\(\int \frac {\sqrt {a+b \log (c (d (e+f x)^p)^q)}}{(g+h x)^{3/2}} \, dx\) [502]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{(g+h x)^{3/2}} \, dx=\text {Int}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{(g+h x)^{3/2}},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/(h*x+g)^(3/2),x)

Rubi [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 32, 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 = {} \[ \int \frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{(g+h x)^{3/2}} \, dx=\int \frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{(g+h x)^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.71 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{(g+h x)^{3/2}} \, dx=\int \frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{(g+h x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.16 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88

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

[In]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/(h*x+g)^(3/2),x)

[Out]

int((a+b*ln(c*(d*(f*x+e)^p)^q))^(1/2)/(h*x+g)^(3/2),x)

Fricas [F(-2)]

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

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^(1/2)/(h*x+g)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [N/A]

Not integrable

Time = 17.42 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{(g+h x)^{3/2}} \, dx=\int \frac {\sqrt {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}}{\left (g + h x\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((a+b*ln(c*(d*(f*x+e)**p)**q))**(1/2)/(h*x+g)**(3/2),x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 11.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{(g+h x)^{3/2}} \, dx=\int { \frac {\sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}}{{\left (h x + g\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^(1/2)/(h*x+g)^(3/2),x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 1.41 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{(g+h x)^{3/2}} \, dx=\int { \frac {\sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}}{{\left (h x + g\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate((a+b*log(c*(d*(f*x+e)^p)^q))^(1/2)/(h*x+g)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.37 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}}{(g+h x)^{3/2}} \, dx=\int \frac {\sqrt {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}}{{\left (g+h\,x\right )}^{3/2}} \,d x \]

[In]

int((a + b*log(c*(d*(e + f*x)^p)^q))^(1/2)/(g + h*x)^(3/2),x)

[Out]

int((a + b*log(c*(d*(e + f*x)^p)^q))^(1/2)/(g + h*x)^(3/2), x)