3.143 \(\int \frac{\log (d (e+f x^m)^r)}{x (a+b \log (c x^n))^2} \, dx\)

Optimal. Leaf size=30 \[ \text{Unintegrable}\left (\frac{\log \left (d \left (e+f x^m\right )^r\right )}{x \left (a+b \log \left (c x^n\right )\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0337653, 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 = {} \[ \int \frac{\log \left (d \left (e+f x^m\right )^r\right )}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 1.77742, size = 0, normalized size = 0. \[ \int \frac{\log \left (d \left (e+f x^m\right )^r\right )}{x \left (a+b \log \left (c x^n\right )\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{\frac{\ln \left ( d \left ( e+f{x}^{m} \right ) ^{r} \right ) }{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*(e+f*x^m)^r)/x/(a+b*ln(c*x^n))^2,x)

[Out]

int(ln(d*(e+f*x^m)^r)/x/(a+b*ln(c*x^n))^2,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((f*x^m + e)^r*d)/(b^2*x*log(c*x^n)^2 + 2*a*b*x*log(c*x^n) + a^2*x), x)

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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(ln(d*(e+f*x**m)**r)/x/(a+b*ln(c*x**n))**2,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((f*x^m + e)^r*d)/((b*log(c*x^n) + a)^2*x), x)