\(\int x (a+b x)^m \log (c x^n) \, dx\) [168]

Optimal result

Integrand size = 15, antiderivative size = 15 \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx=\text {Int}\left (x (a+b x)^m \log \left (c x^n\right ),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 15, 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 = {} \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx=\int x (a+b x)^m \log \left (c x^n\right ) \, dx \]

[In]

Int[x*(a + b*x)^m*Log[c*x^n],x]

[Out]

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(173\) vs. \(2(18)=36\).

Time = 0.16 (sec) , antiderivative size = 173, normalized size of antiderivative = 11.53 \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx=\frac {(a+b x)^m \left (1+\frac {b x}{a}\right )^{-m} \left (-n \left (2 a b x \left (1+\frac {b x}{a}\right )^m+b^2 x^2 \left (1+\frac {b x}{a}\right )^m+a^2 \left (-1+\left (1+\frac {b x}{a}\right )^m\right )\right )+a b (2+m) n x \, _3F_2\left (1,1,-1-m;2,2;-\frac {b x}{a}\right )+\left (a b m x \left (1+\frac {b x}{a}\right )^m+b^2 (1+m) x^2 \left (1+\frac {b x}{a}\right )^m-a^2 \left (-1+\left (1+\frac {b x}{a}\right )^m\right )\right ) \log \left (c x^n\right )\right )}{b^2 (1+m) (2+m)} \]

[In]

Integrate[x*(a + b*x)^m*Log[c*x^n],x]

[Out]

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

Maple [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.33 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx=\int { {\left (b x + a\right )}^{m} x \log \left (c x^{n}\right ) \,d x } \]

[In]

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

[Out]

integral((b*x + a)^m*x*log(c*x^n), x)

Sympy [N/A]

Not integrable

Time = 7.50 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx=\int x \left (a + b x\right )^{m} \log {\left (c x^{n} \right )}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 7.47 \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx=\int { {\left (b x + a\right )}^{m} x \log \left (c x^{n}\right ) \,d x } \]

[In]

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

[Out]

(b^2*(m + 1)*x^2 + a*b*m*x - a^2)*(b*x + a)^m*log(x^n)/((m^2 + 3*m + 2)*b^2) + integrate(-(a*b*m*n*x + (m*n -
(m^2 + 3*m + 2)*log(c) + n)*b^2*x^2 - a^2*n)*(b*x + a)^m/x, x)/((m^2 + 3*m + 2)*b^2)

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx=\int { {\left (b x + a\right )}^{m} x \log \left (c x^{n}\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.13 \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx=\int x\,\ln \left (c\,x^n\right )\,{\left (a+b\,x\right )}^m \,d x \]

[In]

int(x*log(c*x^n)*(a + b*x)^m,x)

[Out]

int(x*log(c*x^n)*(a + b*x)^m, x)