Integrand size = 21, antiderivative size = 31 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\frac {b^5 F^a \Gamma \left (-5,-b (c+d x)^3 \log (F)\right ) \log ^5(F)}{3 d} \] Output:
-1/3*F^a/(d*x+c)^15*Ei(6,-b*(d*x+c)^3*ln(F))/d
Time = 0.34 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\frac {b^5 F^a \Gamma \left (-5,-b (c+d x)^3 \log (F)\right ) \log ^5(F)}{3 d} \] Input:
Integrate[F^(a + b*(c + d*x)^3)/(c + d*x)^16,x]
Output:
(b^5*F^a*Gamma[-5, -(b*(c + d*x)^3*Log[F])]*Log[F]^5)/(3*d)
Time = 0.34 (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 = {2648}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx\) |
\(\Big \downarrow \) 2648 |
\(\displaystyle \frac {b^5 F^a \log ^5(F) \Gamma \left (-5,-b (c+d x)^3 \log (F)\right )}{3 d}\) |
Input:
Int[F^(a + b*(c + d*x)^3)/(c + d*x)^16,x]
Output:
(b^5*F^a*Gamma[-5, -(b*(c + d*x)^3*Log[F])]*Log[F]^5)/(3*d)
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] /; FreeQ[{F , a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]
\[\int \frac {F^{a +b \left (d x +c \right )^{3}}}{\left (d x +c \right )^{16}}d x\]
Input:
int(F^(a+b*(d*x+c)^3)/(d*x+c)^16,x)
Output:
int(F^(a+b*(d*x+c)^3)/(d*x+c)^16,x)
Leaf count of result is larger than twice the leaf count of optimal. 883 vs. \(2 (29) = 58\).
Time = 0.10 (sec) , antiderivative size = 883, normalized size of antiderivative = 28.48 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx =\text {Too large to display} \] Input:
integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^16,x, algorithm="fricas")
Output:
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 + 15*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 + 4 95*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^1 5*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)
\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\int \frac {F^{a + b \left (c + d x\right )^{3}}}{\left (c + d x\right )^{16}}\, dx \] Input:
integrate(F**(a+b*(d*x+c)**3)/(d*x+c)**16,x)
Output:
Integral(F**(a + b*(c + d*x)**3)/(c + d*x)**16, x)
\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{16}} \,d x } \] Input:
integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^16,x, algorithm="maxima")
Output:
integrate(F^((d*x + c)^3*b + a)/(d*x + c)^16, x)
\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{16}} \,d x } \] Input:
integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^16,x, algorithm="giac")
Output:
integrate(F^((d*x + c)^3*b + a)/(d*x + c)^16, x)
Time = 0.52 (sec) , antiderivative size = 136, normalized size of antiderivative = 4.39 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=-\frac {F^a\,b^5\,{\ln \left (F\right )}^5\,\mathrm {expint}\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}{360\,d}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^3}\,b^5\,{\ln \left (F\right )}^5\,\left (\frac {1}{120\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}+\frac {1}{120\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^6}+\frac {1}{60\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^9}+\frac {1}{20\,b^4\,{\ln \left (F\right )}^4\,{\left (c+d\,x\right )}^{12}}+\frac {1}{5\,b^5\,{\ln \left (F\right )}^5\,{\left (c+d\,x\right )}^{15}}\right )}{3\,d} \] Input:
int(F^(a + b*(c + d*x)^3)/(c + d*x)^16,x)
Output:
- (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)
\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{16}} \, dx=\text {too large to display} \] Input:
int(F^(a+b*(d*x+c)^3)/(d*x+c)^16,x)
Output:
(f**(a + b*c**3)*( - 2*f**(3*b*c**2*d*x + 3*b*c*d**2*x**2 + b*d**3*x**3) + 9*int(f**(3*b*c**2*d*x + 3*b*c*d**2*x**2 + b*d**3*x**3)/(log(f)*b*c**19 + 16*log(f)*b*c**18*d*x + 120*log(f)*b*c**17*d**2*x**2 + 560*log(f)*b*c**16 *d**3*x**3 + 1820*log(f)*b*c**15*d**4*x**4 + 4368*log(f)*b*c**14*d**5*x**5 + 8008*log(f)*b*c**13*d**6*x**6 + 11440*log(f)*b*c**12*d**7*x**7 + 12870* log(f)*b*c**11*d**8*x**8 + 11440*log(f)*b*c**10*d**9*x**9 + 8008*log(f)*b* c**9*d**10*x**10 + 4368*log(f)*b*c**8*d**11*x**11 + 1820*log(f)*b*c**7*d** 12*x**12 + 560*log(f)*b*c**6*d**13*x**13 + 120*log(f)*b*c**5*d**14*x**14 + 16*log(f)*b*c**4*d**15*x**15 + log(f)*b*c**3*d**16*x**16 - 5*c**16 - 80*c **15*d*x - 600*c**14*d**2*x**2 - 2800*c**13*d**3*x**3 - 9100*c**12*d**4*x* *4 - 21840*c**11*d**5*x**5 - 40040*c**10*d**6*x**6 - 57200*c**9*d**7*x**7 - 64350*c**8*d**8*x**8 - 57200*c**7*d**9*x**9 - 40040*c**6*d**10*x**10 - 2 1840*c**5*d**11*x**11 - 9100*c**4*d**12*x**12 - 2800*c**3*d**13*x**13 - 60 0*c**2*d**14*x**14 - 80*c*d**15*x**15 - 5*d**16*x**16),x)*log(f)**2*b**2*c **21*d + 135*int(f**(3*b*c**2*d*x + 3*b*c*d**2*x**2 + b*d**3*x**3)/(log(f) *b*c**19 + 16*log(f)*b*c**18*d*x + 120*log(f)*b*c**17*d**2*x**2 + 560*log( f)*b*c**16*d**3*x**3 + 1820*log(f)*b*c**15*d**4*x**4 + 4368*log(f)*b*c**14 *d**5*x**5 + 8008*log(f)*b*c**13*d**6*x**6 + 11440*log(f)*b*c**12*d**7*x** 7 + 12870*log(f)*b*c**11*d**8*x**8 + 11440*log(f)*b*c**10*d**9*x**9 + 8008 *log(f)*b*c**9*d**10*x**10 + 4368*log(f)*b*c**8*d**11*x**11 + 1820*log(...