\(\int \frac {1}{(g+h x)^{3/2} (a+b \log (c (d (e+f x)^p)^q))} \, dx\) [499]

Optimal result

Integrand size = 30, antiderivative size = 30 \[ \int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx=\text {Int}\left (\frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.06 (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 = {} \[ \int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx=\int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx \]

[In]

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

[Out]

Defer[Int][1/((g + h*x)^(3/2)*(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} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 0.80 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx=\int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx \]

[In]

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

[Out]

Integrate[1/((g + h*x)^(3/2)*(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 = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.23 \[ \int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx=\int { \frac {1}{{\left (h x + g\right )}^{\frac {3}{2}} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 15.33 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.90 \[ \int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}\right ) \left (g + h x\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.93 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx=\int { \frac {1}{{\left (h x + g\right )}^{\frac {3}{2}} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.49 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx=\int { \frac {1}{{\left (h x + g\right )}^{\frac {3}{2}} {\left (b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a\right )}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{(g+h x)^{3/2} \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )} \, dx=\int \frac {1}{{\left (g+h\,x\right )}^{3/2}\,\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )\right )} \,d x \]

[In]

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

[Out]

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