\(\int F^{a+b (c+d x)^2} (c+d x)^7 \, dx\) [191]

Optimal result

Integrand size = 21, antiderivative size = 126 \[ \int F^{a+b (c+d x)^2} (c+d x)^7 \, dx=-\frac {3 F^{a+b (c+d x)^2}}{b^4 d \log ^4(F)}+\frac {3 F^{a+b (c+d x)^2} (c+d x)^2}{b^3 d \log ^3(F)}-\frac {3 F^{a+b (c+d x)^2} (c+d x)^4}{2 b^2 d \log ^2(F)}+\frac {F^{a+b (c+d x)^2} (c+d x)^6}{2 b d \log (F)} \] Output:

-3*F^(a+b*(d*x+c)^2)/b^4/d/ln(F)^4+3*F^(a+b*(d*x+c)^2)*(d*x+c)^2/b^3/d/ln( 
F)^3-3/2*F^(a+b*(d*x+c)^2)*(d*x+c)^4/b^2/d/ln(F)^2+1/2*F^(a+b*(d*x+c)^2)*( 
d*x+c)^6/b/d/ln(F)
 

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.57 \[ \int F^{a+b (c+d x)^2} (c+d x)^7 \, dx=\frac {F^{a+b (c+d x)^2} \left (-6+6 b (c+d x)^2 \log (F)-3 b^2 (c+d x)^4 \log ^2(F)+b^3 (c+d x)^6 \log ^3(F)\right )}{2 b^4 d \log ^4(F)} \] Input:

Integrate[F^(a + b*(c + d*x)^2)*(c + d*x)^7,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.86 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.19, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2641, 2641, 2641, 2638}

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.

\(\displaystyle \int (c+d x)^7 F^{a+b (c+d x)^2} \, dx\)

\(\Big \downarrow \) 2641

\(\displaystyle \frac {(c+d x)^6 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {3 \int F^{b (c+d x)^2+a} (c+d x)^5dx}{b \log (F)}\)

\(\Big \downarrow \) 2641

\(\displaystyle \frac {(c+d x)^6 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {3 \left (\frac {(c+d x)^4 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {2 \int F^{b (c+d x)^2+a} (c+d x)^3dx}{b \log (F)}\right )}{b \log (F)}\)

\(\Big \downarrow \) 2641

\(\displaystyle \frac {(c+d x)^6 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {3 \left (\frac {(c+d x)^4 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {2 \left (\frac {(c+d x)^2 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {\int F^{b (c+d x)^2+a} (c+d x)dx}{b \log (F)}\right )}{b \log (F)}\right )}{b \log (F)}\)

\(\Big \downarrow \) 2638

\(\displaystyle \frac {(c+d x)^6 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {3 \left (\frac {(c+d x)^4 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {2 \left (\frac {(c+d x)^2 F^{a+b (c+d x)^2}}{2 b d \log (F)}-\frac {F^{a+b (c+d x)^2}}{2 b^2 d \log ^2(F)}\right )}{b \log (F)}\right )}{b \log (F)}\)

Input:

Int[F^(a + b*(c + d*x)^2)*(c + d*x)^7,x]
 

Output:

(F^(a + b*(c + d*x)^2)*(c + d*x)^6)/(2*b*d*Log[F]) - (3*((F^(a + b*(c + d* 
x)^2)*(c + d*x)^4)/(2*b*d*Log[F]) - (2*(-1/2*F^(a + b*(c + d*x)^2)/(b^2*d* 
Log[F]^2) + (F^(a + b*(c + d*x)^2)*(c + d*x)^2)/(2*b*d*Log[F])))/(b*Log[F] 
)))/(b*Log[F])
 

Defintions of rubi rules used

rule 2638

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(e + f*x)^n*(F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n 
*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] && EqQ 
[d*e - c*f, 0]
 

rule 2641

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(c + d*x)^(m - n + 1)*(F^(a + b*(c + d*x)^n)/(b*d*n*L 
og[F])), x] - Simp[(m - n + 1)/(b*n*Log[F])   Int[(c + d*x)^(m - n)*F^(a + 
b*(c + d*x)^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/ 
n)] && LtQ[0, (m + 1)/n, 5] && IntegerQ[n] && (LtQ[0, n, m + 1] || LtQ[m, n 
, 0])
 

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.90

method	result	size
orering	\(\frac {\left (d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+6 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}+15 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}+20 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}+15 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}+6 \ln \left (F \right )^{3} b^{3} c^{5} d x +\ln \left (F \right )^{3} b^{3} c^{6}-3 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}-12 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}-18 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}-12 \ln \left (F \right )^{2} b^{2} c^{3} d x -3 \ln \left (F \right )^{2} b^{2} c^{4}+6 \ln \left (F \right ) b \,d^{2} x^{2}+12 \ln \left (F \right ) b c d x +6 \ln \left (F \right ) b \,c^{2}-6\right ) F^{a +b \left (d x +c \right )^{2}}}{2 d \,b^{4} \ln \left (F \right )^{4}}\)	\(239\)
gosper	\(\frac {\left (d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+6 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}+15 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}+20 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}+15 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}+6 \ln \left (F \right )^{3} b^{3} c^{5} d x +\ln \left (F \right )^{3} b^{3} c^{6}-3 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}-12 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}-18 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}-12 \ln \left (F \right )^{2} b^{2} c^{3} d x -3 \ln \left (F \right )^{2} b^{2} c^{4}+6 \ln \left (F \right ) b \,d^{2} x^{2}+12 \ln \left (F \right ) b c d x +6 \ln \left (F \right ) b \,c^{2}-6\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 \ln \left (F \right )^{4} b^{4} d}\)	\(249\)
risch	\(\frac {\left (d^{6} x^{6} b^{3} \ln \left (F \right )^{3}+6 c \,d^{5} x^{5} b^{3} \ln \left (F \right )^{3}+15 \ln \left (F \right )^{3} b^{3} c^{2} d^{4} x^{4}+20 \ln \left (F \right )^{3} b^{3} c^{3} d^{3} x^{3}+15 \ln \left (F \right )^{3} b^{3} c^{4} d^{2} x^{2}+6 \ln \left (F \right )^{3} b^{3} c^{5} d x +\ln \left (F \right )^{3} b^{3} c^{6}-3 d^{4} x^{4} b^{2} \ln \left (F \right )^{2}-12 d^{3} c \,x^{3} b^{2} \ln \left (F \right )^{2}-18 \ln \left (F \right )^{2} b^{2} c^{2} d^{2} x^{2}-12 \ln \left (F \right )^{2} b^{2} c^{3} d x -3 \ln \left (F \right )^{2} b^{2} c^{4}+6 \ln \left (F \right ) b \,d^{2} x^{2}+12 \ln \left (F \right ) b c d x +6 \ln \left (F \right ) b \,c^{2}-6\right ) F^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a}}{2 \ln \left (F \right )^{4} b^{4} d}\)	\(249\)
norman	\(\frac {\left (\ln \left (F \right )^{3} b^{3} c^{6}-3 \ln \left (F \right )^{2} b^{2} c^{4}+6 \ln \left (F \right ) b \,c^{2}-6\right ) {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{4} b^{4} d}+\frac {d^{5} x^{6} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right ) b}+\frac {3 c \left (\ln \left (F \right )^{2} b^{2} c^{4}-2 \ln \left (F \right ) b \,c^{2}+2\right ) x \,{\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{3} b^{3}}+\frac {3 d \left (5 \ln \left (F \right )^{2} b^{2} c^{4}-6 \ln \left (F \right ) b \,c^{2}+2\right ) x^{2} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{3} b^{3}}+\frac {3 d^{3} \left (5 \ln \left (F \right ) b \,c^{2}-1\right ) x^{4} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{2} b^{2}}+\frac {3 d^{4} c \,x^{5} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right ) b}+\frac {2 c \,d^{2} \left (5 \ln \left (F \right ) b \,c^{2}-3\right ) x^{3} {\mathrm e}^{\left (a +b \left (d x +c \right )^{2}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{2} b^{2}}\)	\(301\)
parallelrisch	\(\frac {d^{6} F^{a +b \left (d x +c \right )^{2}} x^{6} b^{3} \ln \left (F \right )^{3}+6 c \,d^{5} F^{a +b \left (d x +c \right )^{2}} x^{5} b^{3} \ln \left (F \right )^{3}+15 \ln \left (F \right )^{3} x^{4} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{2} d^{4}+20 \ln \left (F \right )^{3} x^{3} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{3} d^{3}+15 \ln \left (F \right )^{3} x^{2} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{4} d^{2}+6 \ln \left (F \right )^{3} x \,F^{a +b \left (d x +c \right )^{2}} b^{3} c^{5} d +\ln \left (F \right )^{3} F^{a +b \left (d x +c \right )^{2}} b^{3} c^{6}-3 d^{4} F^{a +b \left (d x +c \right )^{2}} x^{4} b^{2} \ln \left (F \right )^{2}-12 d^{3} c \,F^{a +b \left (d x +c \right )^{2}} x^{3} b^{2} \ln \left (F \right )^{2}-18 \ln \left (F \right )^{2} x^{2} F^{a +b \left (d x +c \right )^{2}} b^{2} c^{2} d^{2}-12 \ln \left (F \right )^{2} x \,F^{a +b \left (d x +c \right )^{2}} b^{2} c^{3} d -3 \ln \left (F \right )^{2} F^{a +b \left (d x +c \right )^{2}} b^{2} c^{4}+6 d^{2} F^{a +b \left (d x +c \right )^{2}} x^{2} b \ln \left (F \right )+12 c \,F^{a +b \left (d x +c \right )^{2}} x b \ln \left (F \right ) d +6 \ln \left (F \right ) F^{a +b \left (d x +c \right )^{2}} b \,c^{2}-6 F^{a +b \left (d x +c \right )^{2}}}{2 \ln \left (F \right )^{4} b^{4} d}\)	\(435\)

Input:

int(F^(a+b*(d*x+c)^2)*(d*x+c)^7,x,method=_RETURNVERBOSE)
 

Output:

1/2/d*(d^6*x^6*b^3*ln(F)^3+6*c*d^5*x^5*b^3*ln(F)^3+15*ln(F)^3*b^3*c^2*d^4* 
x^4+20*ln(F)^3*b^3*c^3*d^3*x^3+15*ln(F)^3*b^3*c^4*d^2*x^2+6*ln(F)^3*b^3*c^ 
5*d*x+ln(F)^3*b^3*c^6-3*d^4*x^4*b^2*ln(F)^2-12*d^3*c*x^3*b^2*ln(F)^2-18*ln 
(F)^2*b^2*c^2*d^2*x^2-12*ln(F)^2*b^2*c^3*d*x-3*ln(F)^2*b^2*c^4+6*ln(F)*b*d 
^2*x^2+12*ln(F)*b*c*d*x+6*ln(F)*b*c^2-6)/b^4/ln(F)^4*F^(a+b*(d*x+c)^2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.65 \[ \int F^{a+b (c+d x)^2} (c+d x)^7 \, dx=\frac {{\left ({\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} - 3 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} + 6 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) - 6\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{2 \, b^{4} d \log \left (F\right )^{4}} \] Input:

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^7,x, algorithm="fricas")
 

Output:

1/2*((b^3*d^6*x^6 + 6*b^3*c*d^5*x^5 + 15*b^3*c^2*d^4*x^4 + 20*b^3*c^3*d^3* 
x^3 + 15*b^3*c^4*d^2*x^2 + 6*b^3*c^5*d*x + b^3*c^6)*log(F)^3 - 3*(b^2*d^4* 
x^4 + 4*b^2*c*d^3*x^3 + 6*b^2*c^2*d^2*x^2 + 4*b^2*c^3*d*x + b^2*c^4)*log(F 
)^2 + 6*(b*d^2*x^2 + 2*b*c*d*x + b*c^2)*log(F) - 6)*F^(b*d^2*x^2 + 2*b*c*d 
*x + b*c^2 + a)/(b^4*d*log(F)^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 364 vs. \(2 (112) = 224\).

Time = 0.14 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.89 \[ \int F^{a+b (c+d x)^2} (c+d x)^7 \, dx=\begin {cases} \frac {F^{a + b \left (c + d x\right )^{2}} \left (b^{3} c^{6} \log {\left (F \right )}^{3} + 6 b^{3} c^{5} d x \log {\left (F \right )}^{3} + 15 b^{3} c^{4} d^{2} x^{2} \log {\left (F \right )}^{3} + 20 b^{3} c^{3} d^{3} x^{3} \log {\left (F \right )}^{3} + 15 b^{3} c^{2} d^{4} x^{4} \log {\left (F \right )}^{3} + 6 b^{3} c d^{5} x^{5} \log {\left (F \right )}^{3} + b^{3} d^{6} x^{6} \log {\left (F \right )}^{3} - 3 b^{2} c^{4} \log {\left (F \right )}^{2} - 12 b^{2} c^{3} d x \log {\left (F \right )}^{2} - 18 b^{2} c^{2} d^{2} x^{2} \log {\left (F \right )}^{2} - 12 b^{2} c d^{3} x^{3} \log {\left (F \right )}^{2} - 3 b^{2} d^{4} x^{4} \log {\left (F \right )}^{2} + 6 b c^{2} \log {\left (F \right )} + 12 b c d x \log {\left (F \right )} + 6 b d^{2} x^{2} \log {\left (F \right )} - 6\right )}{2 b^{4} d \log {\left (F \right )}^{4}} & \text {for}\: b^{4} d \log {\left (F \right )}^{4} \neq 0 \\c^{7} x + \frac {7 c^{6} d x^{2}}{2} + 7 c^{5} d^{2} x^{3} + \frac {35 c^{4} d^{3} x^{4}}{4} + 7 c^{3} d^{4} x^{5} + \frac {7 c^{2} d^{5} x^{6}}{2} + c d^{6} x^{7} + \frac {d^{7} x^{8}}{8} & \text {otherwise} \end {cases} \] Input:

integrate(F**(a+b*(d*x+c)**2)*(d*x+c)**7,x)
 

Output:

Piecewise((F**(a + b*(c + d*x)**2)*(b**3*c**6*log(F)**3 + 6*b**3*c**5*d*x* 
log(F)**3 + 15*b**3*c**4*d**2*x**2*log(F)**3 + 20*b**3*c**3*d**3*x**3*log( 
F)**3 + 15*b**3*c**2*d**4*x**4*log(F)**3 + 6*b**3*c*d**5*x**5*log(F)**3 + 
b**3*d**6*x**6*log(F)**3 - 3*b**2*c**4*log(F)**2 - 12*b**2*c**3*d*x*log(F) 
**2 - 18*b**2*c**2*d**2*x**2*log(F)**2 - 12*b**2*c*d**3*x**3*log(F)**2 - 3 
*b**2*d**4*x**4*log(F)**2 + 6*b*c**2*log(F) + 12*b*c*d*x*log(F) + 6*b*d**2 
*x**2*log(F) - 6)/(2*b**4*d*log(F)**4), Ne(b**4*d*log(F)**4, 0)), (c**7*x 
+ 7*c**6*d*x**2/2 + 7*c**5*d**2*x**3 + 35*c**4*d**3*x**4/4 + 7*c**3*d**4*x 
**5 + 7*c**2*d**5*x**6/2 + c*d**6*x**7 + d**7*x**8/8, True))
 

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.89 (sec) , antiderivative size = 2452, normalized size of antiderivative = 19.46 \[ \int F^{a+b (c+d x)^2} (c+d x)^7 \, dx=\text {Too large to display} \] Input:

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^7,x, algorithm="maxima")
 

Output:

-7/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b*c*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F) 
/(b*d^2))) - 1)*log(F)^2/((b*log(F))^(3/2)*d^2*sqrt(-(b*d^2*x + b*c*d)^2*l 
og(F)/(b*d^2))) - F^((b*d^2*x + b*c*d)^2/(b*d^2))*b*log(F)/((b*log(F))^(3/ 
2)*d))*F^a*c^6/sqrt(b*log(F)) + 21/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^2*c^2*( 
erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 1)*log(F)^3/((b*log(F))^( 
5/2)*d^3*sqrt(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))) - 2*F^((b*d^2*x + b*c* 
d)^2/(b*d^2))*b^2*c*log(F)^2/((b*log(F))^(5/2)*d^2) - (b*d^2*x + b*c*d)^3* 
gamma(3/2, -(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))*log(F)^3/((b*log(F))^(5/2) 
*d^5*(-(b*d^2*x + b*c*d)^2*log(F)/(b*d^2))^(3/2)))*F^a*c^5*d/sqrt(b*log(F) 
) - 35/2*(sqrt(pi)*(b*d^2*x + b*c*d)*b^3*c^3*(erf(sqrt(-(b*d^2*x + b*c*d)^ 
2*log(F)/(b*d^2))) - 1)*log(F)^4/((b*log(F))^(7/2)*d^4*sqrt(-(b*d^2*x + b* 
c*d)^2*log(F)/(b*d^2))) - 3*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^3*c^2*log(F) 
^3/((b*log(F))^(7/2)*d^3) - 3*(b*d^2*x + b*c*d)^3*b*c*gamma(3/2, -(b*d^2*x 
 + b*c*d)^2*log(F)/(b*d^2))*log(F)^4/((b*log(F))^(7/2)*d^6*(-(b*d^2*x + b* 
c*d)^2*log(F)/(b*d^2))^(3/2)) + b^2*gamma(2, -(b*d^2*x + b*c*d)^2*log(F)/( 
b*d^2))*log(F)^2/((b*log(F))^(7/2)*d^3))*F^a*c^4*d^2/sqrt(b*log(F)) + 35/2 
*(sqrt(pi)*(b*d^2*x + b*c*d)*b^4*c^4*(erf(sqrt(-(b*d^2*x + b*c*d)^2*log(F) 
/(b*d^2))) - 1)*log(F)^5/((b*log(F))^(9/2)*d^5*sqrt(-(b*d^2*x + b*c*d)^2*l 
og(F)/(b*d^2))) - 4*F^((b*d^2*x + b*c*d)^2/(b*d^2))*b^4*c^3*log(F)^4/((b*l 
og(F))^(9/2)*d^4) - 6*(b*d^2*x + b*c*d)^3*b^2*c^2*gamma(3/2, -(b*d^2*x ...
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.82 \[ \int F^{a+b (c+d x)^2} (c+d x)^7 \, dx=\frac {{\left (b^{3} d^{6} {\left (x + \frac {c}{d}\right )}^{6} \log \left (F\right )^{3} - 3 \, b^{2} d^{4} {\left (x + \frac {c}{d}\right )}^{4} \log \left (F\right )^{2} + 6 \, b d^{2} {\left (x + \frac {c}{d}\right )}^{2} \log \left (F\right ) - 6\right )} e^{\left (b d^{2} x^{2} \log \left (F\right ) + 2 \, b c d x \log \left (F\right ) + b c^{2} \log \left (F\right ) + a \log \left (F\right )\right )}}{2 \, b^{4} d \log \left (F\right )^{4}} \] Input:

integrate(F^(a+b*(d*x+c)^2)*(d*x+c)^7,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

1/2*(b^3*d^6*(x + c/d)^6*log(F)^3 - 3*b^2*d^4*(x + c/d)^4*log(F)^2 + 6*b*d 
^2*(x + c/d)^2*log(F) - 6)*e^(b*d^2*x^2*log(F) + 2*b*c*d*x*log(F) + b*c^2* 
log(F) + a*log(F))/(b^4*d*log(F)^4)
 

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 253, normalized size of antiderivative = 2.01 \[ \int F^{a+b (c+d x)^2} (c+d x)^7 \, dx=\frac {F^{b\,d^2\,x^2}\,F^a\,F^{b\,c^2}\,F^{2\,b\,c\,d\,x}\,\left (b^3\,c^6\,{\ln \left (F\right )}^3+6\,b^3\,c^5\,d\,x\,{\ln \left (F\right )}^3+15\,b^3\,c^4\,d^2\,x^2\,{\ln \left (F\right )}^3+20\,b^3\,c^3\,d^3\,x^3\,{\ln \left (F\right )}^3+15\,b^3\,c^2\,d^4\,x^4\,{\ln \left (F\right )}^3+6\,b^3\,c\,d^5\,x^5\,{\ln \left (F\right )}^3+b^3\,d^6\,x^6\,{\ln \left (F\right )}^3-3\,b^2\,c^4\,{\ln \left (F\right )}^2-12\,b^2\,c^3\,d\,x\,{\ln \left (F\right )}^2-18\,b^2\,c^2\,d^2\,x^2\,{\ln \left (F\right )}^2-12\,b^2\,c\,d^3\,x^3\,{\ln \left (F\right )}^2-3\,b^2\,d^4\,x^4\,{\ln \left (F\right )}^2+6\,b\,c^2\,\ln \left (F\right )+12\,b\,c\,d\,x\,\ln \left (F\right )+6\,b\,d^2\,x^2\,\ln \left (F\right )-6\right )}{2\,b^4\,d\,{\ln \left (F\right )}^4} \] Input:

int(F^(a + b*(c + d*x)^2)*(c + d*x)^7,x)
 

Output:

(F^(b*d^2*x^2)*F^a*F^(b*c^2)*F^(2*b*c*d*x)*(6*b*c^2*log(F) - 3*b^2*c^4*log 
(F)^2 + b^3*c^6*log(F)^3 + 6*b*d^2*x^2*log(F) - 3*b^2*d^4*x^4*log(F)^2 + b 
^3*d^6*x^6*log(F)^3 - 12*b^2*c*d^3*x^3*log(F)^2 + 6*b^3*c*d^5*x^5*log(F)^3 
 - 18*b^2*c^2*d^2*x^2*log(F)^2 + 15*b^3*c^4*d^2*x^2*log(F)^3 + 20*b^3*c^3* 
d^3*x^3*log(F)^3 + 15*b^3*c^2*d^4*x^4*log(F)^3 + 12*b*c*d*x*log(F) - 12*b^ 
2*c^3*d*x*log(F)^2 + 6*b^3*c^5*d*x*log(F)^3 - 6))/(2*b^4*d*log(F)^4)
 

Reduce [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.97 \[ \int F^{a+b (c+d x)^2} (c+d x)^7 \, dx=\frac {f^{b \,d^{2} x^{2}+2 b c d x +b \,c^{2}+a} \left (\mathrm {log}\left (f \right )^{3} b^{3} c^{6}+6 \mathrm {log}\left (f \right )^{3} b^{3} c^{5} d x +15 \mathrm {log}\left (f \right )^{3} b^{3} c^{4} d^{2} x^{2}+20 \mathrm {log}\left (f \right )^{3} b^{3} c^{3} d^{3} x^{3}+15 \mathrm {log}\left (f \right )^{3} b^{3} c^{2} d^{4} x^{4}+6 \mathrm {log}\left (f \right )^{3} b^{3} c \,d^{5} x^{5}+\mathrm {log}\left (f \right )^{3} b^{3} d^{6} x^{6}-3 \mathrm {log}\left (f \right )^{2} b^{2} c^{4}-12 \mathrm {log}\left (f \right )^{2} b^{2} c^{3} d x -18 \mathrm {log}\left (f \right )^{2} b^{2} c^{2} d^{2} x^{2}-12 \mathrm {log}\left (f \right )^{2} b^{2} c \,d^{3} x^{3}-3 \mathrm {log}\left (f \right )^{2} b^{2} d^{4} x^{4}+6 \,\mathrm {log}\left (f \right ) b \,c^{2}+12 \,\mathrm {log}\left (f \right ) b c d x +6 \,\mathrm {log}\left (f \right ) b \,d^{2} x^{2}-6\right )}{2 \mathrm {log}\left (f \right )^{4} b^{4} d} \] Input:

int(F^(a+b*(d*x+c)^2)*(d*x+c)^7,x)
 

Output:

(f**(a + b*c**2 + 2*b*c*d*x + b*d**2*x**2)*(log(f)**3*b**3*c**6 + 6*log(f) 
**3*b**3*c**5*d*x + 15*log(f)**3*b**3*c**4*d**2*x**2 + 20*log(f)**3*b**3*c 
**3*d**3*x**3 + 15*log(f)**3*b**3*c**2*d**4*x**4 + 6*log(f)**3*b**3*c*d**5 
*x**5 + log(f)**3*b**3*d**6*x**6 - 3*log(f)**2*b**2*c**4 - 12*log(f)**2*b* 
*2*c**3*d*x - 18*log(f)**2*b**2*c**2*d**2*x**2 - 12*log(f)**2*b**2*c*d**3* 
x**3 - 3*log(f)**2*b**2*d**4*x**4 + 6*log(f)*b*c**2 + 12*log(f)*b*c*d*x + 
6*log(f)*b*d**2*x**2 - 6))/(2*log(f)**4*b**4*d)