\(\int \frac {(a+b \log (c (d+e x)^n))^{3/2}}{(f+g x)^3} \, dx\) [116]

Optimal result

Integrand size = 26, antiderivative size = 26 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^3} \, dx=-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 g (f+g x)^2}+\frac {3 b e n \text {Int}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(d+e x) (f+g x)^2},x\right )}{4 g} \]

[Out]

-1/2*(a+b*ln(c*(e*x+d)^n))^(3/2)/g/(g*x+f)^2+3/4*b*e*n*Unintegrable((a+b*ln(c*(e*x+d)^n))^(1/2)/(e*x+d)/(g*x+f
)^2,x)/g

Rubi [N/A]

Not integrable

Time = 0.15 (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 = {} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^3} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^3} \, dx \]

[In]

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

[Out]

-1/2*(a + b*Log[c*(d + e*x)^n])^(3/2)/(g*(f + g*x)^2) + (3*b*e*n*Defer[Int][Sqrt[a + b*Log[c*(d + e*x)^n]]/((d
+ e*x)*(f + g*x)^2), x])/(4*g)

Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{2 g (f+g x)^2}+\frac {(3 b e n) \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(d+e x) (f+g x)^2} \, dx}{4 g} \\ \end{align*}

Mathematica [N/A]

Not integrable

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

[In]

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

[Out]

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

Maple [N/A]

Not integrable

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

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}}{{\left (g x + f\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^3} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}}{{\left (g x + f\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.47 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{(f+g x)^3} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{3/2}}{{\left (f+g\,x\right )}^3} \,d x \]

[In]

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

[Out]

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