3.414 \(\int \frac{x (a+b \log (c x^n))}{(d+e x^r)^2} \, dx\)

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\frac{x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2},x\right ) \]

[Out]

Unintegrable[(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2, x]

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Rubi [A]  time = 0.0397893, 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 = {} \[ \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

[Out]

Defer[Int][(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2, x]

Rubi steps

\begin{align*} \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx &=\int \frac{x \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.233625, size = 140, normalized size = 6.09 \[ \frac{x^2 \left (-b n (r-2) \left (d+e x^r\right ) \, _3F_2\left (1,\frac{2}{r},\frac{2}{r};1+\frac{2}{r},1+\frac{2}{r};-\frac{e x^r}{d}\right )+2 \left (d+e x^r\right ) \, _2F_1\left (1,\frac{2}{r};\frac{r+2}{r};-\frac{e x^r}{d}\right ) \left (a (r-2)+b (r-2) \log \left (c x^n\right )-b n\right )+4 d \left (a+b \log \left (c x^n\right )\right )\right )}{4 d^2 r \left (d+e x^r\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x*(a + b*Log[c*x^n]))/(d + e*x^r)^2,x]

[Out]

(x^2*(-(b*n*(-2 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 2/r, 2/r}, {1 + 2/r, 1 + 2/r}, -((e*x^r)/d)]) + 4*d*(a
+ b*Log[c*x^n]) + 2*(d + e*x^r)*Hypergeometric2F1[1, 2/r, (2 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-2 + r) + b*(-
2 + r)*Log[c*x^n])))/(4*d^2*r*(d + e*x^r))

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Maple [A]  time = 0.709, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{ \left ( d+e{x}^{r} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*ln(c*x^n))/(d+e*x^r)^2,x)

[Out]

int(x*(a+b*ln(c*x^n))/(d+e*x^r)^2,x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x}{{\left (e x^{r} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n))/(d+e*x^r)^2,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)*x/(e*x^r + d)^2, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b x \log \left (c x^{n}\right ) + a x}{e^{2} x^{2 \, r} + 2 \, d e x^{r} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n))/(d+e*x^r)^2,x, algorithm="fricas")

[Out]

integral((b*x*log(c*x^n) + a*x)/(e^2*x^(2*r) + 2*d*e*x^r + d^2), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x \left (a + b \log{\left (c x^{n} \right )}\right )}{\left (d + e x^{r}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*ln(c*x**n))/(d+e*x**r)**2,x)

[Out]

Integral(x*(a + b*log(c*x**n))/(d + e*x**r)**2, x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x}{{\left (e x^{r} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*log(c*x^n))/(d+e*x^r)^2,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x/(e*x^r + d)^2, x)