3.627 \(\int \frac{a+b \log (c (d+e x^m)^n)}{x \log ^3(f x^p)} \, dx\)

Optimal. Leaf size=68 \[ \frac{b e m n \text{Unintegrable}\left (\frac{x^{m-1}}{\left (d+e x^m\right ) \log ^2\left (f x^p\right )},x\right )}{2 p}-\frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{2 p \log ^2\left (f x^p\right )} \]

[Out]

-(a + b*Log[c*(d + e*x^m)^n])/(2*p*Log[f*x^p]^2) + (b*e*m*n*Unintegrable[x^(-1 + m)/((d + e*x^m)*Log[f*x^p]^2)
, x])/(2*p)

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Rubi [A]  time = 0.116714, 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 = {} \[ \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^3),x]

[Out]

-(a + b*Log[c*(d + e*x^m)^n])/(2*p*Log[f*x^p]^2) + (b*e*m*n*Defer[Int][x^(-1 + m)/((d + e*x^m)*Log[f*x^p]^2),
x])/(2*p)

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx &=-\frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{2 p \log ^2\left (f x^p\right )}+\frac{(b e m n) \int \frac{x^{-1+m}}{\left (d+e x^m\right ) \log ^2\left (f x^p\right )} \, dx}{2 p}\\ \end{align*}

Mathematica [A]  time = 10.6062, size = 0, normalized size = 0. \[ \int \frac{a+b \log \left (c \left (d+e x^m\right )^n\right )}{x \log ^3\left (f x^p\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^3),x]

[Out]

Integrate[(a + b*Log[c*(d + e*x^m)^n])/(x*Log[f*x^p]^3), x]

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Maple [A]  time = 0.468, 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 ) ^{3}}}\, 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)^3,x)

[Out]

int((a+b*ln(c*(d+e*x^m)^n))/x/ln(f*x^p)^3,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \,{\left (2 \, d e m^{2} n \int \frac{x^{m}}{2 \,{\left (e^{2} p^{2} x x^{2 \, m} \log \left (f\right ) + 2 \, d e p^{2} x x^{m} \log \left (f\right ) + d^{2} p^{2} x \log \left (f\right ) +{\left (e^{2} p^{2} x x^{2 \, m} + 2 \, d e p^{2} x x^{m} + d^{2} p^{2} x\right )} \log \left (x^{p}\right )\right )}}\,{d x} - \frac{e m n x^{m} \log \left (x^{p}\right ) + d p \log \left (c\right ) +{\left (e m n \log \left (f\right ) + e p \log \left (c\right )\right )} x^{m} +{\left (e p x^{m} + d p\right )} \log \left ({\left (e x^{m} + d\right )}^{n}\right )}{e p^{2} x^{m} \log \left (f\right )^{2} + d p^{2} \log \left (f\right )^{2} +{\left (e p^{2} x^{m} + d p^{2}\right )} \log \left (x^{p}\right )^{2} + 2 \,{\left (e p^{2} x^{m} \log \left (f\right ) + d p^{2} \log \left (f\right )\right )} \log \left (x^{p}\right )}\right )} b - \frac{a}{2 \, p \log \left (f x^{p}\right )^{2}} \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)^3,x, algorithm="maxima")

[Out]

1/2*(2*d*e*m^2*n*integrate(1/2*x^m/(e^2*p^2*x*x^(2*m)*log(f) + 2*d*e*p^2*x*x^m*log(f) + d^2*p^2*x*log(f) + (e^
2*p^2*x*x^(2*m) + 2*d*e*p^2*x*x^m + d^2*p^2*x)*log(x^p)), x) - (e*m*n*x^m*log(x^p) + d*p*log(c) + (e*m*n*log(f
) + e*p*log(c))*x^m + (e*p*x^m + d*p)*log((e*x^m + d)^n))/(e*p^2*x^m*log(f)^2 + d*p^2*log(f)^2 + (e*p^2*x^m +
d*p^2)*log(x^p)^2 + 2*(e*p^2*x^m*log(f) + d*p^2*log(f))*log(x^p)))*b - 1/2*a/(p*log(f*x^p)^2)

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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 )^{3}}, 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)^3,x, algorithm="fricas")

[Out]

integral((b*log((e*x^m + d)^n*c) + a)/(x*log(f*x^p)^3), 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)**3,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 )^{3}}\,{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)^3,x, algorithm="giac")

[Out]

integrate((b*log((e*x^m + d)^n*c) + a)/(x*log(f*x^p)^3), x)