\(\int f^{c (a+b x)^2} x^m \, dx\) [241]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int f^{c (a+b x)^2} x^m \, dx=\text {Int}\left (f^{a^2 c+2 a b c x+b^2 c x^2} x^m,x\right ) \]

[Out]

Unintegrable(f^(b^2*c*x^2+2*a*b*c*x+a^2*c)*x^m,x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int f^{c (a+b x)^2} x^m \, dx=\int f^{c (a+b x)^2} x^m \, dx \]

[In]

Int[f^(c*(a + b*x)^2)*x^m,x]

[Out]

Defer[Int][f^(a^2*c + 2*a*b*c*x + b^2*c*x^2)*x^m, x]

Rubi steps \begin{align*} \text {integral}& = \int f^{a^2 c+2 a b c x+b^2 c x^2} x^m \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int f^{c (a+b x)^2} x^m \, dx=\int f^{c (a+b x)^2} x^m \, dx \]

[In]

Integrate[f^(c*(a + b*x)^2)*x^m,x]

[Out]

Integrate[f^(c*(a + b*x)^2)*x^m, x]

Maple [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.87 \[ \int f^{c (a+b x)^2} x^m \, dx=\int { f^{{\left (b x + a\right )}^{2} c} x^{m} \,d x } \]

[In]

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

[Out]

integral(f^(b^2*c*x^2 + 2*a*b*c*x + a^2*c)*x^m, x)

Sympy [N/A]

Not integrable

Time = 1.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.93 \[ \int f^{c (a+b x)^2} x^m \, dx=\int f^{c \left (a + b x\right )^{2}} x^{m}\, dx \]

[In]

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

[Out]

Integral(f**(c*(a + b*x)**2)*x**m, x)

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int f^{c (a+b x)^2} x^m \, dx=\int { f^{{\left (b x + a\right )}^{2} c} x^{m} \,d x } \]

[In]

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

[Out]

integrate(f^((b*x + a)^2*c)*x^m, x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int f^{c (a+b x)^2} x^m \, dx=\int { f^{{\left (b x + a\right )}^{2} c} x^{m} \,d x } \]

[In]

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

[Out]

integrate(f^((b*x + a)^2*c)*x^m, x)

Mupad [N/A]

Not integrable

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int f^{c (a+b x)^2} x^m \, dx=\int f^{c\,{\left (a+b\,x\right )}^2}\,x^m \,d x \]

[In]

int(f^(c*(a + b*x)^2)*x^m,x)

[Out]

int(f^(c*(a + b*x)^2)*x^m, x)