\(\int \frac {\sqrt {a+b \log (c x^n)}}{(d+e x)^3} \, dx\) [129]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx=-\frac {\sqrt {a+b \log \left (c x^n\right )}}{2 e (d+e x)^2}+\frac {b n \text {Int}\left (\frac {1}{x (d+e x)^2 \sqrt {a+b \log \left (c x^n\right )}},x\right )}{4 e} \]

[Out]

-1/2*(a+b*ln(c*x^n))^(1/2)/e/(e*x+d)^2+1/4*b*n*Unintegrable(1/x/(e*x+d)^2/(a+b*ln(c*x^n))^(1/2),x)/e

Rubi [N/A]

Not integrable

Time = 0.13 (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 = {} \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx=\int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx \]

[In]

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

[Out]

-1/2*Sqrt[a + b*Log[c*x^n]]/(e*(d + e*x)^2) + (b*n*Defer[Int][1/(x*(d + e*x)^2*Sqrt[a + b*Log[c*x^n]]), x])/(4
*e)

Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {a+b \log \left (c x^n\right )}}{2 e (d+e x)^2}+\frac {(b n) \int \frac {1}{x (d+e x)^2 \sqrt {a+b \log \left (c x^n\right )}} \, dx}{4 e} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 13.81 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx=\int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

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

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.73 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx=\int \frac {\sqrt {a + b \log {\left (c x^{n} \right )}}}{\left (d + e x\right )^{3}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx=\int { \frac {\sqrt {b \log \left (c x^{n}\right ) + a}}{{\left (e x + d\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx=\int { \frac {\sqrt {b \log \left (c x^{n}\right ) + a}}{{\left (e x + d\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {a+b \log \left (c x^n\right )}}{(d+e x)^3} \, dx=\int \frac {\sqrt {a+b\,\ln \left (c\,x^n\right )}}{{\left (d+e\,x\right )}^3} \,d x \]

[In]

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

[Out]

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