\(\int \frac {\sqrt {a+b \log (c x^n)}}{(d+e x)^2} \, dx\) [128]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx=\frac {x \sqrt {a+b \log \left (c x^n\right )}}{d (d+e x)}-\frac {b n \text {Int}\left (\frac {1}{(d+e x) \sqrt {a+b \log \left (c x^n\right )}},x\right )}{2 d} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx=\int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.64 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx=\int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

integrate((a+b*log(c*x^n))^(1/2)/(e*x+d)^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 = 0.84 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx=\int \frac {\sqrt {a + b \log {\left (c x^{n} \right )}}}{\left (d + e x\right )^{2}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx=\int { \frac {\sqrt {b \log \left (c x^{n}\right ) + a}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx=\int { \frac {\sqrt {b \log \left (c x^{n}\right ) + a}}{{\left (e x + d\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^2} \, dx=\int \frac {\sqrt {a+b\,\ln \left (c\,x^n\right )}}{{\left (d+e\,x\right )}^2} \,d x \]

[In]

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

[Out]

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