\(\int f^{c (a+b x)^n} x^2 \, dx\) [248]

Optimal result

Integrand size = 15, antiderivative size = 154 \[ \int f^{c (a+b x)^n} x^2 \, dx=-\frac {(a+b x)^3 \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-3/n}}{b^3 n}+\frac {2 a (a+b x)^2 \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-2/n}}{b^3 n}-\frac {a^2 (a+b x) \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{-1/n}}{b^3 n} \]

[Out]

-(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))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2258, 2239, 2250} \[ \int f^{c (a+b x)^n} x^2 \, dx=-\frac {a^2 (a+b x) \left (-c \log (f) (a+b x)^n\right )^{-1/n} \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right )}{b^3 n}-\frac {(a+b x)^3 \left (-c \log (f) (a+b x)^n\right )^{-3/n} \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (f)\right )}{b^3 n}+\frac {2 a (a+b x)^2 \left (-c \log (f) (a+b x)^n\right )^{-2/n} \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right )}{b^3 n} \]

[In]

Int[f^(c*(a + b*x)^n)*x^2,x]

[Out]

-(((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))

Rule 2239

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_)), x_Symbol] :> Simp[(-F^a)*(c + d*x)*(Gamma[1/n, (-b)*(c + d
*x)^n*Log[F]]/(d*n*((-b)*(c + d*x)^n*Log[F])^(1/n))), x] /; FreeQ[{F, a, b, c, d, n}, x] &&  !IntegerQ[2/n]

Rule 2250

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(-F^a)*((e +
f*x)^(m + 1)/(f*n*((-b)*(c + d*x)^n*Log[F])^((m + 1)/n)))*Gamma[(m + 1)/n, (-b)*(c + d*x)^n*Log[F]], x] /; Fre
eQ[{F, a, b, c, d, e, f, m, n}, x] && EqQ[d*e - c*f, 0]

Rule 2258

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*(u_), x_Symbol] :> Int[ExpandLinearProduct[F^(a + b*(c + d*
x)^n), u, c, d, x], x] /; FreeQ[{F, a, b, c, d, n}, x] && PolynomialQ[u, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^2 f^{c (a+b x)^n}}{b^2}-\frac {2 a f^{c (a+b x)^n} (a+b x)}{b^2}+\frac {f^{c (a+b x)^n} (a+b x)^2}{b^2}\right ) \, dx \\ & = \frac {\int f^{c (a+b x)^n} (a+b x)^2 \, dx}{b^2}-\frac {(2 a) \int f^{c (a+b x)^n} (a+b x) \, dx}{b^2}+\frac {a^2 \int f^{c (a+b x)^n} \, dx}{b^2} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.88 \[ \int f^{c (a+b x)^n} x^2 \, dx=-\frac {(a+b x) \left (-c (a+b x)^n \log (f)\right )^{-3/n} \left ((a+b x)^2 \Gamma \left (\frac {3}{n},-c (a+b x)^n \log (f)\right )+a \left (-c (a+b x)^n \log (f)\right )^{\frac {1}{n}} \left (-2 (a+b x) \Gamma \left (\frac {2}{n},-c (a+b x)^n \log (f)\right )+a \Gamma \left (\frac {1}{n},-c (a+b x)^n \log (f)\right ) \left (-c (a+b x)^n \log (f)\right )^{\frac {1}{n}}\right )\right )}{b^3 n} \]

[In]

Integrate[f^(c*(a + b*x)^n)*x^2,x]

[Out]

-(((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)))

Maple [F]

\[\int f^{c \left (b x +a \right )^{n}} x^{2}d x\]

[In]

int(f^(c*(b*x+a)^n)*x^2,x)

[Out]

int(f^(c*(b*x+a)^n)*x^2,x)

Fricas [F]

\[ \int f^{c (a+b x)^n} x^2 \, dx=\int { f^{{\left (b x + a\right )}^{n} c} x^{2} \,d x } \]

[In]

integrate(f^(c*(b*x+a)^n)*x^2,x, algorithm="fricas")

[Out]

integral(f^((b*x + a)^n*c)*x^2, x)

Sympy [F]

\[ \int f^{c (a+b x)^n} x^2 \, dx=\int f^{c \left (a + b x\right )^{n}} x^{2}\, dx \]

[In]

integrate(f**(c*(b*x+a)**n)*x**2,x)

[Out]

Integral(f**(c*(a + b*x)**n)*x**2, x)

Maxima [F]

\[ \int f^{c (a+b x)^n} x^2 \, dx=\int { f^{{\left (b x + a\right )}^{n} c} x^{2} \,d x } \]

[In]

integrate(f^(c*(b*x+a)^n)*x^2,x, algorithm="maxima")

[Out]

integrate(f^((b*x + a)^n*c)*x^2, x)

Giac [F]

\[ \int f^{c (a+b x)^n} x^2 \, dx=\int { f^{{\left (b x + a\right )}^{n} c} x^{2} \,d x } \]

[In]

integrate(f^(c*(b*x+a)^n)*x^2,x, algorithm="giac")

[Out]

integrate(f^((b*x + a)^n*c)*x^2, x)

Mupad [F(-1)]

Timed out. \[ \int f^{c (a+b x)^n} x^2 \, dx=\int f^{c\,{\left (a+b\,x\right )}^n}\,x^2 \,d x \]

[In]

int(f^(c*(a + b*x)^n)*x^2,x)

[Out]

int(f^(c*(a + b*x)^n)*x^2, x)