3.398 \(\int \frac {\log (x^{-n} (a+b x^n))}{c+d x} \, dx\)

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {\log \left (a x^{-n}+b\right )}{c+d x},x\right ) \]

[Out]

Unintegrable(ln(b+a/(x^n))/(d*x+c),x)

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Rubi [A]  time = 0.03, 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 = {} \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Log[(a + b*x^n)/x^n]/(c + d*x),x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (x^{-n} \left (a+b x^n\right )\right )}{c+d x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\log \left (\frac {b x^{n} + a}{x^{n}}\right )}{d x + c}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log((b*x^n + a)/x^n)/(d*x + c), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {b x^{n} + a}{x^{n}}\right )}{d x + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((b*x^n + a)/x^n)/(d*x + c), x)

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maple [A]  time = 1.68, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (\left (b \,x^{n}+a \right ) x^{-n}\right )}{d x +c}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln((b*x^n+a)/(x^n))/(d*x+c),x)

[Out]

int(ln((b*x^n+a)/(x^n))/(d*x+c),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (\frac {b x^{n} + a}{x^{n}}\right )}{d x + c}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log((b*x^n + a)/x^n)/(d*x + c), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {\ln \left (\frac {a+b\,x^n}{x^n}\right )}{c+d\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log((a + b*x^n)/x^n)/(c + d*x),x)

[Out]

int(log((a + b*x^n)/x^n)/(c + d*x), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log {\left (a x^{- n} + b \right )}}{c + d x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln((a+b*x**n)/(x**n))/(d*x+c),x)

[Out]

Integral(log(a*x**(-n) + b)/(c + d*x), x)

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