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

Optimal. Leaf size=63 \[ \frac{b e m n \text{Unintegrable}\left (\frac{x^{m-1}}{\left (d+e x^m\right ) \log \left (f x^p\right )},x\right )}{p}-\frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{p \log \left (f x^p\right )} \]

[Out]

-((a + b*Log[c*(d + e*x^m)^n])/(p*Log[f*x^p])) + (b*e*m*n*Unintegrable[x^(-1 + m)/((d + e*x^m)*Log[f*x^p]), x]
)/p

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Rubi [A]  time = 0.119513, 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 = {} \[ \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^2\left (f x^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-((a + b*Log[c*(d + e*x^m)^n])/(p*Log[f*x^p])) + (b*e*m*n*Defer[Int][x^(-1 + m)/((d + e*x^m)*Log[f*x^p]), x])/
p

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^2\left (f x^p\right )} \, dx &=-\frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{p \log \left (f x^p\right )}+\frac{(b e m n) \int \frac{x^{-1+m}}{\left (d+e x^m\right ) \log \left (f x^p\right )} \, dx}{p}\\ \end{align*}

Mathematica [A]  time = 2.00025, size = 0, normalized size = 0. \[ \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^2\left (f x^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.48, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c \left ( d+e{x}^{m} \right ) ^{n} \right ) }{x \left ( \ln \left ( f{x}^{p} \right ) \right ) ^{2}}}\, 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)^2,x)

[Out]

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

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

[Out]

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

[Out]

integral((b*log((e*x^m + d)^n*c) + a)/(x*log(f*x^p)^2), 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)**2,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 )^{2}}\,{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)^2,x, algorithm="giac")

[Out]

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