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

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

[Out]

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

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Rubi [A]  time = 0.0628515, 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 = {} \[ \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.238609, size = 140, normalized size = 5.6 \[ \frac{x^3 \left (-b n (r-3) \left (d+e x^r\right ) \, _3F_2\left (1,\frac{3}{r},\frac{3}{r};1+\frac{3}{r},1+\frac{3}{r};-\frac{e x^r}{d}\right )+3 \left (d+e x^r\right ) \, _2F_1\left (1,\frac{3}{r};\frac{r+3}{r};-\frac{e x^r}{d}\right ) \left (a (r-3)+b (r-3) \log \left (c x^n\right )-b n\right )+9 d \left (a+b \log \left (c x^n\right )\right )\right )}{9 d^2 r \left (d+e x^r\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(x^3*(-(b*n*(-3 + r)*(d + e*x^r)*HypergeometricPFQ[{1, 3/r, 3/r}, {1 + 3/r, 1 + 3/r}, -((e*x^r)/d)]) + 9*d*(a
+ b*Log[c*x^n]) + 3*(d + e*x^r)*Hypergeometric2F1[1, 3/r, (3 + r)/r, -((e*x^r)/d)]*(-(b*n) + a*(-3 + r) + b*(-
3 + r)*Log[c*x^n])))/(9*d^2*r*(d + e*x^r))

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Maple [A]  time = 0.744, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) }{ \left ( d+e{x}^{r} \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^2*(a+b*ln(c*x^n))/(d+e*x^r)^2,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^{2}}{{\left (e x^{r} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^2*log(c*x^n) + a*x^2)/(e^2*x^(2*r) + 2*d*e*x^r + d^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(x**2*(a+b*ln(c*x**n))/(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{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{2}}{{\left (e x^{r} + d\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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