46, normalized size = 0.82 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(1+(2+3*x)^7+(2+3*x)^14)^2,x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 1.60, size = 46, normalized size = 0.82 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x + 2)^6*((3*x + 2)^7 + (3*x + 2)^14 + 1)^2,x)

[Out]

(3*x + 2)^7/21 + (3*x + 2)^14/21 + (3*x + 2)^21/21 + (3*x + 2)^28/42 + (3*x + 2)^35/105

________________________________________________________________________________________