3.168 \(\int x (a+b x)^m \log (c x^n) \, dx\)

Optimal. Leaf size=18 \[ \text {Int}\left (x (a+b x)^m \log \left (c x^n\right ),x\right ) \]

[Out]

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

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Rubi [A]  time = 0.02, 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 = {} \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 173, normalized size = 9.61 \[ \frac {(a+b x)^m \left (\frac {b x}{a}+1\right )^{-m} \left (a b (m+2) n x \, _3F_2\left (1,1,-m-1;2,2;-\frac {b x}{a}\right )+\left (-a^2 \left (\left (\frac {b x}{a}+1\right )^m-1\right )+b^2 (m+1) x^2 \left (\frac {b x}{a}+1\right )^m+a b m x \left (\frac {b x}{a}+1\right )^m\right ) \log \left (c x^n\right )-n \left (a^2 \left (\left (\frac {b x}{a}+1\right )^m-1\right )+b^2 x^2 \left (\frac {b x}{a}+1\right )^m+2 a b x \left (\frac {b x}{a}+1\right )^m\right )\right )}{b^2 (m+1) (m+2)} \]

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b x + a\right )}^{m} x \log \left (c x^{n}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maple [A]  time = 0.65, size = 0, normalized size = 0.00 \[ \int x \left (b x +a \right )^{m} \ln \left (c \,x^{n}\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (b^{2} {\left (m + 1\right )} x^{2} + a b m x - a^{2}\right )} {\left (b x + a\right )}^{m} \log \left (x^{n}\right )}{{\left (m^{2} + 3 \, m + 2\right )} b^{2}} + \frac {-\frac {{\left (b x + a\right )}^{m + 1} a m n}{m + 1} + \int -\frac {{\left ({\left (m n - {\left (m^{2} + 3 \, m + 2\right )} \log \relax (c) + n\right )} b^{2} x^{2} - a^{2} n\right )} {\left (b x + a\right )}^{m}}{x}\,{d x}}{{\left (m^{2} + 3 \, m + 2\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.06 \[ \int x\,\ln \left (c\,x^n\right )\,{\left (a+b\,x\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*log(c*x^n)*(a + b*x)^m,x)

[Out]

int(x*log(c*x^n)*(a + b*x)^m, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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