\(\int \frac {(a+b \log (c (d (e+f x)^p)^q))^{3/2}}{(g+h x)^2} \, dx\) [469]

Optimal result

Integrand size = 30, antiderivative size = 30 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{(g+h x)^2} \, dx=\text {Int}\left (\frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{(g+h x)^2},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*(d*(f*x+e)^p)^q))^(3/2)/(h*x+g)^2,x)

Rubi [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 30, 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 = {} \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{(g+h x)^2} \, dx=\int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{(g+h x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{(g+h x)^2} \, dx=\int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{(g+h x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 10.70 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{(g+h x)^2} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac {3}{2}}}{{\left (h x + g\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.52 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{(g+h x)^2} \, dx=\int { \frac {{\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}^{\frac {3}{2}}}{{\left (h x + g\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.39 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )^{3/2}}{(g+h x)^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )}^{3/2}}{{\left (g+h\,x\right )}^2} \,d x \]

[In]

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

[Out]

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