Integrand size = 17, antiderivative size = 110 \[ \int F^{c (a+b x)} (d+e x)^3 \, dx=-\frac {6 e^3 F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {6 e^2 F^{c (a+b x)} (d+e x)}{b^3 c^3 \log ^3(F)}-\frac {3 e F^{c (a+b x)} (d+e x)^2}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^3}{b c \log (F)} \] Output:
-6*e^3*F^(c*(b*x+a))/b^4/c^4/ln(F)^4+6*e^2*F^(c*(b*x+a))*(e*x+d)/b^3/c^3/l n(F)^3-3*e*F^(c*(b*x+a))*(e*x+d)^2/b^2/c^2/ln(F)^2+F^(c*(b*x+a))*(e*x+d)^3 /b/c/ln(F)
Time = 0.14 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.71 \[ \int F^{c (a+b x)} (d+e x)^3 \, dx=\frac {F^{c (a+b x)} \left (-6 e^3+6 b c e^2 (d+e x) \log (F)-3 b^2 c^2 e (d+e x)^2 \log ^2(F)+b^3 c^3 (d+e x)^3 \log ^3(F)\right )}{b^4 c^4 \log ^4(F)} \] Input:
Integrate[F^(c*(a + b*x))*(d + e*x)^3,x]
Output:
(F^(c*(a + b*x))*(-6*e^3 + 6*b*c*e^2*(d + e*x)*Log[F] - 3*b^2*c^2*e*(d + e *x)^2*Log[F]^2 + b^3*c^3*(d + e*x)^3*Log[F]^3))/(b^4*c^4*Log[F]^4)
Time = 0.58 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.18, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {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.
| \(\displaystyle \int (d+e x)^3 F^{c (a+b x)} \, dx\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle \frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \int F^{c (a+b x)} (d+e x)^2dx}{b c \log (F)}\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle \frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \left (\frac {(d+e x)^2 F^{c (a+b x)}}{b c \log (F)}-\frac {2 e \int F^{c (a+b x)} (d+e x)dx}{b c \log (F)}\right )}{b c \log (F)}\) |
| \(\Big \downarrow \) 2607 |
| \(\displaystyle \frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \left (\frac {(d+e x)^2 F^{c (a+b x)}}{b c \log (F)}-\frac {2 e \left (\frac {(d+e x) F^{c (a+b x)}}{b c \log (F)}-\frac {e \int F^{c (a+b x)}dx}{b c \log (F)}\right )}{b c \log (F)}\right )}{b c \log (F)}\) |
| \(\Big \downarrow \) 2624 |
| \(\displaystyle \frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)}-\frac {3 e \left (\frac {(d+e x)^2 F^{c (a+b x)}}{b c \log (F)}-\frac {2 e \left (\frac {(d+e x) F^{c (a+b x)}}{b c \log (F)}-\frac {e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}\right )}{b c \log (F)}\right )}{b c \log (F)}\) |
Input:
Int[F^(c*(a + b*x))*(d + e*x)^3,x]
Output:
(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])
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 = 0.06 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.50
| method | result | size |
| gosper | \(\frac {\left (e^{3} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+3 \ln \left (F \right )^{3} b^{3} c^{3} d \,e^{2} x^{2}+3 \ln \left (F \right )^{3} b^{3} c^{3} d^{2} e x +\ln \left (F \right )^{3} b^{3} c^{3} d^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2} e^{3} x^{2}-6 \ln \left (F \right )^{2} b^{2} c^{2} d \,e^{2} x -3 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) b c \,e^{3} x +6 \ln \left (F \right ) b c d \,e^{2}-6 e^{3}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}\) | \(165\) |
| risch | \(\frac {\left (e^{3} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+3 \ln \left (F \right )^{3} b^{3} c^{3} d \,e^{2} x^{2}+3 \ln \left (F \right )^{3} b^{3} c^{3} d^{2} e x +\ln \left (F \right )^{3} b^{3} c^{3} d^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2} e^{3} x^{2}-6 \ln \left (F \right )^{2} b^{2} c^{2} d \,e^{2} x -3 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) b c \,e^{3} x +6 \ln \left (F \right ) b c d \,e^{2}-6 e^{3}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}\) | \(165\) |
| orering | \(\frac {\left (e^{3} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+3 \ln \left (F \right )^{3} b^{3} c^{3} d \,e^{2} x^{2}+3 \ln \left (F \right )^{3} b^{3} c^{3} d^{2} e x +\ln \left (F \right )^{3} b^{3} c^{3} d^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2} e^{3} x^{2}-6 \ln \left (F \right )^{2} b^{2} c^{2} d \,e^{2} x -3 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) b c \,e^{3} x +6 \ln \left (F \right ) b c d \,e^{2}-6 e^{3}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}\) | \(165\) |
| norman | \(\frac {\left (\ln \left (F \right )^{3} b^{3} c^{3} d^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) b c d \,e^{2}-6 e^{3}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}+\frac {e^{3} x^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right ) b c}+\frac {3 e \left (\ln \left (F \right )^{2} b^{2} c^{2} d^{2}-2 \ln \left (F \right ) b c e d +2 e^{2}\right ) x \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3} c^{3}}+\frac {3 e^{2} \left (b c d \ln \left (F \right )-e \right ) x^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} c^{2}}\) | \(186\) |
| meijerg | \(\frac {F^{a c} e^{3} \left (6-\frac {\left (-4 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-24 b c x \ln \left (F \right )+24\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{4}\right )}{\ln \left (F \right )^{4} b^{4} c^{4}}-\frac {3 F^{a c} e^{2} d \left (2-\frac {\left (3 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-6 b c x \ln \left (F \right )+6\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{3}\right )}{b^{3} c^{3} \ln \left (F \right )^{3}}+\frac {3 F^{a c} e \,d^{2} \left (1-\frac {\left (-2 b c x \ln \left (F \right )+2\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{2}\right )}{\ln \left (F \right )^{2} b^{2} c^{2}}-\frac {F^{a c} d^{3} \left (1-{\mathrm e}^{b c x \ln \left (F \right )}\right )}{\ln \left (F \right ) b c}\) | \(199\) |
| parallelrisch | \(\frac {x^{3} F^{c \left (b x +a \right )} e^{3} \ln \left (F \right )^{3} b^{3} c^{3}+3 \ln \left (F \right )^{3} x^{2} F^{c \left (b x +a \right )} b^{3} c^{3} d \,e^{2}+3 \ln \left (F \right )^{3} x \,F^{c \left (b x +a \right )} b^{3} c^{3} d^{2} e +\ln \left (F \right )^{3} F^{c \left (b x +a \right )} b^{3} c^{3} d^{3}-3 \ln \left (F \right )^{2} x^{2} F^{c \left (b x +a \right )} b^{2} c^{2} e^{3}-6 \ln \left (F \right )^{2} x \,F^{c \left (b x +a \right )} b^{2} c^{2} d \,e^{2}-3 \ln \left (F \right )^{2} F^{c \left (b x +a \right )} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) x \,F^{c \left (b x +a \right )} b c \,e^{3}+6 \ln \left (F \right ) F^{c \left (b x +a \right )} b c d \,e^{2}-6 F^{c \left (b x +a \right )} e^{3}}{\ln \left (F \right )^{4} b^{4} c^{4}}\) | \(246\) |
Input:
int(F^(c*(b*x+a))*(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
(e^3*x^3*ln(F)^3*b^3*c^3+3*ln(F)^3*b^3*c^3*d*e^2*x^2+3*ln(F)^3*b^3*c^3*d^2 *e*x+ln(F)^3*b^3*c^3*d^3-3*ln(F)^2*b^2*c^2*e^3*x^2-6*ln(F)^2*b^2*c^2*d*e^2 *x-3*ln(F)^2*b^2*c^2*d^2*e+6*ln(F)*b*c*e^3*x+6*ln(F)*b*c*d*e^2-6*e^3)*F^(c *(b*x+a))/ln(F)^4/b^4/c^4
Time = 0.07 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.34 \[ \int F^{c (a+b x)} (d+e x)^3 \, dx=\frac {{\left ({\left (b^{3} c^{3} e^{3} x^{3} + 3 \, b^{3} c^{3} d e^{2} x^{2} + 3 \, b^{3} c^{3} d^{2} e x + b^{3} c^{3} d^{3}\right )} \log \left (F\right )^{3} - 6 \, e^{3} - 3 \, {\left (b^{2} c^{2} e^{3} x^{2} + 2 \, b^{2} c^{2} d e^{2} x + b^{2} c^{2} d^{2} e\right )} \log \left (F\right )^{2} + 6 \, {\left (b c e^{3} x + b c d e^{2}\right )} \log \left (F\right )\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4}} \] Input:
integrate(F^((b*x+a)*c)*(e*x+d)^3,x, algorithm="fricas")
Output:
((b^3*c^3*e^3*x^3 + 3*b^3*c^3*d*e^2*x^2 + 3*b^3*c^3*d^2*e*x + b^3*c^3*d^3) *log(F)^3 - 6*e^3 - 3*(b^2*c^2*e^3*x^2 + 2*b^2*c^2*d*e^2*x + b^2*c^2*d^2*e )*log(F)^2 + 6*(b*c*e^3*x + b*c*d*e^2)*log(F))*F^(b*c*x + a*c)/(b^4*c^4*lo g(F)^4)
Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (107) = 214\).
Time = 0.09 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.10 \[ \int F^{c (a+b x)} (d+e x)^3 \, dx=\begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{3} c^{3} d^{3} \log {\left (F \right )}^{3} + 3 b^{3} c^{3} d^{2} e x \log {\left (F \right )}^{3} + 3 b^{3} c^{3} d e^{2} x^{2} \log {\left (F \right )}^{3} + b^{3} c^{3} e^{3} x^{3} \log {\left (F \right )}^{3} - 3 b^{2} c^{2} d^{2} e \log {\left (F \right )}^{2} - 6 b^{2} c^{2} d e^{2} x \log {\left (F \right )}^{2} - 3 b^{2} c^{2} e^{3} x^{2} \log {\left (F \right )}^{2} + 6 b c d e^{2} \log {\left (F \right )} + 6 b c e^{3} x \log {\left (F \right )} - 6 e^{3}\right )}{b^{4} c^{4} \log {\left (F \right )}^{4}} & \text {for}\: b^{4} c^{4} \log {\left (F \right )}^{4} \neq 0 \\d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4} & \text {otherwise} \end {cases} \] Input:
integrate(F**((b*x+a)*c)*(e*x+d)**3,x)
Output:
Piecewise((F**(c*(a + b*x))*(b**3*c**3*d**3*log(F)**3 + 3*b**3*c**3*d**2*e *x*log(F)**3 + 3*b**3*c**3*d*e**2*x**2*log(F)**3 + b**3*c**3*e**3*x**3*log (F)**3 - 3*b**2*c**2*d**2*e*log(F)**2 - 6*b**2*c**2*d*e**2*x*log(F)**2 - 3 *b**2*c**2*e**3*x**2*log(F)**2 + 6*b*c*d*e**2*log(F) + 6*b*c*e**3*x*log(F) - 6*e**3)/(b**4*c**4*log(F)**4), Ne(b**4*c**4*log(F)**4, 0)), (d**3*x + 3 *d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4, True))
Time = 0.04 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.87 \[ \int F^{c (a+b x)} (d+e x)^3 \, dx=\frac {F^{b c x + a c} d^{3}}{b c \log \left (F\right )} + \frac {3 \, {\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} d^{2} e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac {3 \, {\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} d e^{2}}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac {{\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} e^{3}}{b^{4} c^{4} \log \left (F\right )^{4}} \] Input:
integrate(F^((b*x+a)*c)*(e*x+d)^3,x, algorithm="maxima")
Output:
F^(b*c*x + a*c)*d^3/(b*c*log(F)) + 3*(F^(a*c)*b*c*x*log(F) - F^(a*c))*F^(b *c*x)*d^2*e/(b^2*c^2*log(F)^2) + 3*(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*e^2/(b^3*c^3*log(F)^3) + (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*l og(F) - 6*F^(a*c))*F^(b*c*x)*e^3/(b^4*c^4*log(F)^4)
Result contains complex when optimal does not.
Time = 0.20 (sec) , antiderivative size = 4706, normalized size of antiderivative = 42.78 \[ \int F^{c (a+b x)} (d+e x)^3 \, dx=\text {Too large to display} \] Input:
integrate(F^((b*x+a)*c)*(e*x+d)^3,x, algorithm="giac")
Output:
-(((3*pi^2*b^3*c^3*e^3*x^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*e^3*x^3*log (abs(F)) + 2*b^3*c^3*e^3*x^3*log(abs(F))^3 + 9*pi^2*b^3*c^3*d*e^2*x^2*log( abs(F))*sgn(F) - 9*pi^2*b^3*c^3*d*e^2*x^2*log(abs(F)) + 6*b^3*c^3*d*e^2*x^ 2*log(abs(F))^3 + 9*pi^2*b^3*c^3*d^2*e*x*log(abs(F))*sgn(F) - 9*pi^2*b^3*c ^3*d^2*e*x*log(abs(F)) + 6*b^3*c^3*d^2*e*x*log(abs(F))^3 + 3*pi^2*b^3*c^3* d^3*log(abs(F))*sgn(F) - 3*pi^2*b^3*c^3*d^3*log(abs(F)) + 2*b^3*c^3*d^3*lo g(abs(F))^3 - 3*pi^2*b^2*c^2*e^3*x^2*sgn(F) + 3*pi^2*b^2*c^2*e^3*x^2 - 6*b ^2*c^2*e^3*x^2*log(abs(F))^2 - 6*pi^2*b^2*c^2*d*e^2*x*sgn(F) + 6*pi^2*b^2* c^2*d*e^2*x - 12*b^2*c^2*d*e^2*x*log(abs(F))^2 - 3*pi^2*b^2*c^2*d^2*e*sgn( F) + 3*pi^2*b^2*c^2*d^2*e - 6*b^2*c^2*d^2*e*log(abs(F))^2 + 12*b*c*e^3*x*l og(abs(F)) + 12*b*c*d*e^2*log(abs(F)) - 12*e^3)*(pi^4*b^4*c^4*sgn(F) - 6*p i^2*b^4*c^4*log(abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F ))^2 - 2*b^4*c^4*log(abs(F))^4)/((pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log (abs(F))^2*sgn(F) - pi^4*b^4*c^4 + 6*pi^2*b^4*c^4*log(abs(F))^2 - 2*b^4*c^ 4*log(abs(F))^4)^2 + 16*(pi^3*b^4*c^4*log(abs(F))*sgn(F) - pi*b^4*c^4*log( abs(F))^3*sgn(F) - pi^3*b^4*c^4*log(abs(F)) + pi*b^4*c^4*log(abs(F))^3)^2) - 4*(pi^3*b^3*c^3*e^3*x^3*sgn(F) - 3*pi*b^3*c^3*e^3*x^3*log(abs(F))^2*sgn (F) - pi^3*b^3*c^3*e^3*x^3 + 3*pi*b^3*c^3*e^3*x^3*log(abs(F))^2 + 3*pi^3*b ^3*c^3*d*e^2*x^2*sgn(F) - 9*pi*b^3*c^3*d*e^2*x^2*log(abs(F))^2*sgn(F) - 3* pi^3*b^3*c^3*d*e^2*x^2 + 9*pi*b^3*c^3*d*e^2*x^2*log(abs(F))^2 + 3*pi^3*...
Time = 22.99 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.50 \[ \int F^{c (a+b x)} (d+e x)^3 \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (b^3\,c^3\,d^3\,{\ln \left (F\right )}^3+3\,b^3\,c^3\,d^2\,e\,x\,{\ln \left (F\right )}^3+3\,b^3\,c^3\,d\,e^2\,x^2\,{\ln \left (F\right )}^3+b^3\,c^3\,e^3\,x^3\,{\ln \left (F\right )}^3-3\,b^2\,c^2\,d^2\,e\,{\ln \left (F\right )}^2-6\,b^2\,c^2\,d\,e^2\,x\,{\ln \left (F\right )}^2-3\,b^2\,c^2\,e^3\,x^2\,{\ln \left (F\right )}^2+6\,b\,c\,d\,e^2\,\ln \left (F\right )+6\,b\,c\,e^3\,x\,\ln \left (F\right )-6\,e^3\right )}{b^4\,c^4\,{\ln \left (F\right )}^4} \] Input:
int(F^(c*(a + b*x))*(d + e*x)^3,x)
Output:
(F^(a*c + b*c*x)*(b^3*c^3*d^3*log(F)^3 - 6*e^3 + 6*b*c*e^3*x*log(F) - 3*b^ 2*c^2*d^2*e*log(F)^2 - 3*b^2*c^2*e^3*x^2*log(F)^2 + b^3*c^3*e^3*x^3*log(F) ^3 + 6*b*c*d*e^2*log(F) - 6*b^2*c^2*d*e^2*x*log(F)^2 + 3*b^3*c^3*d^2*e*x*l og(F)^3 + 3*b^3*c^3*d*e^2*x^2*log(F)^3))/(b^4*c^4*log(F)^4)
Time = 0.17 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.50 \[ \int F^{c (a+b x)} (d+e x)^3 \, dx=\frac {f^{b c x +a c} \left (\mathrm {log}\left (f \right )^{3} b^{3} c^{3} d^{3}+3 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d^{2} e x +3 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d \,e^{2} x^{2}+\mathrm {log}\left (f \right )^{3} b^{3} c^{3} e^{3} x^{3}-3 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d^{2} e -6 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d \,e^{2} x -3 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e^{3} x^{2}+6 \,\mathrm {log}\left (f \right ) b c d \,e^{2}+6 \,\mathrm {log}\left (f \right ) b c \,e^{3} x -6 e^{3}\right )}{\mathrm {log}\left (f \right )^{4} b^{4} c^{4}} \] Input:
int(F^((b*x+a)*c)*(e*x+d)^3,x)
Output:
(f**(a*c + b*c*x)*(log(f)**3*b**3*c**3*d**3 + 3*log(f)**3*b**3*c**3*d**2*e *x + 3*log(f)**3*b**3*c**3*d*e**2*x**2 + log(f)**3*b**3*c**3*e**3*x**3 - 3 *log(f)**2*b**2*c**2*d**2*e - 6*log(f)**2*b**2*c**2*d*e**2*x - 3*log(f)**2 *b**2*c**2*e**3*x**2 + 6*log(f)*b*c*d*e**2 + 6*log(f)*b*c*e**3*x - 6*e**3) )/(log(f)**4*b**4*c**4)