Integrand size = 15, antiderivative size = 184 \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=-\frac {a f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{b^4}+\frac {a c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{b^4}-\frac {a^3 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^4}+\frac {a^2 (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{b^4}+\frac {(a+b x)^4 \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3}}{3 b^4} \]
-a*f^(c/(b*x+a)^3)*(b*x+a)^3/b^4+a*c*Ei(c*ln(f)/(b*x+a)^3)*ln(f)/b^4-1/3*a ^3*(b*x+a)*GAMMA(-1/3,-c*ln(f)/(b*x+a)^3)*(-c*ln(f)/(b*x+a)^3)^(1/3)/b^4+a ^2*(b*x+a)^2*GAMMA(-2/3,-c*ln(f)/(b*x+a)^3)*(-c*ln(f)/(b*x+a)^3)^(2/3)/b^4 +1/3*(b*x+a)^4*GAMMA(-4/3,-c*ln(f)/(b*x+a)^3)*(-c*ln(f)/(b*x+a)^3)^(4/3)/b ^4
Time = 0.24 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.91 \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\frac {3 a c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)-(a+b x) \left (a^3 \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}+c \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \log (f) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}+3 a (a+b x) \left (f^{\frac {c}{(a+b x)^3}} (a+b x)-a \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}\right )\right )}{3 b^4} \]
(3*a*c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f] - (a + b*x)*(a^3*Gamma [-1/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3) + c*Ga mma[-4/3, -((c*Log[f])/(a + b*x)^3)]*Log[f]*(-((c*Log[f])/(a + b*x)^3))^(1 /3) + 3*a*(a + b*x)*(f^(c/(a + b*x)^3)*(a + b*x) - a*Gamma[-2/3, -((c*Log[ f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))))/(3*b^4)
Time = 0.36 (sec) , antiderivative size = 184, 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.
\(\displaystyle \int x^3 f^{\frac {c}{(a+b x)^3}} \, dx\) |
\(\Big \downarrow \) 2656 |
\(\displaystyle \int \left (-\frac {a^3 f^{\frac {c}{(a+b x)^3}}}{b^3}+\frac {3 a^2 (a+b x) f^{\frac {c}{(a+b x)^3}}}{b^3}+\frac {(a+b x)^3 f^{\frac {c}{(a+b x)^3}}}{b^3}-\frac {3 a (a+b x)^2 f^{\frac {c}{(a+b x)^3}}}{b^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^3 (a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^4}+\frac {a^2 (a+b x)^2 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{b^4}+\frac {a c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right )}{b^4}-\frac {a (a+b x)^3 f^{\frac {c}{(a+b x)^3}}}{b^4}+\frac {(a+b x)^4 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{4/3} \Gamma \left (-\frac {4}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^4}\) |
-((a*f^(c/(a + b*x)^3)*(a + b*x)^3)/b^4) + (a*c*ExpIntegralEi[(c*Log[f])/( a + b*x)^3]*Log[f])/b^4 - (a^3*(a + b*x)*Gamma[-1/3, -((c*Log[f])/(a + b*x )^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^4) + (a^2*(a + b*x)^2*Gamma[ -2/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))/b^4 + ((a + b*x)^4*Gamma[-4/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x )^3))^(4/3))/(3*b^4)
3.3.33.3.1 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]
\[\int f^{\frac {c}{\left (b x +a \right )^{3}}} x^{3}d x\]
Time = 0.09 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.20 \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=-\frac {6 \, a^{2} b^{2} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - 4 \, a c {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) - {\left (4 \, a^{3} b - 3 \, b c \log \left (f\right )\right )} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - {\left (b^{4} x^{4} - a^{4} + 3 \, {\left (b c x + a c\right )} \log \left (f\right )\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{4 \, b^{4}} \]
-1/4*(6*a^2*b^2*(-c*log(f)/b^3)^(2/3)*gamma(1/3, -c*log(f)/(b^3*x^3 + 3*a* b^2*x^2 + 3*a^2*b*x + a^3)) - 4*a*c*Ei(c*log(f)/(b^3*x^3 + 3*a*b^2*x^2 + 3 *a^2*b*x + a^3))*log(f) - (4*a^3*b - 3*b*c*log(f))*(-c*log(f)/b^3)^(1/3)*g amma(2/3, -c*log(f)/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)) - (b^4*x^4 - a^4 + 3*(b*c*x + a*c)*log(f))*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)))/b^4
\[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int f^{\frac {c}{\left (a + b x\right )^{3}}} x^{3}\, dx \]
\[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{3} \,d x } \]
1/4*(b^3*x^4 + 3*c*x*log(f))*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3 ))/b^3 - integrate(3/4*(4*a*b^3*c*x^3*log(f) + 6*a^2*b^2*c*x^2*log(f) + a^ 4*c*log(f) + (4*a^3*b*c*log(f) - 3*b*c^2*log(f)^2)*x)*f^(c/(b^3*x^3 + 3*a* b^2*x^2 + 3*a^2*b*x + a^3))/(b^7*x^4 + 4*a*b^6*x^3 + 6*a^2*b^5*x^2 + 4*a^3 *b^4*x + a^4*b^3), x)
\[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{3} \,d x } \]
Timed out. \[ \int f^{\frac {c}{(a+b x)^3}} x^3 \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^3}}\,x^3 \,d x \]