3.326 \(\int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx\)
Optimal. Leaf size=113 \[ \frac{F^{a+\frac{b}{(c+d x)^2}} \left (60 b^2 \log ^2(F) (c+d x)^6-20 b^3 \log ^3(F) (c+d x)^4+5 b^4 \log ^4(F) (c+d x)^2-b^5 \log ^5(F)-120 b \log (F) (c+d x)^8+120 (c+d x)^{10}\right )}{2 b^6 d \log ^6(F) (c+d x)^{10}} \]
[Out]
(F^(a + b/(c + d*x)^2)*(120*(c + d*x)^10 - 120*b*(c + d*x)^8*Log[F] + 60*b^2*(c + d*x)^6*Log[F]^2 - 20*b^3*(c
+ d*x)^4*Log[F]^3 + 5*b^4*(c + d*x)^2*Log[F]^4 - b^5*Log[F]^5))/(2*b^6*d*(c + d*x)^10*Log[F]^6)
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Rubi [C] time = 0.0456741, antiderivative size = 31, normalized size of antiderivative =
0.27, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules
used = {2218} \[ \frac{F^a \text{Gamma}\left (6,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
[In]
Int[F^(a + b/(c + d*x)^2)/(c + d*x)^13,x]
[Out]
(F^a*Gamma[6, -((b*Log[F])/(c + d*x)^2)])/(2*b^6*d*Log[F]^6)
Rule 2218
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> -Simp[(F^a*(e + f*
x)^(m + 1)*Gamma[(m + 1)/n, -(b*(c + d*x)^n*Log[F])])/(f*n*(-(b*(c + d*x)^n*Log[F]))^((m + 1)/n)), x] /; FreeQ
[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
Rubi steps
\begin{align*} \int \frac{F^{a+\frac{b}{(c+d x)^2}}}{(c+d x)^{13}} \, dx &=\frac{F^a \Gamma \left (6,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 b^6 d \log ^6(F)}\\ \end{align*}
Mathematica [C] time = 0.0073282, size = 31, normalized size = 0.27 \[ \frac{F^a \text{Gamma}\left (6,-\frac{b \log (F)}{(c+d x)^2}\right )}{2 b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(a + b/(c + d*x)^2)/(c + d*x)^13,x]
[Out]
(F^a*Gamma[6, -((b*Log[F])/(c + d*x)^2)])/(2*b^6*d*Log[F]^6)
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Maple [B] time = 0.189, size = 797, normalized size = 7.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(F^(a+b/(d*x+c)^2)/(d*x+c)^13,x)
[Out]
(720*d^10*c/ln(F)^6/b^6*x^11*exp((a+b/(d*x+c)^2)*ln(F))-1/2*(b^5*ln(F)^5-5*ln(F)^4*b^4*c^2+20*ln(F)^3*b^3*c^4-
60*ln(F)^2*b^2*c^6+120*ln(F)*b*c^8-120*c^10)*c^2/b^6/ln(F)^6/d*exp((a+b/(d*x+c)^2)*ln(F))-c*(b^5*ln(F)^5-10*ln
(F)^4*b^4*c^2+60*ln(F)^3*b^3*c^4-240*ln(F)^2*b^2*c^6+600*ln(F)*b*c^8-720*c^10)/b^6/ln(F)^6*x*exp((a+b/(d*x+c)^
2)*ln(F))-1/2*d*(b^5*ln(F)^5-30*ln(F)^4*b^4*c^2+300*ln(F)^3*b^3*c^4-1680*ln(F)^2*b^2*c^6+5400*ln(F)*b*c^8-7920
*c^10)/ln(F)^6/b^6*x^2*exp((a+b/(d*x+c)^2)*ln(F))+5/2*d^3*(b^4*ln(F)^4-60*ln(F)^3*b^3*c^2+840*ln(F)^2*b^2*c^4-
5040*ln(F)*b*c^6+11880*c^8)/ln(F)^6/b^6*x^4*exp((a+b/(d*x+c)^2)*ln(F))-10*d^5*(ln(F)^3*b^3-84*ln(F)^2*b^2*c^2+
1260*ln(F)*b*c^4-5544*c^6)/ln(F)^6/b^6*x^6*exp((a+b/(d*x+c)^2)*ln(F))+30*d^7*(ln(F)^2*b^2-90*ln(F)*b*c^2+990*c
^4)/ln(F)^6/b^6*x^8*exp((a+b/(d*x+c)^2)*ln(F))-60*d^9*(b*ln(F)-66*c^2)/ln(F)^6/b^6*x^10*exp((a+b/(d*x+c)^2)*ln
(F))+10*c*d^2*(b^4*ln(F)^4-20*ln(F)^3*b^3*c^2+168*ln(F)^2*b^2*c^4-720*ln(F)*b*c^6+1320*c^8)/ln(F)^6/b^6*x^3*ex
p((a+b/(d*x+c)^2)*ln(F))-60*c*d^4*(ln(F)^3*b^3-28*ln(F)^2*b^2*c^2+252*ln(F)*b*c^4-792*c^6)/ln(F)^6/b^6*x^5*exp
((a+b/(d*x+c)^2)*ln(F))+240*c*d^6*(ln(F)^2*b^2-30*ln(F)*b*c^2+198*c^4)/ln(F)^6/b^6*x^7*exp((a+b/(d*x+c)^2)*ln(
F))-600*c*d^8*(b*ln(F)-22*c^2)/ln(F)^6/b^6*x^9*exp((a+b/(d*x+c)^2)*ln(F))+60*d^11/ln(F)^6/b^6*x^12*exp((a+b/(d
*x+c)^2)*ln(F)))/(d*x+c)^12
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Maxima [B] time = 1.07428, size = 999, normalized size = 8.84 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^13,x, algorithm="maxima")
[Out]
1/2*(120*F^a*d^10*x^10 + 1200*F^a*c*d^9*x^9 + 120*F^a*c^10 - 120*F^a*b*c^8*log(F) + 60*F^a*b^2*c^6*log(F)^2 -
20*F^a*b^3*c^4*log(F)^3 + 5*F^a*b^4*c^2*log(F)^4 - F^a*b^5*log(F)^5 + 120*(45*F^a*c^2*d^8 - F^a*b*d^8*log(F))*
x^8 + 960*(15*F^a*c^3*d^7 - F^a*b*c*d^7*log(F))*x^7 + 60*(420*F^a*c^4*d^6 - 56*F^a*b*c^2*d^6*log(F) + F^a*b^2*
d^6*log(F)^2)*x^6 + 120*(252*F^a*c^5*d^5 - 56*F^a*b*c^3*d^5*log(F) + 3*F^a*b^2*c*d^5*log(F)^2)*x^5 + 20*(1260*
F^a*c^6*d^4 - 420*F^a*b*c^4*d^4*log(F) + 45*F^a*b^2*c^2*d^4*log(F)^2 - F^a*b^3*d^4*log(F)^3)*x^4 + 80*(180*F^a
*c^7*d^3 - 84*F^a*b*c^5*d^3*log(F) + 15*F^a*b^2*c^3*d^3*log(F)^2 - F^a*b^3*c*d^3*log(F)^3)*x^3 + 5*(1080*F^a*c
^8*d^2 - 672*F^a*b*c^6*d^2*log(F) + 180*F^a*b^2*c^4*d^2*log(F)^2 - 24*F^a*b^3*c^2*d^2*log(F)^3 + F^a*b^4*d^2*l
og(F)^4)*x^2 + 10*(120*F^a*c^9*d - 96*F^a*b*c^7*d*log(F) + 36*F^a*b^2*c^5*d*log(F)^2 - 8*F^a*b^3*c^3*d*log(F)^
3 + F^a*b^4*c*d*log(F)^4)*x)*F^(b/(d^2*x^2 + 2*c*d*x + c^2))/(b^6*d^11*x^10*log(F)^6 + 10*b^6*c*d^10*x^9*log(F
)^6 + 45*b^6*c^2*d^9*x^8*log(F)^6 + 120*b^6*c^3*d^8*x^7*log(F)^6 + 210*b^6*c^4*d^7*x^6*log(F)^6 + 252*b^6*c^5*
d^6*x^5*log(F)^6 + 210*b^6*c^6*d^5*x^4*log(F)^6 + 120*b^6*c^7*d^4*x^3*log(F)^6 + 45*b^6*c^8*d^3*x^2*log(F)^6 +
10*b^6*c^9*d^2*x*log(F)^6 + b^6*c^10*d*log(F)^6)
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Fricas [B] time = 2.17798, size = 1283, normalized size = 11.35 \begin{align*} \frac{{\left (120 \, d^{10} x^{10} + 1200 \, c d^{9} x^{9} + 5400 \, c^{2} d^{8} x^{8} + 14400 \, c^{3} d^{7} x^{7} + 25200 \, c^{4} d^{6} x^{6} + 30240 \, c^{5} d^{5} x^{5} + 25200 \, c^{6} d^{4} x^{4} + 14400 \, c^{7} d^{3} x^{3} + 5400 \, c^{8} d^{2} x^{2} + 1200 \, c^{9} d x + 120 \, c^{10} - b^{5} \log \left (F\right )^{5} + 5 \,{\left (b^{4} d^{2} x^{2} + 2 \, b^{4} c d x + b^{4} c^{2}\right )} \log \left (F\right )^{4} - 20 \,{\left (b^{3} d^{4} x^{4} + 4 \, b^{3} c d^{3} x^{3} + 6 \, b^{3} c^{2} d^{2} x^{2} + 4 \, b^{3} c^{3} d x + b^{3} c^{4}\right )} \log \left (F\right )^{3} + 60 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} - 120 \,{\left (b d^{8} x^{8} + 8 \, b c d^{7} x^{7} + 28 \, b c^{2} d^{6} x^{6} + 56 \, b c^{3} d^{5} x^{5} + 70 \, b c^{4} d^{4} x^{4} + 56 \, b c^{5} d^{3} x^{3} + 28 \, b c^{6} d^{2} x^{2} + 8 \, b c^{7} d x + b c^{8}\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^{6} d^{11} x^{10} + 10 \, b^{6} c d^{10} x^{9} + 45 \, b^{6} c^{2} d^{9} x^{8} + 120 \, b^{6} c^{3} d^{8} x^{7} + 210 \, b^{6} c^{4} d^{7} x^{6} + 252 \, b^{6} c^{5} d^{6} x^{5} + 210 \, b^{6} c^{6} d^{5} x^{4} + 120 \, b^{6} c^{7} d^{4} x^{3} + 45 \, b^{6} c^{8} d^{3} x^{2} + 10 \, b^{6} c^{9} d^{2} x + b^{6} c^{10} d\right )} \log \left (F\right )^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^13,x, algorithm="fricas")
[Out]
1/2*(120*d^10*x^10 + 1200*c*d^9*x^9 + 5400*c^2*d^8*x^8 + 14400*c^3*d^7*x^7 + 25200*c^4*d^6*x^6 + 30240*c^5*d^5
*x^5 + 25200*c^6*d^4*x^4 + 14400*c^7*d^3*x^3 + 5400*c^8*d^2*x^2 + 1200*c^9*d*x + 120*c^10 - b^5*log(F)^5 + 5*(
b^4*d^2*x^2 + 2*b^4*c*d*x + b^4*c^2)*log(F)^4 - 20*(b^3*d^4*x^4 + 4*b^3*c*d^3*x^3 + 6*b^3*c^2*d^2*x^2 + 4*b^3*
c^3*d*x + b^3*c^4)*log(F)^3 + 60*(b^2*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + 20*b^2*c^3*d^3*x^3 + 15
*b^2*c^4*d^2*x^2 + 6*b^2*c^5*d*x + b^2*c^6)*log(F)^2 - 120*(b*d^8*x^8 + 8*b*c*d^7*x^7 + 28*b*c^2*d^6*x^6 + 56*
b*c^3*d^5*x^5 + 70*b*c^4*d^4*x^4 + 56*b*c^5*d^3*x^3 + 28*b*c^6*d^2*x^2 + 8*b*c^7*d*x + b*c^8)*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^6*d^11*x^10 + 10*b^6*c*d^10*x^9 + 45*b^6*c^2*d^9
*x^8 + 120*b^6*c^3*d^8*x^7 + 210*b^6*c^4*d^7*x^6 + 252*b^6*c^5*d^6*x^5 + 210*b^6*c^6*d^5*x^4 + 120*b^6*c^7*d^4
*x^3 + 45*b^6*c^8*d^3*x^2 + 10*b^6*c^9*d^2*x + b^6*c^10*d)*log(F)^6)
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Sympy [B] time = 0.763467, size = 745, normalized size = 6.59 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F**(a+b/(d*x+c)**2)/(d*x+c)**13,x)
[Out]
F**(a + b/(c + d*x)**2)*(-b**5*log(F)**5 + 5*b**4*c**2*log(F)**4 + 10*b**4*c*d*x*log(F)**4 + 5*b**4*d**2*x**2*
log(F)**4 - 20*b**3*c**4*log(F)**3 - 80*b**3*c**3*d*x*log(F)**3 - 120*b**3*c**2*d**2*x**2*log(F)**3 - 80*b**3*
c*d**3*x**3*log(F)**3 - 20*b**3*d**4*x**4*log(F)**3 + 60*b**2*c**6*log(F)**2 + 360*b**2*c**5*d*x*log(F)**2 + 9
00*b**2*c**4*d**2*x**2*log(F)**2 + 1200*b**2*c**3*d**3*x**3*log(F)**2 + 900*b**2*c**2*d**4*x**4*log(F)**2 + 36
0*b**2*c*d**5*x**5*log(F)**2 + 60*b**2*d**6*x**6*log(F)**2 - 120*b*c**8*log(F) - 960*b*c**7*d*x*log(F) - 3360*
b*c**6*d**2*x**2*log(F) - 6720*b*c**5*d**3*x**3*log(F) - 8400*b*c**4*d**4*x**4*log(F) - 6720*b*c**3*d**5*x**5*
log(F) - 3360*b*c**2*d**6*x**6*log(F) - 960*b*c*d**7*x**7*log(F) - 120*b*d**8*x**8*log(F) + 120*c**10 + 1200*c
**9*d*x + 5400*c**8*d**2*x**2 + 14400*c**7*d**3*x**3 + 25200*c**6*d**4*x**4 + 30240*c**5*d**5*x**5 + 25200*c**
4*d**6*x**6 + 14400*c**3*d**7*x**7 + 5400*c**2*d**8*x**8 + 1200*c*d**9*x**9 + 120*d**10*x**10)/(2*b**6*c**10*d
*log(F)**6 + 20*b**6*c**9*d**2*x*log(F)**6 + 90*b**6*c**8*d**3*x**2*log(F)**6 + 240*b**6*c**7*d**4*x**3*log(F)
**6 + 420*b**6*c**6*d**5*x**4*log(F)**6 + 504*b**6*c**5*d**6*x**5*log(F)**6 + 420*b**6*c**4*d**7*x**6*log(F)**
6 + 240*b**6*c**3*d**8*x**7*log(F)**6 + 90*b**6*c**2*d**9*x**8*log(F)**6 + 20*b**6*c*d**10*x**9*log(F)**6 + 2*
b**6*d**11*x**10*log(F)**6)
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{F^{a + \frac{b}{{\left (d x + c\right )}^{2}}}}{{\left (d x + c\right )}^{13}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b/(d*x+c)^2)/(d*x+c)^13,x, algorithm="giac")
[Out]
integrate(F^(a + b/(d*x + c)^2)/(d*x + c)^13, x)