\(\int \frac {(a+b \log (c (d+e x)^n))^3 (f+g \log (h (i+j x)^m))}{x^2} \, dx\) [400]

Optimal result

Integrand size = 34, antiderivative size = 34 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\text {Int}\left (\frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2},x\right ) \]

[Out]

Unintegrable((a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m))/x^2,x)

Rubi [N/A]

Not integrable

Time = 0.03 (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 = {} \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx \]

[In]

Int[((a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]))/x^2,x]

[Out]

Defer[Int][((a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]))/x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 2.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx \]

[In]

Integrate[((a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]))/x^2,x]

[Out]

Integrate[((a + b*Log[c*(d + e*x)^n])^3*(f + g*Log[h*(i + j*x)^m]))/x^2, x]

Maple [N/A]

Not integrable

Time = 0.12 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00

\[\int \frac {{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{3} \left (f +g \ln \left (h \left (j x +i \right )^{m}\right )\right )}{x^{2}}d x\]

[In]

int((a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m))/x^2,x)

[Out]

int((a+b*ln(c*(e*x+d)^n))^3*(f+g*ln(h*(j*x+i)^m))/x^2,x)

Fricas [N/A]

Not integrable

Time = 0.32 (sec) , antiderivative size = 135, normalized size of antiderivative = 3.97 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x^{2}} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^3*(f+g*log(h*(j*x+i)^m))/x^2,x, algorithm="fricas")

[Out]

integral((b^3*f*log((e*x + d)^n*c)^3 + 3*a*b^2*f*log((e*x + d)^n*c)^2 + 3*a^2*b*f*log((e*x + d)^n*c) + a^3*f +
(b^3*g*log((e*x + d)^n*c)^3 + 3*a*b^2*g*log((e*x + d)^n*c)^2 + 3*a^2*b*g*log((e*x + d)^n*c) + a^3*g)*log((j*x
+ i)^m*h))/x^2, x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\text {Timed out} \]

[In]

integrate((a+b*ln(c*(e*x+d)**n))**3*(f+g*ln(h*(j*x+i)**m))/x**2,x)

[Out]

Timed out

Maxima [N/A]

Not integrable

Time = 0.56 (sec) , antiderivative size = 335, normalized size of antiderivative = 9.85 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x^{2}} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^3*(f+g*log(h*(j*x+i)^m))/x^2,x, algorithm="maxima")

[Out]

-3*a^2*b*e*f*n*(log(e*x + d)/d - log(x)/d) - 3*a^2*b*f*log((e*x + d)^n*c)/x - a^3*f/x + integrate(((g*log(h) +
f)*b^3*log((e*x + d)^n)^3 + (g*log(h) + f)*b^3*log(c)^3 + 3*(g*log(h) + f)*a*b^2*log(c)^2 + 3*a^2*b*g*log(c)*
log(h) + a^3*g*log(h) + 3*((g*log(h) + f)*b^3*log(c) + (g*log(h) + f)*a*b^2)*log((e*x + d)^n)^2 + 3*((g*log(h)
+ f)*b^3*log(c)^2 + 2*(g*log(h) + f)*a*b^2*log(c) + a^2*b*g*log(h))*log((e*x + d)^n) + (b^3*g*log((e*x + d)^n
)^3 + b^3*g*log(c)^3 + 3*a*b^2*g*log(c)^2 + 3*a^2*b*g*log(c) + a^3*g + 3*(b^3*g*log(c) + a*b^2*g)*log((e*x + d
)^n)^2 + 3*(b^3*g*log(c)^2 + 2*a*b^2*g*log(c) + a^2*b*g)*log((e*x + d)^n))*log((j*x + i)^m))/x^2, x)

Giac [N/A]

Not integrable

Time = 0.59 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{3} {\left (g \log \left ({\left (j x + i\right )}^{m} h\right ) + f\right )}}{x^{2}} \,d x } \]

[In]

integrate((a+b*log(c*(e*x+d)^n))^3*(f+g*log(h*(j*x+i)^m))/x^2,x, algorithm="giac")

[Out]

integrate((b*log((e*x + d)^n*c) + a)^3*(g*log((j*x + i)^m*h) + f)/x^2, x)

Mupad [N/A]

Not integrable

Time = 2.74 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.06 \[ \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^3\,\left (f+g\,\ln \left (h\,{\left (i+j\,x\right )}^m\right )\right )}{x^2} \,d x \]

[In]

int(((a + b*log(c*(d + e*x)^n))^3*(f + g*log(h*(i + j*x)^m)))/x^2,x)

[Out]

int(((a + b*log(c*(d + e*x)^n))^3*(f + g*log(h*(i + j*x)^m)))/x^2, x)