\(\int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx\) [292]

Optimal result

Integrand size = 21, antiderivative size = 31 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\frac {b^5 F^a \Gamma \left (-5,-b (c+d x)^3 \log (F)\right ) \log ^5(F)}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (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} \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\frac {b^5 F^a \log ^5(F) \Gamma \left (-5,-b (c+d x)^3 \log (F)\right )}{3 d} \]

[In]

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

[Out]

(b^5*F^a*Gamma[-5, -(b*(c + d*x)^3*Log[F])]*Log[F]^5)/(3*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,-b (c+d x)^3 \log (F)\right ) \log ^5(F)}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\frac {b^5 F^a \Gamma \left (-5,-b (c+d x)^3 \log (F)\right ) \log ^5(F)}{3 d} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {F^{a +b \left (d x +c \right )^{3}}}{\left (d x +c \right )^{16}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 883 vs. \(2 (29) = 58\).

Time = 0.26 (sec) , antiderivative size = 883, normalized size of antiderivative = 28.48 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\frac {{\left (b^{5} d^{15} x^{15} + 15 \, b^{5} c d^{14} x^{14} + 105 \, b^{5} c^{2} d^{13} x^{13} + 455 \, b^{5} c^{3} d^{12} x^{12} + 1365 \, b^{5} c^{4} d^{11} x^{11} + 3003 \, b^{5} c^{5} d^{10} x^{10} + 5005 \, b^{5} c^{6} d^{9} x^{9} + 6435 \, b^{5} c^{7} d^{8} x^{8} + 6435 \, b^{5} c^{8} d^{7} x^{7} + 5005 \, b^{5} c^{9} d^{6} x^{6} + 3003 \, b^{5} c^{10} d^{5} x^{5} + 1365 \, b^{5} c^{11} d^{4} x^{4} + 455 \, b^{5} c^{12} d^{3} x^{3} + 105 \, b^{5} c^{13} d^{2} x^{2} + 15 \, b^{5} c^{14} d x + b^{5} c^{15}\right )} F^{a} {\rm Ei}\left ({\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right ) \log \left (F\right )^{5} - {\left ({\left (b^{4} d^{12} x^{12} + 12 \, b^{4} c d^{11} x^{11} + 66 \, b^{4} c^{2} d^{10} x^{10} + 220 \, b^{4} c^{3} d^{9} x^{9} + 495 \, b^{4} c^{4} d^{8} x^{8} + 792 \, b^{4} c^{5} d^{7} x^{7} + 924 \, b^{4} c^{6} d^{6} x^{6} + 792 \, b^{4} c^{7} d^{5} x^{5} + 495 \, b^{4} c^{8} d^{4} x^{4} + 220 \, b^{4} c^{9} d^{3} x^{3} + 66 \, b^{4} c^{10} d^{2} x^{2} + 12 \, b^{4} c^{11} d x + b^{4} c^{12}\right )} \log \left (F\right )^{4} + {\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} \log \left (F\right )^{3} + 2 \, {\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} + 6 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) + 24\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{360 \, {\left (d^{16} x^{15} + 15 \, c d^{15} x^{14} + 105 \, c^{2} d^{14} x^{13} + 455 \, c^{3} d^{13} x^{12} + 1365 \, c^{4} d^{12} x^{11} + 3003 \, c^{5} d^{11} x^{10} + 5005 \, c^{6} d^{10} x^{9} + 6435 \, c^{7} d^{9} x^{8} + 6435 \, c^{8} d^{8} x^{7} + 5005 \, c^{9} d^{7} x^{6} + 3003 \, c^{10} d^{6} x^{5} + 1365 \, c^{11} d^{5} x^{4} + 455 \, c^{12} d^{4} x^{3} + 105 \, c^{13} d^{3} x^{2} + 15 \, c^{14} d^{2} x + c^{15} d\right )}} \]

[In]

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

[Out]

1/360*((b^5*d^15*x^15 + 15*b^5*c*d^14*x^14 + 105*b^5*c^2*d^13*x^13 + 455*b^5*c^3*d^12*x^12 + 1365*b^5*c^4*d^11
*x^11 + 3003*b^5*c^5*d^10*x^10 + 5005*b^5*c^6*d^9*x^9 + 6435*b^5*c^7*d^8*x^8 + 6435*b^5*c^8*d^7*x^7 + 5005*b^5
*c^9*d^6*x^6 + 3003*b^5*c^10*d^5*x^5 + 1365*b^5*c^11*d^4*x^4 + 455*b^5*c^12*d^3*x^3 + 105*b^5*c^13*d^2*x^2 + 1
5*b^5*c^14*d*x + b^5*c^15)*F^a*Ei((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(F))*log(F)^5 - ((b^4*d
^12*x^12 + 12*b^4*c*d^11*x^11 + 66*b^4*c^2*d^10*x^10 + 220*b^4*c^3*d^9*x^9 + 495*b^4*c^4*d^8*x^8 + 792*b^4*c^5
*d^7*x^7 + 924*b^4*c^6*d^6*x^6 + 792*b^4*c^7*d^5*x^5 + 495*b^4*c^8*d^4*x^4 + 220*b^4*c^9*d^3*x^3 + 66*b^4*c^10
*d^2*x^2 + 12*b^4*c^11*d*x + b^4*c^12)*log(F)^4 + (b^3*d^9*x^9 + 9*b^3*c*d^8*x^8 + 36*b^3*c^2*d^7*x^7 + 84*b^3
*c^3*d^6*x^6 + 126*b^3*c^4*d^5*x^5 + 126*b^3*c^5*d^4*x^4 + 84*b^3*c^6*d^3*x^3 + 36*b^3*c^7*d^2*x^2 + 9*b^3*c^8
*d*x + b^3*c^9)*log(F)^3 + 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 + 6*(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3)*log(F)
+ 24)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a))/(d^16*x^15 + 15*c*d^15*x^14 + 105*c^2*d^14*x^13
+ 455*c^3*d^13*x^12 + 1365*c^4*d^12*x^11 + 3003*c^5*d^11*x^10 + 5005*c^6*d^10*x^9 + 6435*c^7*d^9*x^8 + 6435*c
^8*d^8*x^7 + 5005*c^9*d^7*x^6 + 3003*c^10*d^6*x^5 + 1365*c^11*d^5*x^4 + 455*c^12*d^4*x^3 + 105*c^13*d^3*x^2 +
15*c^14*d^2*x + c^15*d)

Sympy [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\int \frac {F^{a + b \left (c + d x\right )^{3}}}{\left (c + d x\right )^{16}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{16}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{16}} \,d x } \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.99 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.39 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=-\frac {F^a\,b^5\,{\ln \left (F\right )}^5\,\mathrm {expint}\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}{360\,d}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^3}\,b^5\,{\ln \left (F\right )}^5\,\left (\frac {1}{120\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}+\frac {1}{120\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^6}+\frac {1}{60\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^9}+\frac {1}{20\,b^4\,{\ln \left (F\right )}^4\,{\left (c+d\,x\right )}^{12}}+\frac {1}{5\,b^5\,{\ln \left (F\right )}^5\,{\left (c+d\,x\right )}^{15}}\right )}{3\,d} \]

[In]

int(F^(a + b*(c + d*x)^3)/(c + d*x)^16,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*(1/(120*b*log
(F)*(c + d*x)^3) + 1/(120*b^2*log(F)^2*(c + d*x)^6) + 1/(60*b^3*log(F)^3*(c + d*x)^9) + 1/(20*b^4*log(F)^4*(c
+ d*x)^12) + 1/(5*b^5*log(F)^5*(c + d*x)^15)))/(3*d)