3.170 \(\int \frac{(a+b x)^m \log (c x^n)}{x} \, dx\)

Optimal. Leaf size=19 \[ \text{Unintegrable}\left (\frac{(a+b x)^m \log \left (c x^n\right )}{x},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0291745, 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 = {} \[ \int \frac{(a+b x)^m \log \left (c x^n\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((a + b*x)^m*Log[c*x^n])/x,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0541875, size = 89, normalized size = 4.68 \[ \frac{\left (\frac{a}{b x}+1\right )^{-m} (a+b x)^m \left (m \log \left (c x^n\right ) \, _2F_1\left (-m,-m;1-m;-\frac{a}{b x}\right )-n \, _3F_2\left (-m,-m,-m;1-m,1-m;-\frac{a}{b x}\right )\right )}{m^2} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((a + b*x)^m*(-(n*HypergeometricPFQ[{-m, -m, -m}, {1 - m, 1 - m}, -(a/(b*x))]) + m*Hypergeometric2F1[-m, -m, 1
- m, -(a/(b*x))]*Log[c*x^n]))/(m^2*(1 + a/(b*x))^m)

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Maple [A]  time = 0.555, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( bx+a \right ) ^{m}\ln \left ( c{x}^{n} \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x + a)^m*log(c*x^n)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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