\(\int \frac {\sqrt {a+b \log (c (d+e x)^n)}}{f+g x} \, dx\) [108]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 3.58 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{f+g x} \, dx=\int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{f+g x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{f+g x} \, dx=\int \frac {\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}{f + g x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{f+g x} \, dx=\int { \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{g x + f} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{f+g x} \, dx=\int { \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{g x + f} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.27 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{f+g x} \, dx=\int \frac {\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}}{f+g\,x} \,d x \]

[In]

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

[Out]

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