\(\int \frac {\sqrt {f+g x}}{a+b \log (c (d+e x)^n)} \, dx\) [153]

Optimal result

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

[Out]

Unintegrable((g*x+f)^(1/2)/(a+b*ln(c*(e*x+d)^n)),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 = {} \[ \int \frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.13 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {\sqrt {g x + f}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.70 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {\sqrt {f + g x}}{a + b \log {\left (c \left (d + e x\right )^{n} \right )}}\, dx \]

[In]

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

[Out]

Integral(sqrt(f + g*x)/(a + b*log(c*(d + e*x)**n)), x)

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.69 \[ \int \frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {\sqrt {g x + f}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

2/3*(g*x + f)^(3/2)/(b*g*log((e*x + d)^n) + b*g*log(c) + a*g) + integrate(2/3*(b*e*g*n*x + b*e*f*n)*sqrt(g*x +
f)/(b^2*d*g*log(c)^2 + 2*a*b*d*g*log(c) + a^2*d*g + (b^2*e*g*x + b^2*d*g)*log((e*x + d)^n)^2 + (b^2*e*g*log(c
)^2 + 2*a*b*e*g*log(c) + a^2*e*g)*x + 2*(b^2*d*g*log(c) + a*b*d*g + (b^2*e*g*log(c) + a*b*e*g)*x)*log((e*x + d
)^n)), x)

Giac [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \frac {\sqrt {g x + f}}{b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.26 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {f+g x}}{a+b \log \left (c (d+e x)^n\right )} \, dx=\int \frac {\sqrt {f+g\,x}}{a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]

[In]

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

[Out]

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