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

Optimal. Leaf size=36 \[ \text{Unintegrable}\left (\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0396986, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (396+j x)^m\right )\right )}{x^2} \, dx &=\int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (396+j x)^m\right )\right )}{x^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.865298, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2 \left (f+g \log \left (h (i+j x)^m\right )\right )}{x^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 1.424, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2} \left ( f+g\ln \left ( h \left ( jx+i \right ) ^{m} \right ) \right ) }{{x}^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -2 \, a b e f n{\left (\frac{\log \left (e x + d\right )}{d} - \frac{\log \left (x\right )}{d}\right )} - \frac{2 \, a b f \log \left ({\left (e x + d\right )}^{n} c\right )}{x} - \frac{a^{2} f}{x} + \int \frac{{\left (g \log \left (h\right ) + f\right )} b^{2} \log \left ({\left (e x + d\right )}^{n}\right )^{2} +{\left (g \log \left (h\right ) + f\right )} b^{2} \log \left (c\right )^{2} + 2 \, a b g \log \left (c\right ) \log \left (h\right ) + a^{2} g \log \left (h\right ) + 2 \,{\left ({\left (g \log \left (h\right ) + f\right )} b^{2} \log \left (c\right ) + a b g \log \left (h\right )\right )} \log \left ({\left (e x + d\right )}^{n}\right ) +{\left (b^{2} g \log \left ({\left (e x + d\right )}^{n}\right )^{2} + b^{2} g \log \left (c\right )^{2} + 2 \, a b g \log \left (c\right ) + a^{2} g + 2 \,{\left (b^{2} g \log \left (c\right ) + a b g\right )} \log \left ({\left (e x + d\right )}^{n}\right )\right )} \log \left ({\left (j x + i\right )}^{m}\right )}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} f \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b f \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} f +{\left (b^{2} g \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b g \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} g\right )} \log \left ({\left (j x + i\right )}^{m} h\right )}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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