3.409 \(\int \frac{a+b \log (c x^n)}{x^3 (d+e x^r)} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )},x\right ) \]

[Out]

Unintegrable[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)), x]

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Rubi [A]  time = 0.0640177, 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 = {} \[ \int \frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)),x]

[Out]

Defer[Int][(a + b*Log[c*x^n])/(x^3*(d + e*x^r)), x]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )} \, dx &=\int \frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )} \, dx\\ \end{align*}

Mathematica [A]  time = 0.104514, size = 86, normalized size = 3.44 \[ -\frac{b n \, _3F_2\left (1,-\frac{2}{r},-\frac{2}{r};1-\frac{2}{r},1-\frac{2}{r};-\frac{e x^r}{d}\right )+2 \, _2F_1\left (1,-\frac{2}{r};\frac{r-2}{r};-\frac{e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{4 d x^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)),x]

[Out]

-(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*x^2)

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Maple [A]  time = 0.706, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\ln \left ( c{x}^{n} \right ) }{{x}^{3} \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^3/(d+e*x^r),x)

[Out]

int((a+b*ln(c*x^n))/x^3/(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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(d+e*x^r),x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)/((e*x^r + d)*x^3), 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^{3} x^{r} + d x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(d+e*x^r),x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(e*x^3*x^r + d*x^3), 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^{3} \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**3/(d+e*x**r),x)

[Out]

Integral((a + b*log(c*x**n))/(x**3*(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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(d+e*x^r),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((e*x^r + d)*x^3), x)