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

Optimal. Leaf size=17 \[ \text{Unintegrable}\left (x (a+b x)^m \log \left (c x^n\right ),x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0189418, 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 = {} \[ \int x (a+b x)^m \log \left (c x^n\right ) \, dx \]

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*}

Mathematica [A]  time = 0.2398, size = 173, normalized size = 10.18 \[ \frac{(a+b x)^m \left (\frac{b x}{a}+1\right )^{-m} \left (a b (m+2) n x \, _3F_2\left (1,1,-m-1;2,2;-\frac{b x}{a}\right )+\left (-a^2 \left (\left (\frac{b x}{a}+1\right )^m-1\right )+b^2 (m+1) x^2 \left (\frac{b x}{a}+1\right )^m+a b m x \left (\frac{b x}{a}+1\right )^m\right ) \log \left (c x^n\right )-n \left (a^2 \left (\left (\frac{b x}{a}+1\right )^m-1\right )+b^2 x^2 \left (\frac{b x}{a}+1\right )^m+2 a b x \left (\frac{b x}{a}+1\right )^m\right )\right )}{b^2 (m+1) (m+2)} \]

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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Maple [A]  time = 0.618, size = 0, normalized size = 0. \begin{align*} \int x \left ( bx+a \right ) ^{m}\ln \left ( c{x}^{n} \right ) \, dx \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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

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

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

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)