3.351 \(\int \frac {F^{a+\frac {b}{(c+d x)^3}}}{(c+d x)^{16}} \, dx\)

Optimal. Leaf size=96 \[ -\frac {F^{a+\frac {b}{(c+d x)^3}} \left (b^4 \log ^4(F)-4 b^3 \log ^3(F) (c+d x)^3+12 b^2 \log ^2(F) (c+d x)^6-24 b \log (F) (c+d x)^9+24 (c+d x)^{12}\right )}{3 b^5 d \log ^5(F) (c+d x)^{12}} \]

[Out]

-1/3*F^(a+b/(d*x+c)^3)*(24*(d*x+c)^12-24*b*(d*x+c)^9*ln(F)+12*b^2*(d*x+c)^6*ln(F)^2-4*b^3*(d*x+c)^3*ln(F)^3+b^
4*ln(F)^4)/b^5/d/(d*x+c)^12/ln(F)^5

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Rubi [C]  time = 0.04, antiderivative size = 31, normalized size of antiderivative = 0.32, 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} \[ -\frac {F^a \text {Gamma}\left (5,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Int[F^(a + b/(c + d*x)^3)/(c + d*x)^16,x]

[Out]

-(F^a*Gamma[5, -((b*Log[F])/(c + d*x)^3)])/(3*b^5*d*Log[F]^5)

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)^{16}} \, dx &=-\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 b^5 d \log ^5(F)}\\ \end {align*}

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Mathematica [C]  time = 0.01, size = 31, normalized size = 0.32 \[ -\frac {F^a \Gamma \left (5,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 b^5 d \log ^5(F)} \]

Antiderivative was successfully verified.

[In]

Integrate[F^(a + b/(c + d*x)^3)/(c + d*x)^16,x]

[Out]

-1/3*(F^a*Gamma[5, -((b*Log[F])/(c + d*x)^3)])/(b^5*d*Log[F]^5)

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fricas [B]  time = 0.56, size = 621, normalized size = 6.47 \[ -\frac {{\left (24 \, d^{12} x^{12} + 288 \, c d^{11} x^{11} + 1584 \, c^{2} d^{10} x^{10} + 5280 \, c^{3} d^{9} x^{9} + 11880 \, c^{4} d^{8} x^{8} + 19008 \, c^{5} d^{7} x^{7} + 22176 \, c^{6} d^{6} x^{6} + 19008 \, c^{7} d^{5} x^{5} + 11880 \, c^{8} d^{4} x^{4} + 5280 \, c^{9} d^{3} x^{3} + 1584 \, c^{10} d^{2} x^{2} + 288 \, c^{11} d x + 24 \, c^{12} + b^{4} \log \relax (F)^{4} - 4 \, {\left (b^{3} d^{3} x^{3} + 3 \, b^{3} c d^{2} x^{2} + 3 \, b^{3} c^{2} d x + b^{3} c^{3}\right )} \log \relax (F)^{3} + 12 \, {\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \relax (F)^{2} - 24 \, {\left (b d^{9} x^{9} + 9 \, b c d^{8} x^{8} + 36 \, b c^{2} d^{7} x^{7} + 84 \, b c^{3} d^{6} x^{6} + 126 \, b c^{4} d^{5} x^{5} + 126 \, b c^{5} d^{4} x^{4} + 84 \, b c^{6} d^{3} x^{3} + 36 \, b c^{7} d^{2} x^{2} + 9 \, b c^{8} d x + b c^{9}\right )} \log \relax (F)\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{5} d^{13} x^{12} + 12 \, b^{5} c d^{12} x^{11} + 66 \, b^{5} c^{2} d^{11} x^{10} + 220 \, b^{5} c^{3} d^{10} x^{9} + 495 \, b^{5} c^{4} d^{9} x^{8} + 792 \, b^{5} c^{5} d^{8} x^{7} + 924 \, b^{5} c^{6} d^{7} x^{6} + 792 \, b^{5} c^{7} d^{6} x^{5} + 495 \, b^{5} c^{8} d^{5} x^{4} + 220 \, b^{5} c^{9} d^{4} x^{3} + 66 \, b^{5} c^{10} d^{3} x^{2} + 12 \, b^{5} c^{11} d^{2} x + b^{5} c^{12} d\right )} \log \relax (F)^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x, algorithm="fricas")

[Out]

-1/3*(24*d^12*x^12 + 288*c*d^11*x^11 + 1584*c^2*d^10*x^10 + 5280*c^3*d^9*x^9 + 11880*c^4*d^8*x^8 + 19008*c^5*d
^7*x^7 + 22176*c^6*d^6*x^6 + 19008*c^7*d^5*x^5 + 11880*c^8*d^4*x^4 + 5280*c^9*d^3*x^3 + 1584*c^10*d^2*x^2 + 28
8*c^11*d*x + 24*c^12 + b^4*log(F)^4 - 4*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(F)^3 + 1
2*(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 - 24*(b*d^9*x^9 + 9*b*c*d^8*x^8 + 36*b*c^2*d^7*x^7 + 84*b*c^3*d^6*x^6 + 126*b*c^4*d^5*x^
5 + 126*b*c^5*d^4*x^4 + 84*b*c^6*d^3*x^3 + 36*b*c^7*d^2*x^2 + 9*b*c^8*d*x + b*c^9)*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^5*d^13*x^12 + 12*b^5*c*d^
12*x^11 + 66*b^5*c^2*d^11*x^10 + 220*b^5*c^3*d^10*x^9 + 495*b^5*c^4*d^9*x^8 + 792*b^5*c^5*d^8*x^7 + 924*b^5*c^
6*d^7*x^6 + 792*b^5*c^7*d^6*x^5 + 495*b^5*c^8*d^5*x^4 + 220*b^5*c^9*d^4*x^3 + 66*b^5*c^10*d^3*x^2 + 12*b^5*c^1
1*d^2*x + b^5*c^12*d)*log(F)^5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {F^{a + \frac {b}{{\left (d x + c\right )}^{3}}}}{{\left (d x + c\right )}^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x, algorithm="giac")

[Out]

integrate(F^(a + b/(d*x + c)^3)/(d*x + c)^16, x)

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maple [B]  time = 0.31, size = 889, normalized size = 9.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a+1/(d*x+c)^3*b)/(d*x+c)^16,x)

[Out]

(-d*c*(840*c^12-528*ln(F)*b*c^9+144*b^2*c^6*ln(F)^2-20*ln(F)^3*b^3*c^3+b^4*ln(F)^4)/b^5/ln(F)^5*x^2*exp((a+1/(
d*x+c)^3*b)*ln(F))+4*c^2*d^3*(-2730*c^9+990*b*c^6*ln(F)-126*b^2*c^3*ln(F)^2+5*b^3*ln(F)^3)/ln(F)^5/b^5*x^4*exp
((a+1/(d*x+c)^3*b)*ln(F))+8*c*d^4*(-3003*c^9+792*b*c^6*ln(F)-63*b^2*c^3*ln(F)^2+b^3*ln(F)^3)/ln(F)^5/b^5*x^5*e
xp((a+1/(d*x+c)^3*b)*ln(F))-72*c^2*d^6*(715*c^6-88*b*c^3*ln(F)+2*b^2*ln(F)^2)/ln(F)^5/b^5*x^7*exp((a+1/(d*x+c)
^3*b)*ln(F))-36*c*d^7*(1430*c^6-110*b*c^3*ln(F)+b^2*ln(F)^2)/ln(F)^5/b^5*x^8*exp((a+1/(d*x+c)^3*b)*ln(F))+264*
c^2*d^9*(-91*c^3+2*b*ln(F))/ln(F)^5/b^5*x^10*exp((a+1/(d*x+c)^3*b)*ln(F))+24*c*d^10*(-455*c^3+4*b*ln(F))/ln(F)
^5/b^5*x^11*exp((a+1/(d*x+c)^3*b)*ln(F))-4*d^8*(10010*c^6-440*b*c^3*ln(F)+b^2*ln(F)^2)/ln(F)^5/b^5*x^9*exp((a+
1/(d*x+c)^3*b)*ln(F))+8*d^11*(-455*c^3+b*ln(F))/ln(F)^5/b^5*x^12*exp((a+1/(d*x+c)^3*b)*ln(F))-840*d^12*c^2/ln(
F)^5/b^5*x^13*exp((a+1/(d*x+c)^3*b)*ln(F))-120*d^13*c/ln(F)^5/b^5*x^14*exp((a+1/(d*x+c)^3*b)*ln(F))-8*d^14/ln(
F)^5/b^5*x^15*exp((a+1/(d*x+c)^3*b)*ln(F))-1/3*(24*c^12-24*ln(F)*b*c^9+12*b^2*c^6*ln(F)^2-4*ln(F)^3*b^3*c^3+b^
4*ln(F)^4)*c^3/b^5/ln(F)^5/d*exp((a+1/(d*x+c)^3*b)*ln(F))-c^2*(120*c^12-96*ln(F)*b*c^9+36*b^2*c^6*ln(F)^2-8*ln
(F)^3*b^3*c^3+b^4*ln(F)^4)/b^5/ln(F)^5*x*exp((a+1/(d*x+c)^3*b)*ln(F))-1/3*d^2*(10920*c^12-5280*ln(F)*b*c^9+100
8*b^2*c^6*ln(F)^2-80*ln(F)^3*b^3*c^3+b^4*ln(F)^4)/ln(F)^5/b^5*x^3*exp((a+1/(d*x+c)^3*b)*ln(F))+4/3*d^5*(-30030
*c^9+5544*b*c^6*ln(F)-252*b^2*c^3*ln(F)^2+b^3*ln(F)^3)/ln(F)^5/b^5*x^6*exp((a+1/(d*x+c)^3*b)*ln(F)))/(d*x+c)^1
5

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maxima [B]  time = 1.16, size = 770, normalized size = 8.02 \[ -\frac {{\left (24 \, F^{a} d^{12} x^{12} + 288 \, F^{a} c d^{11} x^{11} + 1584 \, F^{a} c^{2} d^{10} x^{10} + 24 \, F^{a} c^{12} - 24 \, F^{a} b c^{9} \log \relax (F) + 12 \, F^{a} b^{2} c^{6} \log \relax (F)^{2} + 24 \, {\left (220 \, F^{a} c^{3} d^{9} - F^{a} b d^{9} \log \relax (F)\right )} x^{9} - 4 \, F^{a} b^{3} c^{3} \log \relax (F)^{3} + 216 \, {\left (55 \, F^{a} c^{4} d^{8} - F^{a} b c d^{8} \log \relax (F)\right )} x^{8} + F^{a} b^{4} \log \relax (F)^{4} + 864 \, {\left (22 \, F^{a} c^{5} d^{7} - F^{a} b c^{2} d^{7} \log \relax (F)\right )} x^{7} + 12 \, {\left (1848 \, F^{a} c^{6} d^{6} - 168 \, F^{a} b c^{3} d^{6} \log \relax (F) + F^{a} b^{2} d^{6} \log \relax (F)^{2}\right )} x^{6} + 72 \, {\left (264 \, F^{a} c^{7} d^{5} - 42 \, F^{a} b c^{4} d^{5} \log \relax (F) + F^{a} b^{2} c d^{5} \log \relax (F)^{2}\right )} x^{5} + 36 \, {\left (330 \, F^{a} c^{8} d^{4} - 84 \, F^{a} b c^{5} d^{4} \log \relax (F) + 5 \, F^{a} b^{2} c^{2} d^{4} \log \relax (F)^{2}\right )} x^{4} + 4 \, {\left (1320 \, F^{a} c^{9} d^{3} - 504 \, F^{a} b c^{6} d^{3} \log \relax (F) + 60 \, F^{a} b^{2} c^{3} d^{3} \log \relax (F)^{2} - F^{a} b^{3} d^{3} \log \relax (F)^{3}\right )} x^{3} + 12 \, {\left (132 \, F^{a} c^{10} d^{2} - 72 \, F^{a} b c^{7} d^{2} \log \relax (F) + 15 \, F^{a} b^{2} c^{4} d^{2} \log \relax (F)^{2} - F^{a} b^{3} c d^{2} \log \relax (F)^{3}\right )} x^{2} + 12 \, {\left (24 \, F^{a} c^{11} d - 18 \, F^{a} b c^{8} d \log \relax (F) + 6 \, F^{a} b^{2} c^{5} d \log \relax (F)^{2} - F^{a} b^{3} c^{2} d \log \relax (F)^{3}\right )} x\right )} F^{\frac {b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{3 \, {\left (b^{5} d^{13} x^{12} \log \relax (F)^{5} + 12 \, b^{5} c d^{12} x^{11} \log \relax (F)^{5} + 66 \, b^{5} c^{2} d^{11} x^{10} \log \relax (F)^{5} + 220 \, b^{5} c^{3} d^{10} x^{9} \log \relax (F)^{5} + 495 \, b^{5} c^{4} d^{9} x^{8} \log \relax (F)^{5} + 792 \, b^{5} c^{5} d^{8} x^{7} \log \relax (F)^{5} + 924 \, b^{5} c^{6} d^{7} x^{6} \log \relax (F)^{5} + 792 \, b^{5} c^{7} d^{6} x^{5} \log \relax (F)^{5} + 495 \, b^{5} c^{8} d^{5} x^{4} \log \relax (F)^{5} + 220 \, b^{5} c^{9} d^{4} x^{3} \log \relax (F)^{5} + 66 \, b^{5} c^{10} d^{3} x^{2} \log \relax (F)^{5} + 12 \, b^{5} c^{11} d^{2} x \log \relax (F)^{5} + b^{5} c^{12} d \log \relax (F)^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F^(a+b/(d*x+c)^3)/(d*x+c)^16,x, algorithm="maxima")

[Out]

-1/3*(24*F^a*d^12*x^12 + 288*F^a*c*d^11*x^11 + 1584*F^a*c^2*d^10*x^10 + 24*F^a*c^12 - 24*F^a*b*c^9*log(F) + 12
*F^a*b^2*c^6*log(F)^2 + 24*(220*F^a*c^3*d^9 - F^a*b*d^9*log(F))*x^9 - 4*F^a*b^3*c^3*log(F)^3 + 216*(55*F^a*c^4
*d^8 - F^a*b*c*d^8*log(F))*x^8 + F^a*b^4*log(F)^4 + 864*(22*F^a*c^5*d^7 - F^a*b*c^2*d^7*log(F))*x^7 + 12*(1848
*F^a*c^6*d^6 - 168*F^a*b*c^3*d^6*log(F) + F^a*b^2*d^6*log(F)^2)*x^6 + 72*(264*F^a*c^7*d^5 - 42*F^a*b*c^4*d^5*l
og(F) + F^a*b^2*c*d^5*log(F)^2)*x^5 + 36*(330*F^a*c^8*d^4 - 84*F^a*b*c^5*d^4*log(F) + 5*F^a*b^2*c^2*d^4*log(F)
^2)*x^4 + 4*(1320*F^a*c^9*d^3 - 504*F^a*b*c^6*d^3*log(F) + 60*F^a*b^2*c^3*d^3*log(F)^2 - F^a*b^3*d^3*log(F)^3)
*x^3 + 12*(132*F^a*c^10*d^2 - 72*F^a*b*c^7*d^2*log(F) + 15*F^a*b^2*c^4*d^2*log(F)^2 - F^a*b^3*c*d^2*log(F)^3)*
x^2 + 12*(24*F^a*c^11*d - 18*F^a*b*c^8*d*log(F) + 6*F^a*b^2*c^5*d*log(F)^2 - F^a*b^3*c^2*d*log(F)^3)*x)*F^(b/(
d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(b^5*d^13*x^12*log(F)^5 + 12*b^5*c*d^12*x^11*log(F)^5 + 66*b^5*c^2*d
^11*x^10*log(F)^5 + 220*b^5*c^3*d^10*x^9*log(F)^5 + 495*b^5*c^4*d^9*x^8*log(F)^5 + 792*b^5*c^5*d^8*x^7*log(F)^
5 + 924*b^5*c^6*d^7*x^6*log(F)^5 + 792*b^5*c^7*d^6*x^5*log(F)^5 + 495*b^5*c^8*d^5*x^4*log(F)^5 + 220*b^5*c^9*d
^4*x^3*log(F)^5 + 66*b^5*c^10*d^3*x^2*log(F)^5 + 12*b^5*c^11*d^2*x*log(F)^5 + b^5*c^12*d*log(F)^5)

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mupad [B]  time = 5.15, size = 620, normalized size = 6.46 \[ -\frac {F^a\,F^{\frac {b}{c^3+3\,c^2\,d\,x+3\,c\,d^2\,x^2+d^3\,x^3}}\,\left (\frac {b^4\,{\ln \relax (F)}^4-4\,b^3\,c^3\,{\ln \relax (F)}^3+12\,b^2\,c^6\,{\ln \relax (F)}^2-24\,b\,c^9\,\ln \relax (F)+24\,c^{12}}{3\,b^5\,d^{13}\,{\ln \relax (F)}^5}+\frac {8\,x^{12}}{b^5\,d\,{\ln \relax (F)}^5}+\frac {96\,c\,x^{11}}{b^5\,d^2\,{\ln \relax (F)}^5}+\frac {528\,c^2\,x^{10}}{b^5\,d^3\,{\ln \relax (F)}^5}-\frac {4\,x^3\,\left (b^3\,{\ln \relax (F)}^3-60\,b^2\,c^3\,{\ln \relax (F)}^2+504\,b\,c^6\,\ln \relax (F)-1320\,c^9\right )}{3\,b^5\,d^{10}\,{\ln \relax (F)}^5}+\frac {4\,x^6\,\left (b^2\,{\ln \relax (F)}^2-168\,b\,c^3\,\ln \relax (F)+1848\,c^6\right )}{b^5\,d^7\,{\ln \relax (F)}^5}-\frac {8\,x^9\,\left (b\,\ln \relax (F)-220\,c^3\right )}{b^5\,d^4\,{\ln \relax (F)}^5}-\frac {4\,c^2\,x\,\left (b^3\,{\ln \relax (F)}^3-6\,b^2\,c^3\,{\ln \relax (F)}^2+18\,b\,c^6\,\ln \relax (F)-24\,c^9\right )}{b^5\,d^{12}\,{\ln \relax (F)}^5}-\frac {4\,c\,x^2\,\left (b^3\,{\ln \relax (F)}^3-15\,b^2\,c^3\,{\ln \relax (F)}^2+72\,b\,c^6\,\ln \relax (F)-132\,c^9\right )}{b^5\,d^{11}\,{\ln \relax (F)}^5}+\frac {24\,c\,x^5\,\left (b^2\,{\ln \relax (F)}^2-42\,b\,c^3\,\ln \relax (F)+264\,c^6\right )}{b^5\,d^8\,{\ln \relax (F)}^5}-\frac {72\,c\,x^8\,\left (b\,\ln \relax (F)-55\,c^3\right )}{b^5\,d^5\,{\ln \relax (F)}^5}+\frac {12\,c^2\,x^4\,\left (5\,b^2\,{\ln \relax (F)}^2-84\,b\,c^3\,\ln \relax (F)+330\,c^6\right )}{b^5\,d^9\,{\ln \relax (F)}^5}-\frac {288\,c^2\,x^7\,\left (b\,\ln \relax (F)-22\,c^3\right )}{b^5\,d^6\,{\ln \relax (F)}^5}\right )}{x^{12}+\frac {c^{12}}{d^{12}}+\frac {12\,c\,x^{11}}{d}+\frac {12\,c^{11}\,x}{d^{11}}+\frac {66\,c^2\,x^{10}}{d^2}+\frac {220\,c^3\,x^9}{d^3}+\frac {495\,c^4\,x^8}{d^4}+\frac {792\,c^5\,x^7}{d^5}+\frac {924\,c^6\,x^6}{d^6}+\frac {792\,c^7\,x^5}{d^7}+\frac {495\,c^8\,x^4}{d^8}+\frac {220\,c^9\,x^3}{d^9}+\frac {66\,c^{10}\,x^2}{d^{10}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(F^(a + b/(c + d*x)^3)/(c + d*x)^16,x)

[Out]

-(F^a*F^(b/(c^3 + d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x))*((b^4*log(F)^4 + 24*c^12 - 24*b*c^9*log(F) - 4*b^3*c^3*l
og(F)^3 + 12*b^2*c^6*log(F)^2)/(3*b^5*d^13*log(F)^5) + (8*x^12)/(b^5*d*log(F)^5) + (96*c*x^11)/(b^5*d^2*log(F)
^5) + (528*c^2*x^10)/(b^5*d^3*log(F)^5) - (4*x^3*(b^3*log(F)^3 - 1320*c^9 + 504*b*c^6*log(F) - 60*b^2*c^3*log(
F)^2))/(3*b^5*d^10*log(F)^5) + (4*x^6*(b^2*log(F)^2 + 1848*c^6 - 168*b*c^3*log(F)))/(b^5*d^7*log(F)^5) - (8*x^
9*(b*log(F) - 220*c^3))/(b^5*d^4*log(F)^5) - (4*c^2*x*(b^3*log(F)^3 - 24*c^9 + 18*b*c^6*log(F) - 6*b^2*c^3*log
(F)^2))/(b^5*d^12*log(F)^5) - (4*c*x^2*(b^3*log(F)^3 - 132*c^9 + 72*b*c^6*log(F) - 15*b^2*c^3*log(F)^2))/(b^5*
d^11*log(F)^5) + (24*c*x^5*(b^2*log(F)^2 + 264*c^6 - 42*b*c^3*log(F)))/(b^5*d^8*log(F)^5) - (72*c*x^8*(b*log(F
) - 55*c^3))/(b^5*d^5*log(F)^5) + (12*c^2*x^4*(5*b^2*log(F)^2 + 330*c^6 - 84*b*c^3*log(F)))/(b^5*d^9*log(F)^5)
- (288*c^2*x^7*(b*log(F) - 22*c^3))/(b^5*d^6*log(F)^5)))/(x^12 + c^12/d^12 + (12*c*x^11)/d + (12*c^11*x)/d^11
+ (66*c^2*x^10)/d^2 + (220*c^3*x^9)/d^3 + (495*c^4*x^8)/d^4 + (792*c^5*x^7)/d^5 + (924*c^6*x^6)/d^6 + (792*c^
7*x^5)/d^7 + (495*c^8*x^4)/d^8 + (220*c^9*x^3)/d^9 + (66*c^10*x^2)/d^10)

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sympy [B]  time = 1.09, size = 760, normalized size = 7.92 \[ \frac {F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (- b^{4} \log {\relax (F )}^{4} + 4 b^{3} c^{3} \log {\relax (F )}^{3} + 12 b^{3} c^{2} d x \log {\relax (F )}^{3} + 12 b^{3} c d^{2} x^{2} \log {\relax (F )}^{3} + 4 b^{3} d^{3} x^{3} \log {\relax (F )}^{3} - 12 b^{2} c^{6} \log {\relax (F )}^{2} - 72 b^{2} c^{5} d x \log {\relax (F )}^{2} - 180 b^{2} c^{4} d^{2} x^{2} \log {\relax (F )}^{2} - 240 b^{2} c^{3} d^{3} x^{3} \log {\relax (F )}^{2} - 180 b^{2} c^{2} d^{4} x^{4} \log {\relax (F )}^{2} - 72 b^{2} c d^{5} x^{5} \log {\relax (F )}^{2} - 12 b^{2} d^{6} x^{6} \log {\relax (F )}^{2} + 24 b c^{9} \log {\relax (F )} + 216 b c^{8} d x \log {\relax (F )} + 864 b c^{7} d^{2} x^{2} \log {\relax (F )} + 2016 b c^{6} d^{3} x^{3} \log {\relax (F )} + 3024 b c^{5} d^{4} x^{4} \log {\relax (F )} + 3024 b c^{4} d^{5} x^{5} \log {\relax (F )} + 2016 b c^{3} d^{6} x^{6} \log {\relax (F )} + 864 b c^{2} d^{7} x^{7} \log {\relax (F )} + 216 b c d^{8} x^{8} \log {\relax (F )} + 24 b d^{9} x^{9} \log {\relax (F )} - 24 c^{12} - 288 c^{11} d x - 1584 c^{10} d^{2} x^{2} - 5280 c^{9} d^{3} x^{3} - 11880 c^{8} d^{4} x^{4} - 19008 c^{7} d^{5} x^{5} - 22176 c^{6} d^{6} x^{6} - 19008 c^{5} d^{7} x^{7} - 11880 c^{4} d^{8} x^{8} - 5280 c^{3} d^{9} x^{9} - 1584 c^{2} d^{10} x^{10} - 288 c d^{11} x^{11} - 24 d^{12} x^{12}\right )}{3 b^{5} c^{12} d \log {\relax (F )}^{5} + 36 b^{5} c^{11} d^{2} x \log {\relax (F )}^{5} + 198 b^{5} c^{10} d^{3} x^{2} \log {\relax (F )}^{5} + 660 b^{5} c^{9} d^{4} x^{3} \log {\relax (F )}^{5} + 1485 b^{5} c^{8} d^{5} x^{4} \log {\relax (F )}^{5} + 2376 b^{5} c^{7} d^{6} x^{5} \log {\relax (F )}^{5} + 2772 b^{5} c^{6} d^{7} x^{6} \log {\relax (F )}^{5} + 2376 b^{5} c^{5} d^{8} x^{7} \log {\relax (F )}^{5} + 1485 b^{5} c^{4} d^{9} x^{8} \log {\relax (F )}^{5} + 660 b^{5} c^{3} d^{10} x^{9} \log {\relax (F )}^{5} + 198 b^{5} c^{2} d^{11} x^{10} \log {\relax (F )}^{5} + 36 b^{5} c d^{12} x^{11} \log {\relax (F )}^{5} + 3 b^{5} d^{13} x^{12} \log {\relax (F )}^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(F**(a+b/(d*x+c)**3)/(d*x+c)**16,x)

[Out]

F**(a + b/(c + d*x)**3)*(-b**4*log(F)**4 + 4*b**3*c**3*log(F)**3 + 12*b**3*c**2*d*x*log(F)**3 + 12*b**3*c*d**2
*x**2*log(F)**3 + 4*b**3*d**3*x**3*log(F)**3 - 12*b**2*c**6*log(F)**2 - 72*b**2*c**5*d*x*log(F)**2 - 180*b**2*
c**4*d**2*x**2*log(F)**2 - 240*b**2*c**3*d**3*x**3*log(F)**2 - 180*b**2*c**2*d**4*x**4*log(F)**2 - 72*b**2*c*d
**5*x**5*log(F)**2 - 12*b**2*d**6*x**6*log(F)**2 + 24*b*c**9*log(F) + 216*b*c**8*d*x*log(F) + 864*b*c**7*d**2*
x**2*log(F) + 2016*b*c**6*d**3*x**3*log(F) + 3024*b*c**5*d**4*x**4*log(F) + 3024*b*c**4*d**5*x**5*log(F) + 201
6*b*c**3*d**6*x**6*log(F) + 864*b*c**2*d**7*x**7*log(F) + 216*b*c*d**8*x**8*log(F) + 24*b*d**9*x**9*log(F) - 2
4*c**12 - 288*c**11*d*x - 1584*c**10*d**2*x**2 - 5280*c**9*d**3*x**3 - 11880*c**8*d**4*x**4 - 19008*c**7*d**5*
x**5 - 22176*c**6*d**6*x**6 - 19008*c**5*d**7*x**7 - 11880*c**4*d**8*x**8 - 5280*c**3*d**9*x**9 - 1584*c**2*d*
*10*x**10 - 288*c*d**11*x**11 - 24*d**12*x**12)/(3*b**5*c**12*d*log(F)**5 + 36*b**5*c**11*d**2*x*log(F)**5 + 1
98*b**5*c**10*d**3*x**2*log(F)**5 + 660*b**5*c**9*d**4*x**3*log(F)**5 + 1485*b**5*c**8*d**5*x**4*log(F)**5 + 2
376*b**5*c**7*d**6*x**5*log(F)**5 + 2772*b**5*c**6*d**7*x**6*log(F)**5 + 2376*b**5*c**5*d**8*x**7*log(F)**5 +
1485*b**5*c**4*d**9*x**8*log(F)**5 + 660*b**5*c**3*d**10*x**9*log(F)**5 + 198*b**5*c**2*d**11*x**10*log(F)**5
+ 36*b**5*c*d**12*x**11*log(F)**5 + 3*b**5*d**13*x**12*log(F)**5)

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