\(\int \frac {1}{\sqrt {g+h x} \sqrt {a+b \log (c (d (e+f x)^p)^q)}} \, dx\) [504]

Optimal result

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

[Out]

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

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.21 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int \frac {1}{\sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate(1/(h*x+g)^(1/2)/(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 = 3.85 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.97 \[ \int \frac {1}{\sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int \frac {1}{\sqrt {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}} \sqrt {g + h x}}\, dx \]

[In]

integrate(1/(h*x+g)**(1/2)/(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))*sqrt(g + h*x)), x)

Maxima [N/A]

Not integrable

Time = 11.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int { \frac {1}{\sqrt {h x + g} \sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int { \frac {1}{\sqrt {h x + g} \sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.34 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}} \, dx=\int \frac {1}{\sqrt {g+h\,x}\,\sqrt {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}} \,d x \]

[In]

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

[Out]

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