Integrand size = 21, antiderivative size = 88 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\frac {F^{a+b (c+d x)^3} \left (24-24 b (c+d x)^3 \log (F)+12 b^2 (c+d x)^6 \log ^2(F)-4 b^3 (c+d x)^9 \log ^3(F)+b^4 (c+d x)^{12} \log ^4(F)\right )}{3 b^5 d \log ^5(F)} \] Output:
1/3*F^(a+b*(d*x+c)^3)*(24-24*b*(d*x+c)^3*ln(F)+12*b^2*(d*x+c)^6*ln(F)^2-4* b^3*(d*x+c)^9*ln(F)^3+b^4*(d*x+c)^12*ln(F)^4)/b^5/d/ln(F)^5
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.33 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.35 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\frac {F^a \Gamma \left (5,-b (c+d x)^3 \log (F)\right )}{3 b^5 d \log ^5(F)} \] Input:
Integrate[F^(a + b*(c + d*x)^3)*(c + d*x)^14,x]
Output:
(F^a*Gamma[5, -(b*(c + d*x)^3*Log[F])])/(3*b^5*d*Log[F]^5)
Time = 0.46 (sec) , antiderivative size = 88, 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 = {2647}
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.
| \(\displaystyle \int (c+d x)^{14} F^{a+b (c+d x)^3} \, dx\) |
| \(\Big \downarrow \) 2647 |
| \(\displaystyle \frac {F^{a+b (c+d x)^3} \left (b^4 \log ^4(F) (c+d x)^{12}-4 b^3 \log ^3(F) (c+d x)^9+12 b^2 \log ^2(F) (c+d x)^6-24 b \log (F) (c+d x)^3+24\right )}{3 b^5 d \log ^5(F)}\) |
Input:
Int[F^(a + b*(c + d*x)^3)*(c + d*x)^14,x]
Output:
(F^(a + b*(c + d*x)^3)*(24 - 24*b*(c + d*x)^3*Log[F] + 12*b^2*(c + d*x)^6* Log[F]^2 - 4*b^3*(c + d*x)^9*Log[F]^3 + b^4*(c + d*x)^12*Log[F]^4))/(3*b^5 *d*Log[F]^5)
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ .), x_Symbol] :> With[{p = Simplify[(m + 1)/n]}, Simp[(-F^a)*((f/d)^m/(d*n* ((-b)*Log[F])^p))*Simplify[FunctionExpand[Gamma[p, (-b)*(c + d*x)^n*Log[F]] ]], x] /; IGtQ[p, 0]] /; FreeQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0] && !TrueQ[$UseGamma]
Leaf count of result is larger than twice the leaf count of optimal. \(561\) vs. \(2(86)=172\).
Time = 2.86 (sec) , antiderivative size = 562, normalized size of antiderivative = 6.39
| method | result | size |
| orering | \(\frac {\left (24-4 \ln \left (F \right )^{3} b^{3} c^{9}+12 \ln \left (F \right )^{2} b^{2} c^{6}-24 \ln \left (F \right ) b \,d^{3} x^{3}-24 \ln \left (F \right ) b \,c^{3}+d^{12} x^{12} \ln \left (F \right )^{4} b^{4}-4 d^{9} x^{9} \ln \left (F \right )^{3} b^{3}+12 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}-72 \ln \left (F \right ) b c \,d^{2} x^{2}-72 \ln \left (F \right ) b \,c^{2} d x +12 c \,d^{11} x^{11} \ln \left (F \right )^{4} b^{4}+66 c^{2} d^{10} x^{10} \ln \left (F \right )^{4} b^{4}+220 \ln \left (F \right )^{4} b^{4} c^{3} d^{9} x^{9}+495 \ln \left (F \right )^{4} b^{4} c^{4} d^{8} x^{8}+\ln \left (F \right )^{4} b^{4} c^{12}+792 \ln \left (F \right )^{4} b^{4} c^{5} d^{7} x^{7}+924 \ln \left (F \right )^{4} b^{4} c^{6} d^{6} x^{6}+792 \ln \left (F \right )^{4} b^{4} c^{7} d^{5} x^{5}+495 \ln \left (F \right )^{4} b^{4} c^{8} d^{4} x^{4}+220 \ln \left (F \right )^{4} b^{4} c^{9} d^{3} x^{3}-36 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}+66 \ln \left (F \right )^{4} b^{4} c^{10} d^{2} x^{2}-144 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}+12 \ln \left (F \right )^{4} b^{4} c^{11} d x -336 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}-504 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}-504 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}-336 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}-144 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}-36 \ln \left (F \right )^{3} b^{3} c^{8} d x +72 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+180 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+240 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+180 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+72 \ln \left (F \right )^{2} b^{2} c^{5} d x \right ) F^{a +b \left (d x +c \right )^{3}}}{3 d \ln \left (F \right )^{5} b^{5}}\) | \(562\) |
| gosper | \(\frac {\left (24-4 \ln \left (F \right )^{3} b^{3} c^{9}+12 \ln \left (F \right )^{2} b^{2} c^{6}-24 \ln \left (F \right ) b \,d^{3} x^{3}-24 \ln \left (F \right ) b \,c^{3}+d^{12} x^{12} \ln \left (F \right )^{4} b^{4}-4 d^{9} x^{9} \ln \left (F \right )^{3} b^{3}+12 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}-72 \ln \left (F \right ) b c \,d^{2} x^{2}-72 \ln \left (F \right ) b \,c^{2} d x +12 c \,d^{11} x^{11} \ln \left (F \right )^{4} b^{4}+66 c^{2} d^{10} x^{10} \ln \left (F \right )^{4} b^{4}+220 \ln \left (F \right )^{4} b^{4} c^{3} d^{9} x^{9}+495 \ln \left (F \right )^{4} b^{4} c^{4} d^{8} x^{8}+\ln \left (F \right )^{4} b^{4} c^{12}+792 \ln \left (F \right )^{4} b^{4} c^{5} d^{7} x^{7}+924 \ln \left (F \right )^{4} b^{4} c^{6} d^{6} x^{6}+792 \ln \left (F \right )^{4} b^{4} c^{7} d^{5} x^{5}+495 \ln \left (F \right )^{4} b^{4} c^{8} d^{4} x^{4}+220 \ln \left (F \right )^{4} b^{4} c^{9} d^{3} x^{3}-36 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}+66 \ln \left (F \right )^{4} b^{4} c^{10} d^{2} x^{2}-144 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}+12 \ln \left (F \right )^{4} b^{4} c^{11} d x -336 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}-504 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}-504 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}-336 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}-144 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}-36 \ln \left (F \right )^{3} b^{3} c^{8} d x +72 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+180 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+240 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+180 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+72 \ln \left (F \right )^{2} b^{2} c^{5} d x \right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 d \ln \left (F \right )^{5} b^{5}}\) | \(584\) |
| risch | \(\frac {\left (24-4 \ln \left (F \right )^{3} b^{3} c^{9}+12 \ln \left (F \right )^{2} b^{2} c^{6}-24 \ln \left (F \right ) b \,d^{3} x^{3}-24 \ln \left (F \right ) b \,c^{3}+d^{12} x^{12} \ln \left (F \right )^{4} b^{4}-4 d^{9} x^{9} \ln \left (F \right )^{3} b^{3}+12 d^{6} x^{6} \ln \left (F \right )^{2} b^{2}-72 \ln \left (F \right ) b c \,d^{2} x^{2}-72 \ln \left (F \right ) b \,c^{2} d x +12 c \,d^{11} x^{11} \ln \left (F \right )^{4} b^{4}+66 c^{2} d^{10} x^{10} \ln \left (F \right )^{4} b^{4}+220 \ln \left (F \right )^{4} b^{4} c^{3} d^{9} x^{9}+495 \ln \left (F \right )^{4} b^{4} c^{4} d^{8} x^{8}+\ln \left (F \right )^{4} b^{4} c^{12}+792 \ln \left (F \right )^{4} b^{4} c^{5} d^{7} x^{7}+924 \ln \left (F \right )^{4} b^{4} c^{6} d^{6} x^{6}+792 \ln \left (F \right )^{4} b^{4} c^{7} d^{5} x^{5}+495 \ln \left (F \right )^{4} b^{4} c^{8} d^{4} x^{4}+220 \ln \left (F \right )^{4} b^{4} c^{9} d^{3} x^{3}-36 c \,d^{8} x^{8} \ln \left (F \right )^{3} b^{3}+66 \ln \left (F \right )^{4} b^{4} c^{10} d^{2} x^{2}-144 c^{2} d^{7} x^{7} \ln \left (F \right )^{3} b^{3}+12 \ln \left (F \right )^{4} b^{4} c^{11} d x -336 \ln \left (F \right )^{3} b^{3} c^{3} d^{6} x^{6}-504 \ln \left (F \right )^{3} b^{3} c^{4} d^{5} x^{5}-504 \ln \left (F \right )^{3} b^{3} c^{5} d^{4} x^{4}-336 \ln \left (F \right )^{3} b^{3} c^{6} d^{3} x^{3}-144 \ln \left (F \right )^{3} b^{3} c^{7} d^{2} x^{2}-36 \ln \left (F \right )^{3} b^{3} c^{8} d x +72 c \,d^{5} x^{5} \ln \left (F \right )^{2} b^{2}+180 c^{2} d^{4} x^{4} \ln \left (F \right )^{2} b^{2}+240 \ln \left (F \right )^{2} b^{2} c^{3} d^{3} x^{3}+180 \ln \left (F \right )^{2} b^{2} c^{4} d^{2} x^{2}+72 \ln \left (F \right )^{2} b^{2} c^{5} d x \right ) F^{b \,d^{3} x^{3}+3 b c \,d^{2} x^{2}+3 b \,c^{2} d x +b \,c^{3}+a}}{3 d \ln \left (F \right )^{5} b^{5}}\) | \(584\) |
| norman | \(\frac {\left (\ln \left (F \right )^{4} b^{4} c^{12}-4 \ln \left (F \right )^{3} b^{3} c^{9}+12 \ln \left (F \right )^{2} b^{2} c^{6}-24 \ln \left (F \right ) b \,c^{3}+24\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 d \ln \left (F \right )^{5} b^{5}}+\frac {d^{11} x^{12} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right ) b}+\frac {4 c^{2} \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 ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {4 d^{2} \left (55 \ln \left (F \right )^{3} b^{3} c^{9}-84 \ln \left (F \right )^{2} b^{2} c^{6}+60 \ln \left (F \right ) b \,c^{3}-6\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{4} b^{4}}+\frac {4 d^{5} \left (77 \ln \left (F \right )^{2} b^{2} c^{6}-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 )^{3} b^{3}}+\frac {4 d^{8} \left (55 \ln \left (F \right ) b \,c^{3}-1\right ) x^{9} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{3 \ln \left (F \right )^{2} b^{2}}+\frac {22 d^{9} c^{2} x^{10} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {4 d^{10} c \,x^{11} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {2 c d \left (11 \ln \left (F \right )^{3} b^{3} c^{9}-24 \ln \left (F \right )^{2} b^{2} c^{6}+30 \ln \left (F \right ) b \,c^{3}-12\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {24 c \,d^{4} \left (11 \ln \left (F \right )^{2} b^{2} c^{6}-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 )^{3} b^{3}}+\frac {3 c \,d^{7} \left (55 \ln \left (F \right ) b \,c^{3}-4\right ) x^{8} {\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 (55 \ln \left (F \right )^{2} b^{2} c^{6}-56 \ln \left (F \right ) b \,c^{3}+20\right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {24 c^{2} d^{6} \left (11 \ln \left (F \right ) b \,c^{3}-2\right ) x^{7} {\mathrm e}^{\left (a +b \left (d x +c \right )^{3}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}\) | \(640\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(1005\) |
Input:
int(F^(a+b*(d*x+c)^3)*(d*x+c)^14,x,method=_RETURNVERBOSE)
Output:
1/3*(24-4*ln(F)^3*b^3*c^9+12*ln(F)^2*b^2*c^6-24*ln(F)*b*d^3*x^3-24*ln(F)*b *c^3+d^12*x^12*ln(F)^4*b^4-4*d^9*x^9*ln(F)^3*b^3+12*d^6*x^6*ln(F)^2*b^2-72 *ln(F)*b*c*d^2*x^2-72*ln(F)*b*c^2*d*x+12*c*d^11*x^11*ln(F)^4*b^4+66*c^2*d^ 10*x^10*ln(F)^4*b^4+220*ln(F)^4*b^4*c^3*d^9*x^9+495*ln(F)^4*b^4*c^4*d^8*x^ 8+ln(F)^4*b^4*c^12+792*ln(F)^4*b^4*c^5*d^7*x^7+924*ln(F)^4*b^4*c^6*d^6*x^6 +792*ln(F)^4*b^4*c^7*d^5*x^5+495*ln(F)^4*b^4*c^8*d^4*x^4+220*ln(F)^4*b^4*c ^9*d^3*x^3-36*c*d^8*x^8*ln(F)^3*b^3+66*ln(F)^4*b^4*c^10*d^2*x^2-144*c^2*d^ 7*x^7*ln(F)^3*b^3+12*ln(F)^4*b^4*c^11*d*x-336*ln(F)^3*b^3*c^3*d^6*x^6-504* ln(F)^3*b^3*c^4*d^5*x^5-504*ln(F)^3*b^3*c^5*d^4*x^4-336*ln(F)^3*b^3*c^6*d^ 3*x^3-144*ln(F)^3*b^3*c^7*d^2*x^2-36*ln(F)^3*b^3*c^8*d*x+72*c*d^5*x^5*ln(F )^2*b^2+180*c^2*d^4*x^4*ln(F)^2*b^2+240*ln(F)^2*b^2*c^3*d^3*x^3+180*ln(F)^ 2*b^2*c^4*d^2*x^2+72*ln(F)^2*b^2*c^5*d*x)/d/ln(F)^5/b^5*F^(a+b*(d*x+c)^3)
Leaf count of result is larger than twice the leaf count of optimal. 474 vs. \(2 (86) = 172\).
Time = 0.10 (sec) , antiderivative size = 474, normalized size of antiderivative = 5.39 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\frac {{\left ({\left (b^{4} d^{12} x^{12} + 12 \, b^{4} c d^{11} x^{11} + 66 \, b^{4} c^{2} d^{10} x^{10} + 220 \, b^{4} c^{3} d^{9} x^{9} + 495 \, b^{4} c^{4} d^{8} x^{8} + 792 \, b^{4} c^{5} d^{7} x^{7} + 924 \, b^{4} c^{6} d^{6} x^{6} + 792 \, b^{4} c^{7} d^{5} x^{5} + 495 \, b^{4} c^{8} d^{4} x^{4} + 220 \, b^{4} c^{9} d^{3} x^{3} + 66 \, b^{4} c^{10} d^{2} x^{2} + 12 \, b^{4} c^{11} d x + b^{4} c^{12}\right )} \log \left (F\right )^{4} - 4 \, {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} - 24 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) + 24\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{3 \, b^{5} d \log \left (F\right )^{5}} \] Input:
integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^14,x, algorithm="fricas")
Output:
1/3*((b^4*d^12*x^12 + 12*b^4*c*d^11*x^11 + 66*b^4*c^2*d^10*x^10 + 220*b^4* c^3*d^9*x^9 + 495*b^4*c^4*d^8*x^8 + 792*b^4*c^5*d^7*x^7 + 924*b^4*c^6*d^6* x^6 + 792*b^4*c^7*d^5*x^5 + 495*b^4*c^8*d^4*x^4 + 220*b^4*c^9*d^3*x^3 + 66 *b^4*c^10*d^2*x^2 + 12*b^4*c^11*d*x + b^4*c^12)*log(F)^4 - 4*(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 + 12*(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 - 24*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 )*log(F) + 24)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a)/(b^ 5*d*log(F)^5)
Leaf count of result is larger than twice the leaf count of optimal. 821 vs. \(2 (87) = 174\).
Time = 0.25 (sec) , antiderivative size = 821, normalized size of antiderivative = 9.33 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx =\text {Too large to display} \] Input:
integrate(F**(a+b*(d*x+c)**3)*(d*x+c)**14,x)
Output:
Piecewise((F**(a + b*(c + d*x)**3)*(b**4*c**12*log(F)**4 + 12*b**4*c**11*d *x*log(F)**4 + 66*b**4*c**10*d**2*x**2*log(F)**4 + 220*b**4*c**9*d**3*x**3 *log(F)**4 + 495*b**4*c**8*d**4*x**4*log(F)**4 + 792*b**4*c**7*d**5*x**5*l og(F)**4 + 924*b**4*c**6*d**6*x**6*log(F)**4 + 792*b**4*c**5*d**7*x**7*log (F)**4 + 495*b**4*c**4*d**8*x**8*log(F)**4 + 220*b**4*c**3*d**9*x**9*log(F )**4 + 66*b**4*c**2*d**10*x**10*log(F)**4 + 12*b**4*c*d**11*x**11*log(F)** 4 + b**4*d**12*x**12*log(F)**4 - 4*b**3*c**9*log(F)**3 - 36*b**3*c**8*d*x* log(F)**3 - 144*b**3*c**7*d**2*x**2*log(F)**3 - 336*b**3*c**6*d**3*x**3*lo g(F)**3 - 504*b**3*c**5*d**4*x**4*log(F)**3 - 504*b**3*c**4*d**5*x**5*log( F)**3 - 336*b**3*c**3*d**6*x**6*log(F)**3 - 144*b**3*c**2*d**7*x**7*log(F) **3 - 36*b**3*c*d**8*x**8*log(F)**3 - 4*b**3*d**9*x**9*log(F)**3 + 12*b**2 *c**6*log(F)**2 + 72*b**2*c**5*d*x*log(F)**2 + 180*b**2*c**4*d**2*x**2*log (F)**2 + 240*b**2*c**3*d**3*x**3*log(F)**2 + 180*b**2*c**2*d**4*x**4*log(F )**2 + 72*b**2*c*d**5*x**5*log(F)**2 + 12*b**2*d**6*x**6*log(F)**2 - 24*b* c**3*log(F) - 72*b*c**2*d*x*log(F) - 72*b*c*d**2*x**2*log(F) - 24*b*d**3*x **3*log(F) + 24)/(3*b**5*d*log(F)**5), Ne(b**5*d*log(F)**5, 0)), (c**14*x + 7*c**13*d*x**2 + 91*c**12*d**2*x**3/3 + 91*c**11*d**3*x**4 + 1001*c**10* d**4*x**5/5 + 1001*c**9*d**5*x**6/3 + 429*c**8*d**6*x**7 + 429*c**7*d**7*x **8 + 1001*c**6*d**8*x**9/3 + 1001*c**5*d**9*x**10/5 + 91*c**4*d**10*x**11 + 91*c**3*d**11*x**12/3 + 7*c**2*d**12*x**13 + c*d**13*x**14 + d**14*x...
Leaf count of result is larger than twice the leaf count of optimal. 874 vs. \(2 (86) = 172\).
Time = 0.18 (sec) , antiderivative size = 874, normalized size of antiderivative = 9.93 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx =\text {Too large to display} \] Input:
integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^14,x, algorithm="maxima")
Output:
1/3*(F^(b*c^3 + a)*b^4*d^12*x^12*log(F)^4 + 12*F^(b*c^3 + a)*b^4*c*d^11*x^ 11*log(F)^4 + 66*F^(b*c^3 + a)*b^4*c^2*d^10*x^10*log(F)^4 + F^(b*c^3 + a)* b^4*c^12*log(F)^4 - 4*F^(b*c^3 + a)*b^3*c^9*log(F)^3 + 12*F^(b*c^3 + a)*b^ 2*c^6*log(F)^2 + 4*(55*F^(b*c^3 + a)*b^4*c^3*d^9*log(F)^4 - F^(b*c^3 + a)* b^3*d^9*log(F)^3)*x^9 + 9*(55*F^(b*c^3 + a)*b^4*c^4*d^8*log(F)^4 - 4*F^(b* c^3 + a)*b^3*c*d^8*log(F)^3)*x^8 + 72*(11*F^(b*c^3 + a)*b^4*c^5*d^7*log(F) ^4 - 2*F^(b*c^3 + a)*b^3*c^2*d^7*log(F)^3)*x^7 + 12*(77*F^(b*c^3 + a)*b^4* c^6*d^6*log(F)^4 - 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 + 72*(11*F^(b*c^3 + a)*b^4*c^7*d^5*log(F)^4 - 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 - 24 *F^(b*c^3 + a)*b*c^3*log(F) + 9*(55*F^(b*c^3 + a)*b^4*c^8*d^4*log(F)^4 - 5 6*F^(b*c^3 + a)*b^3*c^5*d^4*log(F)^3 + 20*F^(b*c^3 + a)*b^2*c^2*d^4*log(F) ^2)*x^4 + 4*(55*F^(b*c^3 + a)*b^4*c^9*d^3*log(F)^4 - 84*F^(b*c^3 + a)*b^3* c^6*d^3*log(F)^3 + 60*F^(b*c^3 + a)*b^2*c^3*d^3*log(F)^2 - 6*F^(b*c^3 + a) *b*d^3*log(F))*x^3 + 6*(11*F^(b*c^3 + a)*b^4*c^10*d^2*log(F)^4 - 24*F^(b*c ^3 + a)*b^3*c^7*d^2*log(F)^3 + 30*F^(b*c^3 + a)*b^2*c^4*d^2*log(F)^2 - 12* F^(b*c^3 + a)*b*c*d^2*log(F))*x^2 + 12*(F^(b*c^3 + a)*b^4*c^11*d*log(F)^4 - 3*F^(b*c^3 + a)*b^3*c^8*d*log(F)^3 + 6*F^(b*c^3 + a)*b^2*c^5*d*log(F)^2 - 6*F^(b*c^3 + a)*b*c^2*d*log(F))*x + 24*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^5*d*log(F)^5)
Exception generated. \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(F^(a+b*(d*x+c)^3)*(d*x+c)^14,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Polynomial exponent overflow. Error : Bad Argument Value
Time = 0.39 (sec) , antiderivative size = 487, normalized size of antiderivative = 5.53 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx=F^{b\,d^3\,x^3}\,F^{3\,b\,c^2\,d\,x}\,F^a\,F^{b\,c^3}\,F^{3\,b\,c\,d^2\,x^2}\,\left (\frac {b^4\,c^{12}\,{\ln \left (F\right )}^4-4\,b^3\,c^9\,{\ln \left (F\right )}^3+12\,b^2\,c^6\,{\ln \left (F\right )}^2-24\,b\,c^3\,\ln \left (F\right )+24}{3\,b^5\,d\,{\ln \left (F\right )}^5}+\frac {d^{11}\,x^{12}}{3\,b\,\ln \left (F\right )}+\frac {4\,c\,d^{10}\,x^{11}}{b\,\ln \left (F\right )}+\frac {4\,d^2\,x^3\,\left (55\,b^3\,c^9\,{\ln \left (F\right )}^3-84\,b^2\,c^6\,{\ln \left (F\right )}^2+60\,b\,c^3\,\ln \left (F\right )-6\right )}{3\,b^4\,{\ln \left (F\right )}^4}+\frac {4\,d^5\,x^6\,\left (77\,b^2\,c^6\,{\ln \left (F\right )}^2-28\,b\,c^3\,\ln \left (F\right )+1\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {4\,d^8\,x^9\,\left (55\,b\,c^3\,\ln \left (F\right )-1\right )}{3\,b^2\,{\ln \left (F\right )}^2}+\frac {22\,c^2\,d^9\,x^{10}}{b\,\ln \left (F\right )}+\frac {4\,c^2\,x\,\left (b^3\,c^9\,{\ln \left (F\right )}^3-3\,b^2\,c^6\,{\ln \left (F\right )}^2+6\,b\,c^3\,\ln \left (F\right )-6\right )}{b^4\,{\ln \left (F\right )}^4}+\frac {3\,c^2\,d^3\,x^4\,\left (55\,b^2\,c^6\,{\ln \left (F\right )}^2-56\,b\,c^3\,\ln \left (F\right )+20\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {24\,c^2\,d^6\,x^7\,\left (11\,b\,c^3\,\ln \left (F\right )-2\right )}{b^2\,{\ln \left (F\right )}^2}+\frac {2\,c\,d\,x^2\,\left (11\,b^3\,c^9\,{\ln \left (F\right )}^3-24\,b^2\,c^6\,{\ln \left (F\right )}^2+30\,b\,c^3\,\ln \left (F\right )-12\right )}{b^4\,{\ln \left (F\right )}^4}+\frac {24\,c\,d^4\,x^5\,\left (11\,b^2\,c^6\,{\ln \left (F\right )}^2-7\,b\,c^3\,\ln \left (F\right )+1\right )}{b^3\,{\ln \left (F\right )}^3}+\frac {3\,c\,d^7\,x^8\,\left (55\,b\,c^3\,\ln \left (F\right )-4\right )}{b^2\,{\ln \left (F\right )}^2}\right ) \] Input:
int(F^(a + b*(c + d*x)^3)*(c + d*x)^14,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)*((12*b^2*c^6 *log(F)^2 - 24*b*c^3*log(F) - 4*b^3*c^9*log(F)^3 + b^4*c^12*log(F)^4 + 24) /(3*b^5*d*log(F)^5) + (d^11*x^12)/(3*b*log(F)) + (4*c*d^10*x^11)/(b*log(F) ) + (4*d^2*x^3*(60*b*c^3*log(F) - 84*b^2*c^6*log(F)^2 + 55*b^3*c^9*log(F)^ 3 - 6))/(3*b^4*log(F)^4) + (4*d^5*x^6*(77*b^2*c^6*log(F)^2 - 28*b*c^3*log( F) + 1))/(b^3*log(F)^3) + (4*d^8*x^9*(55*b*c^3*log(F) - 1))/(3*b^2*log(F)^ 2) + (22*c^2*d^9*x^10)/(b*log(F)) + (4*c^2*x*(6*b*c^3*log(F) - 3*b^2*c^6*l og(F)^2 + b^3*c^9*log(F)^3 - 6))/(b^4*log(F)^4) + (3*c^2*d^3*x^4*(55*b^2*c ^6*log(F)^2 - 56*b*c^3*log(F) + 20))/(b^3*log(F)^3) + (24*c^2*d^6*x^7*(11* b*c^3*log(F) - 2))/(b^2*log(F)^2) + (2*c*d*x^2*(30*b*c^3*log(F) - 24*b^2*c ^6*log(F)^2 + 11*b^3*c^9*log(F)^3 - 12))/(b^4*log(F)^4) + (24*c*d^4*x^5*(1 1*b^2*c^6*log(F)^2 - 7*b*c^3*log(F) + 1))/(b^3*log(F)^3) + (3*c*d^7*x^8*(5 5*b*c^3*log(F) - 4))/(b^2*log(F)^2))
Time = 0.18 (sec) , antiderivative size = 583, normalized size of antiderivative = 6.62 \[ \int F^{a+b (c+d x)^3} (c+d x)^{14} \, dx =\text {Too large to display} \] Input:
int(F^(a+b*(d*x+c)^3)*(d*x+c)^14,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)**4 *b**4*c**12 + 12*log(f)**4*b**4*c**11*d*x + 66*log(f)**4*b**4*c**10*d**2*x **2 + 220*log(f)**4*b**4*c**9*d**3*x**3 + 495*log(f)**4*b**4*c**8*d**4*x** 4 + 792*log(f)**4*b**4*c**7*d**5*x**5 + 924*log(f)**4*b**4*c**6*d**6*x**6 + 792*log(f)**4*b**4*c**5*d**7*x**7 + 495*log(f)**4*b**4*c**4*d**8*x**8 + 220*log(f)**4*b**4*c**3*d**9*x**9 + 66*log(f)**4*b**4*c**2*d**10*x**10 + 1 2*log(f)**4*b**4*c*d**11*x**11 + log(f)**4*b**4*d**12*x**12 - 4*log(f)**3* b**3*c**9 - 36*log(f)**3*b**3*c**8*d*x - 144*log(f)**3*b**3*c**7*d**2*x**2 - 336*log(f)**3*b**3*c**6*d**3*x**3 - 504*log(f)**3*b**3*c**5*d**4*x**4 - 504*log(f)**3*b**3*c**4*d**5*x**5 - 336*log(f)**3*b**3*c**3*d**6*x**6 - 1 44*log(f)**3*b**3*c**2*d**7*x**7 - 36*log(f)**3*b**3*c*d**8*x**8 - 4*log(f )**3*b**3*d**9*x**9 + 12*log(f)**2*b**2*c**6 + 72*log(f)**2*b**2*c**5*d*x + 180*log(f)**2*b**2*c**4*d**2*x**2 + 240*log(f)**2*b**2*c**3*d**3*x**3 + 180*log(f)**2*b**2*c**2*d**4*x**4 + 72*log(f)**2*b**2*c*d**5*x**5 + 12*log (f)**2*b**2*d**6*x**6 - 24*log(f)*b*c**3 - 72*log(f)*b*c**2*d*x - 72*log(f )*b*c*d**2*x**2 - 24*log(f)*b*d**3*x**3 + 24))/(3*log(f)**5*b**5*d)