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

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

[Out]

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

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Rubi [A]  time = 0.0622788, 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 = {} \[ \int \frac{a+b \log \left (c x^n\right )}{x^2 \left (d+e x^r\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.201404, size = 135, normalized size = 5.4 \[ \frac{-b n (r+1) \left (d+e x^r\right ) \, _3F_2\left (1,-\frac{1}{r},-\frac{1}{r};1-\frac{1}{r},1-\frac{1}{r};-\frac{e x^r}{d}\right )-\left (d+e x^r\right ) \, _2F_1\left (1,-\frac{1}{r};\frac{r-1}{r};-\frac{e x^r}{d}\right ) \left (a r+a+b (r+1) \log \left (c x^n\right )-b n\right )+d \left (a+b \log \left (c x^n\right )\right )}{d^2 r x \left (d+e x^r\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [A]  time = 0.75, 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 ) ^{2}}}\, 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)^2,x)

[Out]

int((a+b*ln(c*x^n))/x^2/(d+e*x^r)^2,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 )}^{2} 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)^2,x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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)**2,x)

[Out]

Timed out

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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 )}^{2} 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)^2,x, algorithm="giac")

[Out]

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