\(\int \frac {1}{(f+g x) \sqrt {a+b \log (c (d+e x)^n)}} \, dx\) [127]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\text {Int}\left (\frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}},x\right ) \]

[Out]

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

[In]

Int[1/((f + g*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]),x]

[Out]

Defer[Int][1/((f + g*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx \]

[In]

Integrate[1/((f + g*x)*Sqrt[a + b*Log[c*(d + e*x)^n]]),x]

[Out]

Integrate[1/((f + g*x)*Sqrt[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

\[\int \frac {1}{\left (g x +f \right ) \sqrt {a +b \ln \left (c \left (e x +d \right )^{n}\right )}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int \frac {1}{\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}} \left (f + g x\right )}\, dx \]

[In]

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

[Out]

Integral(1/(sqrt(a + b*log(c*(d + e*x)**n))*(f + g*x)), x)

Maxima [N/A]

Not integrable

Time = 0.60 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int { \frac {1}{{\left (g x + f\right )} \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int { \frac {1}{{\left (g x + f\right )} \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.37 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int \frac {1}{\left (f+g\,x\right )\,\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}} \,d x \]

[In]

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

[Out]

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