\(\int \frac {1}{\sqrt {f+g x} (a+b \log (c (d+e x)^n))} \, dx\) [154]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {1}{\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx \]

[In]

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

[Out]

Integrate[1/(Sqrt[f + g*x]*(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

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.50 \[ \int \frac {1}{\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {1}{\sqrt {g x + f} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.85 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {1}{\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right ) \sqrt {f + g x}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 174, normalized size of antiderivative = 6.69 \[ \int \frac {1}{\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {1}{\sqrt {g x + f} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int { \frac {1}{\sqrt {g x + f} {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.21 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )} \, dx=\int \frac {1}{\sqrt {f+g\,x}\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )} \,d x \]

[In]

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

[Out]

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