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

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

[Out]

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

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Rubi [A]  time = 0.0411681, 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 = {} \[ \int \frac{x \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0997893, size = 87, normalized size = 3.78 \[ \frac{x^2 \left (2 \, _2F_1\left (1,\frac{2}{r};\frac{r+2}{r};-\frac{e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )-b n \, _3F_2\left (1,\frac{2}{r},\frac{2}{r};1+\frac{2}{r},1+\frac{2}{r};-\frac{e x^r}{d}\right )\right )}{4 d} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.659, size = 0, normalized size = 0. \begin{align*} \int{\frac{x \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{d+e{x}^{r}}}\, 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),x)

[Out]

int(x*(a+b*ln(c*x^n))/(d+e*x^r),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}{e x^{r} + d}\,{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),x, algorithm="maxima")

[Out]

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

[Out]

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

[Out]

Integral(x*(a + b*log(c*x**n))/(d + e*x**r), 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}{e x^{r} + d}\,{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),x, algorithm="giac")

[Out]

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