Integrand size = 21, antiderivative size = 124 \[ \int F^{a+b (c+d x)^3} (c+d x)^{11} \, dx=-\frac {2 F^{a+b (c+d x)^3}}{b^4 d \log ^4(F)}+\frac {2 F^{a+b (c+d x)^3} (c+d x)^3}{b^3 d \log ^3(F)}-\frac {F^{a+b (c+d x)^3} (c+d x)^6}{b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^3} (c+d x)^9}{3 b d \log (F)} \] Output:
-2*F^(a+b*(d*x+c)^3)/b^4/d/ln(F)^4+2*F^(a+b*(d*x+c)^3)*(d*x+c)^3/b^3/d/ln( F)^3-F^(a+b*(d*x+c)^3)*(d*x+c)^6/b^2/d/ln(F)^2+1/3*F^(a+b*(d*x+c)^3)*(d*x+ c)^9/b/d/ln(F)
Time = 0.37 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.60 \[ \int F^{a+b (c+d x)^3} (c+d x)^{11} \, dx=\frac {F^{a+b (c+d x)^3} \left (-3 b^2 (c+d x)^6 \log ^2(F)+b^3 (c+d x)^9 \log ^3(F)-6 \left (1-b (c+d x)^3 \log (F)\right )\right )}{3 b^4 d \log ^4(F)} \] Input:
Integrate[F^(a + b*(c + d*x)^3)*(c + d*x)^11,x]
Output:
(F^(a + b*(c + d*x)^3)*(-3*b^2*(c + d*x)^6*Log[F]^2 + b^3*(c + d*x)^9*Log[ F]^3 - 6*(1 - b*(c + d*x)^3*Log[F])))/(3*b^4*d*Log[F]^4)
Time = 0.93 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.21, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2641, 2641, 2641, 2638}
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.
| \(\displaystyle \int (c+d x)^{11} F^{a+b (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {(c+d x)^9 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {3 \int F^{b (c+d x)^3+a} (c+d x)^8dx}{b \log (F)}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {(c+d x)^9 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {3 \left (\frac {(c+d x)^6 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {2 \int F^{b (c+d x)^3+a} (c+d x)^5dx}{b \log (F)}\right )}{b \log (F)}\) |
| \(\Big \downarrow \) 2641 |
| \(\displaystyle \frac {(c+d x)^9 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {3 \left (\frac {(c+d x)^6 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {2 \left (\frac {(c+d x)^3 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {\int F^{b (c+d x)^3+a} (c+d x)^2dx}{b \log (F)}\right )}{b \log (F)}\right )}{b \log (F)}\) |
| \(\Big \downarrow \) 2638 |
| \(\displaystyle \frac {(c+d x)^9 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {3 \left (\frac {(c+d x)^6 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {2 \left (\frac {(c+d x)^3 F^{a+b (c+d x)^3}}{3 b d \log (F)}-\frac {F^{a+b (c+d x)^3}}{3 b^2 d \log ^2(F)}\right )}{b \log (F)}\right )}{b \log (F)}\) |
Input:
Int[F^(a + b*(c + d*x)^3)*(c + d*x)^11,x]
Output:
(F^(a + b*(c + d*x)^3)*(c + d*x)^9)/(3*b*d*Log[F]) - (3*((F^(a + b*(c + d* x)^3)*(c + d*x)^6)/(3*b*d*Log[F]) - (2*(-1/3*F^(a + b*(c + d*x)^3)/(b^2*d* Log[F]^2) + (F^(a + b*(c + d*x)^3)*(c + d*x)^3)/(3*b*d*Log[F])))/(b*Log[F] )))/(b*Log[F])
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ .), x_Symbol] :> Simp[(e + f*x)^n*(F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n *Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] && EqQ [d*e - c*f, 0]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ .), x_Symbol] :> Simp[(c + d*x)^(m - n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*L og[F])), x] - Simp[(m - n + 1)/(b*n*Log[F]) Int[(c + d*x)^(m - n)*F^(a + b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/ n)] && LtQ[0, (m + 1)/n, 5] && IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n , 0])
Leaf count of result is larger than twice the leaf count of optimal. \(342\) vs. \(2(122)=244\).
Time = 1.24 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.77
| method | result | size |
| orering | \(\frac {\left (d^{9} x^{9} \ln \left (F \right )^{3} b^{3}+9 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}+36 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}+84 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}+126 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}+126 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}+84 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}+36 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}+9 \ln \left (F \right )^{3} b^{3} c^{8} d x -3 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+\ln \left (F \right )^{3} b^{3} c^{9}-18 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}-45 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}-60 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}-45 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}-18 \ln \left (F \right )^{2} b^{2} c^{5} d x -3 \ln \left (F \right )^{2} b^{2} c^{6}+6 \ln \left (F \right ) b \,d^{3} x^{3}+18 \ln \left (F \right ) b c \,d^{2} x^{2}+18 \ln \left (F \right ) b \,c^{2} d x +6 \ln \left (F \right ) b \,c^{3}-6\right ) F^{a +b \left (d x +c \right )^{3}}}{3 d \,b^{4} \ln \left (F \right )^{4}}\) | \(343\) |
| gosper | \(\frac {\left (d^{9} x^{9} \ln \left (F \right )^{3} b^{3}+9 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}+36 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}+84 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}+126 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}+126 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}+84 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}+36 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}+9 \ln \left (F \right )^{3} b^{3} c^{8} d x -3 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+\ln \left (F \right )^{3} b^{3} c^{9}-18 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}-45 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}-60 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}-45 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}-18 \ln \left (F \right )^{2} b^{2} c^{5} d x -3 \ln \left (F \right )^{2} b^{2} c^{6}+6 \ln \left (F \right ) b \,d^{3} x^{3}+18 \ln \left (F \right ) b c \,d^{2} x^{2}+18 \ln \left (F \right ) b \,c^{2} d x +6 \ln \left (F \right ) b \,c^{3}-6\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 \ln \left (F \right )^{4} b^{4} d}\) | \(365\) |
| risch | \(\frac {\left (d^{9} x^{9} \ln \left (F \right )^{3} b^{3}+9 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}+36 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}+84 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}+126 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}+126 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}+84 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}+36 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}+9 \ln \left (F \right )^{3} b^{3} c^{8} d x -3 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}+\ln \left (F \right )^{3} b^{3} c^{9}-18 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}-45 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}-60 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}-45 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}-18 \ln \left (F \right )^{2} b^{2} c^{5} d x -3 \ln \left (F \right )^{2} b^{2} c^{6}+6 \ln \left (F \right ) b \,d^{3} x^{3}+18 \ln \left (F \right ) b c \,d^{2} x^{2}+18 \ln \left (F \right ) b \,c^{2} d x +6 \ln \left (F \right ) b \,c^{3}-6\right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 \ln \left (F \right )^{4} b^{4} d}\) | \(365\) |
| norman | \(\frac {d^{5} \left (28 \ln \left (F \right ) b \,c^{3}-1\right ) x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {\left (\ln \left (F \right )^{3} b^{3} c^{9}-3 \ln \left (F \right )^{2} b^{2} c^{6}+6 \ln \left (F \right ) b \,c^{3}-6\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{4} b^{4} d}+\frac {d^{8} x^{9} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right ) b}+\frac {3 c^{2} \left (\ln \left (F \right )^{2} b^{2} c^{6}-2 \ln \left (F \right ) b \,c^{3}+2\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {2 d^{2} \left (14 \ln \left (F \right )^{2} b^{2} c^{6}-10 \ln \left (F \right ) b \,c^{3}+1\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {12 d^{6} c^{2} x^{7} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {3 d^{7} c \,x^{8} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {3 c d \left (4 \ln \left (F \right )^{2} b^{2} c^{6}-5 \ln \left (F \right ) b \,c^{3}+2\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {6 c \,d^{4} \left (7 \ln \left (F \right ) b \,c^{3}-1\right ) x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}+\frac {3 c^{2} d^{3} \left (14 \ln \left (F \right ) b \,c^{3}-5\right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}\) | \(431\) |
| parallelrisch | \(\frac {d^{9} F^{a +b \left (d x +c \right )^{3}} x^{9} b^{3} \ln \left (F \right )^{3}+9 c \,d^{8} F^{a +b \left (d x +c \right )^{3}} x^{8} b^{3} \ln \left (F \right )^{3}+36 c^{2} d^{7} F^{a +b \left (d x +c \right )^{3}} x^{7} b^{3} \ln \left (F \right )^{3}+84 \ln \left (F \right )^{3} x^{6} F^{a +b \left (d x +c \right )^{3}} b^{3} c^{3} d^{6}+126 \ln \left (F \right )^{3} x^{5} F^{a +b \left (d x +c \right )^{3}} b^{3} c^{4} d^{5}+126 \ln \left (F \right )^{3} x^{4} F^{a +b \left (d x +c \right )^{3}} b^{3} c^{5} d^{4}+84 \ln \left (F \right )^{3} x^{3} F^{a +b \left (d x +c \right )^{3}} b^{3} c^{6} d^{3}+36 \ln \left (F \right )^{3} x^{2} F^{a +b \left (d x +c \right )^{3}} b^{3} c^{7} d^{2}+9 \ln \left (F \right )^{3} x \,F^{a +b \left (d x +c \right )^{3}} b^{3} c^{8} d -3 d^{6} F^{a +b \left (d x +c \right )^{3}} x^{6} b^{2} \ln \left (F \right )^{2}+\ln \left (F \right )^{3} F^{a +b \left (d x +c \right )^{3}} b^{3} c^{9}-18 c \,d^{5} F^{a +b \left (d x +c \right )^{3}} x^{5} b^{2} \ln \left (F \right )^{2}-45 c^{2} d^{4} F^{a +b \left (d x +c \right )^{3}} x^{4} b^{2} \ln \left (F \right )^{2}-60 \ln \left (F \right )^{2} x^{3} F^{a +b \left (d x +c \right )^{3}} b^{2} c^{3} d^{3}-45 \ln \left (F \right )^{2} x^{2} F^{a +b \left (d x +c \right )^{3}} b^{2} c^{4} d^{2}-18 \ln \left (F \right )^{2} x \,F^{a +b \left (d x +c \right )^{3}} b^{2} c^{5} d -3 \ln \left (F \right )^{2} F^{a +b \left (d x +c \right )^{3}} b^{2} c^{6}+6 d^{3} F^{a +b \left (d x +c \right )^{3}} x^{3} b \ln \left (F \right )+18 c \,d^{2} F^{a +b \left (d x +c \right )^{3}} x^{2} b \ln \left (F \right )+18 c^{2} F^{a +b \left (d x +c \right )^{3}} x b \ln \left (F \right ) d +6 \ln \left (F \right ) F^{a +b \left (d x +c \right )^{3}} b \,c^{3}-6 F^{a +b \left (d x +c \right )^{3}}}{3 \ln \left (F \right )^{4} b^{4} d}\) | \(617\) |
Input:
int(F^(a+b*(d*x+c)^3)*(d*x+c)^11,x,method=_RETURNVERBOSE)
Output:
1/3/d*(d^9*x^9*ln(F)^3*b^3+9*c*d^8*x^8*ln(F)^3*b^3+36*c^2*d^7*x^7*ln(F)^3* b^3+84*ln(F)^3*b^3*c^3*d^6*x^6+126*ln(F)^3*b^3*c^4*d^5*x^5+126*ln(F)^3*b^3 *c^5*d^4*x^4+84*ln(F)^3*b^3*c^6*d^3*x^3+36*ln(F)^3*b^3*c^7*d^2*x^2+9*ln(F) ^3*b^3*c^8*d*x-3*d^6*x^6*ln(F)^2*b^2+ln(F)^3*b^3*c^9-18*c*d^5*x^5*ln(F)^2* b^2-45*c^2*d^4*x^4*ln(F)^2*b^2-60*ln(F)^2*b^2*c^3*d^3*x^3-45*ln(F)^2*b^2*c ^4*d^2*x^2-18*ln(F)^2*b^2*c^5*d*x-3*ln(F)^2*b^2*c^6+6*ln(F)*b*d^3*x^3+18*l n(F)*b*c*d^2*x^2+18*ln(F)*b*c^2*d*x+6*ln(F)*b*c^3-6)/b^4/ln(F)^4*F^(a+b*(d *x+c)^3)
Leaf count of result is larger than twice the leaf count of optimal. 302 vs. \(2 (122) = 244\).
Time = 0.08 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.44 \[ \int F^{a+b (c+d x)^3} (c+d x)^{11} \, dx=\frac {{\left ({\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \left (F\right )^{3} - 3 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + 6 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) - 6\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{4} d \log \left (F\right )^{4}} \] Input:
integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^11,x, algorithm="fricas")
Output:
1/3*((b^3*d^9*x^9 + 9*b^3*c*d^8*x^8 + 36*b^3*c^2*d^7*x^7 + 84*b^3*c^3*d^6* x^6 + 126*b^3*c^4*d^5*x^5 + 126*b^3*c^5*d^4*x^4 + 84*b^3*c^6*d^3*x^3 + 36* b^3*c^7*d^2*x^2 + 9*b^3*c^8*d*x + b^3*c^9)*log(F)^3 - 3*(b^2*d^6*x^6 + 6*b ^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + 20*b^2*c^3*d^3*x^3 + 15*b^2*c^4*d^2*x^ 2 + 6*b^2*c^5*d*x + b^2*c^6)*log(F)^2 + 6*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b *c^2*d*x + b*c^3)*log(F) - 6)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)/(b^4*d*log(F)^4)
Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (109) = 218\).
Time = 0.20 (sec) , antiderivative size = 536, normalized size of antiderivative = 4.32 \[ \int F^{a+b (c+d x)^3} (c+d x)^{11} \, dx=\begin {cases} \frac {F^{a + b \left (c + d x\right )^{3}} \left (b^{3} c^{9} \log {\left (F \right )}^{3} + 9 b^{3} c^{8} d x \log {\left (F \right )}^{3} + 36 b^{3} c^{7} d^{2} x^{2} \log {\left (F \right )}^{3} + 84 b^{3} c^{6} d^{3} x^{3} \log {\left (F \right )}^{3} + 126 b^{3} c^{5} d^{4} x^{4} \log {\left (F \right )}^{3} + 126 b^{3} c^{4} d^{5} x^{5} \log {\left (F \right )}^{3} + 84 b^{3} c^{3} d^{6} x^{6} \log {\left (F \right )}^{3} + 36 b^{3} c^{2} d^{7} x^{7} \log {\left (F \right )}^{3} + 9 b^{3} c d^{8} x^{8} \log {\left (F \right )}^{3} + b^{3} d^{9} x^{9} \log {\left (F \right )}^{3} - 3 b^{2} c^{6} \log {\left (F \right )}^{2} - 18 b^{2} c^{5} d x \log {\left (F \right )}^{2} - 45 b^{2} c^{4} d^{2} x^{2} \log {\left (F \right )}^{2} - 60 b^{2} c^{3} d^{3} x^{3} \log {\left (F \right )}^{2} - 45 b^{2} c^{2} d^{4} x^{4} \log {\left (F \right )}^{2} - 18 b^{2} c d^{5} x^{5} \log {\left (F \right )}^{2} - 3 b^{2} d^{6} x^{6} \log {\left (F \right )}^{2} + 6 b c^{3} \log {\left (F \right )} + 18 b c^{2} d x \log {\left (F \right )} + 18 b c d^{2} x^{2} \log {\left (F \right )} + 6 b d^{3} x^{3} \log {\left (F \right )} - 6\right )}{3 b^{4} d \log {\left (F \right )}^{4}} & \text {for}\: b^{4} d \log {\left (F \right )}^{4} \neq 0 \\c^{11} x + \frac {11 c^{10} d x^{2}}{2} + \frac {55 c^{9} d^{2} x^{3}}{3} + \frac {165 c^{8} d^{3} x^{4}}{4} + 66 c^{7} d^{4} x^{5} + 77 c^{6} d^{5} x^{6} + 66 c^{5} d^{6} x^{7} + \frac {165 c^{4} d^{7} x^{8}}{4} + \frac {55 c^{3} d^{8} x^{9}}{3} + \frac {11 c^{2} d^{9} x^{10}}{2} + c d^{10} x^{11} + \frac {d^{11} x^{12}}{12} & \text {otherwise} \end {cases} \] Input:
integrate(F**(a+b*(d*x+c)**3)*(d*x+c)**11,x)
Output:
Piecewise((F**(a + b*(c + d*x)**3)*(b**3*c**9*log(F)**3 + 9*b**3*c**8*d*x* log(F)**3 + 36*b**3*c**7*d**2*x**2*log(F)**3 + 84*b**3*c**6*d**3*x**3*log( F)**3 + 126*b**3*c**5*d**4*x**4*log(F)**3 + 126*b**3*c**4*d**5*x**5*log(F) **3 + 84*b**3*c**3*d**6*x**6*log(F)**3 + 36*b**3*c**2*d**7*x**7*log(F)**3 + 9*b**3*c*d**8*x**8*log(F)**3 + b**3*d**9*x**9*log(F)**3 - 3*b**2*c**6*lo g(F)**2 - 18*b**2*c**5*d*x*log(F)**2 - 45*b**2*c**4*d**2*x**2*log(F)**2 - 60*b**2*c**3*d**3*x**3*log(F)**2 - 45*b**2*c**2*d**4*x**4*log(F)**2 - 18*b **2*c*d**5*x**5*log(F)**2 - 3*b**2*d**6*x**6*log(F)**2 + 6*b*c**3*log(F) + 18*b*c**2*d*x*log(F) + 18*b*c*d**2*x**2*log(F) + 6*b*d**3*x**3*log(F) - 6 )/(3*b**4*d*log(F)**4), Ne(b**4*d*log(F)**4, 0)), (c**11*x + 11*c**10*d*x* *2/2 + 55*c**9*d**2*x**3/3 + 165*c**8*d**3*x**4/4 + 66*c**7*d**4*x**5 + 77 *c**6*d**5*x**6 + 66*c**5*d**6*x**7 + 165*c**4*d**7*x**8/4 + 55*c**3*d**8* x**9/3 + 11*c**2*d**9*x**10/2 + c*d**10*x**11 + d**11*x**12/12, True))
Leaf count of result is larger than twice the leaf count of optimal. 555 vs. \(2 (122) = 244\).
Time = 0.17 (sec) , antiderivative size = 555, normalized size of antiderivative = 4.48 \[ \int F^{a+b (c+d x)^3} (c+d x)^{11} \, dx=\frac {{\left (F^{b c^{3} + a} b^{3} d^{9} x^{9} \log \left (F\right )^{3} + 9 \, F^{b c^{3} + a} b^{3} c d^{8} x^{8} \log \left (F\right )^{3} + 36 \, F^{b c^{3} + a} b^{3} c^{2} d^{7} x^{7} \log \left (F\right )^{3} + F^{b c^{3} + a} b^{3} c^{9} \log \left (F\right )^{3} - 3 \, F^{b c^{3} + a} b^{2} c^{6} \log \left (F\right )^{2} + 3 \, {\left (28 \, F^{b c^{3} + a} b^{3} c^{3} d^{6} \log \left (F\right )^{3} - F^{b c^{3} + a} b^{2} d^{6} \log \left (F\right )^{2}\right )} x^{6} + 18 \, {\left (7 \, F^{b c^{3} + a} b^{3} c^{4} d^{5} \log \left (F\right )^{3} - F^{b c^{3} + a} b^{2} c d^{5} \log \left (F\right )^{2}\right )} x^{5} + 6 \, F^{b c^{3} + a} b c^{3} \log \left (F\right ) + 9 \, {\left (14 \, F^{b c^{3} + a} b^{3} c^{5} d^{4} \log \left (F\right )^{3} - 5 \, F^{b c^{3} + a} b^{2} c^{2} d^{4} \log \left (F\right )^{2}\right )} x^{4} + 6 \, {\left (14 \, F^{b c^{3} + a} b^{3} c^{6} d^{3} \log \left (F\right )^{3} - 10 \, F^{b c^{3} + a} b^{2} c^{3} d^{3} \log \left (F\right )^{2} + F^{b c^{3} + a} b d^{3} \log \left (F\right )\right )} x^{3} + 9 \, {\left (4 \, F^{b c^{3} + a} b^{3} c^{7} d^{2} \log \left (F\right )^{3} - 5 \, F^{b c^{3} + a} b^{2} c^{4} d^{2} \log \left (F\right )^{2} + 2 \, F^{b c^{3} + a} b c d^{2} \log \left (F\right )\right )} x^{2} + 9 \, {\left (F^{b c^{3} + a} b^{3} c^{8} d \log \left (F\right )^{3} - 2 \, F^{b c^{3} + a} b^{2} c^{5} d \log \left (F\right )^{2} + 2 \, F^{b c^{3} + a} b c^{2} d \log \left (F\right )\right )} x - 6 \, F^{b c^{3} + a}\right )} e^{\left (b d^{3} x^{3} \log \left (F\right ) + 3 \, b c d^{2} x^{2} \log \left (F\right ) + 3 \, b c^{2} d x \log \left (F\right )\right )}}{3 \, b^{4} d \log \left (F\right )^{4}} \] Input:
integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^11,x, algorithm="maxima")
Output:
1/3*(F^(b*c^3 + a)*b^3*d^9*x^9*log(F)^3 + 9*F^(b*c^3 + a)*b^3*c*d^8*x^8*lo g(F)^3 + 36*F^(b*c^3 + a)*b^3*c^2*d^7*x^7*log(F)^3 + F^(b*c^3 + a)*b^3*c^9 *log(F)^3 - 3*F^(b*c^3 + a)*b^2*c^6*log(F)^2 + 3*(28*F^(b*c^3 + a)*b^3*c^3 *d^6*log(F)^3 - F^(b*c^3 + a)*b^2*d^6*log(F)^2)*x^6 + 18*(7*F^(b*c^3 + a)* b^3*c^4*d^5*log(F)^3 - F^(b*c^3 + a)*b^2*c*d^5*log(F)^2)*x^5 + 6*F^(b*c^3 + a)*b*c^3*log(F) + 9*(14*F^(b*c^3 + a)*b^3*c^5*d^4*log(F)^3 - 5*F^(b*c^3 + a)*b^2*c^2*d^4*log(F)^2)*x^4 + 6*(14*F^(b*c^3 + a)*b^3*c^6*d^3*log(F)^3 - 10*F^(b*c^3 + a)*b^2*c^3*d^3*log(F)^2 + F^(b*c^3 + a)*b*d^3*log(F))*x^3 + 9*(4*F^(b*c^3 + a)*b^3*c^7*d^2*log(F)^3 - 5*F^(b*c^3 + a)*b^2*c^4*d^2*lo g(F)^2 + 2*F^(b*c^3 + a)*b*c*d^2*log(F))*x^2 + 9*(F^(b*c^3 + a)*b^3*c^8*d* log(F)^3 - 2*F^(b*c^3 + a)*b^2*c^5*d*log(F)^2 + 2*F^(b*c^3 + a)*b*c^2*d*lo g(F))*x - 6*F^(b*c^3 + a))*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3* b*c^2*d*x*log(F))/(b^4*d*log(F)^4)
Leaf count of result is larger than twice the leaf count of optimal. 1320 vs. \(2 (122) = 244\).
Time = 0.60 (sec) , antiderivative size = 1320, normalized size of antiderivative = 10.65 \[ \int F^{a+b (c+d x)^3} (c+d x)^{11} \, dx=\text {Too large to display} \] Input:
integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^11,x, algorithm="giac")
Output:
1/3*(b^3*d^9*x^9*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x* log(F) + b*c^3*log(F) + a*log(F))*log(F)^3 + 9*b^3*c*d^8*x^8*e^(b*d^3*x^3* log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log( F))*log(F)^3 + 36*b^3*c^2*d^7*x^7*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log( F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F)^3 + 84*b^3*c^3*d ^6*x^6*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b *c^3*log(F) + a*log(F))*log(F)^3 + 126*b^3*c^4*d^5*x^5*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*lo g(F)^3 + 126*b^3*c^5*d^4*x^4*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F)^3 + 84*b^3*c^6*d^3*x^ 3*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3* log(F) + a*log(F))*log(F)^3 + 36*b^3*c^7*d^2*x^2*e^(b*d^3*x^3*log(F) + 3*b *c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F)^3 - 3*b^2*d^6*x^6*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x* log(F) + b*c^3*log(F) + a*log(F))*log(F)^2 + 9*b^3*c^8*d*x*e^(b*d^3*x^3*lo g(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F) )*log(F)^3 - 18*b^2*c*d^5*x^5*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a*log(F))*log(F)^2 + b^3*c^9*e^(b*d^3 *x^3*log(F) + 3*b*c*d^2*x^2*log(F) + 3*b*c^2*d*x*log(F) + b*c^3*log(F) + a *log(F))*log(F)^3 - 45*b^2*c^2*d^4*x^4*e^(b*d^3*x^3*log(F) + 3*b*c*d^2*...
Time = 0.28 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.60 \[ \int F^{a+b (c+d x)^3} (c+d x)^{11} \, dx=F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (\frac {b^3\,c^9\,{\ln \left (F\right )}^3-3\,b^2\,c^6\,{\ln \left (F\right )}^2+6\,b\,c^3\,\ln \left (F\right )-6}{3\,b^4\,d\,{\ln \left (F\right )}^4}+\frac {d^8\,x^9}{3\,b\,\ln \left (F\right )}+\frac {3\,c\,d^7\,x^8}{b\,\ln \left (F\right )}+\frac {2\,d^2\,x^3\,\left (14\,b^2\,c^6\,{\ln \left (F\right )}^2-10\,b\,c^3\,\ln \left (F\right )+1\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {d^5\,x^6\,\left (28\,b\,c^3\,\ln \left (F\right )-1\right )}{b^2\,{\ln \left (F\right )}^2}+\frac {12\,c^2\,d^6\,x^7}{b\,\ln \left (F\right )}+\frac {3\,c^2\,x\,\left (b^2\,c^6\,{\ln \left (F\right )}^2-2\,b\,c^3\,\ln \left (F\right )+2\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {3\,c^2\,d^3\,x^4\,\left (14\,b\,c^3\,\ln \left (F\right )-5\right )}{b^2\,{\ln \left (F\right )}^2}+\frac {3\,c\,d\,x^2\,\left (4\,b^2\,c^6\,{\ln \left (F\right )}^2-5\,b\,c^3\,\ln \left (F\right )+2\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {6\,c\,d^4\,x^5\,\left (7\,b\,c^3\,\ln \left (F\right )-1\right )}{b^2\,{\ln \left (F\right )}^2}\right ) \] Input:
int(F^(a + b*(c + d*x)^3)*(c + d*x)^11,x)
Output:
F^(b*d^3*x^3)*F^(3*b*c^2*d*x)*F^a*F^(b*c^3)*F^(3*b*c*d^2*x^2)*((6*b*c^3*lo g(F) - 3*b^2*c^6*log(F)^2 + b^3*c^9*log(F)^3 - 6)/(3*b^4*d*log(F)^4) + (d^ 8*x^9)/(3*b*log(F)) + (3*c*d^7*x^8)/(b*log(F)) + (2*d^2*x^3*(14*b^2*c^6*lo g(F)^2 - 10*b*c^3*log(F) + 1))/(b^3*log(F)^3) + (d^5*x^6*(28*b*c^3*log(F) - 1))/(b^2*log(F)^2) + (12*c^2*d^6*x^7)/(b*log(F)) + (3*c^2*x*(b^2*c^6*log (F)^2 - 2*b*c^3*log(F) + 2))/(b^3*log(F)^3) + (3*c^2*d^3*x^4*(14*b*c^3*log (F) - 5))/(b^2*log(F)^2) + (3*c*d*x^2*(4*b^2*c^6*log(F)^2 - 5*b*c^3*log(F) + 2))/(b^3*log(F)^3) + (6*c*d^4*x^5*(7*b*c^3*log(F) - 1))/(b^2*log(F)^2))
Time = 0.16 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.94 \[ \int F^{a+b (c+d x)^3} (c+d x)^{11} \, dx=\frac {f^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a} \left (\mathrm {log}\left (f \right )^{3} b^{3} c^{9}+9 \mathrm {log}\left (f \right )^{3} b^{3} c^{8} d x +36 \mathrm {log}\left (f \right )^{3} b^{3} c^{7} d^{2} x^{2}+84 \mathrm {log}\left (f \right )^{3} b^{3} c^{6} d^{3} x^{3}+126 \mathrm {log}\left (f \right )^{3} b^{3} c^{5} d^{4} x^{4}+126 \mathrm {log}\left (f \right )^{3} b^{3} c^{4} d^{5} x^{5}+84 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d^{6} x^{6}+36 \mathrm {log}\left (f \right )^{3} b^{3} c^{2} d^{7} x^{7}+9 \mathrm {log}\left (f \right )^{3} b^{3} c \,d^{8} x^{8}+\mathrm {log}\left (f \right )^{3} b^{3} d^{9} x^{9}-3 \mathrm {log}\left (f \right )^{2} b^{2} c^{6}-18 \mathrm {log}\left (f \right )^{2} b^{2} c^{5} d x -45 \mathrm {log}\left (f \right )^{2} b^{2} c^{4} d^{2} x^{2}-60 \mathrm {log}\left (f \right )^{2} b^{2} c^{3} d^{3} x^{3}-45 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d^{4} x^{4}-18 \mathrm {log}\left (f \right )^{2} b^{2} c \,d^{5} x^{5}-3 \mathrm {log}\left (f \right )^{2} b^{2} d^{6} x^{6}+6 \,\mathrm {log}\left (f \right ) b \,c^{3}+18 \,\mathrm {log}\left (f \right ) b \,c^{2} d x +18 \,\mathrm {log}\left (f \right ) b c \,d^{2} x^{2}+6 \,\mathrm {log}\left (f \right ) b \,d^{3} x^{3}-6\right )}{3 \mathrm {log}\left (f \right )^{4} b^{4} d} \] Input:
int(F^(a+b*(d*x+c)^3)*(d*x+c)^11,x)
Output:
(f**(a + b*c**3 + 3*b*c**2*d*x + 3*b*c*d**2*x**2 + b*d**3*x**3)*(log(f)**3 *b**3*c**9 + 9*log(f)**3*b**3*c**8*d*x + 36*log(f)**3*b**3*c**7*d**2*x**2 + 84*log(f)**3*b**3*c**6*d**3*x**3 + 126*log(f)**3*b**3*c**5*d**4*x**4 + 1 26*log(f)**3*b**3*c**4*d**5*x**5 + 84*log(f)**3*b**3*c**3*d**6*x**6 + 36*l og(f)**3*b**3*c**2*d**7*x**7 + 9*log(f)**3*b**3*c*d**8*x**8 + log(f)**3*b* *3*d**9*x**9 - 3*log(f)**2*b**2*c**6 - 18*log(f)**2*b**2*c**5*d*x - 45*log (f)**2*b**2*c**4*d**2*x**2 - 60*log(f)**2*b**2*c**3*d**3*x**3 - 45*log(f)* *2*b**2*c**2*d**4*x**4 - 18*log(f)**2*b**2*c*d**5*x**5 - 3*log(f)**2*b**2* d**6*x**6 + 6*log(f)*b*c**3 + 18*log(f)*b*c**2*d*x + 18*log(f)*b*c*d**2*x* *2 + 6*log(f)*b*d**3*x**3 - 6))/(3*log(f)**4*b**4*d)