\(\int F^{c (a+b x)} (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3) \, dx\) [13]

Optimal result

Integrand size = 37, antiderivative size = 110 \[ \int F^{c (a+b x)} \left (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3\right ) \, dx=-\frac {6 e^3 F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {6 e^2 F^{c (a+b x)} (d+e x)}{b^3 c^3 \log ^3(F)}-\frac {3 e F^{c (a+b x)} (d+e x)^2}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} (d+e x)^3}{b c \log (F)} \]

[Out]

-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/ln(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)

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {2218, 2207, 2225} \[ \int F^{c (a+b x)} \left (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3\right ) \, dx=-\frac {6 e^3 F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}+\frac {6 e^2 (d+e x) F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}-\frac {3 e (d+e x)^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {(d+e x)^3 F^{c (a+b x)}}{b c \log (F)} \]

[In]

Int[F^(c*(a + b*x))*(d^3 + 3*d^2*e*x + 3*d*e^2*x^2 + e^3*x^3),x]

[Out]

(-6*e^3*F^(c*(a + b*x)))/(b^4*c^4*Log[F]^4) + (6*e^2*F^(c*(a + b*x))*(d + e*x))/(b^3*c^3*Log[F]^3) - (3*e*F^(c
*(a + b*x))*(d + e*x)^2)/(b^2*c^2*Log[F]^2) + (F^(c*(a + b*x))*(d + e*x)^3)/(b*c*Log[F])

Rule 2207

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] - Dist[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]

Rule 2218

Int[((a_.) + (b_.)*((F_)^((g_.)*(v_)))^(n_.))^(p_.)*(u_)^(m_.), x_Symbol] :> Int[NormalizePowerOfLinear[u, x]^
m*(a + b*(F^(g*ExpandToSum[v, x]))^n)^p, x] /; FreeQ[{F, a, b, g, n, p}, x] && LinearQ[v, x] && PowerOfLinearQ
[u, x] &&  !(LinearMatchQ[v, x] && PowerOfLinearMatchQ[u, x]) && IntegerQ[m]

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rubi steps \begin{align*} \text {integral}& = \int F^{c (a+b x)} (d+e x)^3 \, dx \\ & = \frac {F^{c (a+b x)} (d+e x)^3}{b c \log (F)}-\frac {(3 e) \int F^{c (a+b x)} (d+e x)^2 \, dx}{b c \log (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)}+\frac {\left (6 e^2\right ) \int F^{c (a+b x)} (d+e x) \, dx}{b^2 c^2 \log ^2(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)}-\frac {\left (6 e^3\right ) \int F^{c (a+b x)} \, dx}{b^3 c^3 \log ^3(F)} \\ & = -\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)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.71 \[ \int F^{c (a+b x)} \left (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3\right ) \, dx=\frac {F^{c (a+b x)} \left (-6 e^3+6 b c e^2 (d+e x) \log (F)-3 b^2 c^2 e (d+e x)^2 \log ^2(F)+b^3 c^3 (d+e x)^3 \log ^3(F)\right )}{b^4 c^4 \log ^4(F)} \]

[In]

Integrate[F^(c*(a + b*x))*(d^3 + 3*d^2*e*x + 3*d*e^2*x^2 + e^3*x^3),x]

[Out]

(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)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.50

method	result	size
gosper	\(\frac {\left (e^{3} x^{3} c^{3} b^{3} \ln \left (F \right )^{3}+3 \ln \left (F \right )^{3} b^{3} c^{3} d \,e^{2} x^{2}+3 \ln \left (F \right )^{3} b^{3} c^{3} d^{2} e x +c^{3} b^{3} \ln \left (F \right )^{3} d^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2} e^{3} x^{2}-6 \ln \left (F \right )^{2} b^{2} c^{2} d \,e^{2} x -3 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) b c \,e^{3} x +6 \ln \left (F \right ) b c d \,e^{2}-6 e^{3}\right ) F^{c \left (b x +a \right )}}{c^{4} b^{4} \ln \left (F \right )^{4}}\)	\(165\)
risch	\(\frac {\left (e^{3} x^{3} c^{3} b^{3} \ln \left (F \right )^{3}+3 \ln \left (F \right )^{3} b^{3} c^{3} d \,e^{2} x^{2}+3 \ln \left (F \right )^{3} b^{3} c^{3} d^{2} e x +c^{3} b^{3} \ln \left (F \right )^{3} d^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2} e^{3} x^{2}-6 \ln \left (F \right )^{2} b^{2} c^{2} d \,e^{2} x -3 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) b c \,e^{3} x +6 \ln \left (F \right ) b c d \,e^{2}-6 e^{3}\right ) F^{c \left (b x +a \right )}}{c^{4} b^{4} \ln \left (F \right )^{4}}\)	\(165\)
norman	\(\frac {\left (c^{3} b^{3} \ln \left (F \right )^{3} d^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) b c d \,e^{2}-6 e^{3}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{4} b^{4} \ln \left (F \right )^{4}}+\frac {e^{3} x^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c b \ln \left (F \right )}+\frac {3 e \left (\ln \left (F \right )^{2} b^{2} c^{2} d^{2}-2 \ln \left (F \right ) b c e d +2 e^{2}\right ) x \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{3} b^{3} \ln \left (F \right )^{3}}+\frac {3 e^{2} \left (\ln \left (F \right ) b c d -e \right ) x^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{2} b^{2} \ln \left (F \right )^{2}}\)	\(186\)
meijerg	\(\frac {F^{c a} e^{3} \left (6-\frac {\left (-4 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-24 b c x \ln \left (F \right )+24\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{4}\right )}{c^{4} b^{4} \ln \left (F \right )^{4}}-\frac {3 F^{c a} e^{2} d \left (2-\frac {\left (3 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-6 b c x \ln \left (F \right )+6\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{3}\right )}{c^{3} b^{3} \ln \left (F \right )^{3}}+\frac {3 F^{c a} e \,d^{2} \left (1-\frac {\left (-2 b c x \ln \left (F \right )+2\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{2}\right )}{c^{2} b^{2} \ln \left (F \right )^{2}}-\frac {F^{c a} d^{3} \left (1-{\mathrm e}^{b c x \ln \left (F \right )}\right )}{c b \ln \left (F \right )}\)	\(199\)
parallelrisch	\(\frac {x^{3} F^{c \left (b x +a \right )} e^{3} c^{3} b^{3} \ln \left (F \right )^{3}+3 \ln \left (F \right )^{3} x^{2} F^{c \left (b x +a \right )} b^{3} c^{3} d \,e^{2}+3 \ln \left (F \right )^{3} x \,F^{c \left (b x +a \right )} b^{3} c^{3} d^{2} e +\ln \left (F \right )^{3} F^{c \left (b x +a \right )} b^{3} c^{3} d^{3}-3 \ln \left (F \right )^{2} x^{2} F^{c \left (b x +a \right )} b^{2} c^{2} e^{3}-6 \ln \left (F \right )^{2} x \,F^{c \left (b x +a \right )} b^{2} c^{2} d \,e^{2}-3 \ln \left (F \right )^{2} F^{c \left (b x +a \right )} b^{2} c^{2} d^{2} e +6 \ln \left (F \right ) x \,F^{c \left (b x +a \right )} b c \,e^{3}+6 \ln \left (F \right ) F^{c \left (b x +a \right )} b c d \,e^{2}-6 F^{c \left (b x +a \right )} e^{3}}{c^{4} b^{4} \ln \left (F \right )^{4}}\)	\(246\)

[In]

int(F^(c*(b*x+a))*(e^3*x^3+3*d*e^2*x^2+3*d^2*e*x+d^3),x,method=_RETURNVERBOSE)

[Out]

(e^3*x^3*c^3*b^3*ln(F)^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+c^3*b^3*ln(F)^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))/c^4/b^4/ln(F)^4

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.34 \[ \int F^{c (a+b x)} \left (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3\right ) \, dx=\frac {{\left ({\left (b^{3} c^{3} e^{3} x^{3} + 3 \, b^{3} c^{3} d e^{2} x^{2} + 3 \, b^{3} c^{3} d^{2} e x + b^{3} c^{3} d^{3}\right )} \log \left (F\right )^{3} - 6 \, e^{3} - 3 \, {\left (b^{2} c^{2} e^{3} x^{2} + 2 \, b^{2} c^{2} d e^{2} x + b^{2} c^{2} d^{2} e\right )} \log \left (F\right )^{2} + 6 \, {\left (b c e^{3} x + b c d e^{2}\right )} \log \left (F\right )\right )} F^{b c x + a c}}{b^{4} c^{4} \log \left (F\right )^{4}} \]

[In]

integrate(F^(c*(b*x+a))*(e^3*x^3+3*d*e^2*x^2+3*d^2*e*x+d^3),x, algorithm="fricas")

[Out]

((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*
log(F)^4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 231 vs. \(2 (107) = 214\).

Time = 0.10 (sec) , antiderivative size = 231, normalized size of antiderivative = 2.10 \[ \int F^{c (a+b x)} \left (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3\right ) \, dx=\begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{3} c^{3} d^{3} \log {\left (F \right )}^{3} + 3 b^{3} c^{3} d^{2} e x \log {\left (F \right )}^{3} + 3 b^{3} c^{3} d e^{2} x^{2} \log {\left (F \right )}^{3} + b^{3} c^{3} e^{3} x^{3} \log {\left (F \right )}^{3} - 3 b^{2} c^{2} d^{2} e \log {\left (F \right )}^{2} - 6 b^{2} c^{2} d e^{2} x \log {\left (F \right )}^{2} - 3 b^{2} c^{2} e^{3} x^{2} \log {\left (F \right )}^{2} + 6 b c d e^{2} \log {\left (F \right )} + 6 b c e^{3} x \log {\left (F \right )} - 6 e^{3}\right )}{b^{4} c^{4} \log {\left (F \right )}^{4}} & \text {for}\: b^{4} c^{4} \log {\left (F \right )}^{4} \neq 0 \\d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4} & \text {otherwise} \end {cases} \]

[In]

integrate(F**(c*(b*x+a))*(e**3*x**3+3*d*e**2*x**2+3*d**2*e*x+d**3),x)

[Out]

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))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.87 \[ \int F^{c (a+b x)} \left (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3\right ) \, dx=\frac {F^{b c x + a c} d^{3}}{b c \log \left (F\right )} + \frac {3 \, {\left (F^{a c} b c x \log \left (F\right ) - F^{a c}\right )} F^{b c x} d^{2} e}{b^{2} c^{2} \log \left (F\right )^{2}} + \frac {3 \, {\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} d e^{2}}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac {{\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} e^{3}}{b^{4} c^{4} \log \left (F\right )^{4}} \]

[In]

integrate(F^(c*(b*x+a))*(e^3*x^3+3*d*e^2*x^2+3*d^2*e*x+d^3),x, algorithm="maxima")

[Out]

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*log(F) - 6*F^(a*c))*F^(b*c*x)*e^3/(
b^4*c^4*log(F)^4)

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.41 (sec) , antiderivative size = 4706, normalized size of antiderivative = 42.78 \[ \int F^{c (a+b x)} \left (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3\right ) \, dx=\text {Too large to display} \]

[In]

integrate(F^(c*(b*x+a))*(e^3*x^3+3*d*e^2*x^2+3*d^2*e*x+d^3),x, algorithm="giac")

[Out]

-(((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*log(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*log(abs(F)) + 12*b*c*d*e^2*log(
abs(F)) - 12*e^3)*(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*l
og(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*b^3*c^3*d^2*e*x*sgn(F) - 9*pi*b^3*c^3*d^2*e*x*log(abs(F))^2*sgn
(F) - 3*pi^3*b^3*c^3*d^2*e*x + 9*pi*b^3*c^3*d^2*e*x*log(abs(F))^2 + pi^3*b^3*c^3*d^3*sgn(F) - 3*pi*b^3*c^3*d^3
*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3*d^3 + 3*pi*b^3*c^3*d^3*log(abs(F))^2 + 6*pi*b^2*c^2*e^3*x^2*log(abs(F))*s
gn(F) - 6*pi*b^2*c^2*e^3*x^2*log(abs(F)) + 12*pi*b^2*c^2*d*e^2*x*log(abs(F))*sgn(F) - 12*pi*b^2*c^2*d*e^2*x*lo
g(abs(F)) + 6*pi*b^2*c^2*d^2*e*log(abs(F))*sgn(F) - 6*pi*b^2*c^2*d^2*e*log(abs(F)) - 6*pi*b*c*e^3*x*sgn(F) + 6
*pi*b*c*e^3*x - 6*pi*b*c*d*e^2*sgn(F) + 6*pi*b*c*d*e^2)*(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)/((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))*cos(-1/2*pi*b*c*x*sgn(F) + 1/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c) - ((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*b^3*c^3*d^2*e*x*sgn(F) - 9*pi*b^3*c^3*d^2*e*x*log(abs(F))^2*sgn(F) - 3*
pi^3*b^3*c^3*d^2*e*x + 9*pi*b^3*c^3*d^2*e*x*log(abs(F))^2 + pi^3*b^3*c^3*d^3*sgn(F) - 3*pi*b^3*c^3*d^3*log(abs
(F))^2*sgn(F) - pi^3*b^3*c^3*d^3 + 3*pi*b^3*c^3*d^3*log(abs(F))^2 + 6*pi*b^2*c^2*e^3*x^2*log(abs(F))*sgn(F) -
6*pi*b^2*c^2*e^3*x^2*log(abs(F)) + 12*pi*b^2*c^2*d*e^2*x*log(abs(F))*sgn(F) - 12*pi*b^2*c^2*d*e^2*x*log(abs(F)
) + 6*pi*b^2*c^2*d^2*e*log(abs(F))*sgn(F) - 6*pi*b^2*c^2*d^2*e*log(abs(F)) - 6*pi*b*c*e^3*x*sgn(F) + 6*pi*b*c*
e^3*x - 6*pi*b*c*d*e^2*sgn(F) + 6*pi*b*c*d*e^2)*(pi^4*b^4*c^4*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - p
i^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*l
og(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*(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*l
og(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*log(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*log(abs(F)) + 12*b*c*d*e^
2*log(abs(F)) - 12*e^3)*(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)/((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))*sin(-1/2*pi*b*c*x*sgn(F) + 1
/2*pi*b*c*x - 1/2*pi*a*c*sgn(F) + 1/2*pi*a*c))*e^(b*c*x*log(abs(F)) + a*c*log(abs(F))) - 1/2*I*((pi^3*b^3*c^3*
e^3*x^3*sgn(F) + 3*I*pi^2*b^3*c^3*e^3*x^3*log(abs(F))*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*I*pi^2*b^3*c^3*e^3*x^3*log(abs(F)) + 3*pi*b^3*c^3*e^3*x^3*log(abs(F))^2 + 2*I*b^3*c^3*e^
3*x^3*log(abs(F))^3 + 3*pi^3*b^3*c^3*d*e^2*x^2*sgn(F) + 9*I*pi^2*b^3*c^3*d*e^2*x^2*log(abs(F))*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*I*pi^2*b^3*c^3*d*e^2*x^2*log(abs(F)) + 9*
pi*b^3*c^3*d*e^2*x^2*log(abs(F))^2 + 6*I*b^3*c^3*d*e^2*x^2*log(abs(F))^3 + 3*pi^3*b^3*c^3*d^2*e*x*sgn(F) + 9*I
*pi^2*b^3*c^3*d^2*e*x*log(abs(F))*sgn(F) - 9*pi*b^3*c^3*d^2*e*x*log(abs(F))^2*sgn(F) - 3*pi^3*b^3*c^3*d^2*e*x
- 9*I*pi^2*b^3*c^3*d^2*e*x*log(abs(F)) + 9*pi*b^3*c^3*d^2*e*x*log(abs(F))^2 + 6*I*b^3*c^3*d^2*e*x*log(abs(F))^
3 + pi^3*b^3*c^3*d^3*sgn(F) + 3*I*pi^2*b^3*c^3*d^3*log(abs(F))*sgn(F) - 3*pi*b^3*c^3*d^3*log(abs(F))^2*sgn(F)
- pi^3*b^3*c^3*d^3 - 3*I*pi^2*b^3*c^3*d^3*log(abs(F)) + 3*pi*b^3*c^3*d^3*log(abs(F))^2 + 2*I*b^3*c^3*d^3*log(a
bs(F))^3 - 3*I*pi^2*b^2*c^2*e^3*x^2*sgn(F) + 6*pi*b^2*c^2*e^3*x^2*log(abs(F))*sgn(F) + 3*I*pi^2*b^2*c^2*e^3*x^
2 - 6*pi*b^2*c^2*e^3*x^2*log(abs(F)) - 6*I*b^2*c^2*e^3*x^2*log(abs(F))^2 - 6*I*pi^2*b^2*c^2*d*e^2*x*sgn(F) + 1
2*pi*b^2*c^2*d*e^2*x*log(abs(F))*sgn(F) + 6*I*pi^2*b^2*c^2*d*e^2*x - 12*pi*b^2*c^2*d*e^2*x*log(abs(F)) - 12*I*
b^2*c^2*d*e^2*x*log(abs(F))^2 - 3*I*pi^2*b^2*c^2*d^2*e*sgn(F) + 6*pi*b^2*c^2*d^2*e*log(abs(F))*sgn(F) + 3*I*pi
^2*b^2*c^2*d^2*e - 6*pi*b^2*c^2*d^2*e*log(abs(F)) - 6*I*b^2*c^2*d^2*e*log(abs(F))^2 - 6*pi*b*c*e^3*x*sgn(F) +
6*pi*b*c*e^3*x + 12*I*b*c*e^3*x*log(abs(F)) - 6*pi*b*c*d*e^2*sgn(F) + 6*pi*b*c*d*e^2 + 12*I*b*c*d*e^2*log(abs(
F)) - 12*I*e^3)*e^(1/2*I*pi*b*c*x*sgn(F) - 1/2*I*pi*b*c*x + 1/2*I*pi*a*c*sgn(F) - 1/2*I*pi*a*c)/(pi^4*b^4*c^4*
sgn(F) + 4*I*pi^3*b^4*c^4*log(abs(F))*sgn(F) - 6*pi^2*b^4*c^4*log(abs(F))^2*sgn(F) - 4*I*pi*b^4*c^4*log(abs(F)
)^3*sgn(F) - pi^4*b^4*c^4 - 4*I*pi^3*b^4*c^4*log(abs(F)) + 6*pi^2*b^4*c^4*log(abs(F))^2 + 4*I*pi*b^4*c^4*log(a
bs(F))^3 - 2*b^4*c^4*log(abs(F))^4) + (pi^3*b^3*c^3*e^3*x^3*sgn(F) - 3*I*pi^2*b^3*c^3*e^3*x^3*log(abs(F))*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*I*pi^2*b^3*c^3*e^3*x^3*log(abs(F)) +
3*pi*b^3*c^3*e^3*x^3*log(abs(F))^2 - 2*I*b^3*c^3*e^3*x^3*log(abs(F))^3 + 3*pi^3*b^3*c^3*d*e^2*x^2*sgn(F) - 9*
I*pi^2*b^3*c^3*d*e^2*x^2*log(abs(F))*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*I*pi^2*b^3*c^3*d*e^2*x^2*log(abs(F)) + 9*pi*b^3*c^3*d*e^2*x^2*log(abs(F))^2 - 6*I*b^3*c^3*d*e^2*x^2
*log(abs(F))^3 + 3*pi^3*b^3*c^3*d^2*e*x*sgn(F) - 9*I*pi^2*b^3*c^3*d^2*e*x*log(abs(F))*sgn(F) - 9*pi*b^3*c^3*d^
2*e*x*log(abs(F))^2*sgn(F) - 3*pi^3*b^3*c^3*d^2*e*x + 9*I*pi^2*b^3*c^3*d^2*e*x*log(abs(F)) + 9*pi*b^3*c^3*d^2*
e*x*log(abs(F))^2 - 6*I*b^3*c^3*d^2*e*x*log(abs(F))^3 + pi^3*b^3*c^3*d^3*sgn(F) - 3*I*pi^2*b^3*c^3*d^3*log(abs
(F))*sgn(F) - 3*pi*b^3*c^3*d^3*log(abs(F))^2*sgn(F) - pi^3*b^3*c^3*d^3 + 3*I*pi^2*b^3*c^3*d^3*log(abs(F)) + 3*
pi*b^3*c^3*d^3*log(abs(F))^2 - 2*I*b^3*c^3*d^3*log(abs(F))^3 + 3*I*pi^2*b^2*c^2*e^3*x^2*sgn(F) + 6*pi*b^2*c^2*
e^3*x^2*log(abs(F))*sgn(F) - 3*I*pi^2*b^2*c^2*e^3*x^2 - 6*pi*b^2*c^2*e^3*x^2*log(abs(F)) + 6*I*b^2*c^2*e^3*x^2
*log(abs(F))^2 + 6*I*pi^2*b^2*c^2*d*e^2*x*sgn(F) + 12*pi*b^2*c^2*d*e^2*x*log(abs(F))*sgn(F) - 6*I*pi^2*b^2*c^2
*d*e^2*x - 12*pi*b^2*c^2*d*e^2*x*log(abs(F)) + 12*I*b^2*c^2*d*e^2*x*log(abs(F))^2 + 3*I*pi^2*b^2*c^2*d^2*e*sgn
(F) + 6*pi*b^2*c^2*d^2*e*log(abs(F))*sgn(F) - 3*I*pi^2*b^2*c^2*d^2*e - 6*pi*b^2*c^2*d^2*e*log(abs(F)) + 6*I*b^
2*c^2*d^2*e*log(abs(F))^2 - 6*pi*b*c*e^3*x*sgn(F) + 6*pi*b*c*e^3*x - 12*I*b*c*e^3*x*log(abs(F)) - 6*pi*b*c*d*e
^2*sgn(F) + 6*pi*b*c*d*e^2 - 12*I*b*c*d*e^2*log(abs(F)) + 12*I*e^3)*e^(-1/2*I*pi*b*c*x*sgn(F) + 1/2*I*pi*b*c*x
- 1/2*I*pi*a*c*sgn(F) + 1/2*I*pi*a*c)/(pi^4*b^4*c^4*sgn(F) - 4*I*pi^3*b^4*c^4*log(abs(F))*sgn(F) - 6*pi^2*b^4
*c^4*log(abs(F))^2*sgn(F) + 4*I*pi*b^4*c^4*log(abs(F))^3*sgn(F) - pi^4*b^4*c^4 + 4*I*pi^3*b^4*c^4*log(abs(F))
+ 6*pi^2*b^4*c^4*log(abs(F))^2 - 4*I*pi*b^4*c^4*log(abs(F))^3 - 2*b^4*c^4*log(abs(F))^4))*e^(b*c*x*log(abs(F))
+ a*c*log(abs(F)))

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.50 \[ \int F^{c (a+b x)} \left (d^3+3 d^2 e x+3 d e^2 x^2+e^3 x^3\right ) \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (b^3\,c^3\,d^3\,{\ln \left (F\right )}^3+3\,b^3\,c^3\,d^2\,e\,x\,{\ln \left (F\right )}^3+3\,b^3\,c^3\,d\,e^2\,x^2\,{\ln \left (F\right )}^3+b^3\,c^3\,e^3\,x^3\,{\ln \left (F\right )}^3-3\,b^2\,c^2\,d^2\,e\,{\ln \left (F\right )}^2-6\,b^2\,c^2\,d\,e^2\,x\,{\ln \left (F\right )}^2-3\,b^2\,c^2\,e^3\,x^2\,{\ln \left (F\right )}^2+6\,b\,c\,d\,e^2\,\ln \left (F\right )+6\,b\,c\,e^3\,x\,\ln \left (F\right )-6\,e^3\right )}{b^4\,c^4\,{\ln \left (F\right )}^4} \]

[In]

int(F^(c*(a + b*x))*(d^3 + e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x),x)

[Out]

(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
*log(F)^3 + 3*b^3*c^3*d*e^2*x^2*log(F)^3))/(b^4*c^4*log(F)^4)