3.416 \(\int \frac{a+b \log (c x^n)}{x^3 (d+e x^r)^2} \, dx\)

Optimal. Leaf size=25 \[ \text{Unintegrable}\left (\frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2},x\right ) \]

[Out]

Unintegrable[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2), x]

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Rubi [A]  time = 0.0629914, 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 = {} \[ \int \frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2),x]

[Out]

Defer[Int][(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2), x]

Rubi steps

\begin{align*} \int \frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx &=\int \frac{a+b \log \left (c x^n\right )}{x^3 \left (d+e x^r\right )^2} \, dx\\ \end{align*}

Mathematica [A]  time = 0.23461, size = 139, normalized size = 5.56 \[ -\frac{b n (r+2) \left (d+e x^r\right ) \, _3F_2\left (1,-\frac{2}{r},-\frac{2}{r};1-\frac{2}{r},1-\frac{2}{r};-\frac{e x^r}{d}\right )+2 \left (d+e x^r\right ) \, _2F_1\left (1,-\frac{2}{r};\frac{r-2}{r};-\frac{e x^r}{d}\right ) \left (a (r+2)+b (r+2) \log \left (c x^n\right )-b n\right )-4 d \left (a+b \log \left (c x^n\right )\right )}{4 d^2 r x^2 \left (d+e x^r\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x^r)^2),x]

[Out]

-(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*Lo
g[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*x^2*(d + e*x^r))

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Maple [A]  time = 0.642, 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 ) ^{2}}}\, 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)^2,x)

[Out]

int((a+b*ln(c*x^n))/x^3/(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^{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)^2,x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)/((e*x^r + d)^2*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^{2} x^{3} x^{2 \, r} + 2 \, d e x^{3} x^{r} + d^{2} 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)^2,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(e^2*x^3*x^(2*r) + 2*d*e*x^3*x^r + d^2*x^3), 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**3/(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^{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)^2,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((e*x^r + d)^2*x^3), x)