Integrand size = 21, antiderivative size = 49 \[ \int F^{a+b (c+d x)^2} (c+d x)^{12} \, dx=-\frac {F^a (c+d x)^{13} \Gamma \left (\frac {13}{2},-b (c+d x)^2 \log (F)\right )}{2 d \left (-b (c+d x)^2 \log (F)\right )^{13/2}} \]
-1/2*F^a*(d*x+c)^13*(524288/5621533568633696205238621875*GAMMA(51/2,-b*(d* x+c)^2*ln(F))-524288/5621533568633696205238621875*(-b*(d*x+c)^2*ln(F))^(49 /2)*exp(b*(d*x+c)^2*ln(F))-262144/114725174870075432759971875*(-b*(d*x+c)^ 2*ln(F))^(47/2)*exp(b*(d*x+c)^2*ln(F))-131072/2440961167448413462978125*(- b*(d*x+c)^2*ln(F))^(45/2)*exp(b*(d*x+c)^2*ln(F))-65536/5424358149885363251 0625*(-b*(d*x+c)^2*ln(F))^(43/2)*exp(b*(d*x+c)^2*ln(F))-32768/126147863950 8224011875*(-b*(d*x+c)^2*ln(F))^(41/2)*exp(b*(d*x+c)^2*ln(F))-16384/307677 71695322536875*(-b*(d*x+c)^2*ln(F))^(39/2)*exp(b*(d*x+c)^2*ln(F))-8192/788 917222956988125*(-b*(d*x+c)^2*ln(F))^(37/2)*exp(b*(d*x+c)^2*ln(F))-4096/21 322087106945625*(-b*(d*x+c)^2*ln(F))^(35/2)*exp(b*(d*x+c)^2*ln(F))-2048/60 9202488769875*(-b*(d*x+c)^2*ln(F))^(33/2)*exp(b*(d*x+c)^2*ln(F))-1024/1846 0681477875*(-b*(d*x+c)^2*ln(F))^(31/2)*exp(b*(d*x+c)^2*ln(F))-512/59550585 4125*(-b*(d*x+c)^2*ln(F))^(29/2)*exp(b*(d*x+c)^2*ln(F))-256/20534684625*(- b*(d*x+c)^2*ln(F))^(27/2)*exp(b*(d*x+c)^2*ln(F))-128/760543875*(-b*(d*x+c) ^2*ln(F))^(25/2)*exp(b*(d*x+c)^2*ln(F))-64/30421755*(-b*(d*x+c)^2*ln(F))^( 23/2)*exp(b*(d*x+c)^2*ln(F))-32/1322685*(-b*(d*x+c)^2*ln(F))^(21/2)*exp(b* (d*x+c)^2*ln(F))-16/62985*(-b*(d*x+c)^2*ln(F))^(19/2)*exp(b*(d*x+c)^2*ln(F ))-8/3315*(-b*(d*x+c)^2*ln(F))^(17/2)*exp(b*(d*x+c)^2*ln(F))-4/195*(-b*(d* x+c)^2*ln(F))^(15/2)*exp(b*(d*x+c)^2*ln(F))-2/13*(-b*(d*x+c)^2*ln(F))^(13/ 2)*exp(b*(d*x+c)^2*ln(F)))/d/(-b*(d*x+c)^2*ln(F))^(13/2)
Time = 0.66 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00 \[ \int F^{a+b (c+d x)^2} (c+d x)^{12} \, dx=-\frac {F^a (c+d x)^{13} \Gamma \left (\frac {13}{2},-b (c+d x)^2 \log (F)\right )}{2 d \left (-b (c+d x)^2 \log (F)\right )^{13/2}} \]
-1/2*(F^a*(c + d*x)^13*Gamma[13/2, -(b*(c + d*x)^2*Log[F])])/(d*(-(b*(c + d*x)^2*Log[F]))^(13/2))
Time = 0.21 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2648}
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.
| \(\displaystyle \int (c+d x)^{12} F^{a+b (c+d x)^2} \, dx\) |
| \(\Big \downarrow \) 2648 |
| \(\displaystyle -\frac {F^a (c+d x)^{13} \Gamma \left (\frac {13}{2},-b (c+d x)^2 \log (F)\right )}{2 d \left (-b \log (F) (c+d x)^2\right )^{13/2}}\) |
-1/2*(F^a*(c + d*x)^13*Gamma[13/2, -(b*(c + d*x)^2*Log[F])])/(d*(-(b*(c + d*x)^2*Log[F]))^(13/2))
3.3.67.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ .), x_Symbol] :> Simp[(-F^a)*((e + f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[ F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; FreeQ[{F , a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1910\) vs. \(2(578)=1156\).
Time = 2.83 (sec) , antiderivative size = 1911, normalized size of antiderivative = 39.00
-10395/64*F^(b*c^2)*F^a/d*c/ln(F)^6/b^6*F^(b*d^2*x^2)*F^(2*b*c*d*x)+3465/3 2*F^(b*c^2)*F^a/d*c^3/ln(F)^5/b^5*F^(b*d^2*x^2)*F^(2*b*c*d*x)-693/16*F^(b* c^2)*F^a/d*c^5/ln(F)^4/b^4*F^(b*d^2*x^2)*F^(2*b*c*d*x)+1/2*F^(b*c^2)*F^a/d *c^11/ln(F)/b*F^(b*d^2*x^2)*F^(2*b*c*d*x)-11/4*F^(b*c^2)*F^a/d*c^9/ln(F)^2 /b^2*F^(b*d^2*x^2)*F^(2*b*c*d*x)+99/8*F^(b*c^2)*F^a/d*c^7/ln(F)^3/b^3*F^(b *d^2*x^2)*F^(2*b*c*d*x)+1/2*F^(b*c^2)*F^a*d^10/ln(F)/b*x^11*F^(b*d^2*x^2)* F^(2*b*c*d*x)-11/4*F^(b*c^2)*F^a*d^8/ln(F)^2/b^2*x^9*F^(b*d^2*x^2)*F^(2*b* c*d*x)+3465/8*F^(b*c^2)*F^a*d^3*c^3/ln(F)^3/b^3*x^4*F^(b*d^2*x^2)*F^(2*b*c *d*x)-3465/8*F^(b*c^2)*F^a*d*c^3/ln(F)^4/b^4*x^2*F^(b*d^2*x^2)*F^(2*b*c*d* x)-10395/64*F^(b*c^2)*F^a/ln(F)^6/b^6*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)-693/16 *F^(b*c^2)*F^a*d^4/ln(F)^4/b^4*x^5*F^(b*d^2*x^2)*F^(2*b*c*d*x)+3465/32*F^( b*c^2)*F^a*d^2/ln(F)^5/b^5*x^3*F^(b*d^2*x^2)*F^(2*b*c*d*x)+99/8*F^(b*c^2)* F^a*d^6/ln(F)^3/b^3*x^7*F^(b*d^2*x^2)*F^(2*b*c*d*x)-3465/16*F^(b*c^2)*F^a* c^4/ln(F)^4/b^4*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)+11/2*F^(b*c^2)*F^a*c^10/ln(F )/b*x*F^(b*d^2*x^2)*F^(2*b*c*d*x)-99/4*F^(b*c^2)*F^a*c^8/ln(F)^2/b^2*x*F^( b*d^2*x^2)*F^(2*b*c*d*x)+693/8*F^(b*c^2)*F^a*c^6/ln(F)^3/b^3*x*F^(b*d^2*x^ 2)*F^(2*b*c*d*x)+10395/32*F^(b*c^2)*F^a*c^2/ln(F)^5/b^5*x*F^(b*d^2*x^2)*F^ (2*b*c*d*x)+165/2*F^(b*c^2)*F^a*d^7*c^3/ln(F)/b*x^8*F^(b*d^2*x^2)*F^(2*b*c *d*x)-693/2*F^(b*c^2)*F^a*d^4*c^4/ln(F)^2/b^2*x^5*F^(b*d^2*x^2)*F^(2*b*c*d *x)+55/2*F^(b*c^2)*F^a*d^8*c^2/ln(F)/b*x^9*F^(b*d^2*x^2)*F^(2*b*c*d*x)-...
Time = 0.09 (sec) , antiderivative size = 617, normalized size of antiderivative = 12.59 \[ \int F^{a+b (c+d x)^2} (c+d x)^{12} \, dx=-\frac {10395 \, \sqrt {\pi } \sqrt {-b d^{2} \log \left (F\right )} F^{a} \operatorname {erf}\left (\frac {\sqrt {-b d^{2} \log \left (F\right )} {\left (d x + c\right )}}{d}\right ) - 2 \, {\left (32 \, {\left (b^{6} d^{12} x^{11} + 11 \, b^{6} c d^{11} x^{10} + 55 \, b^{6} c^{2} d^{10} x^{9} + 165 \, b^{6} c^{3} d^{9} x^{8} + 330 \, b^{6} c^{4} d^{8} x^{7} + 462 \, b^{6} c^{5} d^{7} x^{6} + 462 \, b^{6} c^{6} d^{6} x^{5} + 330 \, b^{6} c^{7} d^{5} x^{4} + 165 \, b^{6} c^{8} d^{4} x^{3} + 55 \, b^{6} c^{9} d^{3} x^{2} + 11 \, b^{6} c^{10} d^{2} x + b^{6} c^{11} d\right )} \log \left (F\right )^{6} - 176 \, {\left (b^{5} d^{10} x^{9} + 9 \, b^{5} c d^{9} x^{8} + 36 \, b^{5} c^{2} d^{8} x^{7} + 84 \, b^{5} c^{3} d^{7} x^{6} + 126 \, b^{5} c^{4} d^{6} x^{5} + 126 \, b^{5} c^{5} d^{5} x^{4} + 84 \, b^{5} c^{6} d^{4} x^{3} + 36 \, b^{5} c^{7} d^{3} x^{2} + 9 \, b^{5} c^{8} d^{2} x + b^{5} c^{9} d\right )} \log \left (F\right )^{5} + 792 \, {\left (b^{4} d^{8} x^{7} + 7 \, b^{4} c d^{7} x^{6} + 21 \, b^{4} c^{2} d^{6} x^{5} + 35 \, b^{4} c^{3} d^{5} x^{4} + 35 \, b^{4} c^{4} d^{4} x^{3} + 21 \, b^{4} c^{5} d^{3} x^{2} + 7 \, b^{4} c^{6} d^{2} x + b^{4} c^{7} d\right )} \log \left (F\right )^{4} - 2772 \, {\left (b^{3} d^{6} x^{5} + 5 \, b^{3} c d^{5} x^{4} + 10 \, b^{3} c^{2} d^{4} x^{3} + 10 \, b^{3} c^{3} d^{3} x^{2} + 5 \, b^{3} c^{4} d^{2} x + b^{3} c^{5} d\right )} \log \left (F\right )^{3} + 6930 \, {\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x + b^{2} c^{3} d\right )} \log \left (F\right )^{2} - 10395 \, {\left (b d^{2} x + b c d\right )} \log \left (F\right )\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{128 \, b^{7} d^{2} \log \left (F\right )^{7}} \]
-1/128*(10395*sqrt(pi)*sqrt(-b*d^2*log(F))*F^a*erf(sqrt(-b*d^2*log(F))*(d* x + c)/d) - 2*(32*(b^6*d^12*x^11 + 11*b^6*c*d^11*x^10 + 55*b^6*c^2*d^10*x^ 9 + 165*b^6*c^3*d^9*x^8 + 330*b^6*c^4*d^8*x^7 + 462*b^6*c^5*d^7*x^6 + 462* b^6*c^6*d^6*x^5 + 330*b^6*c^7*d^5*x^4 + 165*b^6*c^8*d^4*x^3 + 55*b^6*c^9*d ^3*x^2 + 11*b^6*c^10*d^2*x + b^6*c^11*d)*log(F)^6 - 176*(b^5*d^10*x^9 + 9* b^5*c*d^9*x^8 + 36*b^5*c^2*d^8*x^7 + 84*b^5*c^3*d^7*x^6 + 126*b^5*c^4*d^6* x^5 + 126*b^5*c^5*d^5*x^4 + 84*b^5*c^6*d^4*x^3 + 36*b^5*c^7*d^3*x^2 + 9*b^ 5*c^8*d^2*x + b^5*c^9*d)*log(F)^5 + 792*(b^4*d^8*x^7 + 7*b^4*c*d^7*x^6 + 2 1*b^4*c^2*d^6*x^5 + 35*b^4*c^3*d^5*x^4 + 35*b^4*c^4*d^4*x^3 + 21*b^4*c^5*d ^3*x^2 + 7*b^4*c^6*d^2*x + b^4*c^7*d)*log(F)^4 - 2772*(b^3*d^6*x^5 + 5*b^3 *c*d^5*x^4 + 10*b^3*c^2*d^4*x^3 + 10*b^3*c^3*d^3*x^2 + 5*b^3*c^4*d^2*x + b ^3*c^5*d)*log(F)^3 + 6930*(b^2*d^4*x^3 + 3*b^2*c*d^3*x^2 + 3*b^2*c^2*d^2*x + b^2*c^3*d)*log(F)^2 - 10395*(b*d^2*x + b*c*d)*log(F))*F^(b*d^2*x^2 + 2* b*c*d*x + b*c^2 + a))/(b^7*d^2*log(F)^7)
\[ \int F^{a+b (c+d x)^2} (c+d x)^{12} \, dx=\int F^{a + b \left (c + d x\right )^{2}} \left (c + d x\right )^{12}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 6135 vs. \(2 (559) = 1118\).
Time = 1.99 (sec) , antiderivative size = 6135, normalized size of antiderivative = 125.20 \[ \int F^{a+b (c+d x)^2} (c+d x)^{12} \, dx=\text {Too large to display} \]
-6*(sqrt(pi)*(b*d^2*x + b*c*d)*b*c*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/( b*d^2))) - 1)*log(F)^2/((b*log(F))^(3/2)*d^2*sqrt(-(b*d^2*x + b*c*d)^2*log (F)/(b*d^2))) - F^((b*d^2*x + b*c*d)^2/(b*d^2))*b*log(F)/((b*log(F))^(3/2) *d))*F^a*c^11/sqrt(b*log(F)) + 33*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*(erf (sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^3/((b*log(F))^(5/2 )*d^3*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 2*F^((b*d^2*x + b*c*d)^ 2/(b*d^2))*b^2*c*log(F)^2/((b*log(F))^(5/2)*d^2) - (b*d^2*x + b*c*d)^3*gam ma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/((b*log(F))^(5/2)*d^ 5*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)))*F^a*c^10*d/sqrt(b*log(F)) - 110*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*(erf(sqrt(-(b*d^2*x + b*c*d)^2*l og(F)/(b*d^2))) - 1)*log(F)^4/((b*log(F))^(7/2)*d^4*sqrt(-(b*d^2*x + b*c*d )^2*log(F)/(b*d^2))) - 3*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^3*c^2*log(F)^3/ ((b*log(F))^(7/2)*d^3) - 3*(b*d^2*x + b*c*d)^3*b*c*gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*log(F))^(7/2)*d^6*(-(b*d^2*x + b*c*d )^2*log(F)/(b*d^2))^(3/2)) + b^2*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d ^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*c^9*d^2/sqrt(b*log(F)) + 495/2*( sqrt(pi)*(b*d^2*x + b*c*d)*b^4*c^4*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/( b*d^2))) - 1)*log(F)^5/((b*log(F))^(9/2)*d^5*sqrt(-(b*d^2*x + b*c*d)^2*log (F)/(b*d^2))) - 4*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^4*c^3*log(F)^4/((b*log (F))^(9/2)*d^4) - 6*(b*d^2*x + b*c*d)^3*b^2*c^2*gamma(3/2, -(b*d^2*x + ...
Time = 0.16 (sec) , antiderivative size = 195, normalized size of antiderivative = 3.98 \[ \int F^{a+b (c+d x)^2} (c+d x)^{12} \, dx=\frac {{\left (32 \, b^{5} d^{10} {\left (x + \frac {c}{d}\right )}^{11} \log \left (F\right )^{5} - 176 \, b^{4} d^{8} {\left (x + \frac {c}{d}\right )}^{9} \log \left (F\right )^{4} + 792 \, b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{7} \log \left (F\right )^{3} - 2772 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{5} \log \left (F\right )^{2} + 6930 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{3} \log \left (F\right ) - 10395 \, x - \frac {10395 \, c}{d}\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{64 \, b^{6} \log \left (F\right )^{6}} - \frac {10395 \, \sqrt {\pi } F^{a} \operatorname {erf}\left (-\sqrt {-b \log \left (F\right )} d {\left (x + \frac {c}{d}\right )}\right )}{128 \, \sqrt {-b \log \left (F\right )} b^{6} d \log \left (F\right )^{6}} \]
1/64*(32*b^5*d^10*(x + c/d)^11*log(F)^5 - 176*b^4*d^8*(x + c/d)^9*log(F)^4 + 792*b^3*d^6*(x + c/d)^7*log(F)^3 - 2772*b^2*d^4*(x + c/d)^5*log(F)^2 + 6930*b*d^2*(x + c/d)^3*log(F) - 10395*x - 10395*c/d)*e^(b*d^2*x^2*log(F) + 2*b*c*d*x*log(F) + b*c^2*log(F) + a*log(F))/(b^6*log(F)^6) - 10395/128*sq rt(pi)*F^a*erf(-sqrt(-b*log(F))*d*(x + c/d))/(sqrt(-b*log(F))*b^6*d*log(F) ^6)
Time = 0.72 (sec) , antiderivative size = 209, normalized size of antiderivative = 4.27 \[ \int F^{a+b (c+d x)^2} (c+d x)^{12} \, dx=\frac {\frac {F^a\,\left (\frac {10395\,\sqrt {\pi }\,\mathrm {erfi}\left (\frac {b\,\ln \left (F\right )\,\left (c+d\,x\right )}{\sqrt {b\,\ln \left (F\right )}}\right )}{128}-\frac {10395\,F^{b\,{\left (c+d\,x\right )}^2}\,\sqrt {b\,\ln \left (F\right )}\,\left (c+d\,x\right )}{64}\right )}{\sqrt {b\,\ln \left (F\right )}}-\frac {693\,F^{a+b\,{\left (c+d\,x\right )}^2}\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^5}{16}+\frac {99\,F^{a+b\,{\left (c+d\,x\right )}^2}\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^7}{8}-\frac {11\,F^{a+b\,{\left (c+d\,x\right )}^2}\,b^4\,{\ln \left (F\right )}^4\,{\left (c+d\,x\right )}^9}{4}+\frac {F^{a+b\,{\left (c+d\,x\right )}^2}\,b^5\,{\ln \left (F\right )}^5\,{\left (c+d\,x\right )}^{11}}{2}+\frac {3465\,F^{a+b\,{\left (c+d\,x\right )}^2}\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}{32}}{b^6\,d\,{\ln \left (F\right )}^6} \]
((F^a*((10395*pi^(1/2)*erfi((b*log(F)*(c + d*x))/(b*log(F))^(1/2)))/128 - (10395*F^(b*(c + d*x)^2)*(b*log(F))^(1/2)*(c + d*x))/64))/(b*log(F))^(1/2) - (693*F^(a + b*(c + d*x)^2)*b^2*log(F)^2*(c + d*x)^5)/16 + (99*F^(a + b* (c + d*x)^2)*b^3*log(F)^3*(c + d*x)^7)/8 - (11*F^(a + b*(c + d*x)^2)*b^4*l og(F)^4*(c + d*x)^9)/4 + (F^(a + b*(c + d*x)^2)*b^5*log(F)^5*(c + d*x)^11) /2 + (3465*F^(a + b*(c + d*x)^2)*b*log(F)*(c + d*x)^3)/32)/(b^6*d*log(F)^6 )