Integrand size = 30, antiderivative size = 162 \[ \int F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx=\frac {24 F^{c (a+b x)} h}{b^5 c^5 \log ^5(F)}-\frac {6 F^{c (a+b x)} (g+4 h x)}{b^4 c^4 \log ^4(F)}+\frac {2 F^{c (a+b x)} \left (f+3 g x+6 h x^2\right )}{b^3 c^3 \log ^3(F)}-\frac {F^{c (a+b x)} \left (e+2 f x+3 g x^2+4 h x^3\right )}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right )}{b c \log (F)} \] Output:
24*F^(c*(b*x+a))*h/b^5/c^5/ln(F)^5-6*F^(c*(b*x+a))*(4*h*x+g)/b^4/c^4/ln(F) ^4+2*F^(c*(b*x+a))*(6*h*x^2+3*g*x+f)/b^3/c^3/ln(F)^3-F^(c*(b*x+a))*(4*h*x^ 3+3*g*x^2+2*f*x+e)/b^2/c^2/ln(F)^2+F^(c*(b*x+a))*(h*x^4+g*x^3+f*x^2+e*x+d) /b/c/ln(F)
Time = 0.70 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.72 \[ \int F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx=\frac {F^{c (a+b x)} \left (24 h-6 b c (g+4 h x) \log (F)+2 b^2 c^2 (f+3 x (g+2 h x)) \log ^2(F)-b^3 c^3 \left (e+x \left (2 f+3 g x+4 h x^2\right )\right ) \log ^3(F)+b^4 c^4 (d+x (e+x (f+x (g+h x)))) \log ^4(F)\right )}{b^5 c^5 \log ^5(F)} \] Input:
Integrate[F^(c*(a + b*x))*(d + e*x + f*x^2 + g*x^3 + h*x^4),x]
Output:
(F^(c*(a + b*x))*(24*h - 6*b*c*(g + 4*h*x)*Log[F] + 2*b^2*c^2*(f + 3*x*(g + 2*h*x))*Log[F]^2 - b^3*c^3*(e + x*(2*f + 3*g*x + 4*h*x^2))*Log[F]^3 + b^ 4*c^4*(d + x*(e + x*(f + x*(g + h*x))))*Log[F]^4))/(b^5*c^5*Log[F]^5)
Leaf count is larger than twice the leaf count of optimal. \(348\) vs. \(2(162)=324\).
Time = 0.96 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.15, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2626, 2009}
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.
| \(\displaystyle \int F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \int \left (d F^{c (a+b x)}+e x F^{c (a+b x)}+f x^2 F^{c (a+b x)}+g x^3 F^{c (a+b x)}+h x^4 F^{c (a+b x)}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {24 h F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac {6 g F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}-\frac {24 h x F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {2 f F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {6 g x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {12 h x^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {e F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {2 f x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {3 g x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {4 h x^3 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {d F^{c (a+b x)}}{b c \log (F)}+\frac {e x F^{c (a+b x)}}{b c \log (F)}+\frac {f x^2 F^{c (a+b x)}}{b c \log (F)}+\frac {g x^3 F^{c (a+b x)}}{b c \log (F)}+\frac {h x^4 F^{c (a+b x)}}{b c \log (F)}\) |
Input:
Int[F^(c*(a + b*x))*(d + e*x + f*x^2 + g*x^3 + h*x^4),x]
Output:
(24*F^(c*(a + b*x))*h)/(b^5*c^5*Log[F]^5) - (6*F^(c*(a + b*x))*g)/(b^4*c^4 *Log[F]^4) - (24*F^(c*(a + b*x))*h*x)/(b^4*c^4*Log[F]^4) + (2*f*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) + (6*F^(c*(a + b*x))*g*x)/(b^3*c^3*Log[F]^3) + ( 12*F^(c*(a + b*x))*h*x^2)/(b^3*c^3*Log[F]^3) - (e*F^(c*(a + b*x)))/(b^2*c^ 2*Log[F]^2) - (2*f*F^(c*(a + b*x))*x)/(b^2*c^2*Log[F]^2) - (3*F^(c*(a + b* x))*g*x^2)/(b^2*c^2*Log[F]^2) - (4*F^(c*(a + b*x))*h*x^3)/(b^2*c^2*Log[F]^ 2) + (d*F^(c*(a + b*x)))/(b*c*Log[F]) + (e*F^(c*(a + b*x))*x)/(b*c*Log[F]) + (f*F^(c*(a + b*x))*x^2)/(b*c*Log[F]) + (F^(c*(a + b*x))*g*x^3)/(b*c*Log [F]) + (F^(c*(a + b*x))*h*x^4)/(b*c*Log[F])
Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr eeQ[F, x] && PolynomialQ[Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Time = 0.05 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.31
| method | result | size |
| gosper | \(\frac {\left (h \,x^{4} \ln \left (F \right )^{4} b^{4} c^{4}+\ln \left (F \right )^{4} b^{4} c^{4} g \,x^{3}+\ln \left (F \right )^{4} b^{4} c^{4} f \,x^{2}+\ln \left (F \right )^{4} b^{4} c^{4} e x +\ln \left (F \right )^{4} b^{4} c^{4} d -4 \ln \left (F \right )^{3} b^{3} c^{3} h \,x^{3}-3 g \,x^{2} \ln \left (F \right )^{3} b^{3} c^{3}-2 \ln \left (F \right )^{3} b^{3} c^{3} f x -\ln \left (F \right )^{3} b^{3} c^{3} e +12 \ln \left (F \right )^{2} b^{2} c^{2} h \,x^{2}+6 \ln \left (F \right )^{2} b^{2} c^{2} g x +2 f \ln \left (F \right )^{2} b^{2} c^{2}-24 \ln \left (F \right ) b c h x -6 \ln \left (F \right ) b c g +24 h \right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(212\) |
| risch | \(\frac {\left (h \,x^{4} \ln \left (F \right )^{4} b^{4} c^{4}+\ln \left (F \right )^{4} b^{4} c^{4} g \,x^{3}+\ln \left (F \right )^{4} b^{4} c^{4} f \,x^{2}+\ln \left (F \right )^{4} b^{4} c^{4} e x +\ln \left (F \right )^{4} b^{4} c^{4} d -4 \ln \left (F \right )^{3} b^{3} c^{3} h \,x^{3}-3 g \,x^{2} \ln \left (F \right )^{3} b^{3} c^{3}-2 \ln \left (F \right )^{3} b^{3} c^{3} f x -\ln \left (F \right )^{3} b^{3} c^{3} e +12 \ln \left (F \right )^{2} b^{2} c^{2} h \,x^{2}+6 \ln \left (F \right )^{2} b^{2} c^{2} g x +2 f \ln \left (F \right )^{2} b^{2} c^{2}-24 \ln \left (F \right ) b c h x -6 \ln \left (F \right ) b c g +24 h \right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(212\) |
| orering | \(\frac {\left (h \,x^{4} \ln \left (F \right )^{4} b^{4} c^{4}+\ln \left (F \right )^{4} b^{4} c^{4} g \,x^{3}+\ln \left (F \right )^{4} b^{4} c^{4} f \,x^{2}+\ln \left (F \right )^{4} b^{4} c^{4} e x +\ln \left (F \right )^{4} b^{4} c^{4} d -4 \ln \left (F \right )^{3} b^{3} c^{3} h \,x^{3}-3 g \,x^{2} \ln \left (F \right )^{3} b^{3} c^{3}-2 \ln \left (F \right )^{3} b^{3} c^{3} f x -\ln \left (F \right )^{3} b^{3} c^{3} e +12 \ln \left (F \right )^{2} b^{2} c^{2} h \,x^{2}+6 \ln \left (F \right )^{2} b^{2} c^{2} g x +2 f \ln \left (F \right )^{2} b^{2} c^{2}-24 \ln \left (F \right ) b c h x -6 \ln \left (F \right ) b c g +24 h \right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(212\) |
| norman | \(\frac {\left (\ln \left (F \right )^{4} b^{4} c^{4} d -\ln \left (F \right )^{3} b^{3} c^{3} e +2 f \ln \left (F \right )^{2} b^{2} c^{2}-6 \ln \left (F \right ) b c g +24 h \right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}+\frac {\left (\ln \left (F \right ) b c g -4 h \right ) x^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} c^{2}}+\frac {\left (f \ln \left (F \right )^{2} b^{2} c^{2}-3 \ln \left (F \right ) b c g +12 h \right ) x^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3} c^{3}}+\frac {\left (\ln \left (F \right )^{3} b^{3} c^{3} e -2 f \ln \left (F \right )^{2} b^{2} c^{2}+6 \ln \left (F \right ) b c g -24 h \right ) x \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}+\frac {h \,x^{4} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right ) b c}\) | \(236\) |
| meijerg | \(-\frac {F^{a c} h \left (24-\frac {\left (5 b^{4} c^{4} x^{4} \ln \left (F \right )^{4}-20 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+60 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-120 b c x \ln \left (F \right )+120\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{5}\right )}{\ln \left (F \right )^{5} b^{5} c^{5}}+\frac {F^{a c} g \left (6-\frac {\left (-4 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-24 b c x \ln \left (F \right )+24\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{4}\right )}{\ln \left (F \right )^{4} b^{4} c^{4}}-\frac {F^{a c} f \left (2-\frac {\left (3 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-6 b c x \ln \left (F \right )+6\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{3}\right )}{c^{3} b^{3} \ln \left (F \right )^{3}}+\frac {F^{a c} e \left (1-\frac {\left (-2 b c x \ln \left (F \right )+2\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{2}\right )}{\ln \left (F \right )^{2} b^{2} c^{2}}-\frac {F^{a c} d \left (1-{\mathrm e}^{b c x \ln \left (F \right )}\right )}{b c \ln \left (F \right )}\) | \(271\) |
| parallelrisch | \(\frac {x^{4} F^{c \left (b x +a \right )} h \ln \left (F \right )^{4} b^{4} c^{4}+\ln \left (F \right )^{4} x^{3} F^{c \left (b x +a \right )} b^{4} c^{4} g +\ln \left (F \right )^{4} x^{2} F^{c \left (b x +a \right )} b^{4} c^{4} f +\ln \left (F \right )^{4} x \,F^{c \left (b x +a \right )} b^{4} c^{4} e +\ln \left (F \right )^{4} F^{c \left (b x +a \right )} b^{4} c^{4} d -4 \ln \left (F \right )^{3} x^{3} F^{c \left (b x +a \right )} b^{3} c^{3} h -3 \ln \left (F \right )^{3} x^{2} F^{c \left (b x +a \right )} b^{3} c^{3} g -2 \ln \left (F \right )^{3} x \,F^{c \left (b x +a \right )} b^{3} c^{3} f -\ln \left (F \right )^{3} F^{c \left (b x +a \right )} b^{3} c^{3} e +12 \ln \left (F \right )^{2} x^{2} F^{c \left (b x +a \right )} b^{2} c^{2} h +6 \ln \left (F \right )^{2} x \,F^{c \left (b x +a \right )} b^{2} c^{2} g +2 \ln \left (F \right )^{2} F^{c \left (b x +a \right )} b^{2} c^{2} f -24 \ln \left (F \right ) x \,F^{c \left (b x +a \right )} b c h -6 \ln \left (F \right ) F^{c \left (b x +a \right )} b c g +24 F^{c \left (b x +a \right )} h}{\ln \left (F \right )^{5} b^{5} c^{5}}\) | \(338\) |
Input:
int(F^(c*(b*x+a))*(h*x^4+g*x^3+f*x^2+e*x+d),x,method=_RETURNVERBOSE)
Output:
(h*x^4*ln(F)^4*b^4*c^4+ln(F)^4*b^4*c^4*g*x^3+ln(F)^4*b^4*c^4*f*x^2+ln(F)^4 *b^4*c^4*e*x+ln(F)^4*b^4*c^4*d-4*ln(F)^3*b^3*c^3*h*x^3-3*g*x^2*ln(F)^3*b^3 *c^3-2*ln(F)^3*b^3*c^3*f*x-ln(F)^3*b^3*c^3*e+12*ln(F)^2*b^2*c^2*h*x^2+6*ln (F)^2*b^2*c^2*g*x+2*f*ln(F)^2*b^2*c^2-24*ln(F)*b*c*h*x-6*ln(F)*b*c*g+24*h) *F^(c*(b*x+a))/ln(F)^5/b^5/c^5
Time = 0.07 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.12 \[ \int F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx=\frac {{\left ({\left (b^{4} c^{4} h x^{4} + b^{4} c^{4} g x^{3} + b^{4} c^{4} f x^{2} + b^{4} c^{4} e x + b^{4} c^{4} d\right )} \log \left (F\right )^{4} - {\left (4 \, b^{3} c^{3} h x^{3} + 3 \, b^{3} c^{3} g x^{2} + 2 \, b^{3} c^{3} f x + b^{3} c^{3} e\right )} \log \left (F\right )^{3} + 2 \, {\left (6 \, b^{2} c^{2} h x^{2} + 3 \, b^{2} c^{2} g x + b^{2} c^{2} f\right )} \log \left (F\right )^{2} - 6 \, {\left (4 \, b c h x + b c g\right )} \log \left (F\right ) + 24 \, h\right )} F^{b c x + a c}}{b^{5} c^{5} \log \left (F\right )^{5}} \] Input:
integrate(F^((b*x+a)*c)*(h*x^4+g*x^3+f*x^2+e*x+d),x, algorithm="fricas")
Output:
((b^4*c^4*h*x^4 + b^4*c^4*g*x^3 + b^4*c^4*f*x^2 + b^4*c^4*e*x + b^4*c^4*d) *log(F)^4 - (4*b^3*c^3*h*x^3 + 3*b^3*c^3*g*x^2 + 2*b^3*c^3*f*x + b^3*c^3*e )*log(F)^3 + 2*(6*b^2*c^2*h*x^2 + 3*b^2*c^2*g*x + b^2*c^2*f)*log(F)^2 - 6* (4*b*c*h*x + b*c*g)*log(F) + 24*h)*F^(b*c*x + a*c)/(b^5*c^5*log(F)^5)
Time = 0.11 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.75 \[ \int F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx=\begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{4} c^{4} d \log {\left (F \right )}^{4} + b^{4} c^{4} e x \log {\left (F \right )}^{4} + b^{4} c^{4} f x^{2} \log {\left (F \right )}^{4} + b^{4} c^{4} g x^{3} \log {\left (F \right )}^{4} + b^{4} c^{4} h x^{4} \log {\left (F \right )}^{4} - b^{3} c^{3} e \log {\left (F \right )}^{3} - 2 b^{3} c^{3} f x \log {\left (F \right )}^{3} - 3 b^{3} c^{3} g x^{2} \log {\left (F \right )}^{3} - 4 b^{3} c^{3} h x^{3} \log {\left (F \right )}^{3} + 2 b^{2} c^{2} f \log {\left (F \right )}^{2} + 6 b^{2} c^{2} g x \log {\left (F \right )}^{2} + 12 b^{2} c^{2} h x^{2} \log {\left (F \right )}^{2} - 6 b c g \log {\left (F \right )} - 24 b c h x \log {\left (F \right )} + 24 h\right )}{b^{5} c^{5} \log {\left (F \right )}^{5}} & \text {for}\: b^{5} c^{5} \log {\left (F \right )}^{5} \neq 0 \\d x + \frac {e x^{2}}{2} + \frac {f x^{3}}{3} + \frac {g x^{4}}{4} + \frac {h x^{5}}{5} & \text {otherwise} \end {cases} \] Input:
integrate(F**((b*x+a)*c)*(h*x**4+g*x**3+f*x**2+e*x+d),x)
Output:
Piecewise((F**(c*(a + b*x))*(b**4*c**4*d*log(F)**4 + b**4*c**4*e*x*log(F)* *4 + b**4*c**4*f*x**2*log(F)**4 + b**4*c**4*g*x**3*log(F)**4 + b**4*c**4*h *x**4*log(F)**4 - b**3*c**3*e*log(F)**3 - 2*b**3*c**3*f*x*log(F)**3 - 3*b* *3*c**3*g*x**2*log(F)**3 - 4*b**3*c**3*h*x**3*log(F)**3 + 2*b**2*c**2*f*lo g(F)**2 + 6*b**2*c**2*g*x*log(F)**2 + 12*b**2*c**2*h*x**2*log(F)**2 - 6*b* c*g*log(F) - 24*b*c*h*x*log(F) + 24*h)/(b**5*c**5*log(F)**5), Ne(b**5*c**5 *log(F)**5, 0)), (d*x + e*x**2/2 + f*x**3/3 + g*x**4/4 + h*x**5/5, True))
Time = 0.08 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.80 \[ \int F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx=\frac {F^{b c x + a c} d}{b c \log \left (F\right )} + \frac {{\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac {{\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} f}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac {{\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} g}{b^{4} c^{4} \log \left (F\right )^{4}} + \frac {{\left (F^{a c} b^{4} c^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{a c} b c x \log \left (F\right ) + 24 \, F^{a c}\right )} F^{b c x} h}{b^{5} c^{5} \log \left (F\right )^{5}} \] Input:
integrate(F^((b*x+a)*c)*(h*x^4+g*x^3+f*x^2+e*x+d),x, algorithm="maxima")
Output:
F^(b*c*x + a*c)*d/(b*c*log(F)) + (F^(a*c)*b*c*x*log(F) - F^(a*c))*F^(b*c*x )*e/(b^2*c^2*log(F)^2) + (F^(a*c)*b^2*c^2*x^2*log(F)^2 - 2*F^(a*c)*b*c*x*l og(F) + 2*F^(a*c))*F^(b*c*x)*f/(b^3*c^3*log(F)^3) + (F^(a*c)*b^3*c^3*x^3*l og(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)*g/(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)*h/(b^5*c^5*log(F)^5)
Result contains complex when optimal does not.
Time = 0.22 (sec) , antiderivative size = 7630, normalized size of antiderivative = 47.10 \[ \int F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx=\text {Too large to display} \] Input:
integrate(F^((b*x+a)*c)*(h*x^4+g*x^3+f*x^2+e*x+d),x, algorithm="giac")
Output:
-(((4*pi^3*b^4*c^4*h*x^4*log(abs(F))*sgn(F) - 4*pi*b^4*c^4*h*x^4*log(abs(F ))^3*sgn(F) - 4*pi^3*b^4*c^4*h*x^4*log(abs(F)) + 4*pi*b^4*c^4*h*x^4*log(ab s(F))^3 + 4*pi^3*b^4*c^4*g*x^3*log(abs(F))*sgn(F) - 4*pi*b^4*c^4*g*x^3*log (abs(F))^3*sgn(F) - 4*pi^3*b^4*c^4*g*x^3*log(abs(F)) + 4*pi*b^4*c^4*g*x^3* log(abs(F))^3 + 4*pi^3*b^4*c^4*f*x^2*log(abs(F))*sgn(F) - 4*pi*b^4*c^4*f*x ^2*log(abs(F))^3*sgn(F) - 4*pi^3*b^4*c^4*f*x^2*log(abs(F)) + 4*pi*b^4*c^4* f*x^2*log(abs(F))^3 + 4*pi^3*b^4*c^4*e*x*log(abs(F))*sgn(F) - 4*pi*b^4*c^4 *e*x*log(abs(F))^3*sgn(F) - 4*pi^3*b^4*c^4*e*x*log(abs(F)) + 4*pi*b^4*c^4* e*x*log(abs(F))^3 - 4*pi^3*b^3*c^3*h*x^3*sgn(F) + 4*pi^3*b^4*c^4*d*log(abs (F))*sgn(F) + 12*pi*b^3*c^3*h*x^3*log(abs(F))^2*sgn(F) - 4*pi*b^4*c^4*d*lo g(abs(F))^3*sgn(F) + 4*pi^3*b^3*c^3*h*x^3 - 4*pi^3*b^4*c^4*d*log(abs(F)) - 12*pi*b^3*c^3*h*x^3*log(abs(F))^2 + 4*pi*b^4*c^4*d*log(abs(F))^3 - 3*pi^3 *b^3*c^3*g*x^2*sgn(F) + 9*pi*b^3*c^3*g*x^2*log(abs(F))^2*sgn(F) + 3*pi^3*b ^3*c^3*g*x^2 - 9*pi*b^3*c^3*g*x^2*log(abs(F))^2 - 2*pi^3*b^3*c^3*f*x*sgn(F ) + 6*pi*b^3*c^3*f*x*log(abs(F))^2*sgn(F) + 2*pi^3*b^3*c^3*f*x - 6*pi*b^3* c^3*f*x*log(abs(F))^2 - pi^3*b^3*c^3*e*sgn(F) + 3*pi*b^3*c^3*e*log(abs(F)) ^2*sgn(F) + pi^3*b^3*c^3*e - 3*pi*b^3*c^3*e*log(abs(F))^2 - 24*pi*b^2*c^2* h*x^2*log(abs(F))*sgn(F) + 24*pi*b^2*c^2*h*x^2*log(abs(F)) - 12*pi*b^2*c^2 *g*x*log(abs(F))*sgn(F) + 12*pi*b^2*c^2*g*x*log(abs(F)) - 4*pi*b^2*c^2*f*l og(abs(F))*sgn(F) + 4*pi*b^2*c^2*f*log(abs(F)) + 24*pi*b*c*h*x*sgn(F) -...
Time = 22.64 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.31 \[ \int F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (h\,b^4\,c^4\,x^4\,{\ln \left (F\right )}^4+g\,b^4\,c^4\,x^3\,{\ln \left (F\right )}^4+f\,b^4\,c^4\,x^2\,{\ln \left (F\right )}^4+e\,b^4\,c^4\,x\,{\ln \left (F\right )}^4+d\,b^4\,c^4\,{\ln \left (F\right )}^4-4\,h\,b^3\,c^3\,x^3\,{\ln \left (F\right )}^3-3\,g\,b^3\,c^3\,x^2\,{\ln \left (F\right )}^3-2\,f\,b^3\,c^3\,x\,{\ln \left (F\right )}^3-e\,b^3\,c^3\,{\ln \left (F\right )}^3+12\,h\,b^2\,c^2\,x^2\,{\ln \left (F\right )}^2+6\,g\,b^2\,c^2\,x\,{\ln \left (F\right )}^2+2\,f\,b^2\,c^2\,{\ln \left (F\right )}^2-24\,h\,b\,c\,x\,\ln \left (F\right )-6\,g\,b\,c\,\ln \left (F\right )+24\,h\right )}{b^5\,c^5\,{\ln \left (F\right )}^5} \] Input:
int(F^(c*(a + b*x))*(d + e*x + f*x^2 + g*x^3 + h*x^4),x)
Output:
(F^(a*c + b*c*x)*(24*h - 6*b*c*g*log(F) + b^4*c^4*d*log(F)^4 - b^3*c^3*e*l og(F)^3 + 2*b^2*c^2*f*log(F)^2 + b^4*c^4*f*x^2*log(F)^4 - 3*b^3*c^3*g*x^2* log(F)^3 + b^4*c^4*g*x^3*log(F)^4 + 12*b^2*c^2*h*x^2*log(F)^2 - 4*b^3*c^3* h*x^3*log(F)^3 + b^4*c^4*h*x^4*log(F)^4 - 24*b*c*h*x*log(F) + b^4*c^4*e*x* log(F)^4 - 2*b^3*c^3*f*x*log(F)^3 + 6*b^2*c^2*g*x*log(F)^2))/(b^5*c^5*log( F)^5)
Time = 0.18 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.31 \[ \int F^{c (a+b x)} \left (d+e x+f x^2+g x^3+h x^4\right ) \, dx=\frac {f^{b c x +a c} \left (\mathrm {log}\left (f \right )^{4} b^{4} c^{4} d +\mathrm {log}\left (f \right )^{4} b^{4} c^{4} e x +\mathrm {log}\left (f \right )^{4} b^{4} c^{4} f \,x^{2}+\mathrm {log}\left (f \right )^{4} b^{4} c^{4} g \,x^{3}+\mathrm {log}\left (f \right )^{4} b^{4} c^{4} h \,x^{4}-\mathrm {log}\left (f \right )^{3} b^{3} c^{3} e -2 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} f x -3 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} g \,x^{2}-4 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} h \,x^{3}+2 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} f +6 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} g x +12 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} h \,x^{2}-6 \,\mathrm {log}\left (f \right ) b c g -24 \,\mathrm {log}\left (f \right ) b c h x +24 h \right )}{\mathrm {log}\left (f \right )^{5} b^{5} c^{5}} \] Input:
int(F^((b*x+a)*c)*(h*x^4+g*x^3+f*x^2+e*x+d),x)
Output:
(f**(a*c + b*c*x)*(log(f)**4*b**4*c**4*d + log(f)**4*b**4*c**4*e*x + log(f )**4*b**4*c**4*f*x**2 + log(f)**4*b**4*c**4*g*x**3 + log(f)**4*b**4*c**4*h *x**4 - log(f)**3*b**3*c**3*e - 2*log(f)**3*b**3*c**3*f*x - 3*log(f)**3*b* *3*c**3*g*x**2 - 4*log(f)**3*b**3*c**3*h*x**3 + 2*log(f)**2*b**2*c**2*f + 6*log(f)**2*b**2*c**2*g*x + 12*log(f)**2*b**2*c**2*h*x**2 - 6*log(f)*b*c*g - 24*log(f)*b*c*h*x + 24*h))/(log(f)**5*b**5*c**5)