\(\int F^{c (a+b x)} (d-e x^2)^2 \, dx\) [15]

Optimal result

Integrand size = 20, antiderivative size = 141 \[ \int F^{c (a+b x)} \left (d-e x^2\right )^2 \, dx=\frac {24 e^2 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac {24 e^2 F^{c (a+b x)} x}{b^4 c^4 \log ^4(F)}-\frac {4 e F^{c (a+b x)} \left (d-3 e x^2\right )}{b^3 c^3 \log ^3(F)}+\frac {4 e F^{c (a+b x)} x \left (d-e x^2\right )}{b^2 c^2 \log ^2(F)}+\frac {F^{c (a+b x)} \left (d-e x^2\right )^2}{b c \log (F)} \] Output:

24*e^2*F^(c*(b*x+a))/b^5/c^5/ln(F)^5-24*e^2*F^(c*(b*x+a))*x/b^4/c^4/ln(F)^ 
4-4*e*F^(c*(b*x+a))*(-3*e*x^2+d)/b^3/c^3/ln(F)^3+4*e*F^(c*(b*x+a))*x*(-e*x 
^2+d)/b^2/c^2/ln(F)^2+F^(c*(b*x+a))*(-e*x^2+d)^2/b/c/ln(F)
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71 \[ \int F^{c (a+b x)} \left (d-e x^2\right )^2 \, dx=\frac {F^{c (a+b x)} \left (24 e^2-24 b c e^2 x \log (F)-4 b^2 c^2 e \left (d-3 e x^2\right ) \log ^2(F)+4 b^3 c^3 e x \left (d-e x^2\right ) \log ^3(F)+b^4 c^4 \left (d-e x^2\right )^2 \log ^4(F)\right )}{b^5 c^5 \log ^5(F)} \] Input:

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

Output:

(F^(c*(a + b*x))*(24*e^2 - 24*b*c*e^2*x*Log[F] - 4*b^2*c^2*e*(d - 3*e*x^2) 
*Log[F]^2 + 4*b^3*c^3*e*x*(d - e*x^2)*Log[F]^3 + b^4*c^4*(d - e*x^2)^2*Log 
[F]^4))/(b^5*c^5*Log[F]^5)
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.60, 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.

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

\(\Big \downarrow \) 2626

\(\displaystyle \int \left (d^2 F^{c (a+b x)}-2 d e x^2 F^{c (a+b x)}+e^2 x^4 F^{c (a+b x)}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {24 e^2 F^{c (a+b x)}}{b^5 c^5 \log ^5(F)}-\frac {24 e^2 x F^{c (a+b x)}}{b^4 c^4 \log ^4(F)}-\frac {4 d e F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {12 e^2 x^2 F^{c (a+b x)}}{b^3 c^3 \log ^3(F)}+\frac {4 d e x F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}-\frac {4 e^2 x^3 F^{c (a+b x)}}{b^2 c^2 \log ^2(F)}+\frac {d^2 F^{c (a+b x)}}{b c \log (F)}-\frac {2 d e x^2 F^{c (a+b x)}}{b c \log (F)}+\frac {e^2 x^4 F^{c (a+b x)}}{b c \log (F)}\)

Input:

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

Output:

(24*e^2*F^(c*(a + b*x)))/(b^5*c^5*Log[F]^5) - (24*e^2*F^(c*(a + b*x))*x)/( 
b^4*c^4*Log[F]^4) - (4*d*e*F^(c*(a + b*x)))/(b^3*c^3*Log[F]^3) + (12*e^2*F 
^(c*(a + b*x))*x^2)/(b^3*c^3*Log[F]^3) + (4*d*e*F^(c*(a + b*x))*x)/(b^2*c^ 
2*Log[F]^2) - (4*e^2*F^(c*(a + b*x))*x^3)/(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^2)/(b*c*Log[F]) + (e^2* 
F^(c*(a + b*x))*x^4)/(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 = 0.06 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06

method	result	size
gosper	\(\frac {\left (e^{2} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}-2 \ln \left (F \right )^{4} b^{4} c^{4} d e \,x^{2}+\ln \left (F \right )^{4} b^{4} c^{4} d^{2}-4 e^{2} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+4 \ln \left (F \right )^{3} b^{3} c^{3} d e x +12 \ln \left (F \right )^{2} b^{2} c^{2} e^{2} x^{2}-4 d e \ln \left (F \right )^{2} b^{2} c^{2}-24 \ln \left (F \right ) b c \,e^{2} x +24 e^{2}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\)	\(150\)
risch	\(\frac {\left (e^{2} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}-2 \ln \left (F \right )^{4} b^{4} c^{4} d e \,x^{2}+\ln \left (F \right )^{4} b^{4} c^{4} d^{2}-4 e^{2} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+4 \ln \left (F \right )^{3} b^{3} c^{3} d e x +12 \ln \left (F \right )^{2} b^{2} c^{2} e^{2} x^{2}-4 d e \ln \left (F \right )^{2} b^{2} c^{2}-24 \ln \left (F \right ) b c \,e^{2} x +24 e^{2}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\)	\(150\)
orering	\(\frac {\left (e^{2} x^{4} \ln \left (F \right )^{4} b^{4} c^{4}-2 \ln \left (F \right )^{4} b^{4} c^{4} d e \,x^{2}+\ln \left (F \right )^{4} b^{4} c^{4} d^{2}-4 e^{2} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+4 \ln \left (F \right )^{3} b^{3} c^{3} d e x +12 \ln \left (F \right )^{2} b^{2} c^{2} e^{2} x^{2}-4 d e \ln \left (F \right )^{2} b^{2} c^{2}-24 \ln \left (F \right ) b c \,e^{2} x +24 e^{2}\right ) F^{c \left (b x +a \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}\)	\(150\)
meijerg	\(-\frac {F^{a c} e^{2} \left (24-\frac {\left (5 b^{4} c^{4} x^{4} \ln \left (F \right )^{4}-20 b^{3} c^{3} x^{3} \ln \left (F \right )^{3}+60 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-120 b c x \ln \left (F \right )+120\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{5}\right )}{b^{5} c^{5} \ln \left (F \right )^{5}}+\frac {2 F^{a c} e d \left (2-\frac {\left (3 b^{2} c^{2} x^{2} \ln \left (F \right )^{2}-6 b c x \ln \left (F \right )+6\right ) {\mathrm e}^{b c x \ln \left (F \right )}}{3}\right )}{b^{3} c^{3} \ln \left (F \right )^{3}}-\frac {F^{a c} d^{2} \left (1-{\mathrm e}^{b c x \ln \left (F \right )}\right )}{\ln \left (F \right ) b c}\)	\(172\)
norman	\(\frac {\left (\ln \left (F \right )^{4} b^{4} c^{4} d^{2}-4 d e \ln \left (F \right )^{2} b^{2} c^{2}+24 e^{2}\right ) {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{5} b^{5} c^{5}}+\frac {e^{2} x^{4} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right ) b c}-\frac {4 e^{2} x^{3} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2} c^{2}}-\frac {2 e \left (\ln \left (F \right )^{2} b^{2} c^{2} d -6 e \right ) x^{2} {\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{c^{3} b^{3} \ln \left (F \right )^{3}}+\frac {4 e \left (\ln \left (F \right )^{2} b^{2} c^{2} d -6 e \right ) x \,{\mathrm e}^{c \left (b x +a \right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4} c^{4}}\)	\(194\)
parallelrisch	\(\frac {x^{4} F^{c \left (b x +a \right )} e^{2} \ln \left (F \right )^{4} b^{4} c^{4}-2 \ln \left (F \right )^{4} x^{2} F^{c \left (b x +a \right )} b^{4} c^{4} d e +\ln \left (F \right )^{4} F^{c \left (b x +a \right )} b^{4} c^{4} d^{2}-4 e^{2} F^{c \left (b x +a \right )} x^{3} \ln \left (F \right )^{3} b^{3} c^{3}+4 \ln \left (F \right )^{3} x \,F^{c \left (b x +a \right )} b^{3} c^{3} d e +12 \ln \left (F \right )^{2} x^{2} F^{c \left (b x +a \right )} b^{2} c^{2} e^{2}-4 \ln \left (F \right )^{2} F^{c \left (b x +a \right )} b^{2} c^{2} d e -24 \ln \left (F \right ) x \,F^{c \left (b x +a \right )} b c \,e^{2}+24 F^{c \left (b x +a \right )} e^{2}}{\ln \left (F \right )^{5} b^{5} c^{5}}\)	\(222\)

Input:

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

Output:

(e^2*x^4*ln(F)^4*b^4*c^4-2*ln(F)^4*b^4*c^4*d*e*x^2+ln(F)^4*b^4*c^4*d^2-4*e 
^2*x^3*ln(F)^3*b^3*c^3+4*ln(F)^3*b^3*c^3*d*e*x+12*ln(F)^2*b^2*c^2*e^2*x^2- 
4*d*e*ln(F)^2*b^2*c^2-24*ln(F)*b*c*e^2*x+24*e^2)*F^(c*(b*x+a))/ln(F)^5/b^5 
/c^5
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.01 \[ \int F^{c (a+b x)} \left (d-e x^2\right )^2 \, dx=-\frac {{\left (24 \, b c e^{2} x \log \left (F\right ) - {\left (b^{4} c^{4} e^{2} x^{4} - 2 \, b^{4} c^{4} d e x^{2} + b^{4} c^{4} d^{2}\right )} \log \left (F\right )^{4} + 4 \, {\left (b^{3} c^{3} e^{2} x^{3} - b^{3} c^{3} d e x\right )} \log \left (F\right )^{3} - 4 \, {\left (3 \, b^{2} c^{2} e^{2} x^{2} - b^{2} c^{2} d e\right )} \log \left (F\right )^{2} - 24 \, e^{2}\right )} F^{b c x + a c}}{b^{5} c^{5} \log \left (F\right )^{5}} \] Input:

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

Output:

-(24*b*c*e^2*x*log(F) - (b^4*c^4*e^2*x^4 - 2*b^4*c^4*d*e*x^2 + b^4*c^4*d^2 
)*log(F)^4 + 4*(b^3*c^3*e^2*x^3 - b^3*c^3*d*e*x)*log(F)^3 - 4*(3*b^2*c^2*e 
^2*x^2 - b^2*c^2*d*e)*log(F)^2 - 24*e^2)*F^(b*c*x + a*c)/(b^5*c^5*log(F)^5 
)
 

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.43 \[ \int F^{c (a+b x)} \left (d-e x^2\right )^2 \, dx=\begin {cases} \frac {F^{c \left (a + b x\right )} \left (b^{4} c^{4} d^{2} \log {\left (F \right )}^{4} - 2 b^{4} c^{4} d e x^{2} \log {\left (F \right )}^{4} + b^{4} c^{4} e^{2} x^{4} \log {\left (F \right )}^{4} + 4 b^{3} c^{3} d e x \log {\left (F \right )}^{3} - 4 b^{3} c^{3} e^{2} x^{3} \log {\left (F \right )}^{3} - 4 b^{2} c^{2} d e \log {\left (F \right )}^{2} + 12 b^{2} c^{2} e^{2} x^{2} \log {\left (F \right )}^{2} - 24 b c e^{2} x \log {\left (F \right )} + 24 e^{2}\right )}{b^{5} c^{5} \log {\left (F \right )}^{5}} & \text {for}\: b^{5} c^{5} \log {\left (F \right )}^{5} \neq 0 \\d^{2} x - \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5} & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((F**(c*(a + b*x))*(b**4*c**4*d**2*log(F)**4 - 2*b**4*c**4*d*e*x* 
*2*log(F)**4 + b**4*c**4*e**2*x**4*log(F)**4 + 4*b**3*c**3*d*e*x*log(F)**3 
 - 4*b**3*c**3*e**2*x**3*log(F)**3 - 4*b**2*c**2*d*e*log(F)**2 + 12*b**2*c 
**2*e**2*x**2*log(F)**2 - 24*b*c*e**2*x*log(F) + 24*e**2)/(b**5*c**5*log(F 
)**5), Ne(b**5*c**5*log(F)**5, 0)), (d**2*x - 2*d*e*x**3/3 + e**2*x**5/5, 
True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.30 \[ \int F^{c (a+b x)} \left (d-e x^2\right )^2 \, dx=\frac {F^{b c x + a c} d^{2}}{b c \log \left (F\right )} - \frac {2 \, {\left (F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 2 \, F^{a c} b c x \log \left (F\right ) + 2 \, F^{a c}\right )} F^{b c x} d e}{b^{3} c^{3} \log \left (F\right )^{3}} + \frac {{\left (F^{a c} b^{4} c^{4} x^{4} \log \left (F\right )^{4} - 4 \, F^{a c} b^{3} c^{3} x^{3} \log \left (F\right )^{3} + 12 \, F^{a c} b^{2} c^{2} x^{2} \log \left (F\right )^{2} - 24 \, F^{a c} b c x \log \left (F\right ) + 24 \, F^{a c}\right )} F^{b c x} e^{2}}{b^{5} c^{5} \log \left (F\right )^{5}} \] Input:

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

Output:

F^(b*c*x + a*c)*d^2/(b*c*log(F)) - 2*(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/(b^3*c^3*log(F)^3) + (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^2/(b^5* 
c^5*log(F)^5)
 

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.19 (sec) , antiderivative size = 5426, normalized size of antiderivative = 38.48 \[ \int F^{c (a+b x)} \left (d-e x^2\right )^2 \, dx=\text {Too large to display} \] Input:

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

Output:

-((4*(pi^3*b^4*c^4*e^2*x^4*log(abs(F))*sgn(F) - pi*b^4*c^4*e^2*x^4*log(abs 
(F))^3*sgn(F) - pi^3*b^4*c^4*e^2*x^4*log(abs(F)) + pi*b^4*c^4*e^2*x^4*log( 
abs(F))^3 - 2*pi^3*b^4*c^4*d*e*x^2*log(abs(F))*sgn(F) + 2*pi*b^4*c^4*d*e*x 
^2*log(abs(F))^3*sgn(F) + 2*pi^3*b^4*c^4*d*e*x^2*log(abs(F)) - 2*pi*b^4*c^ 
4*d*e*x^2*log(abs(F))^3 - pi^3*b^3*c^3*e^2*x^3*sgn(F) + pi^3*b^4*c^4*d^2*l 
og(abs(F))*sgn(F) + 3*pi*b^3*c^3*e^2*x^3*log(abs(F))^2*sgn(F) - pi*b^4*c^4 
*d^2*log(abs(F))^3*sgn(F) + pi^3*b^3*c^3*e^2*x^3 - pi^3*b^4*c^4*d^2*log(ab 
s(F)) - 3*pi*b^3*c^3*e^2*x^3*log(abs(F))^2 + pi*b^4*c^4*d^2*log(abs(F))^3 
+ pi^3*b^3*c^3*d*e*x*sgn(F) - 3*pi*b^3*c^3*d*e*x*log(abs(F))^2*sgn(F) - pi 
^3*b^3*c^3*d*e*x + 3*pi*b^3*c^3*d*e*x*log(abs(F))^2 - 6*pi*b^2*c^2*e^2*x^2 
*log(abs(F))*sgn(F) + 6*pi*b^2*c^2*e^2*x^2*log(abs(F)) + 2*pi*b^2*c^2*d*e* 
log(abs(F))*sgn(F) - 2*pi*b^2*c^2*d*e*log(abs(F)) + 6*pi*b*c*e^2*x*sgn(F) 
- 6*pi*b*c*e^2*x)*(pi^5*b^5*c^5*sgn(F) - 10*pi^3*b^5*c^5*log(abs(F))^2*sgn 
(F) + 5*pi*b^5*c^5*log(abs(F))^4*sgn(F) - pi^5*b^5*c^5 + 10*pi^3*b^5*c^5*l 
og(abs(F))^2 - 5*pi*b^5*c^5*log(abs(F))^4)/((pi^5*b^5*c^5*sgn(F) - 10*pi^3 
*b^5*c^5*log(abs(F))^2*sgn(F) + 5*pi*b^5*c^5*log(abs(F))^4*sgn(F) - pi^5*b 
^5*c^5 + 10*pi^3*b^5*c^5*log(abs(F))^2 - 5*pi*b^5*c^5*log(abs(F))^4)^2 + ( 
5*pi^4*b^5*c^5*log(abs(F))*sgn(F) - 10*pi^2*b^5*c^5*log(abs(F))^3*sgn(F) - 
 5*pi^4*b^5*c^5*log(abs(F)) + 10*pi^2*b^5*c^5*log(abs(F))^3 - 2*b^5*c^5*lo 
g(abs(F))^5)^2) - (pi^4*b^4*c^4*e^2*x^4*sgn(F) - 6*pi^2*b^4*c^4*e^2*x^4...
 

Mupad [B] (verification not implemented)

Time = 22.61 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06 \[ \int F^{c (a+b x)} \left (d-e x^2\right )^2 \, dx=\frac {F^{a\,c+b\,c\,x}\,\left (b^4\,c^4\,d^2\,{\ln \left (F\right )}^4-2\,b^4\,c^4\,d\,e\,x^2\,{\ln \left (F\right )}^4+b^4\,c^4\,e^2\,x^4\,{\ln \left (F\right )}^4+4\,b^3\,c^3\,d\,e\,x\,{\ln \left (F\right )}^3-4\,b^3\,c^3\,e^2\,x^3\,{\ln \left (F\right )}^3-4\,b^2\,c^2\,d\,e\,{\ln \left (F\right )}^2+12\,b^2\,c^2\,e^2\,x^2\,{\ln \left (F\right )}^2-24\,b\,c\,e^2\,x\,\ln \left (F\right )+24\,e^2\right )}{b^5\,c^5\,{\ln \left (F\right )}^5} \] Input:

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

Output:

(F^(a*c + b*c*x)*(24*e^2 + b^4*c^4*d^2*log(F)^4 - 24*b*c*e^2*x*log(F) + 12 
*b^2*c^2*e^2*x^2*log(F)^2 - 4*b^3*c^3*e^2*x^3*log(F)^3 + b^4*c^4*e^2*x^4*l 
og(F)^4 - 4*b^2*c^2*d*e*log(F)^2 + 4*b^3*c^3*d*e*x*log(F)^3 - 2*b^4*c^4*d* 
e*x^2*log(F)^4))/(b^5*c^5*log(F)^5)
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.06 \[ \int F^{c (a+b x)} \left (d-e x^2\right )^2 \, dx=\frac {f^{b c x +a c} \left (\mathrm {log}\left (f \right )^{4} b^{4} c^{4} d^{2}-2 \mathrm {log}\left (f \right )^{4} b^{4} c^{4} d e \,x^{2}+\mathrm {log}\left (f \right )^{4} b^{4} c^{4} e^{2} x^{4}+4 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d e x -4 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} e^{2} x^{3}-4 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d e +12 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} e^{2} x^{2}-24 \,\mathrm {log}\left (f \right ) b c \,e^{2} x +24 e^{2}\right )}{\mathrm {log}\left (f \right )^{5} b^{5} c^{5}} \] Input:

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

Output:

(f**(a*c + b*c*x)*(log(f)**4*b**4*c**4*d**2 - 2*log(f)**4*b**4*c**4*d*e*x* 
*2 + log(f)**4*b**4*c**4*e**2*x**4 + 4*log(f)**3*b**3*c**3*d*e*x - 4*log(f 
)**3*b**3*c**3*e**2*x**3 - 4*log(f)**2*b**2*c**2*d*e + 12*log(f)**2*b**2*c 
**2*e**2*x**2 - 24*log(f)*b*c*e**2*x + 24*e**2))/(log(f)**5*b**5*c**5)