3.352 \(\int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{19}} \, dx\)

Optimal. Leaf size=113 \[ \frac {F^{a+\frac {b}{(c+d x)^3}} \left (-b^5 \log ^5(F)+5 b^4 \log ^4(F) (c+d x)^3-20 b^3 \log ^3(F) (c+d x)^6+60 b^2 \log ^2(F) (c+d x)^9-120 b \log (F) (c+d x)^{12}+120 (c+d x)^{15}\right )}{3 b^6 d \log ^6(F) (c+d x)^{15}} \]

[Out]

1/3*F^(a+b/(d*x+c)^3)*(120*(d*x+c)^15-120*b*(d*x+c)^12*ln(F)+60*b^2*(d*x+c)^9*ln(F)^2-20*b^3*(d*x+c)^6*ln(F)^3
+5*b^4*(d*x+c)^3*ln(F)^4-b^5*ln(F)^5)/b^6/d/(d*x+c)^15/ln(F)^6

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Rubi [C]  time = 0.04, 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*}

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Mathematica [C]  time = 0.01, size = 31, normalized size = 0.27 \[ \frac {F^a \Gamma \left (6,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 b^6 d \log ^6(F)} \]

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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fricas [B]  time = 0.58, size = 863, normalized size = 7.64 \[ \frac {{\left (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} + 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} + 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 \relax (F)^{5} + 5 \, {\left (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}\right )} \log \relax (F)^{4} - 20 \, {\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 \relax (F)^{3} + 60 \, {\left (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}\right )} \log \relax (F)^{2} - 120 \, {\left (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}\right )} \log \relax (F)\right )} F^{\frac {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}}}}{3 \, {\left (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\right )} \log \relax (F)^{6}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{19}}\,{d x} \]

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)

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maple [B]  time = 0.04, size = 733, normalized size = 6.49 \[ -\frac {\left (-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}+120 b \,d^{12} x^{12} \ln \relax (F )-163800 c^{4} d^{11} x^{11}+1440 b c \,d^{11} x^{11} \ln \relax (F )-360360 c^{5} d^{10} x^{10}+7920 b \,c^{2} d^{10} x^{10} \ln \relax (F )-600600 c^{6} d^{9} x^{9}+26400 b \,c^{3} d^{9} x^{9} \ln \relax (F )-772200 c^{7} d^{8} x^{8}-60 b^{2} d^{9} x^{9} \ln \relax (F )^{2}+59400 b \,c^{4} d^{8} x^{8} \ln \relax (F )-772200 c^{8} d^{7} x^{7}-540 b^{2} c \,d^{8} x^{8} \ln \relax (F )^{2}+95040 b \,c^{5} d^{7} x^{7} \ln \relax (F )-600600 c^{9} d^{6} x^{6}-2160 b^{2} c^{2} d^{7} x^{7} \ln \relax (F )^{2}+110880 b \,c^{6} d^{6} x^{6} \ln \relax (F )-360360 c^{10} d^{5} x^{5}-5040 b^{2} c^{3} d^{6} x^{6} \ln \relax (F )^{2}+95040 b \,c^{7} d^{5} x^{5} \ln \relax (F )-163800 c^{11} d^{4} x^{4}+20 b^{3} d^{6} x^{6} \ln \relax (F )^{3}-7560 b^{2} c^{4} d^{5} x^{5} \ln \relax (F )^{2}+59400 b \,c^{8} d^{4} x^{4} \ln \relax (F )-54600 c^{12} d^{3} x^{3}+120 b^{3} c \,d^{5} x^{5} \ln \relax (F )^{3}-7560 b^{2} c^{5} d^{4} x^{4} \ln \relax (F )^{2}+26400 b \,c^{9} d^{3} x^{3} \ln \relax (F )-12600 c^{13} d^{2} x^{2}+300 b^{3} c^{2} d^{4} x^{4} \ln \relax (F )^{3}-5040 b^{2} c^{6} d^{3} x^{3} \ln \relax (F )^{2}+7920 b \,c^{10} d^{2} x^{2} \ln \relax (F )-1800 c^{14} d x +400 b^{3} c^{3} d^{3} x^{3} \ln \relax (F )^{3}-2160 b^{2} c^{7} d^{2} x^{2} \ln \relax (F )^{2}+1440 b \,c^{11} d x \ln \relax (F )-120 c^{15}-5 b^{4} d^{3} x^{3} \ln \relax (F )^{4}+300 b^{3} c^{4} d^{2} x^{2} \ln \relax (F )^{3}-540 b^{2} c^{8} d x \ln \relax (F )^{2}+120 b \,c^{12} \ln \relax (F )-15 b^{4} c \,d^{2} x^{2} \ln \relax (F )^{4}+120 b^{3} c^{5} d x \ln \relax (F )^{3}-60 b^{2} c^{9} \ln \relax (F )^{2}-15 b^{4} c^{2} d x \ln \relax (F )^{4}+20 b^{3} c^{6} \ln \relax (F )^{3}-5 b^{4} c^{3} \ln \relax (F )^{4}+b^{5} \ln \relax (F )^{5}\right ) F^{\frac {a \,d^{3} x^{3}+3 a c \,d^{2} x^{2}+3 a \,c^{2} d x +a \,c^{3}+b}{\left (d x +c \right )^{3}}}}{3 \left (d x +c \right )^{15} b^{6} d \ln \relax (F )^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)^3*b)/(d*x+c)^19,x)

[Out]

-1/3*(-120*c^15-15*ln(F)^4*b^4*c^2*d*x+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+
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+110880*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+59400*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+7920*ln(F)*b*c^10*d^2*x^2-2160*ln(
F)^2*b^2*c^7*d^2*x^2-120*d^15*x^15+b^5*ln(F)^5+20*b^3*c^6*ln(F)^3-1800*c^14*d*x-360360*c^10*d^5*x^5-163800*c^1
1*d^4*x^4-54600*c^12*d^3*x^3-12600*c^13*d^2*x^2-163800*c^4*d^11*x^11-360360*c^5*d^10*x^10-600600*c^6*d^9*x^9-7
72200*c^7*d^8*x^8-772200*c^8*d^7*x^7-600600*c^9*d^6*x^6-12600*c^2*d^13*x^13-54600*c^3*d^12*x^12-1800*c*d^14*x^
14+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+120*ln(F)*b*c^12-60*ln(F)^2*b^2*c^9-5*ln
(F)^4*b^4*c^3+20*b^3*d^6*x^6*ln(F)^3+400*b^3*c^3*d^3*x^3*ln(F)^3+300*b^3*c^4*d^2*x^2*ln(F)^3+120*b^3*c^5*d*x*l
n(F)^3+120*b^3*c*d^5*x^5*ln(F)^3+300*b^3*c^2*d^4*x^4*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 = 1.34, size = 1085, normalized size = 9.60 \[ \text {result too large to display} \]

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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mupad [B]  time = 5.77, size = 854, normalized size = 7.56 \[ \frac {F^a\,F^{\frac {b}{c^3+3\,c^2\,d\,x+3\,c\,d^2\,x^2+d^3\,x^3}}\,\left (\frac {40\,x^{15}}{b^6\,d\,{\ln \relax (F)}^6}-\frac {b^5\,{\ln \relax (F)}^5-5\,b^4\,c^3\,{\ln \relax (F)}^4+20\,b^3\,c^6\,{\ln \relax (F)}^3-60\,b^2\,c^9\,{\ln \relax (F)}^2+120\,b\,c^{12}\,\ln \relax (F)-120\,c^{15}}{3\,b^6\,d^{16}\,{\ln \relax (F)}^6}+\frac {600\,c\,x^{14}}{b^6\,d^2\,{\ln \relax (F)}^6}+\frac {4200\,c^2\,x^{13}}{b^6\,d^3\,{\ln \relax (F)}^6}+\frac {5\,x^3\,\left (b^4\,{\ln \relax (F)}^4-80\,b^3\,c^3\,{\ln \relax (F)}^3+1008\,b^2\,c^6\,{\ln \relax (F)}^2-5280\,b\,c^9\,\ln \relax (F)+10920\,c^{12}\right )}{3\,b^6\,d^{13}\,{\ln \relax (F)}^6}-\frac {20\,x^6\,\left (b^3\,{\ln \relax (F)}^3-252\,b^2\,c^3\,{\ln \relax (F)}^2+5544\,b\,c^6\,\ln \relax (F)-30030\,c^9\right )}{3\,b^6\,d^{10}\,{\ln \relax (F)}^6}+\frac {20\,x^9\,\left (b^2\,{\ln \relax (F)}^2-440\,b\,c^3\,\ln \relax (F)+10010\,c^6\right )}{b^6\,d^7\,{\ln \relax (F)}^6}-\frac {40\,x^{12}\,\left (b\,\ln \relax (F)-455\,c^3\right )}{b^6\,d^4\,{\ln \relax (F)}^6}+\frac {5\,c^2\,x\,\left (b^4\,{\ln \relax (F)}^4-8\,b^3\,c^3\,{\ln \relax (F)}^3+36\,b^2\,c^6\,{\ln \relax (F)}^2-96\,b\,c^9\,\ln \relax (F)+120\,c^{12}\right )}{b^6\,d^{15}\,{\ln \relax (F)}^6}+\frac {5\,c\,x^2\,\left (b^4\,{\ln \relax (F)}^4-20\,b^3\,c^3\,{\ln \relax (F)}^3+144\,b^2\,c^6\,{\ln \relax (F)}^2-528\,b\,c^9\,\ln \relax (F)+840\,c^{12}\right )}{b^6\,d^{14}\,{\ln \relax (F)}^6}-\frac {40\,c\,x^5\,\left (b^3\,{\ln \relax (F)}^3-63\,b^2\,c^3\,{\ln \relax (F)}^2+792\,b\,c^6\,\ln \relax (F)-3003\,c^9\right )}{b^6\,d^{11}\,{\ln \relax (F)}^6}+\frac {180\,c\,x^8\,\left (b^2\,{\ln \relax (F)}^2-110\,b\,c^3\,\ln \relax (F)+1430\,c^6\right )}{b^6\,d^8\,{\ln \relax (F)}^6}-\frac {120\,c\,x^{11}\,\left (4\,b\,\ln \relax (F)-455\,c^3\right )}{b^6\,d^5\,{\ln \relax (F)}^6}-\frac {20\,c^2\,x^4\,\left (5\,b^3\,{\ln \relax (F)}^3-126\,b^2\,c^3\,{\ln \relax (F)}^2+990\,b\,c^6\,\ln \relax (F)-2730\,c^9\right )}{b^6\,d^{12}\,{\ln \relax (F)}^6}+\frac {360\,c^2\,x^7\,\left (2\,b^2\,{\ln \relax (F)}^2-88\,b\,c^3\,\ln \relax (F)+715\,c^6\right )}{b^6\,d^9\,{\ln \relax (F)}^6}-\frac {1320\,c^2\,x^{10}\,\left (2\,b\,\ln \relax (F)-91\,c^3\right )}{b^6\,d^6\,{\ln \relax (F)}^6}\right )}{x^{15}+\frac {c^{15}}{d^{15}}+\frac {15\,c\,x^{14}}{d}+\frac {15\,c^{14}\,x}{d^{14}}+\frac {105\,c^2\,x^{13}}{d^2}+\frac {455\,c^3\,x^{12}}{d^3}+\frac {1365\,c^4\,x^{11}}{d^4}+\frac {3003\,c^5\,x^{10}}{d^5}+\frac {5005\,c^6\,x^9}{d^6}+\frac {6435\,c^7\,x^8}{d^7}+\frac {6435\,c^8\,x^7}{d^8}+\frac {5005\,c^9\,x^6}{d^9}+\frac {3003\,c^{10}\,x^5}{d^{10}}+\frac {1365\,c^{11}\,x^4}{d^{11}}+\frac {455\,c^{12}\,x^3}{d^{12}}+\frac {105\,c^{13}\,x^2}{d^{13}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b/(c + d*x)^3)/(c + d*x)^19,x)

[Out]

(F^a*F^(b/(c^3 + d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x))*((40*x^15)/(b^6*d*log(F)^6) - (b^5*log(F)^5 - 120*c^15 +
120*b*c^12*log(F) - 5*b^4*c^3*log(F)^4 + 20*b^3*c^6*log(F)^3 - 60*b^2*c^9*log(F)^2)/(3*b^6*d^16*log(F)^6) + (6
00*c*x^14)/(b^6*d^2*log(F)^6) + (4200*c^2*x^13)/(b^6*d^3*log(F)^6) + (5*x^3*(b^4*log(F)^4 + 10920*c^12 - 5280*
b*c^9*log(F) - 80*b^3*c^3*log(F)^3 + 1008*b^2*c^6*log(F)^2))/(3*b^6*d^13*log(F)^6) - (20*x^6*(b^3*log(F)^3 - 3
0030*c^9 + 5544*b*c^6*log(F) - 252*b^2*c^3*log(F)^2))/(3*b^6*d^10*log(F)^6) + (20*x^9*(b^2*log(F)^2 + 10010*c^
6 - 440*b*c^3*log(F)))/(b^6*d^7*log(F)^6) - (40*x^12*(b*log(F) - 455*c^3))/(b^6*d^4*log(F)^6) + (5*c^2*x*(b^4*
log(F)^4 + 120*c^12 - 96*b*c^9*log(F) - 8*b^3*c^3*log(F)^3 + 36*b^2*c^6*log(F)^2))/(b^6*d^15*log(F)^6) + (5*c*
x^2*(b^4*log(F)^4 + 840*c^12 - 528*b*c^9*log(F) - 20*b^3*c^3*log(F)^3 + 144*b^2*c^6*log(F)^2))/(b^6*d^14*log(F
)^6) - (40*c*x^5*(b^3*log(F)^3 - 3003*c^9 + 792*b*c^6*log(F) - 63*b^2*c^3*log(F)^2))/(b^6*d^11*log(F)^6) + (18
0*c*x^8*(b^2*log(F)^2 + 1430*c^6 - 110*b*c^3*log(F)))/(b^6*d^8*log(F)^6) - (120*c*x^11*(4*b*log(F) - 455*c^3))
/(b^6*d^5*log(F)^6) - (20*c^2*x^4*(5*b^3*log(F)^3 - 2730*c^9 + 990*b*c^6*log(F) - 126*b^2*c^3*log(F)^2))/(b^6*
d^12*log(F)^6) + (360*c^2*x^7*(2*b^2*log(F)^2 + 715*c^6 - 88*b*c^3*log(F)))/(b^6*d^9*log(F)^6) - (1320*c^2*x^1
0*(2*b*log(F) - 91*c^3))/(b^6*d^6*log(F)^6)))/(x^15 + c^15/d^15 + (15*c*x^14)/d + (15*c^14*x)/d^14 + (105*c^2*
x^13)/d^2 + (455*c^3*x^12)/d^3 + (1365*c^4*x^11)/d^4 + (3003*c^5*x^10)/d^5 + (5005*c^6*x^9)/d^6 + (6435*c^7*x^
8)/d^7 + (6435*c^8*x^7)/d^8 + (5005*c^9*x^6)/d^9 + (3003*c^10*x^5)/d^10 + (1365*c^11*x^4)/d^11 + (455*c^12*x^3
)/d^12 + (105*c^13*x^2)/d^13)

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sympy [B]  time = 3.44, size = 1096, normalized size = 9.70 \[ \text {result too large to display} \]

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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