3.352 \(\int \frac{F^{a+\frac{b}{(c+d x)^3}}}{(c+d x)^{19}} \, dx\)
Optimal. Leaf size=113 \[ \frac{F^{a+\frac{b}{(c+d x)^3}} \left (60 b^2 \log ^2(F) (c+d x)^9-20 b^3 \log ^3(F) (c+d x)^6+5 b^4 \log ^4(F) (c+d x)^3-b^5 \log ^5(F)-120 b \log (F) (c+d x)^{12}+120 (c+d x)^{15}\right )}{3 b^6 d \log ^6(F) (c+d x)^{15}} \]
[Out]
(F^(a + b/(c + d*x)^3)*(120*(c + d*x)^15 - 120*b*(c + d*x)^12*Log[F] + 60*b^2*(c + d*x)^9*Log[F]^2 - 20*b^3*(c
+ d*x)^6*Log[F]^3 + 5*b^4*(c + d*x)^3*Log[F]^4 - b^5*Log[F]^5))/(3*b^6*d*(c + d*x)^15*Log[F]^6)
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Rubi [C] time = 0.0449279, 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} \[ \frac{F^a \text{Gamma}\left (6,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
[In]
Int[F^(a + b/(c + d*x)^3)/(c + d*x)^19,x]
[Out]
(F^a*Gamma[6, -((b*Log[F])/(c + d*x)^3)])/(3*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)^3}}}{(c+d x)^{19}} \, dx &=\frac{F^a \Gamma \left (6,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 b^6 d \log ^6(F)}\\ \end{align*}
Mathematica [C] time = 0.0083544, size = 31, normalized size = 0.27 \[ \frac{F^a \text{Gamma}\left (6,-\frac{b \log (F)}{(c+d x)^3}\right )}{3 b^6 d \log ^6(F)} \]
Antiderivative was successfully verified.
[In]
Integrate[F^(a + b/(c + d*x)^3)/(c + d*x)^19,x]
[Out]
(F^a*Gamma[6, -((b*Log[F])/(c + d*x)^3)])/(3*b^6*d*Log[F]^6)
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Maple [B] time = 0.041, size = 733, normalized size = 6.5 \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)^3)/(d*x+c)^19,x)
[Out]
-1/3*(20*ln(F)^3*b^3*c^6-120*d^15*x^15-120*c^15+120*ln(F)*b*d^12*x^12-60*ln(F)^2*b^2*d^9*x^9-5*ln(F)^4*b^4*d^3
*x^3-163800*c^11*d^4*x^4-54600*c^12*d^3*x^3-12600*c^13*d^2*x^2-1800*c^14*d*x+120*ln(F)*b*c^12-60*ln(F)^2*b^2*c
^9-5*ln(F)^4*b^4*c^3-1800*c*d^14*x^14-12600*c^2*d^13*x^13-54600*c^3*d^12*x^12-163800*c^4*d^11*x^11-360360*c^5*
d^10*x^10-600600*c^6*d^9*x^9-772200*c^7*d^8*x^8-772200*c^8*d^7*x^7-600600*c^9*d^6*x^6-360360*c^10*d^5*x^5-2160
*ln(F)^2*b^2*c^7*d^2*x^2+1440*ln(F)*b*c^11*d*x-540*ln(F)^2*b^2*c^8*d*x-15*ln(F)^4*b^4*c*d^2*x^2-15*ln(F)^4*b^4
*c^2*d*x+120*c*d^5*x^5*b^3*ln(F)^3+300*ln(F)^3*b^3*c^2*d^4*x^4+400*ln(F)^3*b^3*c^3*d^3*x^3+300*ln(F)^3*b^3*c^4
*d^2*x^2+120*ln(F)^3*b^3*c^5*d*x+1440*ln(F)*b*c*d^11*x^11+7920*ln(F)*b*c^2*d^10*x^10+26400*ln(F)*b*c^3*d^9*x^9
+59400*ln(F)*b*c^4*d^8*x^8-540*ln(F)^2*b^2*c*d^8*x^8+95040*ln(F)*b*c^5*d^7*x^7-2160*ln(F)^2*b^2*c^2*d^7*x^7+11
0880*ln(F)*b*c^6*d^6*x^6-5040*ln(F)^2*b^2*c^3*d^6*x^6+95040*ln(F)*b*c^7*d^5*x^5-7560*ln(F)^2*b^2*c^4*d^5*x^5+5
9400*ln(F)*b*c^8*d^4*x^4-7560*ln(F)^2*b^2*c^5*d^4*x^4+26400*ln(F)*b*c^9*d^3*x^3-5040*ln(F)^2*b^2*c^6*d^3*x^3+7
920*ln(F)*b*c^10*d^2*x^2+b^5*ln(F)^5+20*d^6*x^6*b^3*ln(F)^3)/ln(F)^6/b^6/d/(d*x+c)^15*F^((a*d^3*x^3+3*a*c*d^2*
x^2+3*a*c^2*d*x+a*c^3+b)/(d*x+c)^3)
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Maxima [B] time = 1.17461, size = 1465, normalized size = 12.96 \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)^3)/(d*x+c)^19,x, algorithm="maxima")
[Out]
1/3*(120*F^a*d^15*x^15 + 1800*F^a*c*d^14*x^14 + 12600*F^a*c^2*d^13*x^13 + 120*F^a*c^15 - 120*F^a*b*c^12*log(F)
+ 60*F^a*b^2*c^9*log(F)^2 + 120*(455*F^a*c^3*d^12 - F^a*b*d^12*log(F))*x^12 - 20*F^a*b^3*c^6*log(F)^3 + 360*(
455*F^a*c^4*d^11 - 4*F^a*b*c*d^11*log(F))*x^11 + 5*F^a*b^4*c^3*log(F)^4 + 3960*(91*F^a*c^5*d^10 - 2*F^a*b*c^2*
d^10*log(F))*x^10 - F^a*b^5*log(F)^5 + 60*(10010*F^a*c^6*d^9 - 440*F^a*b*c^3*d^9*log(F) + F^a*b^2*d^9*log(F)^2
)*x^9 + 540*(1430*F^a*c^7*d^8 - 110*F^a*b*c^4*d^8*log(F) + F^a*b^2*c*d^8*log(F)^2)*x^8 + 1080*(715*F^a*c^8*d^7
- 88*F^a*b*c^5*d^7*log(F) + 2*F^a*b^2*c^2*d^7*log(F)^2)*x^7 + 20*(30030*F^a*c^9*d^6 - 5544*F^a*b*c^6*d^6*log(
F) + 252*F^a*b^2*c^3*d^6*log(F)^2 - F^a*b^3*d^6*log(F)^3)*x^6 + 120*(3003*F^a*c^10*d^5 - 792*F^a*b*c^7*d^5*log
(F) + 63*F^a*b^2*c^4*d^5*log(F)^2 - F^a*b^3*c*d^5*log(F)^3)*x^5 + 60*(2730*F^a*c^11*d^4 - 990*F^a*b*c^8*d^4*lo
g(F) + 126*F^a*b^2*c^5*d^4*log(F)^2 - 5*F^a*b^3*c^2*d^4*log(F)^3)*x^4 + 5*(10920*F^a*c^12*d^3 - 5280*F^a*b*c^9
*d^3*log(F) + 1008*F^a*b^2*c^6*d^3*log(F)^2 - 80*F^a*b^3*c^3*d^3*log(F)^3 + F^a*b^4*d^3*log(F)^4)*x^3 + 15*(84
0*F^a*c^13*d^2 - 528*F^a*b*c^10*d^2*log(F) + 144*F^a*b^2*c^7*d^2*log(F)^2 - 20*F^a*b^3*c^4*d^2*log(F)^3 + F^a*
b^4*c*d^2*log(F)^4)*x^2 + 15*(120*F^a*c^14*d - 96*F^a*b*c^11*d*log(F) + 36*F^a*b^2*c^8*d*log(F)^2 - 8*F^a*b^3*
c^5*d*log(F)^3 + F^a*b^4*c^2*d*log(F)^4)*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(b^6*d^16*x^15*log
(F)^6 + 15*b^6*c*d^15*x^14*log(F)^6 + 105*b^6*c^2*d^14*x^13*log(F)^6 + 455*b^6*c^3*d^13*x^12*log(F)^6 + 1365*b
^6*c^4*d^12*x^11*log(F)^6 + 3003*b^6*c^5*d^11*x^10*log(F)^6 + 5005*b^6*c^6*d^10*x^9*log(F)^6 + 6435*b^6*c^7*d^
9*x^8*log(F)^6 + 6435*b^6*c^8*d^8*x^7*log(F)^6 + 5005*b^6*c^9*d^7*x^6*log(F)^6 + 3003*b^6*c^10*d^6*x^5*log(F)^
6 + 1365*b^6*c^11*d^5*x^4*log(F)^6 + 455*b^6*c^12*d^4*x^3*log(F)^6 + 105*b^6*c^13*d^3*x^2*log(F)^6 + 15*b^6*c^
14*d^2*x*log(F)^6 + b^6*c^15*d*log(F)^6)
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Fricas [B] time = 2.42029, size = 1979, normalized size = 17.51 \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)^3)/(d*x+c)^19,x, algorithm="fricas")
[Out]
1/3*(120*d^15*x^15 + 1800*c*d^14*x^14 + 12600*c^2*d^13*x^13 + 54600*c^3*d^12*x^12 + 163800*c^4*d^11*x^11 + 360
360*c^5*d^10*x^10 + 600600*c^6*d^9*x^9 + 772200*c^7*d^8*x^8 + 772200*c^8*d^7*x^7 + 600600*c^9*d^6*x^6 + 360360
*c^10*d^5*x^5 + 163800*c^11*d^4*x^4 + 54600*c^12*d^3*x^3 + 12600*c^13*d^2*x^2 + 1800*c^14*d*x + 120*c^15 - b^5
*log(F)^5 + 5*(b^4*d^3*x^3 + 3*b^4*c*d^2*x^2 + 3*b^4*c^2*d*x + b^4*c^3)*log(F)^4 - 20*(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 + 60
*(b^2*d^9*x^9 + 9*b^2*c*d^8*x^8 + 36*b^2*c^2*d^7*x^7 + 84*b^2*c^3*d^6*x^6 + 126*b^2*c^4*d^5*x^5 + 126*b^2*c^5*
d^4*x^4 + 84*b^2*c^6*d^3*x^3 + 36*b^2*c^7*d^2*x^2 + 9*b^2*c^8*d*x + b^2*c^9)*log(F)^2 - 120*(b*d^12*x^12 + 12*
b*c*d^11*x^11 + 66*b*c^2*d^10*x^10 + 220*b*c^3*d^9*x^9 + 495*b*c^4*d^8*x^8 + 792*b*c^5*d^7*x^7 + 924*b*c^6*d^6
*x^6 + 792*b*c^7*d^5*x^5 + 495*b*c^8*d^4*x^4 + 220*b*c^9*d^3*x^3 + 66*b*c^10*d^2*x^2 + 12*b*c^11*d*x + b*c^12)
*log(F))*F^((a*d^3*x^3 + 3*a*c*d^2*x^2 + 3*a*c^2*d*x + a*c^3 + b)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(
(b^6*d^16*x^15 + 15*b^6*c*d^15*x^14 + 105*b^6*c^2*d^14*x^13 + 455*b^6*c^3*d^13*x^12 + 1365*b^6*c^4*d^12*x^11 +
3003*b^6*c^5*d^11*x^10 + 5005*b^6*c^6*d^10*x^9 + 6435*b^6*c^7*d^9*x^8 + 6435*b^6*c^8*d^8*x^7 + 5005*b^6*c^9*d
^7*x^6 + 3003*b^6*c^10*d^6*x^5 + 1365*b^6*c^11*d^5*x^4 + 455*b^6*c^12*d^4*x^3 + 105*b^6*c^13*d^3*x^2 + 15*b^6*
c^14*d^2*x + b^6*c^15*d)*log(F)^6)
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Sympy [B] time = 3.4134, size = 1096, normalized size = 9.7 \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)**3)/(d*x+c)**19,x)
[Out]
F**(a + b/(c + d*x)**3)*(-b**5*log(F)**5 + 5*b**4*c**3*log(F)**4 + 15*b**4*c**2*d*x*log(F)**4 + 15*b**4*c*d**2
*x**2*log(F)**4 + 5*b**4*d**3*x**3*log(F)**4 - 20*b**3*c**6*log(F)**3 - 120*b**3*c**5*d*x*log(F)**3 - 300*b**3
*c**4*d**2*x**2*log(F)**3 - 400*b**3*c**3*d**3*x**3*log(F)**3 - 300*b**3*c**2*d**4*x**4*log(F)**3 - 120*b**3*c
*d**5*x**5*log(F)**3 - 20*b**3*d**6*x**6*log(F)**3 + 60*b**2*c**9*log(F)**2 + 540*b**2*c**8*d*x*log(F)**2 + 21
60*b**2*c**7*d**2*x**2*log(F)**2 + 5040*b**2*c**6*d**3*x**3*log(F)**2 + 7560*b**2*c**5*d**4*x**4*log(F)**2 + 7
560*b**2*c**4*d**5*x**5*log(F)**2 + 5040*b**2*c**3*d**6*x**6*log(F)**2 + 2160*b**2*c**2*d**7*x**7*log(F)**2 +
540*b**2*c*d**8*x**8*log(F)**2 + 60*b**2*d**9*x**9*log(F)**2 - 120*b*c**12*log(F) - 1440*b*c**11*d*x*log(F) -
7920*b*c**10*d**2*x**2*log(F) - 26400*b*c**9*d**3*x**3*log(F) - 59400*b*c**8*d**4*x**4*log(F) - 95040*b*c**7*d
**5*x**5*log(F) - 110880*b*c**6*d**6*x**6*log(F) - 95040*b*c**5*d**7*x**7*log(F) - 59400*b*c**4*d**8*x**8*log(
F) - 26400*b*c**3*d**9*x**9*log(F) - 7920*b*c**2*d**10*x**10*log(F) - 1440*b*c*d**11*x**11*log(F) - 120*b*d**1
2*x**12*log(F) + 120*c**15 + 1800*c**14*d*x + 12600*c**13*d**2*x**2 + 54600*c**12*d**3*x**3 + 163800*c**11*d**
4*x**4 + 360360*c**10*d**5*x**5 + 600600*c**9*d**6*x**6 + 772200*c**8*d**7*x**7 + 772200*c**7*d**8*x**8 + 6006
00*c**6*d**9*x**9 + 360360*c**5*d**10*x**10 + 163800*c**4*d**11*x**11 + 54600*c**3*d**12*x**12 + 12600*c**2*d*
*13*x**13 + 1800*c*d**14*x**14 + 120*d**15*x**15)/(3*b**6*c**15*d*log(F)**6 + 45*b**6*c**14*d**2*x*log(F)**6 +
315*b**6*c**13*d**3*x**2*log(F)**6 + 1365*b**6*c**12*d**4*x**3*log(F)**6 + 4095*b**6*c**11*d**5*x**4*log(F)**
6 + 9009*b**6*c**10*d**6*x**5*log(F)**6 + 15015*b**6*c**9*d**7*x**6*log(F)**6 + 19305*b**6*c**8*d**8*x**7*log(
F)**6 + 19305*b**6*c**7*d**9*x**8*log(F)**6 + 15015*b**6*c**6*d**10*x**9*log(F)**6 + 9009*b**6*c**5*d**11*x**1
0*log(F)**6 + 4095*b**6*c**4*d**12*x**11*log(F)**6 + 1365*b**6*c**3*d**13*x**12*log(F)**6 + 315*b**6*c**2*d**1
4*x**13*log(F)**6 + 45*b**6*c*d**15*x**14*log(F)**6 + 3*b**6*d**16*x**15*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 )}^{3}}}}{{\left (d x + c\right )}^{19}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^19,x, algorithm="giac")
[Out]
integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^19, x)