\(\int F^{c (a+b x)} (d-e x^3)^2 \, dx\) [19]

Optimal result

Integrand size = 20, antiderivative size = 203 \[ \int F^{c (a+b x)} \left (d-e x^3\right )^2 \, dx=\frac {720 e^2 F^{c (a+b x)}}{b^7 c^7 \log ^7(F)}-\frac {720 e^2 F^{c (a+b x)} x}{b^6 c^6 \log ^6(F)}+\frac {360 e^2 F^{c (a+b x)} x^2}{b^5 c^5 \log ^5(F)}+\frac {12 e F^{c (a+b x)} \left (d-10 e x^3\right )}{b^4 c^4 \log ^4(F)}-\frac {6 e F^{c (a+b x)} x \left (2 d-5 e x^3\right )}{b^3 c^3 \log ^3(F)}+\frac {6 e F^{c (a+b x)} x^2 \left (d-e x^3\right )}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} \left (d-e x^3\right )^2}{b c \log (F)} \] Output:

720*e^2*F^(c*(b*x+a))/b^7/c^7/ln(F)^7-720*e^2*F^(c*(b*x+a))*x/b^6/c^6/ln(F 
)^6+360*e^2*F^(c*(b*x+a))*x^2/b^5/c^5/ln(F)^5+12*e*F^(c*(b*x+a))*(-10*e*x^ 
3+d)/b^4/c^4/ln(F)^4-6*e*F^(c*(b*x+a))*x*(-5*e*x^3+2*d)/b^3/c^3/ln(F)^3+6* 
e*F^(c*(b*x+a))*x^2*(-e*x^3+d)/b^2/c^2/ln(F)^2+F^(c*(b*x+a))*(-e*x^3+d)^2/ 
b/c/ln(F)
 

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.71 \[ \int F^{c (a+b x)} \left (d-e x^3\right )^2 \, dx=\frac {F^{c (a+b x)} \left (720 e^2-720 b c e^2 x \log (F)+360 b^2 c^2 e^2 x^2 \log ^2(F)+12 b^3 c^3 e \left (d-10 e x^3\right ) \log ^3(F)+6 b^4 c^4 e x \left (-2 d+5 e x^3\right ) \log ^4(F)+6 b^5 c^5 e x^2 \left (d-e x^3\right ) \log ^5(F)+b^6 c^6 \left (d-e x^3\right )^2 \log ^6(F)\right )}{b^7 c^7 \log ^7(F)} \] Input:

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

Output:

(F^(c*(a + b*x))*(720*e^2 - 720*b*c*e^2*x*Log[F] + 360*b^2*c^2*e^2*x^2*Log 
[F]^2 + 12*b^3*c^3*e*(d - 10*e*x^3)*Log[F]^3 + 6*b^4*c^4*e*x*(-2*d + 5*e*x 
^3)*Log[F]^4 + 6*b^5*c^5*e*x^2*(d - e*x^3)*Log[F]^5 + b^6*c^6*(d - e*x^3)^ 
2*Log[F]^6))/(b^7*c^7*Log[F]^7)
 

Rubi [A] (verified)

Time = 0.95 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.51, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.

\(\displaystyle \int \left (d-e x^3\right )^2 F^{c (a+b x)} \, dx\)

\(\Big \downarrow \) 2626

\(\displaystyle \int \left (d^2 F^{c (a+b x)}-2 d e x^3 F^{c (a+b x)}+e^2 x^6 F^{c (a+b x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {720 e^2 F^{c (a+b x)}}{b^7 c^7 \log ^7(F)}-\frac {720 e^2 x F^{c (a+b x)}}{b^6 c^6 \log ^6(F)}+\frac {360 e^2 x^2 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}+\frac {12 d e F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}-\frac {120 e^2 x^3 F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}-\frac {12 d e x F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {30 e^2 x^4 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {6 d e x^2 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {6 e^2 x^5 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {d^2 F^{c (a+b x)}}{b c \log (F)}-\frac {2 d e x^3 F^{c (a+b x)}}{b c \log (F)}+\frac {e^2 x^6 F^{c (a+b x)}}{b c \log (F)}\)

Input:

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

Output:

(720*e^2*F^(c*(a + b*x)))/(b^7*c^7*Log[F]^7) - (720*e^2*F^(c*(a + b*x))*x) 
/(b^6*c^6*Log[F]^6) + (360*e^2*F^(c*(a + b*x))*x^2)/(b^5*c^5*Log[F]^5) + ( 
12*d*e*F^(c*(a + b*x)))/(b^4*c^4*Log[F]^4) - (120*e^2*F^(c*(a + b*x))*x^3) 
/(b^4*c^4*Log[F]^4) - (12*d*e*F^(c*(a + b*x))*x)/(b^3*c^3*Log[F]^3) + (30* 
e^2*F^(c*(a + b*x))*x^4)/(b^3*c^3*Log[F]^3) + (6*d*e*F^(c*(a + b*x))*x^2)/ 
(b^2*c^2*Log[F]^2) - (6*e^2*F^(c*(a + b*x))*x^5)/(b^2*c^2*Log[F]^2) + (d^2 
*F^(c*(a + b*x)))/(b*c*Log[F]) - (2*d*e*F^(c*(a + b*x))*x^3)/(b*c*Log[F]) 
+ (e^2*F^(c*(a + b*x))*x^6)/(b*c*Log[F])
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2626

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]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00

method	result	size
gosper	\(\frac {\left (e^{2} x^{6} \ln \left (F \right )^{6} b^{6} c^{6}-2 \ln \left (F \right )^{6} b^{6} c^{6} d e \,x^{3}-6 e^{2} x^{5} \ln \left (F \right )^{5} b^{5} c^{5}+\ln \left (F \right )^{6} b^{6} c^{6} d^{2}+6 \ln \left (F \right )^{5} b^{5} c^{5} d e \,x^{2}+30 e^{2} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}-12 \ln \left (F \right )^{4} b^{4} c^{4} d e x -120 e^{2} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+12 \ln \left (F \right )^{3} b^{3} c^{3} d e +360 \ln \left (F \right )^{2} b^{2} c^{2} e^{2} x^{2}-720 \ln \left (F \right ) b c \,e^{2} x +720 e^{2}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{7} b^{7} c^{7}}\)	\(203\)
risch	\(\frac {\left (e^{2} x^{6} \ln \left (F \right )^{6} b^{6} c^{6}-2 \ln \left (F \right )^{6} b^{6} c^{6} d e \,x^{3}-6 e^{2} x^{5} \ln \left (F \right )^{5} b^{5} c^{5}+\ln \left (F \right )^{6} b^{6} c^{6} d^{2}+6 \ln \left (F \right )^{5} b^{5} c^{5} d e \,x^{2}+30 e^{2} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}-12 \ln \left (F \right )^{4} b^{4} c^{4} d e x -120 e^{2} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+12 \ln \left (F \right )^{3} b^{3} c^{3} d e +360 \ln \left (F \right )^{2} b^{2} c^{2} e^{2} x^{2}-720 \ln \left (F \right ) b c \,e^{2} x +720 e^{2}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{7} b^{7} c^{7}}\)	\(203\)
orering	\(\frac {\left (e^{2} x^{6} \ln \left (F \right )^{6} b^{6} c^{6}-2 \ln \left (F \right )^{6} b^{6} c^{6} d e \,x^{3}-6 e^{2} x^{5} \ln \left (F \right )^{5} b^{5} c^{5}+\ln \left (F \right )^{6} b^{6} c^{6} d^{2}+6 \ln \left (F \right )^{5} b^{5} c^{5} d e \,x^{2}+30 e^{2} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}-12 \ln \left (F \right )^{4} b^{4} c^{4} d e x -120 e^{2} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+12 \ln \left (F \right )^{3} b^{3} c^{3} d e +360 \ln \left (F \right )^{2} b^{2} c^{2} e^{2} x^{2}-720 \ln \left (F \right ) b c \,e^{2} x +720 e^{2}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{7} b^{7} c^{7}}\)	\(203\)
meijerg	\(-\frac {F^{a c} e^{2} \left (720-\frac {\left (7 b^{6} c^{6} x^{6} \ln \left (F \right )^{6}-42 b^{5} c^{5} x^{5} \ln \left (F \right )^{5}+210 b^{4} c^{4} x^{4} \ln \left (F \right )^{4}-840 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+2520 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-5040 b c x \ln \left (F \right )+5040\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{7}\right )}{b^{7} c^{7} \ln \left (F \right )^{7}}-\frac {2 F^{a c} e d \left (6-\frac {\left (-4 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+12 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-24 b c x \ln \left (F \right )+24\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{4}\right )}{b^{4} c^{4} \ln \left (F \right )^{4}}-\frac {F^{a c} d^{2} \left (1-{\mathrm e}^{b c x \ln \left (F \right )}\right )}{\ln \left (F \right ) b c}\)	\(217\)
norman	\(\frac {\left (\ln \left (F \right )^{6} b^{6} c^{6} d^{2}+12 \ln \left (F \right )^{3} b^{3} c^{3} d e +720 e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{7} b^{7} c^{7}}+\frac {e^{2} x^{6} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right ) b c}+\frac {30 e^{2} x^{4} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{3} b^{3} \ln \left (F \right )^{3}}-\frac {6 e^{2} x^{5} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} c^{2}}+\frac {6 e \left (\ln \left (F \right )^{3} b^{3} c^{3} d +60 e \right ) x^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{5} b^{5} \ln \left (F \right )^{5}}-\frac {12 e \left (\ln \left (F \right )^{3} b^{3} c^{3} d +60 e \right ) x \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{6} b^{6} c^{6}}-\frac {2 e \left (\ln \left (F \right )^{3} b^{3} c^{3} d +60 e \right ) x^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}\)	\(264\)
parallelrisch	\(\frac {x^{6} F^{c \left (b x +a \right )} e^{2} \ln \left (F \right )^{6} b^{6} c^{6}-2 \ln \left (F \right )^{6} x^{3} F^{c \left (b x +a \right )} b^{6} c^{6} d e -6 e^{2} F^{c \left (b x +a \right )} x^{5} \ln \left (F \right )^{5} b^{5} c^{5}+\ln \left (F \right )^{6} F^{c \left (b x +a \right )} b^{6} c^{6} d^{2}+6 \ln \left (F \right )^{5} x^{2} F^{c \left (b x +a \right )} b^{5} c^{5} d e +30 x^{4} F^{c \left (b x +a \right )} e^{2} \ln \left (F \right )^{4} b^{4} c^{4}-12 \ln \left (F \right )^{4} x \,F^{c \left (b x +a \right )} b^{4} c^{4} d e -120 e^{2} F^{c \left (b x +a \right )} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+12 \ln \left (F \right )^{3} F^{c \left (b x +a \right )} b^{3} c^{3} d e +360 \ln \left (F \right )^{2} x^{2} F^{c \left (b x +a \right )} b^{2} c^{2} e^{2}-720 \ln \left (F \right ) x \,F^{c \left (b x +a \right )} b c \,e^{2}+720 F^{c \left (b x +a \right )} e^{2}}{\ln \left (F \right )^{7} b^{7} c^{7}}\)	\(302\)

Input:

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

Output:

(e^2*x^6*ln(F)^6*b^6*c^6-2*ln(F)^6*b^6*c^6*d*e*x^3-6*e^2*x^5*ln(F)^5*b^5*c 
^5+ln(F)^6*b^6*c^6*d^2+6*ln(F)^5*b^5*c^5*d*e*x^2+30*e^2*x^4*ln(F)^4*b^4*c^ 
4-12*ln(F)^4*b^4*c^4*d*e*x-120*e^2*x^3*ln(F)^3*b^3*c^3+12*ln(F)^3*b^3*c^3* 
d*e+360*ln(F)^2*b^2*c^2*e^2*x^2-720*ln(F)*b*c*e^2*x+720*e^2)*F^(c*(b*x+a)) 
/ln(F)^7/b^7/c^7
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 193, normalized size of antiderivative = 0.95 \[ \int F^{c (a+b x)} \left (d-e x^3\right )^2 \, dx=\frac {{\left (360 \, b^{2} c^{2} e^{2} x^{2} \log \left (F\right )^{2} + {\left (b^{6} c^{6} e^{2} x^{6} - 2 \, b^{6} c^{6} d e x^{3} + b^{6} c^{6} d^{2}\right )} \log \left (F\right )^{6} - 720 \, b c e^{2} x \log \left (F\right ) - 6 \, {\left (b^{5} c^{5} e^{2} x^{5} - b^{5} c^{5} d e x^{2}\right )} \log \left (F\right )^{5} + 6 \, {\left (5 \, b^{4} c^{4} e^{2} x^{4} - 2 \, b^{4} c^{4} d e x\right )} \log \left (F\right )^{4} - 12 \, {\left (10 \, b^{3} c^{3} e^{2} x^{3} - b^{3} c^{3} d e\right )} \log \left (F\right )^{3} + 720 \, e^{2}\right )} F^{b c x + a c}}{b^{7} c^{7} \log \left (F\right )^{7}} \] Input:

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

Output:

(360*b^2*c^2*e^2*x^2*log(F)^2 + (b^6*c^6*e^2*x^6 - 2*b^6*c^6*d*e*x^3 + b^6 
*c^6*d^2)*log(F)^6 - 720*b*c*e^2*x*log(F) - 6*(b^5*c^5*e^2*x^5 - b^5*c^5*d 
*e*x^2)*log(F)^5 + 6*(5*b^4*c^4*e^2*x^4 - 2*b^4*c^4*d*e*x)*log(F)^4 - 12*( 
10*b^3*c^3*e^2*x^3 - b^3*c^3*d*e)*log(F)^3 + 720*e^2)*F^(b*c*x + a*c)/(b^7 
*c^7*log(F)^7)
 

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.29 \[ \int F^{c (a+b x)} \left (d-e x^3\right )^2 \, dx=\begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{6} c^{6} d^{2} \log {\left (F \right )}^{6} - 2 b^{6} c^{6} d e x^{3} \log {\left (F \right )}^{6} + b^{6} c^{6} e^{2} x^{6} \log {\left (F \right )}^{6} + 6 b^{5} c^{5} d e x^{2} \log {\left (F \right )}^{5} - 6 b^{5} c^{5} e^{2} x^{5} \log {\left (F \right )}^{5} - 12 b^{4} c^{4} d e x \log {\left (F \right )}^{4} + 30 b^{4} c^{4} e^{2} x^{4} \log {\left (F \right )}^{4} + 12 b^{3} c^{3} d e \log {\left (F \right )}^{3} - 120 b^{3} c^{3} e^{2} x^{3} \log {\left (F \right )}^{3} + 360 b^{2} c^{2} e^{2} x^{2} \log {\left (F \right )}^{2} - 720 b c e^{2} x \log {\left (F \right )} + 720 e^{2}\right )}{b^{7} c^{7} \log {\left (F \right )}^{7}} & \text {for}\: b^{7} c^{7} \log {\left (F \right )}^{7} \neq 0 \\d^{2} x - \frac {d e x^{4}}{2} + \frac {e^{2} x^{7}}{7} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((F**(c*(a + b*x))*(b**6*c**6*d**2*log(F)**6 - 2*b**6*c**6*d*e*x* 
*3*log(F)**6 + b**6*c**6*e**2*x**6*log(F)**6 + 6*b**5*c**5*d*e*x**2*log(F) 
**5 - 6*b**5*c**5*e**2*x**5*log(F)**5 - 12*b**4*c**4*d*e*x*log(F)**4 + 30* 
b**4*c**4*e**2*x**4*log(F)**4 + 12*b**3*c**3*d*e*log(F)**3 - 120*b**3*c**3 
*e**2*x**3*log(F)**3 + 360*b**2*c**2*e**2*x**2*log(F)**2 - 720*b*c*e**2*x* 
log(F) + 720*e**2)/(b**7*c**7*log(F)**7), Ne(b**7*c**7*log(F)**7, 0)), (d* 
*2*x - d*e*x**4/2 + e**2*x**7/7, True))
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.20 \[ \int F^{c (a+b x)} \left (d-e x^3\right )^2 \, dx=\frac {F^{b c x + a c} d^{2}}{b c \log \left (F\right )} - \frac {2 \, {\left (F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} - 3 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} + 6 \, F^{a c} b c x \log \left (F\right ) - 6 \, F^{a c}\right )} F^{b c x} d e}{b^{4} c^{4} \log \left (F\right )^{4}} + \frac {{\left (F^{a c} b^{6} c^{6} x^{6} \log \left (F\right )^{6} - 6 \, F^{a c} b^{5} c^{5} x^{5} \log \left (F\right )^{5} + 30 \, F^{a c} b^{4} c^{4} x^{4} \log \left (F\right )^{4} - 120 \, F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} + 360 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 720 \, F^{a c} b c x \log \left (F\right ) + 720 \, F^{a c}\right )} F^{b c x} e^{2}}{b^{7} c^{7} \log \left (F\right )^{7}} \] Input:

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

Output:

F^(b*c*x + a*c)*d^2/(b*c*log(F)) - 2*(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/(b^4*c^4*log(F)^4) + (F^(a*c)*b^6*c^6*x^6*log(F)^6 - 6*F^(a*c)*b^5*c^5 
*x^5*log(F)^5 + 30*F^(a*c)*b^4*c^4*x^4*log(F)^4 - 120*F^(a*c)*b^3*c^3*x^3* 
log(F)^3 + 360*F^(a*c)*b^2*c^2*x^2*log(F)^2 - 720*F^(a*c)*b*c*x*log(F) + 7 
20*F^(a*c))*F^(b*c*x)*e^2/(b^7*c^7*log(F)^7)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.24 (sec) , antiderivative size = 9878, normalized size of antiderivative = 48.66 \[ \int F^{c (a+b x)} \left (d-e x^3\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

-((2*(3*pi^5*b^6*c^6*e^2*x^6*log(abs(F))*sgn(F) - 10*pi^3*b^6*c^6*e^2*x^6* 
log(abs(F))^3*sgn(F) + 3*pi*b^6*c^6*e^2*x^6*log(abs(F))^5*sgn(F) - 3*pi^5* 
b^6*c^6*e^2*x^6*log(abs(F)) + 10*pi^3*b^6*c^6*e^2*x^6*log(abs(F))^3 - 3*pi 
*b^6*c^6*e^2*x^6*log(abs(F))^5 - 6*pi^5*b^6*c^6*d*e*x^3*log(abs(F))*sgn(F) 
 + 20*pi^3*b^6*c^6*d*e*x^3*log(abs(F))^3*sgn(F) - 6*pi*b^6*c^6*d*e*x^3*log 
(abs(F))^5*sgn(F) + 6*pi^5*b^6*c^6*d*e*x^3*log(abs(F)) - 20*pi^3*b^6*c^6*d 
*e*x^3*log(abs(F))^3 + 6*pi*b^6*c^6*d*e*x^3*log(abs(F))^5 - 3*pi^5*b^5*c^5 
*e^2*x^5*sgn(F) + 30*pi^3*b^5*c^5*e^2*x^5*log(abs(F))^2*sgn(F) - 15*pi*b^5 
*c^5*e^2*x^5*log(abs(F))^4*sgn(F) + 3*pi^5*b^5*c^5*e^2*x^5 - 30*pi^3*b^5*c 
^5*e^2*x^5*log(abs(F))^2 + 15*pi*b^5*c^5*e^2*x^5*log(abs(F))^4 + 3*pi^5*b^ 
6*c^6*d^2*log(abs(F))*sgn(F) - 10*pi^3*b^6*c^6*d^2*log(abs(F))^3*sgn(F) + 
3*pi*b^6*c^6*d^2*log(abs(F))^5*sgn(F) - 3*pi^5*b^6*c^6*d^2*log(abs(F)) + 1 
0*pi^3*b^6*c^6*d^2*log(abs(F))^3 - 3*pi*b^6*c^6*d^2*log(abs(F))^5 + 3*pi^5 
*b^5*c^5*d*e*x^2*sgn(F) - 30*pi^3*b^5*c^5*d*e*x^2*log(abs(F))^2*sgn(F) + 1 
5*pi*b^5*c^5*d*e*x^2*log(abs(F))^4*sgn(F) - 3*pi^5*b^5*c^5*d*e*x^2 + 30*pi 
^3*b^5*c^5*d*e*x^2*log(abs(F))^2 - 15*pi*b^5*c^5*d*e*x^2*log(abs(F))^4 - 6 
0*pi^3*b^4*c^4*e^2*x^4*log(abs(F))*sgn(F) + 60*pi*b^4*c^4*e^2*x^4*log(abs( 
F))^3*sgn(F) + 60*pi^3*b^4*c^4*e^2*x^4*log(abs(F)) - 60*pi*b^4*c^4*e^2*x^4 
*log(abs(F))^3 + 24*pi^3*b^4*c^4*d*e*x*log(abs(F))*sgn(F) - 24*pi*b^4*c^4* 
d*e*x*log(abs(F))^3*sgn(F) - 24*pi^3*b^4*c^4*d*e*x*log(abs(F)) + 24*pi*...
 

Mupad [B] (verification not implemented)

Time = 22.98 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)} \left (d-e x^3\right )^2 \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (b^6\,c^6\,d^2\,{\ln \left (F\right )}^6-2\,b^6\,c^6\,d\,e\,x^3\,{\ln \left (F\right )}^6+b^6\,c^6\,e^2\,x^6\,{\ln \left (F\right )}^6+6\,b^5\,c^5\,d\,e\,x^2\,{\ln \left (F\right )}^5-6\,b^5\,c^5\,e^2\,x^5\,{\ln \left (F\right )}^5-12\,b^4\,c^4\,d\,e\,x\,{\ln \left (F\right )}^4+30\,b^4\,c^4\,e^2\,x^4\,{\ln \left (F\right )}^4+12\,b^3\,c^3\,d\,e\,{\ln \left (F\right )}^3-120\,b^3\,c^3\,e^2\,x^3\,{\ln \left (F\right )}^3+360\,b^2\,c^2\,e^2\,x^2\,{\ln \left (F\right )}^2-720\,b\,c\,e^2\,x\,\ln \left (F\right )+720\,e^2\right )}{b^7\,c^7\,{\ln \left (F\right )}^7} \] Input:

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

Output:

(F^(a*c + b*c*x)*(720*e^2 + b^6*c^6*d^2*log(F)^6 - 720*b*c*e^2*x*log(F) + 
360*b^2*c^2*e^2*x^2*log(F)^2 - 120*b^3*c^3*e^2*x^3*log(F)^3 + 30*b^4*c^4*e 
^2*x^4*log(F)^4 - 6*b^5*c^5*e^2*x^5*log(F)^5 + b^6*c^6*e^2*x^6*log(F)^6 + 
12*b^3*c^3*d*e*log(F)^3 - 12*b^4*c^4*d*e*x*log(F)^4 + 6*b^5*c^5*d*e*x^2*lo 
g(F)^5 - 2*b^6*c^6*d*e*x^3*log(F)^6))/(b^7*c^7*log(F)^7)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.00 \[ \int F^{c (a+b x)} \left (d-e x^3\right )^2 \, dx=\frac {f^{b c x +a c} \left (\mathrm {log}\left (f \right )^{6} b^{6} c^{6} d^{2}-2 \mathrm {log}\left (f \right )^{6} b^{6} c^{6} d e \,x^{3}+\mathrm {log}\left (f \right )^{6} b^{6} c^{6} e^{2} x^{6}+6 \mathrm {log}\left (f \right )^{5} b^{5} c^{5} d e \,x^{2}-6 \mathrm {log}\left (f \right )^{5} b^{5} c^{5} e^{2} x^{5}-12 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d e x +30 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} e^{2} x^{4}+12 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d e -120 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e^{2} x^{3}+360 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e^{2} x^{2}-720 \,\mathrm {log}\left (f \right ) b c \,e^{2} x +720 e^{2}\right )}{\mathrm {log}\left (f \right )^{7} b^{7} c^{7}} \] Input:

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

Output:

(f**(a*c + b*c*x)*(log(f)**6*b**6*c**6*d**2 - 2*log(f)**6*b**6*c**6*d*e*x* 
*3 + log(f)**6*b**6*c**6*e**2*x**6 + 6*log(f)**5*b**5*c**5*d*e*x**2 - 6*lo 
g(f)**5*b**5*c**5*e**2*x**5 - 12*log(f)**4*b**4*c**4*d*e*x + 30*log(f)**4* 
b**4*c**4*e**2*x**4 + 12*log(f)**3*b**3*c**3*d*e - 120*log(f)**3*b**3*c**3 
*e**2*x**3 + 360*log(f)**2*b**2*c**2*e**2*x**2 - 720*log(f)*b*c*e**2*x + 7 
20*e**2))/(log(f)**7*b**7*c**7)