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

Optimal. Leaf size=267 \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2}-\frac{3 b^2 n^2 \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )}{d^3 r^3}+\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{d x^{-r}}{e}\right )}{d^3 r^3}+\frac{b e n x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2 \left (d+e x^r\right )}+\frac{3 b n \log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3 r}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}-\frac{b^2 n^2 \log \left (d+e x^r\right )}{d^3 r^3} \]

[Out]

(b*e*n*x^r*(a + b*Log[c*x^n]))/(d^3*r^2*(d + e*x^r)) + (a + b*Log[c*x^n])^2/(2*d*r*(d + e*x^r)^2) + (a + b*Log
[c*x^n])^2/(d^2*r*(d + e*x^r)) + (3*b*n*(a + b*Log[c*x^n])*Log[1 + d/(e*x^r)])/(d^3*r^2) - ((a + b*Log[c*x^n])
^2*Log[1 + d/(e*x^r)])/(d^3*r) - (b^2*n^2*Log[d + e*x^r])/(d^3*r^3) - (3*b^2*n^2*PolyLog[2, -(d/(e*x^r))])/(d^
3*r^3) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x^r))])/(d^3*r^2) + (2*b^2*n^2*PolyLog[3, -(d/(e*x^r))])/
(d^3*r^3)

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Rubi [A]  time = 0.891979, antiderivative size = 267, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.32, Rules used = {2349, 2345, 2374, 6589, 2338, 2391, 2335, 260} \[ \frac{2 b n \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2}-\frac{3 b^2 n^2 \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )}{d^3 r^3}+\frac{2 b^2 n^2 \text{PolyLog}\left (3,-\frac{d x^{-r}}{e}\right )}{d^3 r^3}+\frac{b e n x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2 \left (d+e x^r\right )}+\frac{3 b n \log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}-\frac{\log \left (\frac{d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3 r}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}-\frac{b^2 n^2 \log \left (d+e x^r\right )}{d^3 r^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*n*x^r*(a + b*Log[c*x^n]))/(d^3*r^2*(d + e*x^r)) + (a + b*Log[c*x^n])^2/(2*d*r*(d + e*x^r)^2) + (a + b*Log
[c*x^n])^2/(d^2*r*(d + e*x^r)) + (3*b*n*(a + b*Log[c*x^n])*Log[1 + d/(e*x^r)])/(d^3*r^2) - ((a + b*Log[c*x^n])
^2*Log[1 + d/(e*x^r)])/(d^3*r) - (b^2*n^2*Log[d + e*x^r])/(d^3*r^3) - (3*b^2*n^2*PolyLog[2, -(d/(e*x^r))])/(d^
3*r^3) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, -(d/(e*x^r))])/(d^3*r^2) + (2*b^2*n^2*PolyLog[3, -(d/(e*x^r))])/
(d^3*r^3)

Rule 2349

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_)^(r_.))^(q_))/(x_), x_Symbol] :> Dist[1/d,
Int[((d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p)/x, x], x] - Dist[e/d, Int[x^(r - 1)*(d + e*x^r)^q*(a + b*Log[c*
x^n])^p, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0] && ILtQ[q, -1]

Rule 2345

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> -Simp[(Log[1 +
d/(e*x^r)]*(a + b*Log[c*x^n])^p)/(d*r), x] + Dist[(b*n*p)/(d*r), Int[(Log[1 + d/(e*x^r)]*(a + b*Log[c*x^n])^(p
- 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(PolyLog[2, -(d*f*x^m)]*(a + b*Log[c*x^n])^p)/m, x] + Dist[(b*n*p)/m, Int[(PolyLog[2, -(d*f*x^m)]*(a + b*Log
[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_))^(q_.), x_Symbol] :
> Simp[(f^m*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^p)/(e*r*(q + 1)), x] - Dist[(b*f^m*n*p)/(e*r*(q + 1)), Int[
((d + e*x^r)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[
m, r - 1] && IGtQ[p, 0] && (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n] && NeQ[q, -1]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2335

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp
[((f*x)^(m + 1)*(d + e*x^r)^(q + 1)*(a + b*Log[c*x^n]))/(d*f*(m + 1)), x] - Dist[(b*n)/(d*(m + 1)), Int[(f*x)^
m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && EqQ[m + r*(q + 1) + 1, 0] && NeQ[
m, -1]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^3} \, dx &=\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^2} \, dx}{d}-\frac{e \int \frac{x^{-1+r} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^r\right )^3} \, dx}{d}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )} \, dx}{d^2}-\frac{e \int \frac{x^{-1+r} \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^r\right )^2} \, dx}{d^2}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )^2} \, dx}{d r}\\ &=\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{d x^{-r}}{e}\right )}{d^3 r}+\frac{(2 b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-r}}{e}\right )}{x} \, dx}{d^3 r}-\frac{(b n) \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx}{d^2 r}-\frac{(2 b n) \int \frac{a+b \log \left (c x^n\right )}{x \left (d+e x^r\right )} \, dx}{d^2 r}+\frac{(b e n) \int \frac{x^{-1+r} \left (a+b \log \left (c x^n\right )\right )}{\left (d+e x^r\right )^2} \, dx}{d^2 r}\\ &=\frac{b e n x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2 \left (d+e x^r\right )}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-r}}{e}\right )}{d^3 r^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{d x^{-r}}{e}\right )}{d^3 r}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{d^3 r^2}-\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1+\frac{d x^{-r}}{e}\right )}{x} \, dx}{d^3 r^2}-\frac{\left (2 b^2 n^2\right ) \int \frac{\log \left (1+\frac{d x^{-r}}{e}\right )}{x} \, dx}{d^3 r^2}-\frac{\left (2 b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{x} \, dx}{d^3 r^2}-\frac{\left (b^2 e n^2\right ) \int \frac{x^{-1+r}}{d+e x^r} \, dx}{d^3 r^2}\\ &=\frac{b e n x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2 \left (d+e x^r\right )}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}+\frac{3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{d x^{-r}}{e}\right )}{d^3 r^2}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{d x^{-r}}{e}\right )}{d^3 r}-\frac{b^2 n^2 \log \left (d+e x^r\right )}{d^3 r^3}-\frac{3 b^2 n^2 \text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{d^3 r^3}+\frac{2 b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{d x^{-r}}{e}\right )}{d^3 r^2}+\frac{2 b^2 n^2 \text{Li}_3\left (-\frac{d x^{-r}}{e}\right )}{d^3 r^3}\\ \end{align*}

Mathematica [A]  time = 0.592054, size = 459, normalized size = 1.72 \[ \frac{4 a b n r \left (\text{PolyLog}\left (2,\frac{e x^r}{d}+1\right )+\left (\log \left (-\frac{e x^r}{d}\right )-r \log (x)\right ) \log \left (d+e x^r\right )+\frac{1}{2} r^2 \log ^2(x)\right )+4 b^2 n r \left (\log \left (c x^n\right )-n \log (x)\right ) \left (\text{PolyLog}\left (2,\frac{e x^r}{d}+1\right )+\left (\log \left (-\frac{e x^r}{d}\right )-r \log (x)\right ) \log \left (d+e x^r\right )+\frac{1}{2} r^2 \log ^2(x)\right )-6 b^2 n^2 \left (\text{PolyLog}\left (2,\frac{e x^r}{d}+1\right )+\left (\log \left (-\frac{e x^r}{d}\right )-r \log (x)\right ) \log \left (d+e x^r\right )+\frac{1}{2} r^2 \log ^2(x)\right )-2 b^2 n^2 \left (-2 \text{PolyLog}\left (3,-\frac{d x^{-r}}{e}\right )-2 r \log (x) \text{PolyLog}\left (2,-\frac{d x^{-r}}{e}\right )+r^2 \log ^2(x) \log \left (\frac{d x^{-r}}{e}+1\right )\right )-2 a^2 r^2 \log \left (d-d x^r\right )+\frac{d^2 r^2 \left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^r\right )^2}+\frac{2 d r \left (a+b \log \left (c x^n\right )\right ) \left (a r+b r \log \left (c x^n\right )-b n\right )}{d+e x^r}+4 a b r^2 \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+6 a b n r \log \left (d-d x^r\right )-2 b^2 r^2 \left (\log \left (c x^n\right )-n \log (x)\right )^2 \log \left (d-d x^r\right )+6 b^2 n r \left (\log \left (c x^n\right )-n \log (x)\right ) \log \left (d-d x^r\right )-2 b^2 n^2 \log \left (d-d x^r\right )}{2 d^3 r^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((d^2*r^2*(a + b*Log[c*x^n])^2)/(d + e*x^r)^2 + (2*d*r*(a + b*Log[c*x^n])*(-(b*n) + a*r + b*r*Log[c*x^n]))/(d
+ e*x^r) - 2*b^2*n^2*Log[d - d*x^r] + 6*a*b*n*r*Log[d - d*x^r] - 2*a^2*r^2*Log[d - d*x^r] + 4*a*b*r^2*(n*Log[x
] - Log[c*x^n])*Log[d - d*x^r] + 6*b^2*n*r*(-(n*Log[x]) + Log[c*x^n])*Log[d - d*x^r] - 2*b^2*r^2*(-(n*Log[x])
+ Log[c*x^n])^2*Log[d - d*x^r] - 6*b^2*n^2*((r^2*Log[x]^2)/2 + (-(r*Log[x]) + Log[-((e*x^r)/d)])*Log[d + e*x^r
] + PolyLog[2, 1 + (e*x^r)/d]) + 4*a*b*n*r*((r^2*Log[x]^2)/2 + (-(r*Log[x]) + Log[-((e*x^r)/d)])*Log[d + e*x^r
] + PolyLog[2, 1 + (e*x^r)/d]) + 4*b^2*n*r*(-(n*Log[x]) + Log[c*x^n])*((r^2*Log[x]^2)/2 + (-(r*Log[x]) + Log[-
((e*x^r)/d)])*Log[d + e*x^r] + PolyLog[2, 1 + (e*x^r)/d]) - 2*b^2*n^2*(r^2*Log[x]^2*Log[1 + d/(e*x^r)] - 2*r*L
og[x]*PolyLog[2, -(d/(e*x^r))] - 2*PolyLog[3, -(d/(e*x^r))]))/(2*d^3*r^3)

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Maple [F]  time = 0.714, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{x \left ( d+e{x}^{r} \right ) ^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{2 \, e x^{r} + 3 \, d}{d^{2} e^{2} r x^{2 \, r} + 2 \, d^{3} e r x^{r} + d^{4} r} + \frac{2 \, \log \left (x\right )}{d^{3}} - \frac{2 \, \log \left (\frac{e x^{r} + d}{e}\right )}{d^{3} r}\right )} + \int \frac{b^{2} \log \left (c\right )^{2} + b^{2} \log \left (x^{n}\right )^{2} + 2 \, a b \log \left (c\right ) + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} \log \left (x^{n}\right )}{e^{3} x x^{3 \, r} + 3 \, d e^{2} x x^{2 \, r} + 3 \, d^{2} e x x^{r} + d^{3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*((2*e*x^r + 3*d)/(d^2*e^2*r*x^(2*r) + 2*d^3*e*r*x^r + d^4*r) + 2*log(x)/d^3 - 2*log((e*x^r + d)/e)/(d^
3*r)) + integrate((b^2*log(c)^2 + b^2*log(x^n)^2 + 2*a*b*log(c) + 2*(b^2*log(c) + a*b)*log(x^n))/(e^3*x*x^(3*r
) + 3*d*e^2*x*x^(2*r) + 3*d^2*e*x*x^r + d^3*x), x)

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Fricas [C]  time = 1.44088, size = 2619, normalized size = 9.81 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(2*b^2*d^2*n^2*r^3*log(x)^3 + 9*b^2*d^2*r^2*log(c)^2 - 6*a*b*d^2*n*r + 9*a^2*d^2*r^2 + 6*(b^2*d^2*n*r^3*lo
g(c) + a*b*d^2*n*r^3)*log(x)^2 + (2*b^2*e^2*n^2*r^3*log(x)^3 + 3*(2*b^2*e^2*n*r^3*log(c) - 3*b^2*e^2*n^2*r^2 +
2*a*b*e^2*n*r^3)*log(x)^2 + 6*(b^2*e^2*r^3*log(c)^2 + b^2*e^2*n^2*r - 3*a*b*e^2*n*r^2 + a^2*e^2*r^3 - (3*b^2*
e^2*n*r^2 - 2*a*b*e^2*r^3)*log(c))*log(x))*x^(2*r) + 2*(2*b^2*d*e*n^2*r^3*log(x)^3 + 3*b^2*d*e*r^2*log(c)^2 -
3*a*b*d*e*n*r + 3*a^2*d*e*r^2 + 6*(b^2*d*e*n*r^3*log(c) - b^2*d*e*n^2*r^2 + a*b*d*e*n*r^3)*log(x)^2 - 3*(b^2*d
*e*n*r - 2*a*b*d*e*r^2)*log(c) + 3*(2*b^2*d*e*r^3*log(c)^2 + b^2*d*e*n^2*r - 4*a*b*d*e*n*r^2 + 2*a^2*d*e*r^3 -
4*(b^2*d*e*n*r^2 - a*b*d*e*r^3)*log(c))*log(x))*x^r - 6*(2*b^2*d^2*n^2*r*log(x) + 2*b^2*d^2*n*r*log(c) - 3*b^
2*d^2*n^2 + 2*a*b*d^2*n*r + (2*b^2*e^2*n^2*r*log(x) + 2*b^2*e^2*n*r*log(c) - 3*b^2*e^2*n^2 + 2*a*b*e^2*n*r)*x^
(2*r) + 2*(2*b^2*d*e*n^2*r*log(x) + 2*b^2*d*e*n*r*log(c) - 3*b^2*d*e*n^2 + 2*a*b*d*e*n*r)*x^r)*dilog(-(e*x^r +
d)/d + 1) - 6*(b^2*d^2*r^2*log(c)^2 + b^2*d^2*n^2 - 3*a*b*d^2*n*r + a^2*d^2*r^2 + (b^2*e^2*r^2*log(c)^2 + b^2
*e^2*n^2 - 3*a*b*e^2*n*r + a^2*e^2*r^2 - (3*b^2*e^2*n*r - 2*a*b*e^2*r^2)*log(c))*x^(2*r) + 2*(b^2*d*e*r^2*log(
c)^2 + b^2*d*e*n^2 - 3*a*b*d*e*n*r + a^2*d*e*r^2 - (3*b^2*d*e*n*r - 2*a*b*d*e*r^2)*log(c))*x^r - (3*b^2*d^2*n*
r - 2*a*b*d^2*r^2)*log(c))*log(e*x^r + d) - 6*(b^2*d^2*n*r - 3*a*b*d^2*r^2)*log(c) + 6*(b^2*d^2*r^3*log(c)^2 +
2*a*b*d^2*r^3*log(c) + a^2*d^2*r^3)*log(x) - 6*(b^2*d^2*n^2*r^2*log(x)^2 + (b^2*e^2*n^2*r^2*log(x)^2 + (2*b^2
*e^2*n*r^2*log(c) - 3*b^2*e^2*n^2*r + 2*a*b*e^2*n*r^2)*log(x))*x^(2*r) + 2*(b^2*d*e*n^2*r^2*log(x)^2 + (2*b^2*
d*e*n*r^2*log(c) - 3*b^2*d*e*n^2*r + 2*a*b*d*e*n*r^2)*log(x))*x^r + (2*b^2*d^2*n*r^2*log(c) - 3*b^2*d^2*n^2*r
+ 2*a*b*d^2*n*r^2)*log(x))*log((e*x^r + d)/d) + 12*(b^2*e^2*n^2*x^(2*r) + 2*b^2*d*e*n^2*x^r + b^2*d^2*n^2)*pol
ylog(3, -e*x^r/d))/(d^3*e^2*r^3*x^(2*r) + 2*d^4*e*r^3*x^r + d^5*r^3)

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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))**2/x/(d+e*x**r)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )}^{2}}{{\left (e x^{r} + d\right )}^{3} x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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