\(\int \frac {(a+b \log (c x^n))^2}{x (d+e x^r)^2} \, dx\) [431]

Optimal result

Integrand size = 25, antiderivative size = 182 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^2} \, dx=\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d r \left (d+e x^r\right )}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d x^{-r}}{e}\right )}{d^2 r^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {d x^{-r}}{e}\right )}{d^2 r}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^2 r^3}+\frac {2 b n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^2 r^2}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d x^{-r}}{e}\right )}{d^2 r^3} \]

[Out]

(a+b*ln(c*x^n))^2/d/r/(d+e*x^r)+2*b*n*(a+b*ln(c*x^n))*ln(1+d/e/(x^r))/d^2/r^2-(a+b*ln(c*x^n))^2*ln(1+d/e/(x^r)
)/d^2/r-2*b^2*n^2*polylog(2,-d/e/(x^r))/d^2/r^3+2*b*n*(a+b*ln(c*x^n))*polylog(2,-d/e/(x^r))/d^2/r^2+2*b^2*n^2*
polylog(3,-d/e/(x^r))/d^2/r^3

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2391, 2379, 2421, 6724, 2376, 2438} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^2} \, dx=\frac {2 b n \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 r^2}+\frac {2 b n \log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^2 r^2}-\frac {\log \left (\frac {d x^{-r}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^2 r}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{d r \left (d+e x^r\right )}-\frac {2 b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )}{d^2 r^3}+\frac {2 b^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d x^{-r}}{e}\right )}{d^2 r^3} \]

[In]

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

[Out]

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

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

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(397\) vs. \(2(182)=364\).

Time = 0.25 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.18 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^2} \, dx=\frac {\frac {d r^2 \left (a+b \log \left (c x^n\right )\right )^2}{d+e x^r}+2 a b n r \log \left (d-d x^r\right )-a^2 r^2 \log \left (d-d x^r\right )+2 a b r^2 \left (n \log (x)-\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )+2 b^2 n r \left (-n \log (x)+\log \left (c x^n\right )\right ) \log \left (d-d x^r\right )-b^2 r^2 \left (-n \log (x)+\log \left (c x^n\right )\right )^2 \log \left (d-d x^r\right )-2 b^2 n^2 \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )\right )+2 a b n r \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )\right )+2 b^2 n r \left (-n \log (x)+\log \left (c x^n\right )\right ) \left (\frac {1}{2} r^2 \log ^2(x)+\left (-r \log (x)+\log \left (-\frac {e x^r}{d}\right )\right ) \log \left (d+e x^r\right )+\operatorname {PolyLog}\left (2,1+\frac {e x^r}{d}\right )\right )-b^2 n^2 \left (r^2 \log ^2(x) \log \left (1+\frac {d x^{-r}}{e}\right )-2 r \log (x) \operatorname {PolyLog}\left (2,-\frac {d x^{-r}}{e}\right )-2 \operatorname {PolyLog}\left (3,-\frac {d x^{-r}}{e}\right )\right )}{d^2 r^3} \]

[In]

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

[Out]

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

Maple [F]

\[\int \frac {{\left (a +b \ln \left (c \,x^{n}\right )\right )}^{2}}{x \left (d +e \,x^{r}\right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (180) = 360\).

Time = 0.27 (sec) , antiderivative size = 600, normalized size of antiderivative = 3.30 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^2} \, dx=\frac {b^{2} d n^{2} r^{3} \log \left (x\right )^{3} + 3 \, b^{2} d r^{2} \log \left (c\right )^{2} + 6 \, a b d r^{2} \log \left (c\right ) + 3 \, a^{2} d r^{2} + 3 \, {\left (b^{2} d n r^{3} \log \left (c\right ) + a b d n r^{3}\right )} \log \left (x\right )^{2} + {\left (b^{2} e n^{2} r^{3} \log \left (x\right )^{3} + 3 \, {\left (b^{2} e n r^{3} \log \left (c\right ) - b^{2} e n^{2} r^{2} + a b e n r^{3}\right )} \log \left (x\right )^{2} + 3 \, {\left (b^{2} e r^{3} \log \left (c\right )^{2} - 2 \, a b e n r^{2} + a^{2} e r^{3} - 2 \, {\left (b^{2} e n r^{2} - a b e r^{3}\right )} \log \left (c\right )\right )} \log \left (x\right )\right )} x^{r} - 6 \, {\left (b^{2} d n^{2} r \log \left (x\right ) + b^{2} d n r \log \left (c\right ) - b^{2} d n^{2} + a b d n r + {\left (b^{2} e n^{2} r \log \left (x\right ) + b^{2} e n r \log \left (c\right ) - b^{2} e n^{2} + a b e n r\right )} x^{r}\right )} {\rm Li}_2\left (-\frac {e x^{r} + d}{d} + 1\right ) - 3 \, {\left (b^{2} d r^{2} \log \left (c\right )^{2} - 2 \, a b d n r + a^{2} d r^{2} + {\left (b^{2} e r^{2} \log \left (c\right )^{2} - 2 \, a b e n r + a^{2} e r^{2} - 2 \, {\left (b^{2} e n r - a b e r^{2}\right )} \log \left (c\right )\right )} x^{r} - 2 \, {\left (b^{2} d n r - a b d r^{2}\right )} \log \left (c\right )\right )} \log \left (e x^{r} + d\right ) + 3 \, {\left (b^{2} d r^{3} \log \left (c\right )^{2} + 2 \, a b d r^{3} \log \left (c\right ) + a^{2} d r^{3}\right )} \log \left (x\right ) - 3 \, {\left (b^{2} d n^{2} r^{2} \log \left (x\right )^{2} + {\left (b^{2} e n^{2} r^{2} \log \left (x\right )^{2} + 2 \, {\left (b^{2} e n r^{2} \log \left (c\right ) - b^{2} e n^{2} r + a b e n r^{2}\right )} \log \left (x\right )\right )} x^{r} + 2 \, {\left (b^{2} d n r^{2} \log \left (c\right ) - b^{2} d n^{2} r + a b d n r^{2}\right )} \log \left (x\right )\right )} \log \left (\frac {e x^{r} + d}{d}\right ) + 6 \, {\left (b^{2} e n^{2} x^{r} + b^{2} d n^{2}\right )} {\rm polylog}\left (3, -\frac {e x^{r}}{d}\right )}{3 \, {\left (d^{2} e r^{3} x^{r} + d^{3} r^{3}\right )}} \]

[In]

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

[Out]

1/3*(b^2*d*n^2*r^3*log(x)^3 + 3*b^2*d*r^2*log(c)^2 + 6*a*b*d*r^2*log(c) + 3*a^2*d*r^2 + 3*(b^2*d*n*r^3*log(c)
+ a*b*d*n*r^3)*log(x)^2 + (b^2*e*n^2*r^3*log(x)^3 + 3*(b^2*e*n*r^3*log(c) - b^2*e*n^2*r^2 + a*b*e*n*r^3)*log(x
)^2 + 3*(b^2*e*r^3*log(c)^2 - 2*a*b*e*n*r^2 + a^2*e*r^3 - 2*(b^2*e*n*r^2 - a*b*e*r^3)*log(c))*log(x))*x^r - 6*
(b^2*d*n^2*r*log(x) + b^2*d*n*r*log(c) - b^2*d*n^2 + a*b*d*n*r + (b^2*e*n^2*r*log(x) + b^2*e*n*r*log(c) - b^2*
e*n^2 + a*b*e*n*r)*x^r)*dilog(-(e*x^r + d)/d + 1) - 3*(b^2*d*r^2*log(c)^2 - 2*a*b*d*n*r + a^2*d*r^2 + (b^2*e*r
^2*log(c)^2 - 2*a*b*e*n*r + a^2*e*r^2 - 2*(b^2*e*n*r - a*b*e*r^2)*log(c))*x^r - 2*(b^2*d*n*r - a*b*d*r^2)*log(
c))*log(e*x^r + d) + 3*(b^2*d*r^3*log(c)^2 + 2*a*b*d*r^3*log(c) + a^2*d*r^3)*log(x) - 3*(b^2*d*n^2*r^2*log(x)^
2 + (b^2*e*n^2*r^2*log(x)^2 + 2*(b^2*e*n*r^2*log(c) - b^2*e*n^2*r + a*b*e*n*r^2)*log(x))*x^r + 2*(b^2*d*n*r^2*
log(c) - b^2*d*n^2*r + a*b*d*n*r^2)*log(x))*log((e*x^r + d)/d) + 6*(b^2*e*n^2*x^r + b^2*d*n^2)*polylog(3, -e*x
^r/d))/(d^2*e*r^3*x^r + d^3*r^3)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x \left (d+e x^r\right )^2} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x\,{\left (d+e\,x^r\right )}^2} \,d x \]

[In]

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

[Out]

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