\(\int \frac {\sqrt {a+b \log (c (d+e x)^n)}}{(f+g x)^2} \, dx\) [109]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx=\frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{(e f-d g) (f+g x)}-\frac {b e n \text {Int}\left (\frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}},x\right )}{2 (e f-d g)} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (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 = {} \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx=\int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx \]

[In]

Int[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^2,x]

[Out]

((d + e*x)*Sqrt[a + b*Log[c*(d + e*x)^n]])/((e*f - d*g)*(f + g*x)) - (b*e*n*Defer[Int][1/((f + g*x)*Sqrt[a + b
*Log[c*(d + e*x)^n]]), x])/(2*(e*f - d*g))

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{(e f-d g) (f+g x)}-\frac {(b e n) \int \frac {1}{(f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{2 (e f-d g)} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx=\int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx \]

[In]

Integrate[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^2,x]

[Out]

Integrate[Sqrt[a + b*Log[c*(d + e*x)^n]]/(f + g*x)^2, x]

Maple [N/A]

Not integrable

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

\[\int \frac {\sqrt {a +b \ln \left (c \left (e x +d \right )^{n}\right )}}{\left (g x +f \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^(1/2)/(g*x+f)^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 = 1.87 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx=\int \frac {\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}{\left (f + g x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx=\int { \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx=\int { \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.38 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^2} \, dx=\int \frac {\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}}{{\left (f+g\,x\right )}^2} \,d x \]

[In]

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

[Out]

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