\(\int \frac {\sqrt {a+b \log (c (d+e x)^n)}}{(f+g x)^{3/2}} \, dx\) [158]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx=-\frac {2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{g \sqrt {f+g x}}+\frac {b e n \text {Int}\left (\frac {1}{(d+e x) \sqrt {f+g x} \sqrt {a+b \log \left (c (d+e x)^n\right )}},x\right )}{g} \]

[Out]

-2*(a+b*ln(c*(e*x+d)^n))^(1/2)/g/(g*x+f)^(1/2)+b*e*n*Unintegrable(1/(e*x+d)/(g*x+f)^(1/2)/(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 = {} \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx=\int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx \]

[In]

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

[Out]

(-2*Sqrt[a + b*Log[c*(d + e*x)^n]])/(g*Sqrt[f + g*x]) + (b*e*n*Defer[Int][1/((d + e*x)*Sqrt[f + g*x]*Sqrt[a +
b*Log[c*(d + e*x)^n]]), x])/g

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

Mathematica [N/A]

Not integrable

Time = 1.49 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx=\int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.86

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 9.71 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx=\int \frac {\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}{\left (f + g x\right )^{\frac {3}{2}}}\, dx \]

[In]

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

[Out]

Integral(sqrt(a + b*log(c*(d + e*x)**n))/(f + g*x)**(3/2), x)

Maxima [N/A]

Not integrable

Time = 0.69 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx=\int { \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.01 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx=\int { \frac {\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}}{{\left (g x + f\right )}^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.36 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{(f+g x)^{3/2}} \, dx=\int \frac {\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}}{{\left (f+g\,x\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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