\(\int \frac {(a+b \log (c (d+e x)^n))^{3/2}}{(f+g x)^2} \, dx\) [115]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^2} \, dx=\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(e f-d g) (f+g x)}-\frac {3 b e n \text {Int}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{f+g x},x\right )}{2 (e f-d g)} \]

[Out]

(e*x+d)*(a+b*ln(c*(e*x+d)^n))^(3/2)/(-d*g+e*f)/(g*x+f)-3/2*b*e*n*Unintegrable((a+b*ln(c*(e*x+d)^n))^(1/2)/(g*x
+f),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 = {} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^2} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^2} \, dx \]

[In]

Int[(a + b*Log[c*(d + e*x)^n])^(3/2)/(f + g*x)^2,x]

[Out]

((d + e*x)*(a + b*Log[c*(d + e*x)^n])^(3/2))/((e*f - d*g)*(f + g*x)) - (3*b*e*n*Defer[Int][Sqrt[a + b*Log[c*(d
+ e*x)^n]]/(f + g*x), x])/(2*(e*f - d*g))

Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(e f-d g) (f+g x)}-\frac {(3 b e n) \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{f+g x} \, dx}{2 (e f-d g)} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.53 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^2} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^2} \, dx \]

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])^(3/2)/(f + g*x)^2,x]

[Out]

Integrate[(a + b*Log[c*(d + e*x)^n])^(3/2)/(f + g*x)^2, x]

Maple [N/A]

Not integrable

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

\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {3}{2}}}{\left (g x +f \right )^{2}}d x\]

[In]

int((a+b*ln(c*(e*x+d)^n))^(3/2)/(g*x+f)^2,x)

[Out]

int((a+b*ln(c*(e*x+d)^n))^(3/2)/(g*x+f)^2,x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^2} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.63 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}}{{\left (g x + f\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}}{{\left (g x + f\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.41 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{3/2}}{{\left (f+g\,x\right )}^2} \,d x \]

[In]

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

[Out]

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