\(\int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^{14} \, dx\) [341]

Optimal result

Integrand size = 21, antiderivative size = 31 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^{14} \, dx=-\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{(c+d x)^3}\right ) \log ^5(F)}{3 d} \]

[Out]

1/3*F^a*(d*x+c)^15*Ei(6,-b*ln(F)/(d*x+c)^3)/d

Rubi [A] (verified)

Time = 0.03 (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 = {2250} \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^{14} \, dx=-\frac {b^5 F^a \log ^5(F) \Gamma \left (-5,-\frac {b \log (F)}{(c+d x)^3}\right )}{3 d} \]

[In]

Int[F^(a + b/(c + d*x)^3)*(c + d*x)^14,x]

[Out]

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

Rule 2250

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] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{(c+d x)^3}\right ) \log ^5(F)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^{14} \, dx=-\frac {b^5 F^a \Gamma \left (-5,-\frac {b \log (F)}{(c+d x)^3}\right ) \log ^5(F)}{3 d} \]

[In]

Integrate[F^(a + b/(c + d*x)^3)*(c + d*x)^14,x]

[Out]

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

Maple [F]

\[\int F^{a +\frac {b}{\left (d x +c \right )^{3}}} \left (d x +c \right )^{14}d x\]

[In]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^14,x)

[Out]

int(F^(a+b/(d*x+c)^3)*(d*x+c)^14,x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 686 vs. \(2 (29) = 58\).

Time = 0.34 (sec) , antiderivative size = 686, normalized size of antiderivative = 22.13 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^{14} \, dx=-\frac {F^{a} b^{5} {\rm Ei}\left (\frac {b \log \left (F\right )}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}\right ) \log \left (F\right )^{5} - {\left (24 \, d^{15} x^{15} + 360 \, c d^{14} x^{14} + 2520 \, c^{2} d^{13} x^{13} + 10920 \, c^{3} d^{12} x^{12} + 32760 \, c^{4} d^{11} x^{11} + 72072 \, c^{5} d^{10} x^{10} + 120120 \, c^{6} d^{9} x^{9} + 154440 \, c^{7} d^{8} x^{8} + 154440 \, c^{8} d^{7} x^{7} + 120120 \, c^{9} d^{6} x^{6} + 72072 \, c^{10} d^{5} x^{5} + 32760 \, c^{11} d^{4} x^{4} + 10920 \, c^{12} d^{3} x^{3} + 2520 \, c^{13} d^{2} x^{2} + 360 \, c^{14} d x + 24 \, c^{15} + {\left (b^{4} d^{3} x^{3} + 3 \, b^{4} c d^{2} x^{2} + 3 \, b^{4} c^{2} d x + b^{4} c^{3}\right )} \log \left (F\right )^{4} + {\left (b^{3} d^{6} x^{6} + 6 \, b^{3} c d^{5} x^{5} + 15 \, b^{3} c^{2} d^{4} x^{4} + 20 \, b^{3} c^{3} d^{3} x^{3} + 15 \, b^{3} c^{4} d^{2} x^{2} + 6 \, b^{3} c^{5} d x + b^{3} c^{6}\right )} \log \left (F\right )^{3} + 2 \, {\left (b^{2} d^{9} x^{9} + 9 \, b^{2} c d^{8} x^{8} + 36 \, b^{2} c^{2} d^{7} x^{7} + 84 \, b^{2} c^{3} d^{6} x^{6} + 126 \, b^{2} c^{4} d^{5} x^{5} + 126 \, b^{2} c^{5} d^{4} x^{4} + 84 \, b^{2} c^{6} d^{3} x^{3} + 36 \, b^{2} c^{7} d^{2} x^{2} + 9 \, b^{2} c^{8} d x + b^{2} c^{9}\right )} \log \left (F\right )^{2} + 6 \, {\left (b d^{12} x^{12} + 12 \, b c d^{11} x^{11} + 66 \, b c^{2} d^{10} x^{10} + 220 \, b c^{3} d^{9} x^{9} + 495 \, b c^{4} d^{8} x^{8} + 792 \, b c^{5} d^{7} x^{7} + 924 \, b c^{6} d^{6} x^{6} + 792 \, b c^{7} d^{5} x^{5} + 495 \, b c^{8} d^{4} x^{4} + 220 \, b c^{9} d^{3} x^{3} + 66 \, b c^{10} d^{2} x^{2} + 12 \, b c^{11} d x + b c^{12}\right )} \log \left (F\right )\right )} F^{\frac {a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x + a c^{3} + b}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}}}}{360 \, d} \]

[In]

integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^14,x, algorithm="fricas")

[Out]

-1/360*(F^a*b^5*Ei(b*log(F)/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))*log(F)^5 - (24*d^15*x^15 + 360*c*d^14*x
^14 + 2520*c^2*d^13*x^13 + 10920*c^3*d^12*x^12 + 32760*c^4*d^11*x^11 + 72072*c^5*d^10*x^10 + 120120*c^6*d^9*x^
9 + 154440*c^7*d^8*x^8 + 154440*c^8*d^7*x^7 + 120120*c^9*d^6*x^6 + 72072*c^10*d^5*x^5 + 32760*c^11*d^4*x^4 + 1
0920*c^12*d^3*x^3 + 2520*c^13*d^2*x^2 + 360*c^14*d*x + 24*c^15 + (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 + (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^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 + 6*(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)))/d

Sympy [F]

\[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^{14} \, dx=\int F^{a + \frac {b}{\left (c + d x\right )^{3}}} \left (c + d x\right )^{14}\, dx \]

[In]

integrate(F**(a+b/(d*x+c)**3)*(d*x+c)**14,x)

[Out]

Integral(F**(a + b/(c + d*x)**3)*(c + d*x)**14, x)

Maxima [F]

\[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^{14} \, dx=\int { {\left (d x + c\right )}^{14} F^{a + \frac {b}{{\left (d x + c\right )}^{3}}} \,d x } \]

[In]

integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^14,x, algorithm="maxima")

[Out]

1/360*(24*F^a*d^14*x^15 + 360*F^a*c*d^13*x^14 + 2520*F^a*c^2*d^12*x^13 + 6*(1820*F^a*c^3*d^11 + F^a*b*d^11*log
(F))*x^12 + 72*(455*F^a*c^4*d^10 + F^a*b*c*d^10*log(F))*x^11 + 396*(182*F^a*c^5*d^9 + F^a*b*c^2*d^9*log(F))*x^
10 + 2*(60060*F^a*c^6*d^8 + 660*F^a*b*c^3*d^8*log(F) + F^a*b^2*d^8*log(F)^2)*x^9 + 18*(8580*F^a*c^7*d^7 + 165*
F^a*b*c^4*d^7*log(F) + F^a*b^2*c*d^7*log(F)^2)*x^8 + 72*(2145*F^a*c^8*d^6 + 66*F^a*b*c^5*d^6*log(F) + F^a*b^2*
c^2*d^6*log(F)^2)*x^7 + (120120*F^a*c^9*d^5 + 5544*F^a*b*c^6*d^5*log(F) + 168*F^a*b^2*c^3*d^5*log(F)^2 + F^a*b
^3*d^5*log(F)^3)*x^6 + 6*(12012*F^a*c^10*d^4 + 792*F^a*b*c^7*d^4*log(F) + 42*F^a*b^2*c^4*d^4*log(F)^2 + F^a*b^
3*c*d^4*log(F)^3)*x^5 + 3*(10920*F^a*c^11*d^3 + 990*F^a*b*c^8*d^3*log(F) + 84*F^a*b^2*c^5*d^3*log(F)^2 + 5*F^a
*b^3*c^2*d^3*log(F)^3)*x^4 + (10920*F^a*c^12*d^2 + 1320*F^a*b*c^9*d^2*log(F) + 168*F^a*b^2*c^6*d^2*log(F)^2 +
20*F^a*b^3*c^3*d^2*log(F)^3 + F^a*b^4*d^2*log(F)^4)*x^3 + 3*(840*F^a*c^13*d + 132*F^a*b*c^10*d*log(F) + 24*F^a
*b^2*c^7*d*log(F)^2 + 5*F^a*b^3*c^4*d*log(F)^3 + F^a*b^4*c*d*log(F)^4)*x^2 + 3*(120*F^a*c^14 + 24*F^a*b*c^11*l
og(F) + 6*F^a*b^2*c^8*log(F)^2 + 2*F^a*b^3*c^5*log(F)^3 + F^a*b^4*c^2*log(F)^4)*x)*F^(b/(d^3*x^3 + 3*c*d^2*x^2
+ 3*c^2*d*x + c^3)) + integrate(-1/120*(24*F^a*b*c^15*log(F) + 6*F^a*b^2*c^12*log(F)^2 - F^a*b^5*d^3*x^3*log(
F)^5 + 2*F^a*b^3*c^9*log(F)^3 - 3*F^a*b^5*c*d^2*x^2*log(F)^5 + F^a*b^4*c^6*log(F)^4 - 3*F^a*b^5*c^2*d*x*log(F)
^5)*F^(b/(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3))/(d^4*x^4 + 4*c*d^3*x^3 + 6*c^2*d^2*x^2 + 4*c^3*d*x + c^4),
x)

Giac [F]

\[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^{14} \, dx=\int { {\left (d x + c\right )}^{14} F^{a + \frac {b}{{\left (d x + c\right )}^{3}}} \,d x } \]

[In]

integrate(F^(a+b/(d*x+c)^3)*(d*x+c)^14,x, algorithm="giac")

[Out]

integrate((d*x + c)^14*F^(a + b/(d*x + c)^3), x)

Mupad [B] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.39 \[ \int F^{a+\frac {b}{(c+d x)^3}} (c+d x)^{14} \, dx=\frac {F^a\,b^5\,{\ln \left (F\right )}^5\,\mathrm {expint}\left (-\frac {b\,\ln \left (F\right )}{{\left (c+d\,x\right )}^3}\right )}{360\,d}+\frac {F^a\,F^{\frac {b}{{\left (c+d\,x\right )}^3}}\,b^5\,{\ln \left (F\right )}^5\,\left (\frac {{\left (c+d\,x\right )}^3}{120\,b\,\ln \left (F\right )}+\frac {{\left (c+d\,x\right )}^6}{120\,b^2\,{\ln \left (F\right )}^2}+\frac {{\left (c+d\,x\right )}^9}{60\,b^3\,{\ln \left (F\right )}^3}+\frac {{\left (c+d\,x\right )}^{12}}{20\,b^4\,{\ln \left (F\right )}^4}+\frac {{\left (c+d\,x\right )}^{15}}{5\,b^5\,{\ln \left (F\right )}^5}\right )}{3\,d} \]

[In]

int(F^(a + b/(c + d*x)^3)*(c + d*x)^14,x)

[Out]

(F^a*b^5*log(F)^5*expint(-(b*log(F))/(c + d*x)^3))/(360*d) + (F^a*F^(b/(c + d*x)^3)*b^5*log(F)^5*((c + d*x)^3/
(120*b*log(F)) + (c + d*x)^6/(120*b^2*log(F)^2) + (c + d*x)^9/(60*b^3*log(F)^3) + (c + d*x)^12/(20*b^4*log(F)^
4) + (c + d*x)^15/(5*b^5*log(F)^5)))/(3*d)