3.7.25 \(\int \frac {a+b \log (c (d+e x^m)^n)}{x \log (f x^p)} \, dx\) [625]

Optimal. Leaf size=42 \[ \frac {a \log \left (\log \left (f x^p\right )\right )}{p}+b \text {Int}\left (\frac {\log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx \end {gather*}

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 \text {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*}

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Mathematica [A]
time = 0.39, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log \left (f x^p\right )} \, dx \end {gather*}

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 = 0.02, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (d +e \,x^{m}\right )^{n}\right )}{x \ln \left (f \,x^{p}\right )}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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(c) + log((d + e^(m*log(x) + 1))^n))/(x*log(f) + x*log(x^p)), x) + a*log(log(f*x^p))/p

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c \left (d + e x^{m}\right )^{n} \right )}}{x \log {\left (f x^{p} \right )}}\, dx \end {gather*}

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]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x^m\right )}^n\right )}{x\,\ln \left (f\,x^p\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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