\(\int \frac {\sqrt {a+b \log (c x^n)}}{d+e x} \, dx\) [127]

Optimal result

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

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (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 = {} \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{d+e x} \, dx=\int \frac {\sqrt {a+b \log \left (c x^n\right )}}{d+e x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 7.97 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{d+e x} \, dx=\int \frac {\sqrt {a+b \log \left (c x^n\right )}}{d+e x} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{d+e x} \, dx=\int \frac {\sqrt {a + b \log {\left (c x^{n} \right )}}}{d + e x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{d+e x} \, dx=\int { \frac {\sqrt {b \log \left (c x^{n}\right ) + a}}{e x + d} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.43 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{d+e x} \, dx=\int { \frac {\sqrt {b \log \left (c x^{n}\right ) + a}}{e x + d} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{d+e x} \, dx=\int \frac {\sqrt {a+b\,\ln \left (c\,x^n\right )}}{d+e\,x} \,d x \]

[In]

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

[Out]

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