\(\int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx\) [224]

Optimal result

Integrand size = 21, antiderivative size = 121 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx=-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}-\frac {b F^{a+b (c+d x)^3} \log (F)}{18 d (c+d x)^6}-\frac {b^2 F^{a+b (c+d x)^3} \log ^2(F)}{18 d (c+d x)^3}+\frac {b^3 F^a \operatorname {ExpIntegralEi}\left (b (c+d x)^3 \log (F)\right ) \log ^3(F)}{18 d} \] Output:

-1/9*F^(a+b*(d*x+c)^3)/d/(d*x+c)^9-1/18*b*F^(a+b*(d*x+c)^3)*ln(F)/d/(d*x+c 
)^6-1/18*b^2*F^(a+b*(d*x+c)^3)*ln(F)^2/d/(d*x+c)^3+1/18*b^3*F^a*Ei(b*(d*x+ 
c)^3*ln(F))*ln(F)^3/d
 

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.66 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx=\frac {F^a \left (b^3 \operatorname {ExpIntegralEi}\left (b (c+d x)^3 \log (F)\right ) \log ^3(F)+\frac {F^{b (c+d x)^3} \left (-2-b (c+d x)^3 \log (F)-b^2 (c+d x)^6 \log ^2(F)\right )}{(c+d x)^9}\right )}{18 d} \] Input:

Integrate[F^(a + b*(c + d*x)^3)/(c + d*x)^10,x]
 

Output:

(F^a*(b^3*ExpIntegralEi[b*(c + d*x)^3*Log[F]]*Log[F]^3 + (F^(b*(c + d*x)^3 
)*(-2 - b*(c + d*x)^3*Log[F] - b^2*(c + d*x)^6*Log[F]^2))/(c + d*x)^9))/(1 
8*d)
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2643, 2643, 2643, 2639}

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.

\(\displaystyle \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx\)

\(\Big \downarrow \) 2643

\(\displaystyle \frac {1}{3} b \log (F) \int \frac {F^{b (c+d x)^3+a}}{(c+d x)^7}dx-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}\)

\(\Big \downarrow \) 2643

\(\displaystyle \frac {1}{3} b \log (F) \left (\frac {1}{2} b \log (F) \int \frac {F^{b (c+d x)^3+a}}{(c+d x)^4}dx-\frac {F^{a+b (c+d x)^3}}{6 d (c+d x)^6}\right )-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}\)

\(\Big \downarrow \) 2643

\(\displaystyle \frac {1}{3} b \log (F) \left (\frac {1}{2} b \log (F) \left (b \log (F) \int \frac {F^{b (c+d x)^3+a}}{c+d x}dx-\frac {F^{a+b (c+d x)^3}}{3 d (c+d x)^3}\right )-\frac {F^{a+b (c+d x)^3}}{6 d (c+d x)^6}\right )-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}\)

\(\Big \downarrow \) 2639

\(\displaystyle \frac {1}{3} b \log (F) \left (\frac {1}{2} b \log (F) \left (\frac {b F^a \log (F) \operatorname {ExpIntegralEi}\left (b (c+d x)^3 \log (F)\right )}{3 d}-\frac {F^{a+b (c+d x)^3}}{3 d (c+d x)^3}\right )-\frac {F^{a+b (c+d x)^3}}{6 d (c+d x)^6}\right )-\frac {F^{a+b (c+d x)^3}}{9 d (c+d x)^9}\)

Input:

Int[F^(a + b*(c + d*x)^3)/(c + d*x)^10,x]
 

Output:

-1/9*F^(a + b*(c + d*x)^3)/(d*(c + d*x)^9) + (b*Log[F]*(-1/6*F^(a + b*(c + 
 d*x)^3)/(d*(c + d*x)^6) + (b*Log[F]*(-1/3*F^(a + b*(c + d*x)^3)/(d*(c + d 
*x)^3) + (b*F^a*ExpIntegralEi[b*(c + d*x)^3*Log[F]]*Log[F])/(3*d)))/2))/3
 

Defintions of rubi rules used

rule 2639

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_ 
Symbol] :> Simp[F^a*(ExpIntegralEi[b*(c + d*x)^n*Log[F]]/(f*n)), x] /; Free 
Q[{F, a, b, c, d, e, f, n}, x] && EqQ[d*e - c*f, 0]
 

rule 2643

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((c_.) + (d_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(c + d*x)^(m + 1)*(F^(a + b*(c + d*x)^n)/(d*(m + 1))) 
, x] - Simp[b*n*(Log[F]/(m + 1))   Int[(c + d*x)^(m + n)*F^(a + b*(c + d*x) 
^n), x], x] /; FreeQ[{F, a, b, c, d}, x] && IntegerQ[2*((m + 1)/n)] && LtQ[ 
-4, (m + 1)/n, 5] && IntegerQ[n] && ((GtQ[n, 0] && LtQ[m, -1]) || (GtQ[-n, 
0] && LeQ[-n, m + 1]))
 

Maple [F]

\[\int \frac {F^{a +b \left (d x +c \right )^{3}}}{\left (d x +c \right )^{10}}d x\]

Input:

int(F^(a+b*(d*x+c)^3)/(d*x+c)^10,x)
 

Output:

int(F^(a+b*(d*x+c)^3)/(d*x+c)^10,x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 431 vs. \(2 (113) = 226\).

Time = 0.08 (sec) , antiderivative size = 431, normalized size of antiderivative = 3.56 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx=\frac {{\left (b^{3} d^{9} x^{9} + 9 \, b^{3} c d^{8} x^{8} + 36 \, b^{3} c^{2} d^{7} x^{7} + 84 \, b^{3} c^{3} d^{6} x^{6} + 126 \, b^{3} c^{4} d^{5} x^{5} + 126 \, b^{3} c^{5} d^{4} x^{4} + 84 \, b^{3} c^{6} d^{3} x^{3} + 36 \, b^{3} c^{7} d^{2} x^{2} + 9 \, b^{3} c^{8} d x + b^{3} c^{9}\right )} F^{a} {\rm Ei}\left ({\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right )\right ) \log \left (F\right )^{3} - {\left ({\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + 20 \, b^{2} c^{3} d^{3} x^{3} + 15 \, b^{2} c^{4} d^{2} x^{2} + 6 \, b^{2} c^{5} d x + b^{2} c^{6}\right )} \log \left (F\right )^{2} + {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (F\right ) + 2\right )} F^{b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a}}{18 \, {\left (d^{10} x^{9} + 9 \, c d^{9} x^{8} + 36 \, c^{2} d^{8} x^{7} + 84 \, c^{3} d^{7} x^{6} + 126 \, c^{4} d^{6} x^{5} + 126 \, c^{5} d^{5} x^{4} + 84 \, c^{6} d^{4} x^{3} + 36 \, c^{7} d^{3} x^{2} + 9 \, c^{8} d^{2} x + c^{9} d\right )}} \] Input:

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^10,x, algorithm="fricas")
 

Output:

1/18*((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)*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)^3 - ((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 + (b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + 
b*c^3)*log(F) + 2)*F^(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a) 
)/(d^10*x^9 + 9*c*d^9*x^8 + 36*c^2*d^8*x^7 + 84*c^3*d^7*x^6 + 126*c^4*d^6* 
x^5 + 126*c^5*d^5*x^4 + 84*c^6*d^4*x^3 + 36*c^7*d^3*x^2 + 9*c^8*d^2*x + c^ 
9*d)
 

Sympy [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx=\int \frac {F^{a + b \left (c + d x\right )^{3}}}{\left (c + d x\right )^{10}}\, dx \] Input:

integrate(F**(a+b*(d*x+c)**3)/(d*x+c)**10,x)
 

Output:

Integral(F**(a + b*(c + d*x)**3)/(c + d*x)**10, x)
 

Maxima [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{10}} \,d x } \] Input:

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^10,x, algorithm="maxima")
 

Output:

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^10, x)
 

Giac [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx=\int { \frac {F^{{\left (d x + c\right )}^{3} b + a}}{{\left (d x + c\right )}^{10}} \,d x } \] Input:

integrate(F^(a+b*(d*x+c)^3)/(d*x+c)^10,x, algorithm="giac")
 

Output:

integrate(F^((d*x + c)^3*b + a)/(d*x + c)^10, x)
 

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.86 \[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx=-\frac {F^a\,b^3\,{\ln \left (F\right )}^3\,\mathrm {expint}\left (-b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3\right )}{18\,d}-\frac {F^a\,F^{b\,{\left (c+d\,x\right )}^3}\,b^3\,{\ln \left (F\right )}^3\,\left (\frac {1}{6\,b\,\ln \left (F\right )\,{\left (c+d\,x\right )}^3}+\frac {1}{6\,b^2\,{\ln \left (F\right )}^2\,{\left (c+d\,x\right )}^6}+\frac {1}{3\,b^3\,{\ln \left (F\right )}^3\,{\left (c+d\,x\right )}^9}\right )}{3\,d} \] Input:

int(F^(a + b*(c + d*x)^3)/(c + d*x)^10,x)
 

Output:

- (F^a*b^3*log(F)^3*expint(-b*log(F)*(c + d*x)^3))/(18*d) - (F^a*F^(b*(c + 
 d*x)^3)*b^3*log(F)^3*(1/(6*b*log(F)*(c + d*x)^3) + 1/(6*b^2*log(F)^2*(c + 
 d*x)^6) + 1/(3*b^3*log(F)^3*(c + d*x)^9)))/(3*d)
 

Reduce [F]

\[ \int \frac {F^{a+b (c+d x)^3}}{(c+d x)^{10}} \, dx=\text {too large to display} \] Input:

int(F^(a+b*(d*x+c)^3)/(d*x+c)^10,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**13 + 
 10*log(f)*b*c**12*d*x + 45*log(f)*b*c**11*d**2*x**2 + 120*log(f)*b*c**10* 
d**3*x**3 + 210*log(f)*b*c**9*d**4*x**4 + 252*log(f)*b*c**8*d**5*x**5 + 21 
0*log(f)*b*c**7*d**6*x**6 + 120*log(f)*b*c**6*d**7*x**7 + 45*log(f)*b*c**5 
*d**8*x**8 + 10*log(f)*b*c**4*d**9*x**9 + log(f)*b*c**3*d**10*x**10 - 3*c* 
*10 - 30*c**9*d*x - 135*c**8*d**2*x**2 - 360*c**7*d**3*x**3 - 630*c**6*d** 
4*x**4 - 756*c**5*d**5*x**5 - 630*c**4*d**6*x**6 - 360*c**3*d**7*x**7 - 13 
5*c**2*d**8*x**8 - 30*c*d**9*x**9 - 3*d**10*x**10),x)*log(f)**2*b**2*c**15 
*d + 81*int(f**(3*b*c**2*d*x + 3*b*c*d**2*x**2 + b*d**3*x**3)/(log(f)*b*c* 
*13 + 10*log(f)*b*c**12*d*x + 45*log(f)*b*c**11*d**2*x**2 + 120*log(f)*b*c 
**10*d**3*x**3 + 210*log(f)*b*c**9*d**4*x**4 + 252*log(f)*b*c**8*d**5*x**5 
 + 210*log(f)*b*c**7*d**6*x**6 + 120*log(f)*b*c**6*d**7*x**7 + 45*log(f)*b 
*c**5*d**8*x**8 + 10*log(f)*b*c**4*d**9*x**9 + log(f)*b*c**3*d**10*x**10 - 
 3*c**10 - 30*c**9*d*x - 135*c**8*d**2*x**2 - 360*c**7*d**3*x**3 - 630*c** 
6*d**4*x**4 - 756*c**5*d**5*x**5 - 630*c**4*d**6*x**6 - 360*c**3*d**7*x**7 
 - 135*c**2*d**8*x**8 - 30*c*d**9*x**9 - 3*d**10*x**10),x)*log(f)**2*b**2* 
c**14*d**2*x + 324*int(f**(3*b*c**2*d*x + 3*b*c*d**2*x**2 + b*d**3*x**3)/( 
log(f)*b*c**13 + 10*log(f)*b*c**12*d*x + 45*log(f)*b*c**11*d**2*x**2 + 120 
*log(f)*b*c**10*d**3*x**3 + 210*log(f)*b*c**9*d**4*x**4 + 252*log(f)*b*...