\(\int F^{f (a+b \log (c (d+e x)^n))^2} (g+h x)^m \, dx\) [609]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx=\text {Int}\left (F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m,x\right ) \]

[Out]

Unintegrable(F^(f*(a+b*ln(c*(e*x+d)^n))^2)*(h*x+g)^m,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 28, 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 = {} \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx=\int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx \]

[In]

Int[F^(f*(a + b*Log[c*(d + e*x)^n])^2)*(g + h*x)^m,x]

[Out]

Defer[Int][F^(f*(a + b*Log[c*(d + e*x)^n])^2)*(g + h*x)^m, x]

Rubi steps \begin{align*} \text {integral}& = \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx=\int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx \]

[In]

Integrate[F^(f*(a + b*Log[c*(d + e*x)^n])^2)*(g + h*x)^m,x]

[Out]

Integrate[F^(f*(a + b*Log[c*(d + e*x)^n])^2)*(g + h*x)^m, x]

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

\[\int F^{f {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{2}} \left (h x +g \right )^{m}d x\]

[In]

int(F^(f*(a+b*ln(c*(e*x+d)^n))^2)*(h*x+g)^m,x)

[Out]

int(F^(f*(a+b*ln(c*(e*x+d)^n))^2)*(h*x+g)^m,x)

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.79 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx=\int { {\left (h x + g\right )}^{m} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f} \,d x } \]

[In]

integrate(F^(f*(a+b*log(c*(e*x+d)^n))^2)*(h*x+g)^m,x, algorithm="fricas")

[Out]

integral((h*x + g)^m*F^(b^2*f*log((e*x + d)^n*c)^2 + 2*a*b*f*log((e*x + d)^n*c) + a^2*f), x)

Sympy [F(-2)]

Exception generated. \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate(F**(f*(a+b*ln(c*(e*x+d)**n))**2)*(h*x+g)**m,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.83 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx=\int { {\left (h x + g\right )}^{m} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f} \,d x } \]

[In]

integrate(F^(f*(a+b*log(c*(e*x+d)^n))^2)*(h*x+g)^m,x, algorithm="maxima")

[Out]

integrate((h*x + g)^m*F^((b*log((e*x + d)^n*c) + a)^2*f), x)

Giac [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx=\int { {\left (h x + g\right )}^{m} F^{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} f} \,d x } \]

[In]

integrate(F^(f*(a+b*log(c*(e*x+d)^n))^2)*(h*x+g)^m,x, algorithm="giac")

[Out]

integrate((h*x + g)^m*F^((b*log((e*x + d)^n*c) + a)^2*f), x)

Mupad [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.11 \[ \int F^{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2} (g+h x)^m \, dx=\int {\mathrm {e}}^{f\,\ln \left (F\right )\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}\,{\left (g+h\,x\right )}^m \,d x \]

[In]

int(F^(f*(a + b*log(c*(d + e*x)^n))^2)*(g + h*x)^m,x)

[Out]

int(exp(f*log(F)*(a + b*log(c*(d + e*x)^n))^2)*(g + h*x)^m, x)