\(\int \sqrt {f+g x} \sqrt {a+b \log (c (d+e x)^n)} \, dx\) [156]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\frac {2 (f+g x)^{3/2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{3 g}-\frac {b e n \text {Int}\left (\frac {(f+g x)^{3/2}}{(d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}},x\right )}{3 g} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.17 (sec) , antiderivative size = 28, 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 = {} \[ \int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 0.87 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.14 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((g*x+f)^(1/2)*(a+b*log(c*(e*x+d)^n))^(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 = 2.66 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int \sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}} \sqrt {f + g x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.65 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \sqrt {g x + f} \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.73 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int { \sqrt {g x + f} \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.35 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int \sqrt {f+g\,x}\,\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]

[In]

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

[Out]

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