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

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0655535, 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 = {} \[ \int \frac{a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.0973344, size = 83, normalized size = 3.32 \[ -\frac{b n \, _3F_2\left (1,-\frac{1}{r},-\frac{1}{r};1-\frac{1}{r},1-\frac{1}{r};-\frac{e x^r}{d}\right )+\, _2F_1\left (1,-\frac{1}{r};\frac{r-1}{r};-\frac{e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{d x} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.687, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{2} \left ( d+e{x}^{r} \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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