Integrand size = 24, antiderivative size = 34 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right ) \, dx=\frac {1}{21} (2+3 x)^7+\frac {1}{42} (2+3 x)^{14}+\frac {1}{63} (2+3 x)^{21} \]
Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right ) \, dx=\frac {1}{21} (2+3 x)^7+\frac {1}{42} (2+3 x)^{14}+\frac {1}{63} (2+3 x)^{21} \]
Time = 0.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {1725, 1690, 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.
\(\displaystyle \int (3 x+2)^6 \left ((3 x+2)^{14}+(3 x+2)^7+1\right ) \, dx\) |
\(\Big \downarrow \) 1725 |
\(\displaystyle \frac {1}{3} \int (3 x+2)^6 \left ((3 x+2)^{14}+(3 x+2)^7+1\right )d(3 x+2)\) |
\(\Big \downarrow \) 1690 |
\(\displaystyle \frac {1}{21} \int \left ((3 x+2)^{14}+3 x+3\right )d(3 x+2)^7\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{21} \left ((3 x+2)^7+\frac {1}{3} (3 x+2)^3+\frac {1}{2} (3 x+2)^2\right )\) |
3.7.63.3.1 Defintions of rubi rules used
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. \(103\) vs. \(2(28)=56\).
Time = 0.56 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.06
method | result | size |
gosper | \(\frac {x \left (2324522934 x^{20}+32543321076 x^{19}+216955473840 x^{18}+916034222880 x^{17}+2748102668640 x^{16}+6229032715584 x^{15}+11073835938816 x^{14}+15819767221203 x^{13}+18456408111708 x^{12}+17772887593188 x^{11}+14218430440032 x^{10}+9479154235824 x^{9}+5266441986624 x^{8}+2430891860544 x^{7}+926214166962 x^{6}+288242703252 x^{5}+72097012008 x^{4}+14148077328 x^{3}+2098628448 x^{2}+221323200 x +14795648\right )}{14}\) | \(104\) |
default | \(1056832 x +15808800 x^{2}+149902032 x^{3}+1010576952 x^{4}+5149786572 x^{5}+20588764518 x^{6}+66158154783 x^{7}+173635132896 x^{8}+376174427616 x^{9}+677082445416 x^{10}+1015602174288 x^{11}+1269491970942 x^{12}+1318314865122 x^{13}+\frac {15819767221203}{14} x^{14}+790988281344 x^{15}+444930908256 x^{16}+196293047760 x^{17}+65431015920 x^{18}+15496819560 x^{19}+2324522934 x^{20}+\frac {1162261467}{7} x^{21}\) | \(105\) |
norman | \(1056832 x +15808800 x^{2}+149902032 x^{3}+1010576952 x^{4}+5149786572 x^{5}+20588764518 x^{6}+66158154783 x^{7}+173635132896 x^{8}+376174427616 x^{9}+677082445416 x^{10}+1015602174288 x^{11}+1269491970942 x^{12}+1318314865122 x^{13}+\frac {15819767221203}{14} x^{14}+790988281344 x^{15}+444930908256 x^{16}+196293047760 x^{17}+65431015920 x^{18}+15496819560 x^{19}+2324522934 x^{20}+\frac {1162261467}{7} x^{21}\) | \(105\) |
risch | \(1056832 x +15808800 x^{2}+149902032 x^{3}+1010576952 x^{4}+5149786572 x^{5}+20588764518 x^{6}+66158154783 x^{7}+173635132896 x^{8}+376174427616 x^{9}+677082445416 x^{10}+1015602174288 x^{11}+1269491970942 x^{12}+1318314865122 x^{13}+\frac {15819767221203}{14} x^{14}+790988281344 x^{15}+444930908256 x^{16}+196293047760 x^{17}+65431015920 x^{18}+15496819560 x^{19}+2324522934 x^{20}+\frac {1162261467}{7} x^{21}\) | \(105\) |
parallelrisch | \(1056832 x +15808800 x^{2}+149902032 x^{3}+1010576952 x^{4}+5149786572 x^{5}+20588764518 x^{6}+66158154783 x^{7}+173635132896 x^{8}+376174427616 x^{9}+677082445416 x^{10}+1015602174288 x^{11}+1269491970942 x^{12}+1318314865122 x^{13}+\frac {15819767221203}{14} x^{14}+790988281344 x^{15}+444930908256 x^{16}+196293047760 x^{17}+65431015920 x^{18}+15496819560 x^{19}+2324522934 x^{20}+\frac {1162261467}{7} x^{21}\) | \(105\) |
1/14*x*(2324522934*x^20+32543321076*x^19+216955473840*x^18+916034222880*x^ 17+2748102668640*x^16+6229032715584*x^15+11073835938816*x^14+1581976722120 3*x^13+18456408111708*x^12+17772887593188*x^11+14218430440032*x^10+9479154 235824*x^9+5266441986624*x^8+2430891860544*x^7+926214166962*x^6+2882427032 52*x^5+72097012008*x^4+14148077328*x^3+2098628448*x^2+221323200*x+14795648 )
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (28) = 56\).
Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.06 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right ) \, dx=\frac {1162261467}{7} \, x^{21} + 2324522934 \, x^{20} + 15496819560 \, x^{19} + 65431015920 \, x^{18} + 196293047760 \, x^{17} + 444930908256 \, x^{16} + 790988281344 \, x^{15} + \frac {15819767221203}{14} \, x^{14} + 1318314865122 \, x^{13} + 1269491970942 \, x^{12} + 1015602174288 \, x^{11} + 677082445416 \, x^{10} + 376174427616 \, x^{9} + 173635132896 \, x^{8} + 66158154783 \, x^{7} + 20588764518 \, x^{6} + 5149786572 \, x^{5} + 1010576952 \, x^{4} + 149902032 \, x^{3} + 15808800 \, x^{2} + 1056832 \, x \]
1162261467/7*x^21 + 2324522934*x^20 + 15496819560*x^19 + 65431015920*x^18 + 196293047760*x^17 + 444930908256*x^16 + 790988281344*x^15 + 158197672212 03/14*x^14 + 1318314865122*x^13 + 1269491970942*x^12 + 1015602174288*x^11 + 677082445416*x^10 + 376174427616*x^9 + 173635132896*x^8 + 66158154783*x^ 7 + 20588764518*x^6 + 5149786572*x^5 + 1010576952*x^4 + 149902032*x^3 + 15 808800*x^2 + 1056832*x
Leaf count of result is larger than twice the leaf count of optimal. 107 vs. \(2 (24) = 48\).
Time = 0.04 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.15 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right ) \, dx=\frac {1162261467 x^{21}}{7} + 2324522934 x^{20} + 15496819560 x^{19} + 65431015920 x^{18} + 196293047760 x^{17} + 444930908256 x^{16} + 790988281344 x^{15} + \frac {15819767221203 x^{14}}{14} + 1318314865122 x^{13} + 1269491970942 x^{12} + 1015602174288 x^{11} + 677082445416 x^{10} + 376174427616 x^{9} + 173635132896 x^{8} + 66158154783 x^{7} + 20588764518 x^{6} + 5149786572 x^{5} + 1010576952 x^{4} + 149902032 x^{3} + 15808800 x^{2} + 1056832 x \]
1162261467*x**21/7 + 2324522934*x**20 + 15496819560*x**19 + 65431015920*x* *18 + 196293047760*x**17 + 444930908256*x**16 + 790988281344*x**15 + 15819 767221203*x**14/14 + 1318314865122*x**13 + 1269491970942*x**12 + 101560217 4288*x**11 + 677082445416*x**10 + 376174427616*x**9 + 173635132896*x**8 + 66158154783*x**7 + 20588764518*x**6 + 5149786572*x**5 + 1010576952*x**4 + 149902032*x**3 + 15808800*x**2 + 1056832*x
Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (28) = 56\).
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 3.06 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right ) \, dx=\frac {1162261467}{7} \, x^{21} + 2324522934 \, x^{20} + 15496819560 \, x^{19} + 65431015920 \, x^{18} + 196293047760 \, x^{17} + 444930908256 \, x^{16} + 790988281344 \, x^{15} + \frac {15819767221203}{14} \, x^{14} + 1318314865122 \, x^{13} + 1269491970942 \, x^{12} + 1015602174288 \, x^{11} + 677082445416 \, x^{10} + 376174427616 \, x^{9} + 173635132896 \, x^{8} + 66158154783 \, x^{7} + 20588764518 \, x^{6} + 5149786572 \, x^{5} + 1010576952 \, x^{4} + 149902032 \, x^{3} + 15808800 \, x^{2} + 1056832 \, x \]
1162261467/7*x^21 + 2324522934*x^20 + 15496819560*x^19 + 65431015920*x^18 + 196293047760*x^17 + 444930908256*x^16 + 790988281344*x^15 + 158197672212 03/14*x^14 + 1318314865122*x^13 + 1269491970942*x^12 + 1015602174288*x^11 + 677082445416*x^10 + 376174427616*x^9 + 173635132896*x^8 + 66158154783*x^ 7 + 20588764518*x^6 + 5149786572*x^5 + 1010576952*x^4 + 149902032*x^3 + 15 808800*x^2 + 1056832*x
Time = 0.35 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.82 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right ) \, dx=\frac {1}{63} \, {\left (3 \, x + 2\right )}^{21} + \frac {1}{42} \, {\left (3 \, x + 2\right )}^{14} + \frac {1}{21} \, {\left (3 \, x + 2\right )}^{7} \]
Time = 8.50 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.85 \[ \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right ) \, dx=\frac {{\left (3\,x+2\right )}^7\,\left (3\,{\left (3\,x+2\right )}^7+2\,{\left (3\,x+2\right )}^{14}+6\right )}{126} \]