Integrand size = 15, antiderivative size = 142 \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3}{3 b^3}-\frac {c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)}{3 b^3}+\frac {a^2 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}}{3 b^3}-\frac {2 a (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^3} \]
1/3*f^(c/(b*x+a)^3)*(b*x+a)^3/b^3-1/3*c*Ei(c*ln(f)/(b*x+a)^3)*ln(f)/b^3+1/ 3*a^2*(b*x+a)*GAMMA(-1/3,-c*ln(f)/(b*x+a)^3)*(-c*ln(f)/(b*x+a)^3)^(1/3)/b^ 3-2/3*a*(b*x+a)^2*GAMMA(-2/3,-c*ln(f)/(b*x+a)^3)*(-c*ln(f)/(b*x+a)^3)^(2/3 )/b^3
Time = 0.05 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.89 \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\frac {f^{\frac {c}{(a+b x)^3}} (a+b x)^3-c \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right ) \log (f)+a^2 (a+b x) \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}}-2 a (a+b x)^2 \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right ) \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3}}{3 b^3} \]
(f^(c/(a + b*x)^3)*(a + b*x)^3 - c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*L og[f] + a^2*(a + b*x)*Gamma[-1/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f]) /(a + b*x)^3))^(1/3) - 2*a*(a + b*x)^2*Gamma[-2/3, -((c*Log[f])/(a + b*x)^ 3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))/(3*b^3)
Time = 0.32 (sec) , antiderivative size = 142, 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.
\(\displaystyle \int x^2 f^{\frac {c}{(a+b x)^3}} \, dx\) |
\(\Big \downarrow \) 2656 |
\(\displaystyle \int \left (\frac {a^2 f^{\frac {c}{(a+b x)^3}}}{b^2}+\frac {(a+b x)^2 f^{\frac {c}{(a+b x)^3}}}{b^2}-\frac {2 a (a+b x) f^{\frac {c}{(a+b x)^3}}}{b^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^2 (a+b x) \sqrt [3]{-\frac {c \log (f)}{(a+b x)^3}} \Gamma \left (-\frac {1}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^3}-\frac {c \log (f) \operatorname {ExpIntegralEi}\left (\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^3}+\frac {(a+b x)^3 f^{\frac {c}{(a+b x)^3}}}{3 b^3}-\frac {2 a (a+b x)^2 \left (-\frac {c \log (f)}{(a+b x)^3}\right )^{2/3} \Gamma \left (-\frac {2}{3},-\frac {c \log (f)}{(a+b x)^3}\right )}{3 b^3}\) |
(f^(c/(a + b*x)^3)*(a + b*x)^3)/(3*b^3) - (c*ExpIntegralEi[(c*Log[f])/(a + b*x)^3]*Log[f])/(3*b^3) + (a^2*(a + b*x)*Gamma[-1/3, -((c*Log[f])/(a + b* x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(1/3))/(3*b^3) - (2*a*(a + b*x)^2*Gamma [-2/3, -((c*Log[f])/(a + b*x)^3)]*(-((c*Log[f])/(a + b*x)^3))^(2/3))/(3*b^ 3)
3.3.34.3.1 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]
\[\int f^{\frac {c}{\left (b x +a \right )^{3}}} x^{2}d x\]
Time = 0.09 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.37 \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\frac {3 \, a b^{2} \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {2}{3}} \Gamma \left (\frac {1}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - 3 \, a^{2} b \left (-\frac {c \log \left (f\right )}{b^{3}}\right )^{\frac {1}{3}} \Gamma \left (\frac {2}{3}, -\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) - c {\rm Ei}\left (\frac {c \log \left (f\right )}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}\right ) \log \left (f\right ) + {\left (b^{3} x^{3} + a^{3}\right )} f^{\frac {c}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}}}}{3 \, b^{3}} \]
1/3*(3*a*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)) - 3*a^2*b*(-c*log(f)/b^3)^(1/3)*gamma(2/3, -c*log (f)/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)) - c*Ei(c*log(f)/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*log(f) + (b^3*x^3 + a^3)*f^(c/(b^3*x^3 + 3 *a*b^2*x^2 + 3*a^2*b*x + a^3)))/b^3
\[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int f^{\frac {c}{\left (a + b x\right )^{3}}} x^{2}\, dx \]
\[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{2} \,d x } \]
1/3*f^(c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*x^3 + b*c*integrate(f^ (c/(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3))*x^3/(b^4*x^4 + 4*a*b^3*x^3 + 6*a^2*b^2*x^2 + 4*a^3*b*x + a^4), x)*log(f)
\[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int { f^{\frac {c}{{\left (b x + a\right )}^{3}}} x^{2} \,d x } \]
Timed out. \[ \int f^{\frac {c}{(a+b x)^3}} x^2 \, dx=\int f^{\frac {c}{{\left (a+b\,x\right )}^3}}\,x^2 \,d x \]