Integrand size = 15, antiderivative size = 291 \[ \int f^{\frac {c}{a+b x}} x^4 \, dx=\frac {a^4 f^{\frac {c}{a+b x}} (a+b x)}{b^5}-\frac {2 a^3 f^{\frac {c}{a+b x}} (a+b x)^2}{b^5}+\frac {2 a^2 f^{\frac {c}{a+b x}} (a+b x)^3}{b^5}-\frac {2 a^3 c f^{\frac {c}{a+b x}} (a+b x) \log (f)}{b^5}+\frac {a^2 c f^{\frac {c}{a+b x}} (a+b x)^2 \log (f)}{b^5}-\frac {a^4 c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f)}{b^5}+\frac {a^2 c^2 f^{\frac {c}{a+b x}} (a+b x) \log ^2(f)}{b^5}+\frac {2 a^3 c^2 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log ^2(f)}{b^5}-\frac {a^2 c^3 \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log ^3(f)}{b^5}-\frac {4 a c^4 \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right ) \log ^4(f)}{b^5}-\frac {c^5 \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right ) \log ^5(f)}{b^5} \]
a^4*f^(c/(b*x+a))*(b*x+a)/b^5-2*a^3*f^(c/(b*x+a))*(b*x+a)^2/b^5+2*a^2*f^(c /(b*x+a))*(b*x+a)^3/b^5-2*a^3*c*f^(c/(b*x+a))*(b*x+a)*ln(f)/b^5+a^2*c*f^(c /(b*x+a))*(b*x+a)^2*ln(f)/b^5-a^4*c*Ei(c*ln(f)/(b*x+a))*ln(f)/b^5+a^2*c^2* f^(c/(b*x+a))*(b*x+a)*ln(f)^2/b^5+2*a^3*c^2*Ei(c*ln(f)/(b*x+a))*ln(f)^2/b^ 5-a^2*c^3*Ei(c*ln(f)/(b*x+a))*ln(f)^3/b^5-4*a*(b*x+a)^4*Ei(5,-c*ln(f)/(b*x +a))/b^5+(b*x+a)^5*Ei(6,-c*ln(f)/(b*x+a))/b^5
Time = 0.12 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.83 \[ \int f^{\frac {c}{a+b x}} x^4 \, dx=\frac {a f^{\frac {c}{a+b x}} \left (24 a^4-154 a^3 c \log (f)+102 a^2 c^2 \log ^2(f)-19 a c^3 \log ^3(f)+c^4 \log ^4(f)\right )}{120 b^5}+\frac {-c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right ) \log (f) \left (120 a^4-240 a^3 c \log (f)+120 a^2 c^2 \log ^2(f)-20 a c^3 \log ^3(f)+c^4 \log ^4(f)\right )+b f^{\frac {c}{a+b x}} x \left (24 b^4 x^4+2 c \left (-48 a^3+18 a^2 b x-8 a b^2 x^2+3 b^3 x^3\right ) \log (f)+2 c^2 \left (43 a^2-7 a b x+b^2 x^2\right ) \log ^2(f)+c^3 (-18 a+b x) \log ^3(f)+c^4 \log ^4(f)\right )}{120 b^5} \]
(a*f^(c/(a + b*x))*(24*a^4 - 154*a^3*c*Log[f] + 102*a^2*c^2*Log[f]^2 - 19* a*c^3*Log[f]^3 + c^4*Log[f]^4))/(120*b^5) + (-(c*ExpIntegralEi[(c*Log[f])/ (a + b*x)]*Log[f]*(120*a^4 - 240*a^3*c*Log[f] + 120*a^2*c^2*Log[f]^2 - 20* a*c^3*Log[f]^3 + c^4*Log[f]^4)) + b*f^(c/(a + b*x))*x*(24*b^4*x^4 + 2*c*(- 48*a^3 + 18*a^2*b*x - 8*a*b^2*x^2 + 3*b^3*x^3)*Log[f] + 2*c^2*(43*a^2 - 7* a*b*x + b^2*x^2)*Log[f]^2 + c^3*(-18*a + b*x)*Log[f]^3 + c^4*Log[f]^4))/(1 20*b^5)
Time = 0.54 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2656, 2009}
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.
| \(\displaystyle \int x^4 f^{\frac {c}{a+b x}} \, dx\) |
| \(\Big \downarrow \) 2656 |
| \(\displaystyle \int \left (\frac {a^4 f^{\frac {c}{a+b x}}}{b^4}-\frac {4 a^3 (a+b x) f^{\frac {c}{a+b x}}}{b^4}+\frac {6 a^2 (a+b x)^2 f^{\frac {c}{a+b x}}}{b^4}+\frac {(a+b x)^4 f^{\frac {c}{a+b x}}}{b^4}-\frac {4 a (a+b x)^3 f^{\frac {c}{a+b x}}}{b^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^4 c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}+\frac {a^4 (a+b x) f^{\frac {c}{a+b x}}}{b^5}+\frac {2 a^3 c^2 \log ^2(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}-\frac {2 a^3 (a+b x)^2 f^{\frac {c}{a+b x}}}{b^5}-\frac {2 a^3 c \log (f) (a+b x) f^{\frac {c}{a+b x}}}{b^5}-\frac {a^2 c^3 \log ^3(f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{a+b x}\right )}{b^5}+\frac {a^2 c^2 \log ^2(f) (a+b x) f^{\frac {c}{a+b x}}}{b^5}+\frac {2 a^2 (a+b x)^3 f^{\frac {c}{a+b x}}}{b^5}+\frac {a^2 c \log (f) (a+b x)^2 f^{\frac {c}{a+b x}}}{b^5}-\frac {c^5 \log ^5(f) \Gamma \left (-5,-\frac {c \log (f)}{a+b x}\right )}{b^5}-\frac {4 a c^4 \log ^4(f) \Gamma \left (-4,-\frac {c \log (f)}{a+b x}\right )}{b^5}\) |
(a^4*f^(c/(a + b*x))*(a + b*x))/b^5 - (2*a^3*f^(c/(a + b*x))*(a + b*x)^2)/ b^5 + (2*a^2*f^(c/(a + b*x))*(a + b*x)^3)/b^5 - (2*a^3*c*f^(c/(a + b*x))*( a + b*x)*Log[f])/b^5 + (a^2*c*f^(c/(a + b*x))*(a + b*x)^2*Log[f])/b^5 - (a ^4*c*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f])/b^5 + (a^2*c^2*f^(c/(a + b*x))*(a + b*x)*Log[f]^2)/b^5 + (2*a^3*c^2*ExpIntegralEi[(c*Log[f])/(a + b *x)]*Log[f]^2)/b^5 - (a^2*c^3*ExpIntegralEi[(c*Log[f])/(a + b*x)]*Log[f]^3 )/b^5 - (4*a*c^4*Gamma[-4, -((c*Log[f])/(a + b*x))]*Log[f]^4)/b^5 - (c^5*G amma[-5, -((c*Log[f])/(a + b*x))]*Log[f]^5)/b^5
3.3.16.3.1 Defintions of rubi rules used
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(Px_), x_Symbol] :> Int[ ExpandLinearProduct[F^(a + b*(c + d*x)^n), Px, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[Px, x]
Time = 0.21 (sec) , antiderivative size = 517, normalized size of antiderivative = 1.78
| method | result | size |
| risch | \(\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{4} c^{4} x}{120 b^{4}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{3} c^{3} x^{2}}{120 b^{3}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} c^{2} x^{3}}{60 b^{2}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right ) c \,x^{4}}{20 b}+\frac {f^{\frac {c}{b x +a}} x^{5}}{5}+\frac {\operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) c^{5} \ln \left (f \right )^{5}}{120 b^{5}}+\frac {f^{\frac {c}{b x +a}} \ln \left (f \right )^{4} a \,c^{4}}{120 b^{5}}-\frac {3 f^{\frac {c}{b x +a}} \ln \left (f \right )^{3} a \,c^{3} x}{20 b^{4}}-\frac {7 f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} a \,c^{2} x^{2}}{60 b^{3}}-\frac {2 f^{\frac {c}{b x +a}} \ln \left (f \right ) a c \,x^{3}}{15 b^{2}}-\frac {\ln \left (f \right )^{4} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a \,c^{4}}{6 b^{5}}-\frac {19 f^{\frac {c}{b x +a}} \ln \left (f \right )^{3} a^{2} c^{3}}{120 b^{5}}+\frac {43 f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} a^{2} c^{2} x}{60 b^{4}}+\frac {3 f^{\frac {c}{b x +a}} \ln \left (f \right ) a^{2} c \,x^{2}}{10 b^{3}}+\frac {\ln \left (f \right )^{3} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a^{2} c^{3}}{b^{5}}+\frac {17 f^{\frac {c}{b x +a}} \ln \left (f \right )^{2} a^{3} c^{2}}{20 b^{5}}-\frac {4 f^{\frac {c}{b x +a}} \ln \left (f \right ) a^{3} c x}{5 b^{4}}-\frac {2 \ln \left (f \right )^{2} \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a^{3} c^{2}}{b^{5}}-\frac {77 f^{\frac {c}{b x +a}} \ln \left (f \right ) a^{4} c}{60 b^{5}}+\frac {\ln \left (f \right ) \operatorname {Ei}_{1}\left (-\frac {c \ln \left (f \right )}{b x +a}\right ) a^{4} c}{b^{5}}+\frac {f^{\frac {c}{b x +a}} a^{5}}{5 b^{5}}\) | \(517\) |
1/120/b^4*f^(c/(b*x+a))*ln(f)^4*c^4*x+1/120/b^3*f^(c/(b*x+a))*ln(f)^3*c^3* x^2+1/60/b^2*f^(c/(b*x+a))*ln(f)^2*c^2*x^3+1/20/b*f^(c/(b*x+a))*ln(f)*c*x^ 4+1/5*f^(c/(b*x+a))*x^5+1/120/b^5*Ei(1,-c*ln(f)/(b*x+a))*c^5*ln(f)^5+1/120 /b^5*f^(c/(b*x+a))*ln(f)^4*a*c^4-3/20/b^4*f^(c/(b*x+a))*ln(f)^3*a*c^3*x-7/ 60/b^3*f^(c/(b*x+a))*ln(f)^2*a*c^2*x^2-2/15/b^2*f^(c/(b*x+a))*ln(f)*a*c*x^ 3-1/6/b^5*ln(f)^4*Ei(1,-c*ln(f)/(b*x+a))*a*c^4-19/120/b^5*f^(c/(b*x+a))*ln (f)^3*a^2*c^3+43/60/b^4*f^(c/(b*x+a))*ln(f)^2*a^2*c^2*x+3/10/b^3*f^(c/(b*x +a))*ln(f)*a^2*c*x^2+1/b^5*ln(f)^3*Ei(1,-c*ln(f)/(b*x+a))*a^2*c^3+17/20/b^ 5*f^(c/(b*x+a))*ln(f)^2*a^3*c^2-4/5/b^4*f^(c/(b*x+a))*ln(f)*a^3*c*x-2/b^5* ln(f)^2*Ei(1,-c*ln(f)/(b*x+a))*a^3*c^2-77/60/b^5*f^(c/(b*x+a))*ln(f)*a^4*c +1/b^5*ln(f)*Ei(1,-c*ln(f)/(b*x+a))*a^4*c+1/5/b^5*f^(c/(b*x+a))*a^5
Time = 0.29 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.84 \[ \int f^{\frac {c}{a+b x}} x^4 \, dx=\frac {{\left (24 \, b^{5} x^{5} + 24 \, a^{5} + {\left (b c^{4} x + a c^{4}\right )} \log \left (f\right )^{4} + {\left (b^{2} c^{3} x^{2} - 18 \, a b c^{3} x - 19 \, a^{2} c^{3}\right )} \log \left (f\right )^{3} + 2 \, {\left (b^{3} c^{2} x^{3} - 7 \, a b^{2} c^{2} x^{2} + 43 \, a^{2} b c^{2} x + 51 \, a^{3} c^{2}\right )} \log \left (f\right )^{2} + 2 \, {\left (3 \, b^{4} c x^{4} - 8 \, a b^{3} c x^{3} + 18 \, a^{2} b^{2} c x^{2} - 48 \, a^{3} b c x - 77 \, a^{4} c\right )} \log \left (f\right )\right )} f^{\frac {c}{b x + a}} - {\left (c^{5} \log \left (f\right )^{5} - 20 \, a c^{4} \log \left (f\right )^{4} + 120 \, a^{2} c^{3} \log \left (f\right )^{3} - 240 \, a^{3} c^{2} \log \left (f\right )^{2} + 120 \, a^{4} c \log \left (f\right )\right )} {\rm Ei}\left (\frac {c \log \left (f\right )}{b x + a}\right )}{120 \, b^{5}} \]
1/120*((24*b^5*x^5 + 24*a^5 + (b*c^4*x + a*c^4)*log(f)^4 + (b^2*c^3*x^2 - 18*a*b*c^3*x - 19*a^2*c^3)*log(f)^3 + 2*(b^3*c^2*x^3 - 7*a*b^2*c^2*x^2 + 4 3*a^2*b*c^2*x + 51*a^3*c^2)*log(f)^2 + 2*(3*b^4*c*x^4 - 8*a*b^3*c*x^3 + 18 *a^2*b^2*c*x^2 - 48*a^3*b*c*x - 77*a^4*c)*log(f))*f^(c/(b*x + a)) - (c^5*l og(f)^5 - 20*a*c^4*log(f)^4 + 120*a^2*c^3*log(f)^3 - 240*a^3*c^2*log(f)^2 + 120*a^4*c*log(f))*Ei(c*log(f)/(b*x + a)))/b^5
\[ \int f^{\frac {c}{a+b x}} x^4 \, dx=\int f^{\frac {c}{a + b x}} x^{4}\, dx \]
\[ \int f^{\frac {c}{a+b x}} x^4 \, dx=\int { f^{\frac {c}{b x + a}} x^{4} \,d x } \]
1/120*(24*b^4*x^5 + 6*b^3*c*x^4*log(f) + 2*(b^2*c^2*log(f)^2 - 8*a*b^2*c*l og(f))*x^3 + (b*c^3*log(f)^3 - 14*a*b*c^2*log(f)^2 + 36*a^2*b*c*log(f))*x^ 2 + (c^4*log(f)^4 - 18*a*c^3*log(f)^3 + 86*a^2*c^2*log(f)^2 - 96*a^3*c*log (f))*x)*f^(c/(b*x + a))/b^4 + integrate(-1/120*(a^2*c^4*log(f)^4 - 18*a^3* c^3*log(f)^3 + 86*a^4*c^2*log(f)^2 - 96*a^5*c*log(f) - (b*c^5*log(f)^5 - 2 0*a*b*c^4*log(f)^4 + 120*a^2*b*c^3*log(f)^3 - 240*a^3*b*c^2*log(f)^2 + 120 *a^4*b*c*log(f))*x)*f^(c/(b*x + a))/(b^6*x^2 + 2*a*b^5*x + a^2*b^4), x)
\[ \int f^{\frac {c}{a+b x}} x^4 \, dx=\int { f^{\frac {c}{b x + a}} x^{4} \,d x } \]
Timed out. \[ \int f^{\frac {c}{a+b x}} x^4 \, dx=\int f^{\frac {c}{a+b\,x}}\,x^4 \,d x \]