\(\int (f+g x)^m \sqrt {a+b \log (c (d+e x)^n)} \, dx\) [166]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int (f+g x)^m \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\text {Int}\left ((f+g x)^m \sqrt {a+b \log \left (c (d+e x)^n\right )},x\right ) \]

[Out]

Unintegrable((g*x+f)^m*(a+b*ln(c*(e*x+d)^n))^(1/2),x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 26, 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 = {} \[ \int (f+g x)^m \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int (f+g x)^m \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int (f+g x)^m \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int (f+g x)^m \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

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

[In]

int((g*x+f)^m*(a+b*ln(c*(e*x+d)^n))^(1/2),x)

[Out]

int((g*x+f)^m*(a+b*ln(c*(e*x+d)^n))^(1/2),x)

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int (f+g x)^m \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} {\left (g x + f\right )}^{m} \,d x } \]

[In]

integrate((g*x+f)^m*(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

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

[In]

integrate((g*x+f)**m*(a+b*ln(c*(e*x+d)**n))**(1/2),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int (f+g x)^m \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} {\left (g x + f\right )}^{m} \,d x } \]

[In]

integrate((g*x+f)^m*(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int (f+g x)^m \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} {\left (g x + f\right )}^{m} \,d x } \]

[In]

integrate((g*x+f)^m*(a+b*log(c*(e*x+d)^n))^(1/2),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int (f+g x)^m \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int {\left (f+g\,x\right )}^m\,\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]

[In]

int((f + g*x)^m*(a + b*log(c*(d + e*x)^n))^(1/2),x)

[Out]

int((f + g*x)^m*(a + b*log(c*(d + e*x)^n))^(1/2), x)