\(\int \sqrt {g+h x} \sqrt {a+b \log (c (d (e+f x)^p)^q)} \, dx\) [500]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.23 (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 = {} \[ \int \sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int \sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \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 = 1.13 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.06 \[ \int \sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int \sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.88

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((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 = 3.56 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.91 \[ \int \sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int \sqrt {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}} \sqrt {g + h x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 11.06 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int { \sqrt {h x + g} \sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int { \sqrt {h x + g} \sqrt {b \log \left (\left ({\left (f x + e\right )}^{p} d\right )^{q} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.33 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94 \[ \int \sqrt {g+h x} \sqrt {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )} \, dx=\int \sqrt {g+h\,x}\,\sqrt {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )} \,d x \]

[In]

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

[Out]

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