3.5.32 \(\int \frac {(a+b \log (c x^n))^2}{x (d+e x^r)^3} \, dx\) [432]

Optimal. Leaf size=267 \[ \frac {b e n x^r \left (a+b \log \left (c x^n\right )\right )}{d^3 r^2 \left (d+e x^r\right )}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d r \left (d+e x^r\right )^2}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d^2 r \left (d+e x^r\right )}+\frac {3 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d^3 r}-\frac {b^2 n^2 \log \left (d+e x^r\right )}{d^3 r^3}-\frac {3 b^2 n^2 \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^2}+\frac {2 b^2 n^2 \text {Li}_3\left (-\frac {d x^{-r}}{e}\right )}{d^3 r^3} \]

[Out]

b*e*n*x^r*(a+b*ln(c*x^n))/d^3/r^2/(d+e*x^r)+1/2*(a+b*ln(c*x^n))^2/d/r/(d+e*x^r)^2+(a+b*ln(c*x^n))^2/d^2/r/(d+e
*x^r)+3*b*n*(a+b*ln(c*x^n))*ln(1+d/e/(x^r))/d^3/r^2-(a+b*ln(c*x^n))^2*ln(1+d/e/(x^r))/d^3/r-b^2*n^2*ln(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*ln(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 = 0.61, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2391, 2379, 2421, 6724, 2376, 2438, 2373, 266} \begin {gather*} \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 {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 {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 {\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}{d^2 r \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 {b^2 n^2 \log \left (d+e x^r\right )}{d^3 r^3} \end {gather*}

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 266

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]

Rule 2373

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 2376

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 2379

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 2391

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 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2438

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 6724

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]

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*}

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Mathematica [A]
time = 0.36, size = 459, normalized size = 1.72 \begin {gather*} \frac {\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 (-b n+a r+b r \log \left (c x^n\right )\right )}{d+e x^r}-2 b^2 n^2 \log \left (d-d x^r\right )+6 a b n r \log \left (d-d x^r\right )-2 a^2 r^2 \log \left (d-d x^r\right )+4 a b r^2 \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+6 b^2 n r \left (-n \log (x)+\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )-2 b^2 r^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2 \log \left (d-d x^r\right )-6 b^2 n^2 \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\text {Li}_2\left (1+\frac {e x^r}{d}\right )\right )+4 a b n r \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\text {Li}_2\left (1+\frac {e x^r}{d}\right )\right )+4 b^2 n r \left (-n \log (x)+\log \left (c x^n\right )\right ) \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\text {Li}_2\left (1+\frac {e x^r}{d}\right )\right )-2 b^2 n^2 \left (r^2 \log ^2(x) \log \left (1+\frac {d x^{-r}}{e}\right )-2 r \log (x) \text {Li}_2\left (-\frac {d x^{-r}}{e}\right )-2 \text {Li}_3\left (-\frac {d x^{-r}}{e}\right )\right )}{2 d^3 r^3} \end {gather*}

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 = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{2}}{x \left (d +e \,x^{r}\right )^{3}}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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*((3*d + 2*e^(r*log(x) + 1))/(d^4*r + 2*d^3*r*e^(r*log(x) + 1) + d^2*r*e^(2*r*log(x) + 2)) + 2*log(x)/d
^3 - 2*log((d + e^(r*log(x) + 1))*e^(-1))/(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))/(d^3*x + 3*d^2*x*e^(r*log(x) + 1) + 3*d*x*e^(2*r*log(x) + 2) + x*e^(3*r*log(x)
+ 3)), x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1162 vs. \(2 (263) = 526\).
time = 0.39, size = 1162, normalized size = 4.35 \begin {gather*} \frac {2 \, b^{2} d^{2} n^{2} r^{3} \log \left (x\right )^{3} + 9 \, b^{2} d^{2} r^{2} \log \left (c\right )^{2} - 6 \, a b d^{2} n r + 9 \, a^{2} d^{2} r^{2} + 6 \, {\left (b^{2} d^{2} n r^{3} \log \left (c\right ) + a b d^{2} n r^{3}\right )} \log \left (x\right )^{2} + {\left (2 \, b^{2} n^{2} r^{3} e^{2} \log \left (x\right )^{3} + 3 \, {\left (2 \, b^{2} n r^{3} e^{2} \log \left (c\right ) - {\left (3 \, b^{2} n^{2} r^{2} - 2 \, a b n r^{3}\right )} e^{2}\right )} \log \left (x\right )^{2} + 6 \, {\left (b^{2} r^{3} e^{2} \log \left (c\right )^{2} - {\left (3 \, b^{2} n r^{2} - 2 \, a b r^{3}\right )} e^{2} \log \left (c\right ) + {\left (b^{2} n^{2} r - 3 \, a b n r^{2} + a^{2} r^{3}\right )} e^{2}\right )} \log \left (x\right )\right )} x^{2 \, r} + 2 \, {\left (2 \, b^{2} d n^{2} r^{3} e \log \left (x\right )^{3} + 3 \, b^{2} d r^{2} e \log \left (c\right )^{2} - 3 \, {\left (b^{2} d n r - 2 \, a b d r^{2}\right )} e \log \left (c\right ) + 6 \, {\left (b^{2} d n r^{3} e \log \left (c\right ) - {\left (b^{2} d n^{2} r^{2} - a b d n r^{3}\right )} e\right )} \log \left (x\right )^{2} - 3 \, {\left (a b d n r - a^{2} d r^{2}\right )} e + 3 \, {\left (2 \, b^{2} d r^{3} e \log \left (c\right )^{2} - 4 \, {\left (b^{2} d n r^{2} - a b d r^{3}\right )} e \log \left (c\right ) + {\left (b^{2} d n^{2} r - 4 \, a b d n r^{2} + 2 \, a^{2} d r^{3}\right )} e\right )} \log \left (x\right )\right )} x^{r} - 6 \, {\left (2 \, b^{2} d^{2} n^{2} r \log \left (x\right ) + 2 \, b^{2} d^{2} n r \log \left (c\right ) - 3 \, b^{2} d^{2} n^{2} + 2 \, a b d^{2} n r + {\left (2 \, b^{2} n^{2} r e^{2} \log \left (x\right ) + 2 \, b^{2} n r e^{2} \log \left (c\right ) - {\left (3 \, b^{2} n^{2} - 2 \, a b n r\right )} e^{2}\right )} x^{2 \, r} + 2 \, {\left (2 \, b^{2} d n^{2} r e \log \left (x\right ) + 2 \, b^{2} d n r e \log \left (c\right ) - {\left (3 \, b^{2} d n^{2} - 2 \, a b d n r\right )} e\right )} x^{r}\right )} {\rm Li}_2\left (-\frac {x^{r} e + d}{d} + 1\right ) - 6 \, {\left (b^{2} d^{2} r^{2} \log \left (c\right )^{2} + b^{2} d^{2} n^{2} - 3 \, a b d^{2} n r + a^{2} d^{2} r^{2} + {\left (b^{2} r^{2} e^{2} \log \left (c\right )^{2} - {\left (3 \, b^{2} n r - 2 \, a b r^{2}\right )} e^{2} \log \left (c\right ) + {\left (b^{2} n^{2} - 3 \, a b n r + a^{2} r^{2}\right )} e^{2}\right )} x^{2 \, r} + 2 \, {\left (b^{2} d r^{2} e \log \left (c\right )^{2} - {\left (3 \, b^{2} d n r - 2 \, a b d r^{2}\right )} e \log \left (c\right ) + {\left (b^{2} d n^{2} - 3 \, a b d n r + a^{2} d r^{2}\right )} e\right )} x^{r} - {\left (3 \, b^{2} d^{2} n r - 2 \, a b d^{2} r^{2}\right )} \log \left (c\right )\right )} \log \left (x^{r} e + d\right ) - 6 \, {\left (b^{2} d^{2} n r - 3 \, a b d^{2} r^{2}\right )} \log \left (c\right ) + 6 \, {\left (b^{2} d^{2} r^{3} \log \left (c\right )^{2} + 2 \, a b d^{2} r^{3} \log \left (c\right ) + a^{2} d^{2} r^{3}\right )} \log \left (x\right ) - 6 \, {\left (b^{2} d^{2} n^{2} r^{2} \log \left (x\right )^{2} + {\left (b^{2} n^{2} r^{2} e^{2} \log \left (x\right )^{2} + {\left (2 \, b^{2} n r^{2} e^{2} \log \left (c\right ) - {\left (3 \, b^{2} n^{2} r - 2 \, a b n r^{2}\right )} e^{2}\right )} \log \left (x\right )\right )} x^{2 \, r} + 2 \, {\left (b^{2} d n^{2} r^{2} e \log \left (x\right )^{2} + {\left (2 \, b^{2} d n r^{2} e \log \left (c\right ) - {\left (3 \, b^{2} d n^{2} r - 2 \, a b d n r^{2}\right )} e\right )} \log \left (x\right )\right )} x^{r} + {\left (2 \, b^{2} d^{2} n r^{2} \log \left (c\right ) - 3 \, b^{2} d^{2} n^{2} r + 2 \, a b d^{2} n r^{2}\right )} \log \left (x\right )\right )} \log \left (\frac {x^{r} e + d}{d}\right ) + 12 \, {\left (2 \, b^{2} d n^{2} x^{r} e + b^{2} d^{2} n^{2} + b^{2} n^{2} x^{2 \, r} e^{2}\right )} {\rm polylog}\left (3, -\frac {x^{r} e}{d}\right )}{6 \, {\left (2 \, d^{4} r^{3} x^{r} e + d^{5} r^{3} + d^{3} r^{3} x^{2 \, r} e^{2}\right )}} \end {gather*}

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*n^2*r^3*e^2*log(x)^3 + 3*(2*b^2*n*r^3*e^2*log(c) - (3*b^2*n^2*r^2 - 2*
a*b*n*r^3)*e^2)*log(x)^2 + 6*(b^2*r^3*e^2*log(c)^2 - (3*b^2*n*r^2 - 2*a*b*r^3)*e^2*log(c) + (b^2*n^2*r - 3*a*b
*n*r^2 + a^2*r^3)*e^2)*log(x))*x^(2*r) + 2*(2*b^2*d*n^2*r^3*e*log(x)^3 + 3*b^2*d*r^2*e*log(c)^2 - 3*(b^2*d*n*r
- 2*a*b*d*r^2)*e*log(c) + 6*(b^2*d*n*r^3*e*log(c) - (b^2*d*n^2*r^2 - a*b*d*n*r^3)*e)*log(x)^2 - 3*(a*b*d*n*r
- a^2*d*r^2)*e + 3*(2*b^2*d*r^3*e*log(c)^2 - 4*(b^2*d*n*r^2 - a*b*d*r^3)*e*log(c) + (b^2*d*n^2*r - 4*a*b*d*n*r
^2 + 2*a^2*d*r^3)*e)*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*n^2*r*e^2*log(x) + 2*b^2*n*r*e^2*log(c) - (3*b^2*n^2 - 2*a*b*n*r)*e^2)*x^(2*r) + 2*(2*b^2*d*n^2
*r*e*log(x) + 2*b^2*d*n*r*e*log(c) - (3*b^2*d*n^2 - 2*a*b*d*n*r)*e)*x^r)*dilog(-(x^r*e + 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*r^2*e^2*log(c)^2 - (3*b^2*n*r - 2*a*b*r^2)*e
^2*log(c) + (b^2*n^2 - 3*a*b*n*r + a^2*r^2)*e^2)*x^(2*r) + 2*(b^2*d*r^2*e*log(c)^2 - (3*b^2*d*n*r - 2*a*b*d*r^
2)*e*log(c) + (b^2*d*n^2 - 3*a*b*d*n*r + a^2*d*r^2)*e)*x^r - (3*b^2*d^2*n*r - 2*a*b*d^2*r^2)*log(c))*log(x^r*e
+ 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*n^2*r^2*e^2*log(x)^2 + (2*b^2*n*r^2*e^2*log(c) - (3*b^2*n^2*r - 2
*a*b*n*r^2)*e^2)*log(x))*x^(2*r) + 2*(b^2*d*n^2*r^2*e*log(x)^2 + (2*b^2*d*n*r^2*e*log(c) - (3*b^2*d*n^2*r - 2*
a*b*d*n*r^2)*e)*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((x^r*e
+ d)/d) + 12*(2*b^2*d*n^2*x^r*e + b^2*d^2*n^2 + b^2*n^2*x^(2*r)*e^2)*polylog(3, -x^r*e/d))/(2*d^4*r^3*x^r*e +
d^5*r^3 + d^3*r^3*x^(2*r)*e^2)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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/((x^r*e + d)^3*x), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x\,{\left (d+e\,x^r\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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