\(\int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log (c (d (e+f x)^p)^q)}} \, dx\) [505]

Optimal result

Integrand size = 32, antiderivative size = 32 \[ \int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\text {Int}\left (\frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (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 = {} \[ \int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx \]

[In]

Int[1/((g + h*x)^(3/2)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]),x]

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx \]

[In]

Integrate[1/((g + h*x)^(3/2)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]),x]

[Out]

Integrate[1/((g + h*x)^(3/2)*Sqrt[a + b*Log[c*(d*(e + f*x)^p)^q]]), x]

Maple [N/A]

Not integrable

Time = 0.19 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88

\[\int \frac {1}{\left (h x +g \right )^{\frac {3}{2}} \sqrt {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 51.96 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int \frac {1}{\sqrt {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}} \left (g + h x\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 11.20 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int { \frac {1}{{\left (h x + g\right )}^{\frac {3}{2}} \sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int { \frac {1}{{\left (h x + g\right )}^{\frac {3}{2}} \sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.40 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {1}{(g+h x)^{3/2} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int \frac {1}{{\left (g+h\,x\right )}^{3/2}\,\sqrt {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}} \,d x \]

[In]

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

[Out]

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