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

Optimal. Leaf size=41 \[ b \text{Unintegrable}\left (\frac{\log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )},x\right )+\frac{a \log \left (\log \left (f x^p\right )\right )}{p} \]

[Out]

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

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Rubi [A]  time = 0.292362, 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 = {} \[ \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx &=\int \left (\frac{a}{x \log \left (f x^p\right )}+\frac{b \log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )}\right ) \, dx\\ &=a \int \frac{1}{x \log \left (f x^p\right )} \, dx+b \int \frac{\log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx\\ &=b \int \frac{\log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx+\frac{a \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,\log \left (f x^p\right )\right )}{p}\\ &=\frac{a \log \left (\log \left (f x^p\right )\right )}{p}+b \int \frac{\log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.54297, size = 0, normalized size = 0. \[ \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.474, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( d+e{x}^{m} \right ) ^{n} \right ) }{x\ln \left ( f{x}^{p} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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