\(\int \frac {(a+b \log (c (d+e \sqrt {x})^2))^p}{x^2} \, dx\) [543]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2} \, dx=\text {Int}\left (\frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2} \, dx=\int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2} \, dx \]

[In]

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

[Out]

2*Defer[Subst][Defer[Int][(a + b*Log[c*(d + e*x)^2])^p/x^3, x], x, Sqrt[x]]

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

Mathematica [N/A]

Not integrable

Time = 0.11 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2} \, dx=\int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 1.62 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e \sqrt {x} + d\right )}^{2} c\right ) + a\right )}^{p}}{x^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.56 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^2\right )\right )^p}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^2\right )\right )}^p}{x^2} \,d x \]

[In]

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

[Out]

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