\(\int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx\) [200]

Optimal result

Integrand size = 21, antiderivative size = 31 \[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx=\frac {b^5 F^a \Gamma \left (-5,-b (c+d x)^2 \log (F)\right ) \log ^5(F)}{2 d} \] Output:

-1/2*F^a/(d*x+c)^10*Ei(6,-b*(d*x+c)^2*ln(F))/d
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx=\frac {b^5 F^a \Gamma \left (-5,-b (c+d x)^2 \log (F)\right ) \log ^5(F)}{2 d} \] Input:

Integrate[F^(a + b*(c + d*x)^2)/(c + d*x)^11,x]
 

Output:

(b^5*F^a*Gamma[-5, -(b*(c + d*x)^2*Log[F])]*Log[F]^5)/(2*d)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {2648}

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.

\(\displaystyle \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx\)

\(\Big \downarrow \) 2648

\(\displaystyle \frac {b^5 F^a \log ^5(F) \Gamma \left (-5,-b (c+d x)^2 \log (F)\right )}{2 d}\)

Input:

Int[F^(a + b*(c + d*x)^2)/(c + d*x)^11,x]
 

Output:

(b^5*F^a*Gamma[-5, -(b*(c + d*x)^2*Log[F])]*Log[F]^5)/(2*d)
 

Defintions of rubi rules used

rule 2648

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(184\) vs. \(2(29)=58\).

Time = 1.00 (sec) , antiderivative size = 185, normalized size of antiderivative = 5.97

method	result	size
risch	\(-\frac {F^{b \left (d x +c \right )^{2}} F^{a}}{10 d \left (d x +c \right )^{10}}-\frac {b \ln \left (F \right ) F^{b \left (d x +c \right )^{2}} F^{a}}{40 d \left (d x +c \right )^{8}}-\frac {b^{2} \ln \left (F \right )^{2} F^{b \left (d x +c \right )^{2}} F^{a}}{120 d \left (d x +c \right )^{6}}-\frac {b^{3} \ln \left (F \right )^{3} F^{b \left (d x +c \right )^{2}} F^{a}}{240 d \left (d x +c \right )^{4}}-\frac {b^{4} \ln \left (F \right )^{4} F^{b \left (d x +c \right )^{2}} F^{a}}{240 d \left (d x +c \right )^{2}}-\frac {b^{5} \ln \left (F \right )^{5} F^{a} \operatorname {expIntegral}_{1}\left (-b \left (d x +c \right )^{2} \ln \left (F \right )\right )}{240 d}\)	\(185\)

Input:

int(F^(a+b*(d*x+c)^2)/(d*x+c)^11,x,method=_RETURNVERBOSE)
 

Output:

-1/10/d/(d*x+c)^10*F^(b*(d*x+c)^2)*F^a-1/40/d*b*ln(F)/(d*x+c)^8*F^(b*(d*x+ 
c)^2)*F^a-1/120/d*b^2*ln(F)^2/(d*x+c)^6*F^(b*(d*x+c)^2)*F^a-1/240/d*b^3*ln 
(F)^3/(d*x+c)^4*F^(b*(d*x+c)^2)*F^a-1/240/d*b^4*ln(F)^4/(d*x+c)^2*F^(b*(d* 
x+c)^2)*F^a-1/240/d*b^5*ln(F)^5*F^a*Ei(1,-b*(d*x+c)^2*ln(F))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 596 vs. \(2 (29) = 58\).

Time = 0.09 (sec) , antiderivative size = 596, normalized size of antiderivative = 19.23 \[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx=\frac {{\left (b^{5} d^{10} x^{10} + 10 \, b^{5} c d^{9} x^{9} + 45 \, b^{5} c^{2} d^{8} x^{8} + 120 \, b^{5} c^{3} d^{7} x^{7} + 210 \, b^{5} c^{4} d^{6} x^{6} + 252 \, b^{5} c^{5} d^{5} x^{5} + 210 \, b^{5} c^{6} d^{4} x^{4} + 120 \, b^{5} c^{7} d^{3} x^{3} + 45 \, b^{5} c^{8} d^{2} x^{2} + 10 \, b^{5} c^{9} d x + b^{5} c^{10}\right )} F^{a} {\rm Ei}\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right )\right ) \log \left (F\right )^{5} - {\left ({\left (b^{4} d^{8} x^{8} + 8 \, b^{4} c d^{7} x^{7} + 28 \, b^{4} c^{2} d^{6} x^{6} + 56 \, b^{4} c^{3} d^{5} x^{5} + 70 \, b^{4} c^{4} d^{4} x^{4} + 56 \, b^{4} c^{5} d^{3} x^{3} + 28 \, b^{4} c^{6} d^{2} x^{2} + 8 \, b^{4} c^{7} d x + b^{4} c^{8}\right )} \log \left (F\right )^{4} + {\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} + 2 \, {\left (b^{2} d^{4} x^{4} + 4 \, b^{2} c d^{3} x^{3} + 6 \, b^{2} c^{2} d^{2} x^{2} + 4 \, b^{2} c^{3} d x + b^{2} c^{4}\right )} \log \left (F\right )^{2} + 6 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2}\right )} \log \left (F\right ) + 24\right )} F^{b d^{2} x^{2} + 2 \, b c d x + b c^{2} + a}}{240 \, {\left (d^{11} x^{10} + 10 \, c d^{10} x^{9} + 45 \, c^{2} d^{9} x^{8} + 120 \, c^{3} d^{8} x^{7} + 210 \, c^{4} d^{7} x^{6} + 252 \, c^{5} d^{6} x^{5} + 210 \, c^{6} d^{5} x^{4} + 120 \, c^{7} d^{4} x^{3} + 45 \, c^{8} d^{3} x^{2} + 10 \, c^{9} d^{2} x + c^{10} d\right )}} \] Input:

integrate(F^(a+b*(d*x+c)^2)/(d*x+c)^11,x, algorithm="fricas")
 

Output:

1/240*((b^5*d^10*x^10 + 10*b^5*c*d^9*x^9 + 45*b^5*c^2*d^8*x^8 + 120*b^5*c^ 
3*d^7*x^7 + 210*b^5*c^4*d^6*x^6 + 252*b^5*c^5*d^5*x^5 + 210*b^5*c^6*d^4*x^ 
4 + 120*b^5*c^7*d^3*x^3 + 45*b^5*c^8*d^2*x^2 + 10*b^5*c^9*d*x + b^5*c^10)* 
F^a*Ei((b*d^2*x^2 + 2*b*c*d*x + b*c^2)*log(F))*log(F)^5 - ((b^4*d^8*x^8 + 
8*b^4*c*d^7*x^7 + 28*b^4*c^2*d^6*x^6 + 56*b^4*c^3*d^5*x^5 + 70*b^4*c^4*d^4 
*x^4 + 56*b^4*c^5*d^3*x^3 + 28*b^4*c^6*d^2*x^2 + 8*b^4*c^7*d*x + b^4*c^8)* 
log(F)^4 + (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 + 2*(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) + 24)*F^(b*d^2*x^2 + 
2*b*c*d*x + b*c^2 + a))/(d^11*x^10 + 10*c*d^10*x^9 + 45*c^2*d^9*x^8 + 120* 
c^3*d^8*x^7 + 210*c^4*d^7*x^6 + 252*c^5*d^6*x^5 + 210*c^6*d^5*x^4 + 120*c^ 
7*d^4*x^3 + 45*c^8*d^3*x^2 + 10*c^9*d^2*x + c^10*d)
 

Sympy [F]

\[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx=\int \frac {F^{a + b \left (c + d x\right )^{2}}}{\left (c + d x\right )^{11}}\, dx \] Input:

integrate(F**(a+b*(d*x+c)**2)/(d*x+c)**11,x)
 

Output:

Integral(F**(a + b*(c + d*x)**2)/(c + d*x)**11, x)
 

Maxima [F]

\[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{11}} \,d x } \] Input:

integrate(F^(a+b*(d*x+c)^2)/(d*x+c)^11,x, algorithm="maxima")
 

Output:

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^11, x)
 

Giac [F]

\[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{2} b + a}}{{\left (d x + c\right )}^{11}} \,d x } \] Input:

integrate(F^(a+b*(d*x+c)^2)/(d*x+c)^11,x, algorithm="giac")
 

Output:

integrate(F^((d*x + c)^2*b + a)/(d*x + c)^11, x)
 

Mupad [B] (verification not implemented)

Time = 0.35 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.39 \[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx=-\frac {F^a\,b^5\,{\ln \left (F\right )}^5\,\mathrm {expint}\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^2\right )}{240\,d}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^2}\,b^5\,{\ln \left (F\right )}^5\,\left (\frac {1}{120\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^2}+\frac {1}{120\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^4}+\frac {1}{60\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^6}+\frac {1}{20\,b^4\,{\ln \left (F\right )}^4\,{\left (c+d\,x\right )}^8}+\frac {1}{5\,b^5\,{\ln \left (F\right )}^5\,{\left (c+d\,x\right )}^{10}}\right )}{2\,d} \] Input:

int(F^(a + b*(c + d*x)^2)/(c + d*x)^11,x)
 

Output:

- (F^a*b^5*log(F)^5*expint(-b*log(F)*(c + d*x)^2))/(240*d) - (F^a*F^(b*(c 
+ d*x)^2)*b^5*log(F)^5*(1/(120*b*log(F)*(c + d*x)^2) + 1/(120*b^2*log(F)^2 
*(c + d*x)^4) + 1/(60*b^3*log(F)^3*(c + d*x)^6) + 1/(20*b^4*log(F)^4*(c + 
d*x)^8) + 1/(5*b^5*log(F)^5*(c + d*x)^10)))/(2*d)
 

Reduce [F]

\[ \int \frac {F^{a+b (c+d x)^2}}{(c+d x)^{11}} \, dx=\text {too large to display} \] Input:

int(F^(a+b*(d*x+c)^2)/(d*x+c)^11,x)
 

Output:

(f**(a + b*c**2)*( - f**(2*b*c*d*x + b*d**2*x**2) + 4*int(f**(2*b*c*d*x + 
b*d**2*x**2)/(log(f)*b*c**13 + 11*log(f)*b*c**12*d*x + 55*log(f)*b*c**11*d 
**2*x**2 + 165*log(f)*b*c**10*d**3*x**3 + 330*log(f)*b*c**9*d**4*x**4 + 46 
2*log(f)*b*c**8*d**5*x**5 + 462*log(f)*b*c**7*d**6*x**6 + 330*log(f)*b*c** 
6*d**7*x**7 + 165*log(f)*b*c**5*d**8*x**8 + 55*log(f)*b*c**4*d**9*x**9 + 1 
1*log(f)*b*c**3*d**10*x**10 + log(f)*b*c**2*d**11*x**11 - 5*c**11 - 55*c** 
10*d*x - 275*c**9*d**2*x**2 - 825*c**8*d**3*x**3 - 1650*c**7*d**4*x**4 - 2 
310*c**6*d**5*x**5 - 2310*c**5*d**6*x**6 - 1650*c**4*d**7*x**7 - 825*c**3* 
d**8*x**8 - 275*c**2*d**9*x**9 - 55*c*d**10*x**10 - 5*d**11*x**11),x)*log( 
f)**2*b**2*c**14*d + 40*int(f**(2*b*c*d*x + b*d**2*x**2)/(log(f)*b*c**13 + 
 11*log(f)*b*c**12*d*x + 55*log(f)*b*c**11*d**2*x**2 + 165*log(f)*b*c**10* 
d**3*x**3 + 330*log(f)*b*c**9*d**4*x**4 + 462*log(f)*b*c**8*d**5*x**5 + 46 
2*log(f)*b*c**7*d**6*x**6 + 330*log(f)*b*c**6*d**7*x**7 + 165*log(f)*b*c** 
5*d**8*x**8 + 55*log(f)*b*c**4*d**9*x**9 + 11*log(f)*b*c**3*d**10*x**10 + 
log(f)*b*c**2*d**11*x**11 - 5*c**11 - 55*c**10*d*x - 275*c**9*d**2*x**2 - 
825*c**8*d**3*x**3 - 1650*c**7*d**4*x**4 - 2310*c**6*d**5*x**5 - 2310*c**5 
*d**6*x**6 - 1650*c**4*d**7*x**7 - 825*c**3*d**8*x**8 - 275*c**2*d**9*x**9 
 - 55*c*d**10*x**10 - 5*d**11*x**11),x)*log(f)**2*b**2*c**13*d**2*x + 180* 
int(f**(2*b*c*d*x + b*d**2*x**2)/(log(f)*b*c**13 + 11*log(f)*b*c**12*d*x + 
 55*log(f)*b*c**11*d**2*x**2 + 165*log(f)*b*c**10*d**3*x**3 + 330*log(f...