\(\int \frac {(a+b \log (c (d+e x)^n))^2 (f+g \log (h (i+j x)^m))}{x} \, dx\) [395]

Optimal result

Integrand size = 34, antiderivative size = 34 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\text {Int}\left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*(e*x+d)^n))^2*(f+g*ln(h*(j*x+i)^m))/x,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 34, 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 = {} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx \]

[In]

Int[((a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]))/x,x]

[Out]

Defer[Int][((a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]))/x, x]

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

Mathematica [N/A]

Not integrable

Time = 0.66 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx \]

[In]

Integrate[((a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]))/x,x]

[Out]

Integrate[((a + b*Log[c*(d + e*x)^n])^2*(f + g*Log[h*(i + j*x)^m]))/x, x]

Maple [N/A]

Not integrable

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00

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

[In]

int((a+b*ln(c*(e*x+d)^n))^2*(f+g*ln(h*(j*x+i)^m))/x,x)

[Out]

int((a+b*ln(c*(e*x+d)^n))^2*(f+g*ln(h*(j*x+i)^m))/x,x)

Fricas [N/A]

Not integrable

Time = 0.36 (sec) , antiderivative size = 93, normalized size of antiderivative = 2.74 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^2*(f+g*log(h*(j*x+i)^m))/x,x, algorithm="fricas")

[Out]

integral((b^2*f*log((e*x + d)^n*c)^2 + 2*a*b*f*log((e*x + d)^n*c) + a^2*f + (b^2*g*log((e*x + d)^n*c)^2 + 2*a*
b*g*log((e*x + d)^n*c) + a^2*g)*log((j*x + i)^m*h))/x, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\text {Timed out} \]

[In]

integrate((a+b*ln(c*(e*x+d)**n))**2*(f+g*ln(h*(j*x+i)**m))/x,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 168, normalized size of antiderivative = 4.94 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^2*(f+g*log(h*(j*x+i)^m))/x,x, algorithm="maxima")

[Out]

a^2*f*log(x) + integrate(((g*log(h) + f)*b^2*log((e*x + d)^n)^2 + (g*log(h) + f)*b^2*log(c)^2 + 2*(g*log(h) +
f)*a*b*log(c) + a^2*g*log(h) + 2*((g*log(h) + f)*b^2*log(c) + (g*log(h) + f)*a*b)*log((e*x + d)^n) + (b^2*g*lo
g((e*x + d)^n)^2 + b^2*g*log(c)^2 + 2*a*b*g*log(c) + a^2*g + 2*(b^2*g*log(c) + a*b*g)*log((e*x + d)^n))*log((j
*x + i)^m))/x, x)

Giac [N/A]

Not integrable

Time = 0.37 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^2*(f+g*log(h*(j*x+i)^m))/x,x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^2*(g*log((j*x + i)^m*h) + f)/x, x)

Mupad [N/A]

Not integrable

Time = 1.88 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right )}{x} \,d x \]

[In]

int(((a + b*log(c*(d + e*x)^n))^2*(f + g*log(h*(i + j*x)^m)))/x,x)

[Out]

int(((a + b*log(c*(d + e*x)^n))^2*(f + g*log(h*(i + j*x)^m)))/x, x)