Integrand size = 48, antiderivative size = 141 \[ \int F^{c (a+b x)} \left (d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4\right ) \, dx=\frac {24 e^4 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac {24 e^3 F^{c (a+b x)} (d+e x)}{b^4 c^4 \log ^4(F)}+\frac {12 e^2 F^{c (a+b x)} (d+e x)^2}{b^3 c^3 \log ^3(F)}-\frac {4 e F^{c (a+b x)} (d+e x)^3}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^4}{b c \log (F)} \] Output:
24*e^4*F^(c*(b*x+a))/b^5/c^5/ln(F)^5-24*e^3*F^(c*(b*x+a))*(e*x+d)/b^4/c^4/ ln(F)^4+12*e^2*F^(c*(b*x+a))*(e*x+d)^2/b^3/c^3/ln(F)^3-4*e*F^(c*(b*x+a))*( e*x+d)^3/b^2/c^2/ln(F)^2+F^(c*(b*x+a))*(e*x+d)^4/b/c/ln(F)
Time = 0.03 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71 \[ \int F^{c (a+b x)} \left (d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4\right ) \, dx=\frac {F^{c (a+b x)} \left (24 e^4-24 b c e^3 (d+e x) \log (F)+12 b^2 c^2 e^2 (d+e x)^2 \log ^2(F)-4 b^3 c^3 e (d+e x)^3 \log ^3(F)+b^4 c^4 (d+e x)^4 \log ^4(F)\right )}{b^5 c^5 \log ^5(F)} \] Input:
Integrate[F^(c*(a + b*x))*(d^4 + 4*d^3*e*x + 6*d^2*e^2*x^2 + 4*d*e^3*x^3 + e^4*x^4),x]
Output:
(F^(c*(a + b*x))*(24*e^4 - 24*b*c*e^3*(d + e*x)*Log[F] + 12*b^2*c^2*e^2*(d + e*x)^2*Log[F]^2 - 4*b^3*c^3*e*(d + e*x)^3*Log[F]^3 + b^4*c^4*(d + e*x)^ 4*Log[F]^4))/(b^5*c^5*Log[F]^5)
Time = 0.79 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2006, 2607, 2607, 2607, 2607, 2624}
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.
| \(\displaystyle \int \left (d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4\right ) F^{c (a+b x)} \, dx\) |
| \(\Big \downarrow \) 2006 |
| \(\displaystyle \int (d+e x)^4 F^{c (a+b x)}dx\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle \frac {(d+e x)^4 F^{c (a+b x)}}{b c \log (F)}-\frac {4 e \int F^{c (a+b x)} (d+e x)^3dx}{b c \log (F)}\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle \frac {(d+e x)^4 F^{c (a+b x)}}{b c \log (F)}-\frac {4 e \left (\frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \int F^{c (a+b x)} (d+e x)^2dx}{b c \log (F)}\right )}{b c \log (F)}\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle \frac {(d+e x)^4 F^{c (a+b x)}}{b c \log (F)}-\frac {4 e \left (\frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \left (\frac {(d+e x)^2 F^{c (a+b x)}}{b c \log (F)}-\frac {2 e \int F^{c (a+b x)} (d+e x)dx}{b c \log (F)}\right )}{b c \log (F)}\right )}{b c \log (F)}\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle \frac {(d+e x)^4 F^{c (a+b x)}}{b c \log (F)}-\frac {4 e \left (\frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \left (\frac {(d+e x)^2 F^{c (a+b x)}}{b c \log (F)}-\frac {2 e \left (\frac {(d+e x) F^{c (a+b x)}}{b c \log (F)}-\frac {e \int F^{c (a+b x)}dx}{b c \log (F)}\right )}{b c \log (F)}\right )}{b c \log (F)}\right )}{b c \log (F)}\) |
| \(\Big \downarrow \) 2624 |
| \(\displaystyle \frac {(d+e x)^4 F^{c (a+b x)}}{b c \log (F)}-\frac {4 e \left (\frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \left (\frac {(d+e x)^2 F^{c (a+b x)}}{b c \log (F)}-\frac {2 e \left (\frac {(d+e x) F^{c (a+b x)}}{b c \log (F)}-\frac {e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}\right )}{b c \log (F)}\right )}{b c \log (F)}\right )}{b c \log (F)}\) |
Input:
Int[F^(c*(a + b*x))*(d^4 + 4*d^3*e*x + 6*d^2*e^2*x^2 + 4*d*e^3*x^3 + e^4*x ^4),x]
Output:
(F^(c*(a + b*x))*(d + e*x)^4)/(b*c*Log[F]) - (4*e*((F^(c*(a + b*x))*(d + e *x)^3)/(b*c*Log[F]) - (3*e*((F^(c*(a + b*x))*(d + e*x)^2)/(b*c*Log[F]) - ( 2*e*(-((e*F^(c*(a + b*x)))/(b^2*c^2*Log[F]^2)) + (F^(c*(a + b*x))*(d + e*x ))/(b*c*Log[F])))/(b*c*Log[F])))/(b*c*Log[F])))/(b*c*Log[F])
Int[(u_.)*(Px_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^Expon[Px , x], x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; PolyQ[Px, x] && GtQ[Expon[P x, x], 1] && NeQ[Coeff[Px, x, 0], 0] && !MatchQ[Px, (a_.)*(v_)^Expon[Px, x ] /; FreeQ[a, x] && LinearQ[v, x]]
Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m _.), x_Symbol] :> Simp[(c + d*x)^m*((b*F^(g*(e + f*x)))^n/(f*g*n*Log[F])), x] - Simp[d*(m/(f*g*n*Log[F])) Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x)))^ n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2* m] && !TrueQ[$UseGamma]
Int[((F_)^(v_))^(n_.), x_Symbol] :> Simp[(F^v)^n/(n*Log[F]*D[v, x]), x] /; FreeQ[{F, n}, x] && LinearQ[v, x]
Time = 0.07 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.84
| method | result | size |
| gosper | \(\frac {\left (e^{4} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}+4 \ln \left (F \right )^{4} b^{4} c^{4} d \,e^{3} x^{3}+6 \ln \left (F \right )^{4} b^{4} c^{4} d^{2} e^{2} x^{2}+4 \ln \left (F \right )^{4} b^{4} c^{4} d^{3} e x +\ln \left (F \right )^{4} b^{4} c^{4} d^{4}-4 \ln \left (F \right )^{3} b^{3} c^{3} e^{4} x^{3}-12 \ln \left (F \right )^{3} b^{3} c^{3} d \,e^{3} x^{2}-12 \ln \left (F \right )^{3} b^{3} c^{3} d^{2} e^{2} x -4 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} e +12 \ln \left (F \right )^{2} b^{2} c^{2} e^{4} x^{2}+24 \ln \left (F \right )^{2} b^{2} c^{2} d \,e^{3} x +12 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e^{2}-24 \ln \left (F \right ) b c \,e^{4} x -24 d \,e^{3} \ln \left (F \right ) b c +24 e^{4}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(260\) |
| risch | \(\frac {\left (e^{4} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}+4 \ln \left (F \right )^{4} b^{4} c^{4} d \,e^{3} x^{3}+6 \ln \left (F \right )^{4} b^{4} c^{4} d^{2} e^{2} x^{2}+4 \ln \left (F \right )^{4} b^{4} c^{4} d^{3} e x +\ln \left (F \right )^{4} b^{4} c^{4} d^{4}-4 \ln \left (F \right )^{3} b^{3} c^{3} e^{4} x^{3}-12 \ln \left (F \right )^{3} b^{3} c^{3} d \,e^{3} x^{2}-12 \ln \left (F \right )^{3} b^{3} c^{3} d^{2} e^{2} x -4 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} e +12 \ln \left (F \right )^{2} b^{2} c^{2} e^{4} x^{2}+24 \ln \left (F \right )^{2} b^{2} c^{2} d \,e^{3} x +12 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e^{2}-24 \ln \left (F \right ) b c \,e^{4} x -24 d \,e^{3} \ln \left (F \right ) b c +24 e^{4}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(260\) |
| norman | \(\frac {\left (\ln \left (F \right )^{4} b^{4} c^{4} d^{4}-4 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} e +12 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e^{2}-24 d \,e^{3} \ln \left (F \right ) b c +24 e^{4}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}+\frac {e^{4} x^{4} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right ) b c}+\frac {4 e \left (\ln \left (F \right )^{3} b^{3} c^{3} d^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) b c d \,e^{2}-6 e^{3}\right ) x \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}+\frac {6 e^{2} \left (\ln \left (F \right )^{2} b^{2} c^{2} d^{2}-2 \ln \left (F \right ) b c e d +2 e^{2}\right ) x^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3} c^{3}}+\frac {4 e^{3} \left (b c d \ln \left (F \right )-e \right ) x^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} c^{2}}\) | \(278\) |
| meijerg | \(-\frac {F^{a c} e^{4} \left (24-\frac {\left (5 b^{4} c^{4} x^{4} \ln \left (F \right )^{4}-20 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+60 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-120 b c x \ln \left (F \right )+120\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{5}\right )}{\ln \left (F \right )^{5} b^{5} c^{5}}+\frac {4 F^{a c} e^{3} d \left (6-\frac {\left (-4 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-24 b c x \ln \left (F \right )+24\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{4}\right )}{b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {6 F^{a c} e^{2} d^{2} \left (2-\frac {\left (3 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-6 b c x \ln \left (F \right )+6\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{3}\right )}{b^{3} c^{3} \ln \left (F \right )^{3}}+\frac {4 F^{a c} e \,d^{3} \left (1-\frac {\left (-2 b c x \ln \left (F \right )+2\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{2}\right )}{\ln \left (F \right )^{2} b^{2} c^{2}}-\frac {F^{a c} d^{4} \left (1-{\mathrm e}^{b c x \ln \left (F \right )}\right )}{\ln \left (F \right ) b c}\) | \(288\) |
| orering | \(\frac {\left (e^{4} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}+4 \ln \left (F \right )^{4} b^{4} c^{4} d \,e^{3} x^{3}+6 \ln \left (F \right )^{4} b^{4} c^{4} d^{2} e^{2} x^{2}+4 \ln \left (F \right )^{4} b^{4} c^{4} d^{3} e x +\ln \left (F \right )^{4} b^{4} c^{4} d^{4}-4 \ln \left (F \right )^{3} b^{3} c^{3} e^{4} x^{3}-12 \ln \left (F \right )^{3} b^{3} c^{3} d \,e^{3} x^{2}-12 \ln \left (F \right )^{3} b^{3} c^{3} d^{2} e^{2} x -4 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} e +12 \ln \left (F \right )^{2} b^{2} c^{2} e^{4} x^{2}+24 \ln \left (F \right )^{2} b^{2} c^{2} d \,e^{3} x +12 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e^{2}-24 \ln \left (F \right ) b c \,e^{4} x -24 d \,e^{3} \ln \left (F \right ) b c +24 e^{4}\right ) F^{c \left (b x +a \right )} \left (e^{4} x^{4}+4 d \,e^{3} x^{3}+6 d^{2} e^{2} x^{2}+4 d^{3} e x +d^{4}\right )}{\ln \left (F \right )^{5} b^{5} c^{5} \left (e x +d \right )^{4}}\) | \(305\) |
| parallelrisch | \(\frac {x^{4} F^{c \left (b x +a \right )} e^{4} \ln \left (F \right )^{4} b^{4} c^{4}+4 \ln \left (F \right )^{4} x^{3} F^{c \left (b x +a \right )} b^{4} c^{4} d \,e^{3}+6 \ln \left (F \right )^{4} x^{2} F^{c \left (b x +a \right )} b^{4} c^{4} d^{2} e^{2}+4 \ln \left (F \right )^{4} x \,F^{c \left (b x +a \right )} b^{4} c^{4} d^{3} e +\ln \left (F \right )^{4} F^{c \left (b x +a \right )} b^{4} c^{4} d^{4}-4 \ln \left (F \right )^{3} x^{3} F^{c \left (b x +a \right )} b^{3} c^{3} e^{4}-12 \ln \left (F \right )^{3} x^{2} F^{c \left (b x +a \right )} b^{3} c^{3} d \,e^{3}-12 \ln \left (F \right )^{3} x \,F^{c \left (b x +a \right )} b^{3} c^{3} d^{2} e^{2}-4 \ln \left (F \right )^{3} F^{c \left (b x +a \right )} b^{3} c^{3} d^{3} e +12 \ln \left (F \right )^{2} x^{2} F^{c \left (b x +a \right )} b^{2} c^{2} e^{4}+24 \ln \left (F \right )^{2} x \,F^{c \left (b x +a \right )} b^{2} c^{2} d \,e^{3}+12 \ln \left (F \right )^{2} F^{c \left (b x +a \right )} b^{2} c^{2} d^{2} e^{2}-24 \ln \left (F \right ) x \,F^{c \left (b x +a \right )} b c \,e^{4}-24 \ln \left (F \right ) F^{c \left (b x +a \right )} b c d \,e^{3}+24 F^{c \left (b x +a \right )} e^{4}}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(386\) |
Input:
int(F^(c*(b*x+a))*(e^4*x^4+4*d*e^3*x^3+6*d^2*e^2*x^2+4*d^3*e*x+d^4),x,meth od=_RETURNVERBOSE)
Output:
(e^4*x^4*ln(F)^4*b^4*c^4+4*ln(F)^4*b^4*c^4*d*e^3*x^3+6*ln(F)^4*b^4*c^4*d^2 *e^2*x^2+4*ln(F)^4*b^4*c^4*d^3*e*x+ln(F)^4*b^4*c^4*d^4-4*ln(F)^3*b^3*c^3*e ^4*x^3-12*ln(F)^3*b^3*c^3*d*e^3*x^2-12*ln(F)^3*b^3*c^3*d^2*e^2*x-4*ln(F)^3 *b^3*c^3*d^3*e+12*ln(F)^2*b^2*c^2*e^4*x^2+24*ln(F)^2*b^2*c^2*d*e^3*x+12*ln (F)^2*b^2*c^2*d^2*e^2-24*ln(F)*b*c*e^4*x-24*d*e^3*ln(F)*b*c+24*e^4)*F^(c*( b*x+a))/ln(F)^5/b^5/c^5
Time = 0.07 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.61 \[ \int F^{c (a+b x)} \left (d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4\right ) \, dx=\frac {{\left ({\left (b^{4} c^{4} e^{4} x^{4} + 4 \, b^{4} c^{4} d e^{3} x^{3} + 6 \, b^{4} c^{4} d^{2} e^{2} x^{2} + 4 \, b^{4} c^{4} d^{3} e x + b^{4} c^{4} d^{4}\right )} \log \left (F\right )^{4} + 24 \, e^{4} - 4 \, {\left (b^{3} c^{3} e^{4} x^{3} + 3 \, b^{3} c^{3} d e^{3} x^{2} + 3 \, b^{3} c^{3} d^{2} e^{2} x + b^{3} c^{3} d^{3} e\right )} \log \left (F\right )^{3} + 12 \, {\left (b^{2} c^{2} e^{4} x^{2} + 2 \, b^{2} c^{2} d e^{3} x + b^{2} c^{2} d^{2} e^{2}\right )} \log \left (F\right )^{2} - 24 \, {\left (b c e^{4} x + b c d e^{3}\right )} \log \left (F\right )\right )} F^{b c x + a c}}{b^{5} c^{5} \log \left (F\right )^{5}} \] Input:
integrate(F^((b*x+a)*c)*(e^4*x^4+4*d*e^3*x^3+6*d^2*e^2*x^2+4*d^3*e*x+d^4), x, algorithm="fricas")
Output:
((b^4*c^4*e^4*x^4 + 4*b^4*c^4*d*e^3*x^3 + 6*b^4*c^4*d^2*e^2*x^2 + 4*b^4*c^ 4*d^3*e*x + b^4*c^4*d^4)*log(F)^4 + 24*e^4 - 4*(b^3*c^3*e^4*x^3 + 3*b^3*c^ 3*d*e^3*x^2 + 3*b^3*c^3*d^2*e^2*x + b^3*c^3*d^3*e)*log(F)^3 + 12*(b^2*c^2* e^4*x^2 + 2*b^2*c^2*d*e^3*x + b^2*c^2*d^2*e^2)*log(F)^2 - 24*(b*c*e^4*x + b*c*d*e^3)*log(F))*F^(b*c*x + a*c)/(b^5*c^5*log(F)^5)
Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (139) = 278\).
Time = 0.11 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.48 \[ \int F^{c (a+b x)} \left (d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4\right ) \, dx=\begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{4} c^{4} d^{4} \log {\left (F \right )}^{4} + 4 b^{4} c^{4} d^{3} e x \log {\left (F \right )}^{4} + 6 b^{4} c^{4} d^{2} e^{2} x^{2} \log {\left (F \right )}^{4} + 4 b^{4} c^{4} d e^{3} x^{3} \log {\left (F \right )}^{4} + b^{4} c^{4} e^{4} x^{4} \log {\left (F \right )}^{4} - 4 b^{3} c^{3} d^{3} e \log {\left (F \right )}^{3} - 12 b^{3} c^{3} d^{2} e^{2} x \log {\left (F \right )}^{3} - 12 b^{3} c^{3} d e^{3} x^{2} \log {\left (F \right )}^{3} - 4 b^{3} c^{3} e^{4} x^{3} \log {\left (F \right )}^{3} + 12 b^{2} c^{2} d^{2} e^{2} \log {\left (F \right )}^{2} + 24 b^{2} c^{2} d e^{3} x \log {\left (F \right )}^{2} + 12 b^{2} c^{2} e^{4} x^{2} \log {\left (F \right )}^{2} - 24 b c d e^{3} \log {\left (F \right )} - 24 b c e^{4} x \log {\left (F \right )} + 24 e^{4}\right )}{b^{5} c^{5} \log {\left (F \right )}^{5}} & \text {for}\: b^{5} c^{5} \log {\left (F \right )}^{5} \neq 0 \\d^{4} x + 2 d^{3} e x^{2} + 2 d^{2} e^{2} x^{3} + d e^{3} x^{4} + \frac {e^{4} x^{5}}{5} & \text {otherwise} \end {cases} \] Input:
integrate(F**((b*x+a)*c)*(e**4*x**4+4*d*e**3*x**3+6*d**2*e**2*x**2+4*d**3* e*x+d**4),x)
Output:
Piecewise((F**(c*(a + b*x))*(b**4*c**4*d**4*log(F)**4 + 4*b**4*c**4*d**3*e *x*log(F)**4 + 6*b**4*c**4*d**2*e**2*x**2*log(F)**4 + 4*b**4*c**4*d*e**3*x **3*log(F)**4 + b**4*c**4*e**4*x**4*log(F)**4 - 4*b**3*c**3*d**3*e*log(F)* *3 - 12*b**3*c**3*d**2*e**2*x*log(F)**3 - 12*b**3*c**3*d*e**3*x**2*log(F)* *3 - 4*b**3*c**3*e**4*x**3*log(F)**3 + 12*b**2*c**2*d**2*e**2*log(F)**2 + 24*b**2*c**2*d*e**3*x*log(F)**2 + 12*b**2*c**2*e**4*x**2*log(F)**2 - 24*b* c*d*e**3*log(F) - 24*b*c*e**4*x*log(F) + 24*e**4)/(b**5*c**5*log(F)**5), N e(b**5*c**5*log(F)**5, 0)), (d**4*x + 2*d**3*e*x**2 + 2*d**2*e**2*x**3 + d *e**3*x**4 + e**4*x**5/5, True))
Leaf count of result is larger than twice the leaf count of optimal. 309 vs. \(2 (141) = 282\).
Time = 0.04 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.19 \[ \int F^{c (a+b x)} \left (d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4\right ) \, dx=\frac {F^{b c x + a c} d^{4}}{b c \log \left (F\right )} + \frac {4 \, {\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} d^{3} e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac {6 \, {\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} d^{2} e^{2}}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac {4 \, {\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} d e^{3}}{b^{4} c^{4} \log \left (F\right )^{4}} + \frac {{\left (F^{a c} b^{4} c^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{a c} b c x \log \left (F\right ) + 24 \, F^{a c}\right )} F^{b c x} e^{4}}{b^{5} c^{5} \log \left (F\right )^{5}} \] Input:
integrate(F^((b*x+a)*c)*(e^4*x^4+4*d*e^3*x^3+6*d^2*e^2*x^2+4*d^3*e*x+d^4), x, algorithm="maxima")
Output:
F^(b*c*x + a*c)*d^4/(b*c*log(F)) + 4*(F^(a*c)*b*c*x*log(F) - F^(a*c))*F^(b *c*x)*d^3*e/(b^2*c^2*log(F)^2) + 6*(F^(a*c)*b^2*c^2*x^2*log(F)^2 - 2*F^(a* c)*b*c*x*log(F) + 2*F^(a*c))*F^(b*c*x)*d^2*e^2/(b^3*c^3*log(F)^3) + 4*(F^( a*c)*b^3*c^3*x^3*log(F)^3 - 3*F^(a*c)*b^2*c^2*x^2*log(F)^2 + 6*F^(a*c)*b*c *x*log(F) - 6*F^(a*c))*F^(b*c*x)*d*e^3/(b^4*c^4*log(F)^4) + (F^(a*c)*b^4*c ^4*x^4*log(F)^4 - 4*F^(a*c)*b^3*c^3*x^3*log(F)^3 + 12*F^(a*c)*b^2*c^2*x^2* log(F)^2 - 24*F^(a*c)*b*c*x*log(F) + 24*F^(a*c))*F^(b*c*x)*e^4/(b^5*c^5*lo g(F)^5)
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 8802, normalized size of antiderivative = 62.43 \[ \int F^{c (a+b x)} \left (d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate(F^((b*x+a)*c)*(e^4*x^4+4*d*e^3*x^3+6*d^2*e^2*x^2+4*d^3*e*x+d^4), x, algorithm="giac")
Output:
-((4*(pi^3*b^4*c^4*e^4*x^4*log(abs(F))*sgn(F) - pi*b^4*c^4*e^4*x^4*log(abs (F))^3*sgn(F) - pi^3*b^4*c^4*e^4*x^4*log(abs(F)) + pi*b^4*c^4*e^4*x^4*log( abs(F))^3 + 4*pi^3*b^4*c^4*d*e^3*x^3*log(abs(F))*sgn(F) - 4*pi*b^4*c^4*d*e ^3*x^3*log(abs(F))^3*sgn(F) - 4*pi^3*b^4*c^4*d*e^3*x^3*log(abs(F)) + 4*pi* b^4*c^4*d*e^3*x^3*log(abs(F))^3 + 6*pi^3*b^4*c^4*d^2*e^2*x^2*log(abs(F))*s gn(F) - 6*pi*b^4*c^4*d^2*e^2*x^2*log(abs(F))^3*sgn(F) - 6*pi^3*b^4*c^4*d^2 *e^2*x^2*log(abs(F)) + 6*pi*b^4*c^4*d^2*e^2*x^2*log(abs(F))^3 + 4*pi^3*b^4 *c^4*d^3*e*x*log(abs(F))*sgn(F) - 4*pi*b^4*c^4*d^3*e*x*log(abs(F))^3*sgn(F ) - 4*pi^3*b^4*c^4*d^3*e*x*log(abs(F)) + 4*pi*b^4*c^4*d^3*e*x*log(abs(F))^ 3 - pi^3*b^3*c^3*e^4*x^3*sgn(F) + pi^3*b^4*c^4*d^4*log(abs(F))*sgn(F) + 3* pi*b^3*c^3*e^4*x^3*log(abs(F))^2*sgn(F) - pi*b^4*c^4*d^4*log(abs(F))^3*sgn (F) + pi^3*b^3*c^3*e^4*x^3 - pi^3*b^4*c^4*d^4*log(abs(F)) - 3*pi*b^3*c^3*e ^4*x^3*log(abs(F))^2 + pi*b^4*c^4*d^4*log(abs(F))^3 - 3*pi^3*b^3*c^3*d*e^3 *x^2*sgn(F) + 9*pi*b^3*c^3*d*e^3*x^2*log(abs(F))^2*sgn(F) + 3*pi^3*b^3*c^3 *d*e^3*x^2 - 9*pi*b^3*c^3*d*e^3*x^2*log(abs(F))^2 - 3*pi^3*b^3*c^3*d^2*e^2 *x*sgn(F) + 9*pi*b^3*c^3*d^2*e^2*x*log(abs(F))^2*sgn(F) + 3*pi^3*b^3*c^3*d ^2*e^2*x - 9*pi*b^3*c^3*d^2*e^2*x*log(abs(F))^2 - pi^3*b^3*c^3*d^3*e*sgn(F ) + 3*pi*b^3*c^3*d^3*e*log(abs(F))^2*sgn(F) + pi^3*b^3*c^3*d^3*e - 3*pi*b^ 3*c^3*d^3*e*log(abs(F))^2 - 6*pi*b^2*c^2*e^4*x^2*log(abs(F))*sgn(F) + 6*pi *b^2*c^2*e^4*x^2*log(abs(F)) - 12*pi*b^2*c^2*d*e^3*x*log(abs(F))*sgn(F)...
Time = 22.89 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.84 \[ \int F^{c (a+b x)} \left (d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4\right ) \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (b^4\,c^4\,d^4\,{\ln \left (F\right )}^4+4\,b^4\,c^4\,d^3\,e\,x\,{\ln \left (F\right )}^4+6\,b^4\,c^4\,d^2\,e^2\,x^2\,{\ln \left (F\right )}^4+4\,b^4\,c^4\,d\,e^3\,x^3\,{\ln \left (F\right )}^4+b^4\,c^4\,e^4\,x^4\,{\ln \left (F\right )}^4-4\,b^3\,c^3\,d^3\,e\,{\ln \left (F\right )}^3-12\,b^3\,c^3\,d^2\,e^2\,x\,{\ln \left (F\right )}^3-12\,b^3\,c^3\,d\,e^3\,x^2\,{\ln \left (F\right )}^3-4\,b^3\,c^3\,e^4\,x^3\,{\ln \left (F\right )}^3+12\,b^2\,c^2\,d^2\,e^2\,{\ln \left (F\right )}^2+24\,b^2\,c^2\,d\,e^3\,x\,{\ln \left (F\right )}^2+12\,b^2\,c^2\,e^4\,x^2\,{\ln \left (F\right )}^2-24\,b\,c\,d\,e^3\,\ln \left (F\right )-24\,b\,c\,e^4\,x\,\ln \left (F\right )+24\,e^4\right )}{b^5\,c^5\,{\ln \left (F\right )}^5} \] Input:
int(F^(c*(a + b*x))*(d^4 + e^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e *x),x)
Output:
(F^(a*c + b*c*x)*(24*e^4 + b^4*c^4*d^4*log(F)^4 - 24*b*c*e^4*x*log(F) - 4* b^3*c^3*d^3*e*log(F)^3 + 12*b^2*c^2*d^2*e^2*log(F)^2 + 12*b^2*c^2*e^4*x^2* log(F)^2 - 4*b^3*c^3*e^4*x^3*log(F)^3 + b^4*c^4*e^4*x^4*log(F)^4 - 24*b*c* d*e^3*log(F) + 6*b^4*c^4*d^2*e^2*x^2*log(F)^4 + 24*b^2*c^2*d*e^3*x*log(F)^ 2 + 4*b^4*c^4*d^3*e*x*log(F)^4 - 12*b^3*c^3*d^2*e^2*x*log(F)^3 - 12*b^3*c^ 3*d*e^3*x^2*log(F)^3 + 4*b^4*c^4*d*e^3*x^3*log(F)^4))/(b^5*c^5*log(F)^5)
Time = 0.18 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.84 \[ \int F^{c (a+b x)} \left (d^4+4 d^3 e x+6 d^2 e^2 x^2+4 d e^3 x^3+e^4 x^4\right ) \, dx=\frac {f^{b c x +a c} \left (\mathrm {log}\left (f \right )^{4} b^{4} c^{4} d^{4}+4 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d^{3} e x +6 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d^{2} e^{2} x^{2}+4 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d \,e^{3} x^{3}+\mathrm {log}\left (f \right )^{4} b^{4} c^{4} e^{4} x^{4}-4 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d^{3} e -12 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d^{2} e^{2} x -12 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d \,e^{3} x^{2}-4 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e^{4} x^{3}+12 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d^{2} e^{2}+24 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d \,e^{3} x +12 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e^{4} x^{2}-24 \,\mathrm {log}\left (f \right ) b c d \,e^{3}-24 \,\mathrm {log}\left (f \right ) b c \,e^{4} x +24 e^{4}\right )}{\mathrm {log}\left (f \right )^{5} b^{5} c^{5}} \] Input:
int(F^((b*x+a)*c)*(e^4*x^4+4*d*e^3*x^3+6*d^2*e^2*x^2+4*d^3*e*x+d^4),x)
Output:
(f**(a*c + b*c*x)*(log(f)**4*b**4*c**4*d**4 + 4*log(f)**4*b**4*c**4*d**3*e *x + 6*log(f)**4*b**4*c**4*d**2*e**2*x**2 + 4*log(f)**4*b**4*c**4*d*e**3*x **3 + log(f)**4*b**4*c**4*e**4*x**4 - 4*log(f)**3*b**3*c**3*d**3*e - 12*lo g(f)**3*b**3*c**3*d**2*e**2*x - 12*log(f)**3*b**3*c**3*d*e**3*x**2 - 4*log (f)**3*b**3*c**3*e**4*x**3 + 12*log(f)**2*b**2*c**2*d**2*e**2 + 24*log(f)* *2*b**2*c**2*d*e**3*x + 12*log(f)**2*b**2*c**2*e**4*x**2 - 24*log(f)*b*c*d *e**3 - 24*log(f)*b*c*e**4*x + 24*e**4))/(log(f)**5*b**5*c**5)