\(\int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx\) [258]

Optimal result

Integrand size = 21, antiderivative size = 126 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^4 d \log ^4(F)}-\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{b^3 d (c+d x)^2 \log ^3(F)}+\frac {3 F^{a+\frac {b}{(c+d x)^2}}}{2 b^2 d (c+d x)^4 \log ^2(F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d (c+d x)^6 \log (F)} \] Output:

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 81, normalized size of antiderivative = 0.64 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {F^{a+\frac {b}{(c+d x)^2}} \left (6 (c+d x)^6-6 b (c+d x)^4 \log (F)+3 b^2 (c+d x)^2 \log ^2(F)-b^3 \log ^3(F)\right )}{2 b^4 d (c+d x)^6 \log ^4(F)} \] Input:

Integrate[F^(a + b/(c + d*x)^2)/(c + d*x)^9,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.75 (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.

\(\displaystyle \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx\)

\(\Big \downarrow \) 2641

\(\displaystyle -\frac {3 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^7}dx}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^6}\)

\(\Big \downarrow \) 2641

\(\displaystyle -\frac {3 \left (-\frac {2 \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^5}dx}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^4}\right )}{b \log (F)}-\frac {F^{a+\frac {b}{(c+d x)^2}}}{2 b d \log (F) (c+d x)^6}\)

\(\Big \downarrow \) 2641

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

\(\Big \downarrow \) 2638

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

Input:

Int[F^(a + b/(c + d*x)^2)/(c + d*x)^9,x]
 

Output:

-1/2*F^(a + b/(c + d*x)^2)/(b*d*(c + d*x)^6*Log[F]) - (3*(-1/2*F^(a + b/(c 
 + d*x)^2)/(b*d*(c + d*x)^4*Log[F]) - (2*(F^(a + b/(c + d*x)^2)/(2*b^2*d*L 
og[F]^2) - F^(a + b/(c + d*x)^2)/(2*b*d*(c + d*x)^2*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 = 1.71 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.71

method	result	size
risch	\(-\frac {\left (-6 d^{6} x^{6}-36 c \,d^{5} x^{5}+6 \ln \left (F \right ) b \,d^{4} x^{4}-90 c^{2} d^{4} x^{4}+24 \ln \left (F \right ) b c \,d^{3} x^{3}-120 c^{3} d^{3} x^{3}-3 \ln \left (F \right )^{2} b^{2} d^{2} x^{2}+36 \ln \left (F \right ) b \,c^{2} d^{2} x^{2}-90 c^{4} d^{2} x^{2}-6 \ln \left (F \right )^{2} b^{2} c d x +24 \ln \left (F \right ) b \,c^{3} d x -36 c^{5} d x +\ln \left (F \right )^{3} b^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2}+6 \ln \left (F \right ) b \,c^{4}-6 c^{6}\right ) F^{\frac {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}{\left (d x +c \right )^{2}}}}{2 b^{4} \ln \left (F \right )^{4} d \left (d x +c \right )^{6}}\)	\(216\)
parallelrisch	\(\frac {6 d^{17} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{6}+36 d^{16} c \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{5}-6 \ln \left (F \right ) x^{4} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,d^{15}+90 d^{15} c^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{4}-24 \ln \left (F \right ) x^{3} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b c \,d^{14}+120 c^{3} d^{14} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{3}+3 \ln \left (F \right )^{2} x^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} d^{13}-36 \ln \left (F \right ) x^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{2} d^{13}+90 d^{13} c^{4} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x^{2}+6 \ln \left (F \right )^{2} x \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} c \,d^{12}-24 \ln \left (F \right ) x \,F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{3} d^{12}+36 c^{5} F^{a +\frac {b}{\left (d x +c \right )^{2}}} x \,d^{12}-\ln \left (F \right )^{3} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{3} d^{11}+3 \ln \left (F \right )^{2} F^{a +\frac {b}{\left (d x +c \right )^{2}}} b^{2} c^{2} d^{11}-6 \ln \left (F \right ) F^{a +\frac {b}{\left (d x +c \right )^{2}}} b \,c^{4} d^{11}+6 c^{6} F^{a +\frac {b}{\left (d x +c \right )^{2}}} d^{11}}{2 \left (d x +c \right )^{6} \ln \left (F \right )^{4} b^{4} d^{12}}\)	\(412\)
norman	\(\frac {\frac {3 d^{7} x^{8} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {c \left (\ln \left (F \right )^{3} b^{3}-6 \ln \left (F \right )^{2} b^{2} c^{2}+18 \ln \left (F \right ) b \,c^{4}-24 c^{6}\right ) x \,{\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{b^{4} \ln \left (F \right )^{4}}-\frac {d \left (\ln \left (F \right )^{3} b^{3}-18 \ln \left (F \right )^{2} b^{2} c^{2}+90 \ln \left (F \right ) b \,c^{4}-168 c^{6}\right ) x^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{4} b^{4}}+\frac {3 d^{3} \left (\ln \left (F \right )^{2} b^{2}-30 \ln \left (F \right ) b \,c^{2}+140 c^{4}\right ) x^{4} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 \ln \left (F \right )^{4} b^{4}}-\frac {3 d^{5} \left (b \ln \left (F \right )-28 c^{2}\right ) x^{6} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}+\frac {24 d^{6} c \,x^{7} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {\left (\ln \left (F \right )^{3} b^{3}-3 \ln \left (F \right )^{2} b^{2} c^{2}+6 \ln \left (F \right ) b \,c^{4}-6 c^{6}\right ) c^{2} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{2 b^{4} \ln \left (F \right )^{4} d}+\frac {6 c \,d^{2} \left (\ln \left (F \right )^{2} b^{2}-10 \ln \left (F \right ) b \,c^{2}+28 c^{4}\right ) x^{3} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}-\frac {6 c \,d^{4} \left (3 b \ln \left (F \right )-28 c^{2}\right ) x^{5} {\mathrm e}^{\left (a +\frac {b}{\left (d x +c \right )^{2}}\right ) \ln \left (F \right )}}{\ln \left (F \right )^{4} b^{4}}}{\left (d x +c \right )^{8}}\)	\(444\)

Input:

int(F^(a+b/(d*x+c)^2)/(d*x+c)^9,x,method=_RETURNVERBOSE)
 

Output:

-1/2*(-6*d^6*x^6-36*c*d^5*x^5+6*ln(F)*b*d^4*x^4-90*c^2*d^4*x^4+24*ln(F)*b* 
c*d^3*x^3-120*c^3*d^3*x^3-3*ln(F)^2*b^2*d^2*x^2+36*ln(F)*b*c^2*d^2*x^2-90* 
c^4*d^2*x^2-6*ln(F)^2*b^2*c*d*x+24*ln(F)*b*c^3*d*x-36*c^5*d*x+ln(F)^3*b^3- 
3*ln(F)^2*b^2*c^2+6*ln(F)*b*c^4-6*c^6)/b^4/ln(F)^4/d/(d*x+c)^6*F^((a*d^2*x 
^2+2*a*c*d*x+a*c^2+b)/(d*x+c)^2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 287 vs. \(2 (122) = 244\).

Time = 0.09 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.28 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {{\left (6 \, d^{6} x^{6} + 36 \, c d^{5} x^{5} + 90 \, c^{2} d^{4} x^{4} + 120 \, c^{3} d^{3} x^{3} + 90 \, c^{4} d^{2} x^{2} + 36 \, c^{5} d x + 6 \, c^{6} - b^{3} \log \left (F\right )^{3} + 3 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (F\right )^{2} - 6 \, {\left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4}\right )} \log \left (F\right )\right )} F^{\frac {a d^{2} x^{2} + 2 \, a c d x + a c^{2} + b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{4} d^{7} x^{6} + 6 \, b^{4} c d^{6} x^{5} + 15 \, b^{4} c^{2} d^{5} x^{4} + 20 \, b^{4} c^{3} d^{4} x^{3} + 15 \, b^{4} c^{4} d^{3} x^{2} + 6 \, b^{4} c^{5} d^{2} x + b^{4} c^{6} d\right )} \log \left (F\right )^{4}} \] Input:

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^9,x, algorithm="fricas")
 

Output:

1/2*(6*d^6*x^6 + 36*c*d^5*x^5 + 90*c^2*d^4*x^4 + 120*c^3*d^3*x^3 + 90*c^4* 
d^2*x^2 + 36*c^5*d*x + 6*c^6 - b^3*log(F)^3 + 3*(b^2*d^2*x^2 + 2*b^2*c*d*x 
 + b^2*c^2)*log(F)^2 - 6*(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4* 
b*c^3*d*x + b*c^4)*log(F))*F^((a*d^2*x^2 + 2*a*c*d*x + a*c^2 + b)/(d^2*x^2 
 + 2*c*d*x + c^2))/((b^4*d^7*x^6 + 6*b^4*c*d^6*x^5 + 15*b^4*c^2*d^5*x^4 + 
20*b^4*c^3*d^4*x^3 + 15*b^4*c^4*d^3*x^2 + 6*b^4*c^5*d^2*x + b^4*c^6*d)*log 
(F)^4)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (112) = 224\).

Time = 0.17 (sec) , antiderivative size = 333, normalized size of antiderivative = 2.64 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {F^{a + \frac {b}{\left (c + d x\right )^{2}}} \left (- b^{3} \log {\left (F \right )}^{3} + 3 b^{2} c^{2} \log {\left (F \right )}^{2} + 6 b^{2} c d x \log {\left (F \right )}^{2} + 3 b^{2} d^{2} x^{2} \log {\left (F \right )}^{2} - 6 b c^{4} \log {\left (F \right )} - 24 b c^{3} d x \log {\left (F \right )} - 36 b c^{2} d^{2} x^{2} \log {\left (F \right )} - 24 b c d^{3} x^{3} \log {\left (F \right )} - 6 b d^{4} x^{4} \log {\left (F \right )} + 6 c^{6} + 36 c^{5} d x + 90 c^{4} d^{2} x^{2} + 120 c^{3} d^{3} x^{3} + 90 c^{2} d^{4} x^{4} + 36 c d^{5} x^{5} + 6 d^{6} x^{6}\right )}{2 b^{4} c^{6} d \log {\left (F \right )}^{4} + 12 b^{4} c^{5} d^{2} x \log {\left (F \right )}^{4} + 30 b^{4} c^{4} d^{3} x^{2} \log {\left (F \right )}^{4} + 40 b^{4} c^{3} d^{4} x^{3} \log {\left (F \right )}^{4} + 30 b^{4} c^{2} d^{5} x^{4} \log {\left (F \right )}^{4} + 12 b^{4} c d^{6} x^{5} \log {\left (F \right )}^{4} + 2 b^{4} d^{7} x^{6} \log {\left (F \right )}^{4}} \] Input:

integrate(F**(a+b/(d*x+c)**2)/(d*x+c)**9,x)
 

Output:

F**(a + b/(c + d*x)**2)*(-b**3*log(F)**3 + 3*b**2*c**2*log(F)**2 + 6*b**2* 
c*d*x*log(F)**2 + 3*b**2*d**2*x**2*log(F)**2 - 6*b*c**4*log(F) - 24*b*c**3 
*d*x*log(F) - 36*b*c**2*d**2*x**2*log(F) - 24*b*c*d**3*x**3*log(F) - 6*b*d 
**4*x**4*log(F) + 6*c**6 + 36*c**5*d*x + 90*c**4*d**2*x**2 + 120*c**3*d**3 
*x**3 + 90*c**2*d**4*x**4 + 36*c*d**5*x**5 + 6*d**6*x**6)/(2*b**4*c**6*d*l 
og(F)**4 + 12*b**4*c**5*d**2*x*log(F)**4 + 30*b**4*c**4*d**3*x**2*log(F)** 
4 + 40*b**4*c**3*d**4*x**3*log(F)**4 + 30*b**4*c**2*d**5*x**4*log(F)**4 + 
12*b**4*c*d**6*x**5*log(F)**4 + 2*b**4*d**7*x**6*log(F)**4)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 349 vs. \(2 (122) = 244\).

Time = 0.04 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.77 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {{\left (6 \, F^{a} d^{6} x^{6} + 36 \, F^{a} c d^{5} x^{5} + 6 \, F^{a} c^{6} - 6 \, F^{a} b c^{4} \log \left (F\right ) + 3 \, F^{a} b^{2} c^{2} \log \left (F\right )^{2} - F^{a} b^{3} \log \left (F\right )^{3} + 6 \, {\left (15 \, F^{a} c^{2} d^{4} - F^{a} b d^{4} \log \left (F\right )\right )} x^{4} + 24 \, {\left (5 \, F^{a} c^{3} d^{3} - F^{a} b c d^{3} \log \left (F\right )\right )} x^{3} + 3 \, {\left (30 \, F^{a} c^{4} d^{2} - 12 \, F^{a} b c^{2} d^{2} \log \left (F\right ) + F^{a} b^{2} d^{2} \log \left (F\right )^{2}\right )} x^{2} + 6 \, {\left (6 \, F^{a} c^{5} d - 4 \, F^{a} b c^{3} d \log \left (F\right ) + F^{a} b^{2} c d \log \left (F\right )^{2}\right )} x\right )} F^{\frac {b}{d^{2} x^{2} + 2 \, c d x + c^{2}}}}{2 \, {\left (b^{4} d^{7} x^{6} \log \left (F\right )^{4} + 6 \, b^{4} c d^{6} x^{5} \log \left (F\right )^{4} + 15 \, b^{4} c^{2} d^{5} x^{4} \log \left (F\right )^{4} + 20 \, b^{4} c^{3} d^{4} x^{3} \log \left (F\right )^{4} + 15 \, b^{4} c^{4} d^{3} x^{2} \log \left (F\right )^{4} + 6 \, b^{4} c^{5} d^{2} x \log \left (F\right )^{4} + b^{4} c^{6} d \log \left (F\right )^{4}\right )}} \] Input:

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^9,x, algorithm="maxima")
 

Output:

1/2*(6*F^a*d^6*x^6 + 36*F^a*c*d^5*x^5 + 6*F^a*c^6 - 6*F^a*b*c^4*log(F) + 3 
*F^a*b^2*c^2*log(F)^2 - F^a*b^3*log(F)^3 + 6*(15*F^a*c^2*d^4 - F^a*b*d^4*l 
og(F))*x^4 + 24*(5*F^a*c^3*d^3 - F^a*b*c*d^3*log(F))*x^3 + 3*(30*F^a*c^4*d 
^2 - 12*F^a*b*c^2*d^2*log(F) + F^a*b^2*d^2*log(F)^2)*x^2 + 6*(6*F^a*c^5*d 
- 4*F^a*b*c^3*d*log(F) + F^a*b^2*c*d*log(F)^2)*x)*F^(b/(d^2*x^2 + 2*c*d*x 
+ c^2))/(b^4*d^7*x^6*log(F)^4 + 6*b^4*c*d^6*x^5*log(F)^4 + 15*b^4*c^2*d^5* 
x^4*log(F)^4 + 20*b^4*c^3*d^4*x^3*log(F)^4 + 15*b^4*c^4*d^3*x^2*log(F)^4 + 
 6*b^4*c^5*d^2*x*log(F)^4 + b^4*c^6*d*log(F)^4)
 

Giac [F]

\[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\int { \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{9}} \,d x } \] Input:

integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^9,x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^9, x)
 

Mupad [B] (verification not implemented)

Time = 0.45 (sec) , antiderivative size = 292, normalized size of antiderivative = 2.32 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {F^a\,F^{\frac {b}{c^2+2\,c\,d\,x+d^2\,x^2}}\,\left (\frac {3\,x^6}{b^4\,d\,{\ln \left (F\right )}^4}-\frac {b^3\,{\ln \left (F\right )}^3-3\,b^2\,c^2\,{\ln \left (F\right )}^2+6\,b\,c^4\,\ln \left (F\right )-6\,c^6}{2\,b^4\,d^7\,{\ln \left (F\right )}^4}+\frac {18\,c\,x^5}{b^4\,d^2\,{\ln \left (F\right )}^4}+\frac {3\,x^2\,\left (b^2\,{\ln \left (F\right )}^2-12\,b\,c^2\,\ln \left (F\right )+30\,c^4\right )}{2\,b^4\,d^5\,{\ln \left (F\right )}^4}-\frac {3\,x^4\,\left (b\,\ln \left (F\right )-15\,c^2\right )}{b^4\,d^3\,{\ln \left (F\right )}^4}-\frac {12\,c\,x^3\,\left (b\,\ln \left (F\right )-5\,c^2\right )}{b^4\,d^4\,{\ln \left (F\right )}^4}+\frac {3\,c\,x\,\left (b^2\,{\ln \left (F\right )}^2-4\,b\,c^2\,\ln \left (F\right )+6\,c^4\right )}{b^4\,d^6\,{\ln \left (F\right )}^4}\right )}{x^6+\frac {c^6}{d^6}+\frac {6\,c\,x^5}{d}+\frac {6\,c^5\,x}{d^5}+\frac {15\,c^2\,x^4}{d^2}+\frac {20\,c^3\,x^3}{d^3}+\frac {15\,c^4\,x^2}{d^4}} \] Input:

int(F^(a + b/(c + d*x)^2)/(c + d*x)^9,x)
 

Output:

(F^a*F^(b/(c^2 + d^2*x^2 + 2*c*d*x))*((3*x^6)/(b^4*d*log(F)^4) - (b^3*log( 
F)^3 - 6*c^6 + 6*b*c^4*log(F) - 3*b^2*c^2*log(F)^2)/(2*b^4*d^7*log(F)^4) + 
 (18*c*x^5)/(b^4*d^2*log(F)^4) + (3*x^2*(b^2*log(F)^2 + 30*c^4 - 12*b*c^2* 
log(F)))/(2*b^4*d^5*log(F)^4) - (3*x^4*(b*log(F) - 15*c^2))/(b^4*d^3*log(F 
)^4) - (12*c*x^3*(b*log(F) - 5*c^2))/(b^4*d^4*log(F)^4) + (3*c*x*(b^2*log( 
F)^2 + 6*c^4 - 4*b*c^2*log(F)))/(b^4*d^6*log(F)^4)))/(x^6 + c^6/d^6 + (6*c 
*x^5)/d + (6*c^5*x)/d^5 + (15*c^2*x^4)/d^2 + (20*c^3*x^3)/d^3 + (15*c^4*x^ 
2)/d^4)
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 282, normalized size of antiderivative = 2.24 \[ \int \frac {F^{a+\frac {b}{(c+d x)^2}}}{(c+d x)^9} \, dx=\frac {f^{\frac {a \,d^{2} x^{2}+2 a c d x +a \,c^{2}+b}{d^{2} x^{2}+2 c d x +c^{2}}} \left (-\mathrm {log}\left (f \right )^{3} b^{3}+3 \mathrm {log}\left (f \right )^{2} b^{2} c^{2}+6 \mathrm {log}\left (f \right )^{2} b^{2} c d x +3 \mathrm {log}\left (f \right )^{2} b^{2} d^{2} x^{2}-6 \,\mathrm {log}\left (f \right ) b \,c^{4}-24 \,\mathrm {log}\left (f \right ) b \,c^{3} d x -36 \,\mathrm {log}\left (f \right ) b \,c^{2} d^{2} x^{2}-24 \,\mathrm {log}\left (f \right ) b c \,d^{3} x^{3}-6 \,\mathrm {log}\left (f \right ) b \,d^{4} x^{4}+6 c^{6}+36 c^{5} d x +90 c^{4} d^{2} x^{2}+120 c^{3} d^{3} x^{3}+90 c^{2} d^{4} x^{4}+36 c \,d^{5} x^{5}+6 d^{6} x^{6}\right )}{2 \mathrm {log}\left (f \right )^{4} b^{4} d \left (d^{6} x^{6}+6 c \,d^{5} x^{5}+15 c^{2} d^{4} x^{4}+20 c^{3} d^{3} x^{3}+15 c^{4} d^{2} x^{2}+6 c^{5} d x +c^{6}\right )} \] Input:

int(F^(a+b/(d*x+c)^2)/(d*x+c)^9,x)
 

Output:

(f**((a*c**2 + 2*a*c*d*x + a*d**2*x**2 + b)/(c**2 + 2*c*d*x + d**2*x**2))* 
( - log(f)**3*b**3 + 3*log(f)**2*b**2*c**2 + 6*log(f)**2*b**2*c*d*x + 3*lo 
g(f)**2*b**2*d**2*x**2 - 6*log(f)*b*c**4 - 24*log(f)*b*c**3*d*x - 36*log(f 
)*b*c**2*d**2*x**2 - 24*log(f)*b*c*d**3*x**3 - 6*log(f)*b*d**4*x**4 + 6*c* 
*6 + 36*c**5*d*x + 90*c**4*d**2*x**2 + 120*c**3*d**3*x**3 + 90*c**2*d**4*x 
**4 + 36*c*d**5*x**5 + 6*d**6*x**6))/(2*log(f)**4*b**4*d*(c**6 + 6*c**5*d* 
x + 15*c**4*d**2*x**2 + 20*c**3*d**3*x**3 + 15*c**2*d**4*x**4 + 6*c*d**5*x 
**5 + d**6*x**6))