Integrand size = 15, antiderivative size = 154 \[ \int F^{c (a+b x)^n} x^2 \, dx=-\frac {(a+b x)^3 \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (F)\right ) \left (-c (a+b x)^n \log (F)\right )^{-3/n}}{b^3 n}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (F)\right ) \left (-c (a+b x)^n \log (F)\right )^{-2/n}}{b^3 n}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (F)\right ) \left (-c (a+b x)^n \log (F)\right )^{-1/n}}{b^3 n} \] Output:
-(b*x+a)^3*GAMMA(3/n,-c*(b*x+a)^n*ln(F))/b^3/n/((-c*(b*x+a)^n*ln(F))^(3/n) )+2*a*(b*x+a)^2*GAMMA(2/n,-c*(b*x+a)^n*ln(F))/b^3/n/((-c*(b*x+a)^n*ln(F))^ (2/n))-a^2*(b*x+a)*GAMMA(1/n,-c*(b*x+a)^n*ln(F))/b^3/n/((-c*(b*x+a)^n*ln(F ))^(1/n))
Time = 0.08 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88 \[ \int F^{c (a+b x)^n} x^2 \, dx=-\frac {(a+b x) \left (-c (a+b x)^n \log (F)\right )^{-3/n} \left ((a+b x)^2 \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (F)\right )+a \left (-c (a+b x)^n \log (F)\right )^{\frac {1}{n}} \left (-2 (a+b x) \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (F)\right )+a \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (F)\right ) \left (-c (a+b x)^n \log (F)\right )^{\frac {1}{n}}\right )\right )}{b^3 n} \] Input:
Integrate[F^(c*(a + b*x)^n)*x^2,x]
Output:
-(((a + b*x)*((a + b*x)^2*Gamma[3/n, -(c*(a + b*x)^n*Log[F])] + a*(-(c*(a + b*x)^n*Log[F]))^n^(-1)*(-2*(a + b*x)*Gamma[2/n, -(c*(a + b*x)^n*Log[F])] + a*Gamma[n^(-1), -(c*(a + b*x)^n*Log[F])]*(-(c*(a + b*x)^n*Log[F]))^n^(- 1))))/(b^3*n*(-(c*(a + b*x)^n*Log[F]))^(3/n)))
Time = 0.53 (sec) , antiderivative size = 154, 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.
\(\displaystyle \int x^2 F^{c (a+b x)^n} \, dx\) |
\(\Big \downarrow \) 2656 |
\(\displaystyle \int \left (\frac {a^2 F^{c (a+b x)^n}}{b^2}+\frac {(a+b x)^2 F^{c (a+b x)^n}}{b^2}-\frac {2 a (a+b x) F^{c (a+b x)^n}}{b^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a^2 (a+b x) \left (-c \log (F) (a+b x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (F)\right )}{b^3 n}-\frac {(a+b x)^3 \left (-c \log (F) (a+b x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (F)\right )}{b^3 n}+\frac {2 a (a+b x)^2 \left (-c \log (F) (a+b x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (F)\right )}{b^3 n}\) |
Input:
Int[F^(c*(a + b*x)^n)*x^2,x]
Output:
-(((a + b*x)^3*Gamma[3/n, -(c*(a + b*x)^n*Log[F])])/(b^3*n*(-(c*(a + b*x)^ n*Log[F]))^(3/n))) + (2*a*(a + b*x)^2*Gamma[2/n, -(c*(a + b*x)^n*Log[F])]) /(b^3*n*(-(c*(a + b*x)^n*Log[F]))^(2/n)) - (a^2*(a + b*x)*Gamma[n^(-1), -( c*(a + b*x)^n*Log[F])])/(b^3*n*(-(c*(a + b*x)^n*Log[F]))^n^(-1))
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]
\[\int F^{c \left (b x +a \right )^{n}} x^{2}d x\]
Input:
int(F^(c*(b*x+a)^n)*x^2,x)
Output:
int(F^(c*(b*x+a)^n)*x^2,x)
\[ \int F^{c (a+b x)^n} x^2 \, dx=\int { F^{{\left (b x + a\right )}^{n} c} x^{2} \,d x } \] Input:
integrate(F^(c*(b*x+a)^n)*x^2,x, algorithm="fricas")
Output:
integral(F^((b*x + a)^n*c)*x^2, x)
\[ \int F^{c (a+b x)^n} x^2 \, dx=\int F^{c \left (a + b x\right )^{n}} x^{2}\, dx \] Input:
integrate(F**(c*(b*x+a)**n)*x**2,x)
Output:
Integral(F**(c*(a + b*x)**n)*x**2, x)
\[ \int F^{c (a+b x)^n} x^2 \, dx=\int { F^{{\left (b x + a\right )}^{n} c} x^{2} \,d x } \] Input:
integrate(F^(c*(b*x+a)^n)*x^2,x, algorithm="maxima")
Output:
integrate(F^((b*x + a)^n*c)*x^2, x)
\[ \int F^{c (a+b x)^n} x^2 \, dx=\int { F^{{\left (b x + a\right )}^{n} c} x^{2} \,d x } \] Input:
integrate(F^(c*(b*x+a)^n)*x^2,x, algorithm="giac")
Output:
integrate(F^((b*x + a)^n*c)*x^2, x)
Timed out. \[ \int F^{c (a+b x)^n} x^2 \, dx=\int F^{c\,{\left (a+b\,x\right )}^n}\,x^2 \,d x \] Input:
int(F^(c*(a + b*x)^n)*x^2,x)
Output:
int(F^(c*(a + b*x)^n)*x^2, x)
\[ \int F^{c (a+b x)^n} x^2 \, dx=\int f^{\left (b x +a \right )^{n} c} x^{2}d x \] Input:
int(F^(c*(b*x+a)^n)*x^2,x)
Output:
int(f**((a + b*x)**n*c)*x**2,x)