\(\int \frac {(a+b x)^m \log (c x^n)}{x} \, dx\) [170]

Optimal result

Integrand size = 17, antiderivative size = 17 \[ \int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx=\text {Int}\left (\frac {(a+b x)^m \log \left (c x^n\right )}{x},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx=\int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx \]

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(89\) vs. \(2(20)=40\).

Time = 0.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 5.24 \[ \int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx=\frac {\left (1+\frac {a}{b x}\right )^{-m} (a+b x)^m \left (-n \, _3F_2\left (-m,-m,-m;1-m,1-m;-\frac {a}{b x}\right )+m \operatorname {Hypergeometric2F1}\left (-m,-m,1-m,-\frac {a}{b x}\right ) \log \left (c x^n\right )\right )}{m^2} \]

[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)

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00

\[\int \frac {\left (b x +a \right )^{m} \ln \left (c \,x^{n}\right )}{x}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} \log \left (c x^{n}\right )}{x} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 4.98 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx=\int \frac {\left (a + b x\right )^{m} \log {\left (c x^{n} \right )}}{x}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx=\int { \frac {{\left (b x + a\right )}^{m} \log \left (c x^{n}\right )}{x} \,d x } \]

[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)

Giac [F(-2)]

Exception generated. \[ \int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:Unable to divide,
perhaps due to rounding error%%%{1,[0,1,0]%%%} / %%%{1,[0,0,1]%%%} Error: Bad Argument Value

Mupad [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.12 \[ \int \frac {(a+b x)^m \log \left (c x^n\right )}{x} \, dx=\int \frac {\ln \left (c\,x^n\right )\,{\left (a+b\,x\right )}^m}{x} \,d x \]

[In]

int((log(c*x^n)*(a + b*x)^m)/x,x)

[Out]

int((log(c*x^n)*(a + b*x)^m)/x, x)