\(\int \frac {(a+b \log (c x^n))^2}{x^3 (d+e x)} \, dx\) [98]

Optimal result

Integrand size = 23, antiderivative size = 204 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=-\frac {b^2 n^2}{4 d x^2}+\frac {2 b^2 e n^2}{d^2 x}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac {2 b e n \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d x^2}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}-\frac {e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}+\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^3}+\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^3} \]

[Out]

-1/4*b^2*n^2/d/x^2+2*b^2*e*n^2/d^2/x-1/2*b*n*(a+b*ln(c*x^n))/d/x^2+2*b*e*n*(a+b*ln(c*x^n))/d^2/x-1/2*(a+b*ln(c
*x^n))^2/d/x^2+e*(a+b*ln(c*x^n))^2/d^2/x-e^2*ln(1+d/e/x)*(a+b*ln(c*x^n))^2/d^3+2*b*e^2*n*(a+b*ln(c*x^n))*polyl
og(2,-d/e/x)/d^3+2*b^2*e^2*n^2*polylog(3,-d/e/x)/d^3

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2380, 2342, 2341, 2379, 2421, 6724} \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=\frac {2 b e^2 n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^3}-\frac {e^2 \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}+\frac {2 b e n \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac {2 b^2 e^2 n^2 \operatorname {PolyLog}\left (3,-\frac {d}{e x}\right )}{d^3}+\frac {2 b^2 e n^2}{d^2 x}-\frac {b^2 n^2}{4 d x^2} \]

[In]

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

[Out]

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

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

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 2380

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(x_)^(m_.))/((d_) + (e_.)*(x_)^(r_.)), x_Symbol] :> Dist[1/d,
Int[x^m*(a + b*Log[c*x^n])^p, x], x] - Dist[e/d, Int[(x^(m + r)*(a + b*Log[c*x^n])^p)/(d + e*x^r), x], x] /;
FreeQ[{a, b, c, d, e, m, n, r}, x] && IGtQ[p, 0] && IGtQ[r, 0] && ILtQ[m, -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 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^3} \, dx}{d}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2 (d+e x)} \, dx}{d} \\ & = -\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d x^2}-\frac {e \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^2} \, dx}{d^2}+\frac {e^2 \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x (d+e x)} \, dx}{d^2}+\frac {(b n) \int \frac {a+b \log \left (c x^n\right )}{x^3} \, dx}{d} \\ & = -\frac {b^2 n^2}{4 d x^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d x^2}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}-\frac {e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}-\frac {(2 b e n) \int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^2}+\frac {\left (2 b e^2 n\right ) \int \frac {\log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{x} \, dx}{d^3} \\ & = -\frac {b^2 n^2}{4 d x^2}+\frac {2 b^2 e n^2}{d^2 x}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac {2 b e n \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d x^2}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}-\frac {e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}+\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^3}-\frac {\left (2 b^2 e^2 n^2\right ) \int \frac {\text {Li}_2\left (-\frac {d}{e x}\right )}{x} \, dx}{d^3} \\ & = -\frac {b^2 n^2}{4 d x^2}+\frac {2 b^2 e n^2}{d^2 x}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{2 d x^2}+\frac {2 b e n \left (a+b \log \left (c x^n\right )\right )}{d^2 x}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 d x^2}+\frac {e \left (a+b \log \left (c x^n\right )\right )^2}{d^2 x}-\frac {e^2 \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{d^3}+\frac {2 b e^2 n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {d}{e x}\right )}{d^3}+\frac {2 b^2 e^2 n^2 \text {Li}_3\left (-\frac {d}{e x}\right )}{d^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=\frac {-\frac {6 d^2 \left (a+b \log \left (c x^n\right )\right )^2}{x^2}+\frac {12 d e \left (a+b \log \left (c x^n\right )\right )^2}{x}+\frac {4 e^2 \left (a+b \log \left (c x^n\right )\right )^3}{b n}+\frac {24 b d e n \left (a+b n+b \log \left (c x^n\right )\right )}{x}-\frac {3 b d^2 n \left (2 a+b n+2 b \log \left (c x^n\right )\right )}{x^2}-12 e^2 \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-24 b e^2 n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{12 d^3} \]

[In]

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

[Out]

((-6*d^2*(a + b*Log[c*x^n])^2)/x^2 + (12*d*e*(a + b*Log[c*x^n])^2)/x + (4*e^2*(a + b*Log[c*x^n])^3)/(b*n) + (2
4*b*d*e*n*(a + b*n + b*Log[c*x^n]))/x - (3*b*d^2*n*(2*a + b*n + 2*b*Log[c*x^n]))/x^2 - 12*e^2*(a + b*Log[c*x^n
])^2*Log[1 + (e*x)/d] - 24*b*e^2*n*((a + b*Log[c*x^n])*PolyLog[2, -((e*x)/d)] - b*n*PolyLog[3, -((e*x)/d)]))/(
12*d^3)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.46 (sec) , antiderivative size = 731, normalized size of antiderivative = 3.58

method	result	size
risch	\(-\frac {b^{2} \ln \left (x^{n}\right )^{2} e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{2 d \,x^{2}}+\frac {b^{2} \ln \left (x^{n}\right )^{2} e^{2} \ln \left (x \right )}{d^{3}}+\frac {b^{2} \ln \left (x^{n}\right )^{2} e}{d^{2} x}+\frac {2 b^{2} n \ln \left (x^{n}\right ) e}{d^{2} x}-\frac {b^{2} n \ln \left (x^{n}\right )}{2 d \,x^{2}}+\frac {2 b^{2} e \,n^{2}}{d^{2} x}-\frac {b^{2} n^{2}}{4 d \,x^{2}}-\frac {b^{2} n \,e^{2} \ln \left (x^{n}\right ) \ln \left (x \right )^{2}}{d^{3}}+\frac {b^{2} e^{2} \ln \left (x \right )^{3} n^{2}}{3 d^{3}}-\frac {2 b^{2} e^{2} \ln \left (x \right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) n^{2}}{d^{3}}-\frac {2 b^{2} e^{2} \ln \left (x \right ) \operatorname {dilog}\left (-\frac {e x}{d}\right ) n^{2}}{d^{3}}+\frac {2 b^{2} n \,e^{2} \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{3}}+\frac {2 b^{2} n \,e^{2} \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{3}}+\frac {b^{2} e^{2} n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{d^{3}}-\frac {b^{2} e^{2} n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{d^{3}}-\frac {2 b^{2} e^{2} n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{d^{3}}+\frac {2 b^{2} e^{2} n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{d^{3}}+\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right ) b \left (-\frac {\ln \left (x^{n}\right ) e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {\ln \left (x^{n}\right )}{2 d \,x^{2}}+\frac {\ln \left (x^{n}\right ) e^{2} \ln \left (x \right )}{d^{3}}+\frac {\ln \left (x^{n}\right ) e}{d^{2} x}-\frac {n \left (\frac {-\frac {2 e}{x}+\frac {d}{2 x^{2}}}{d^{2}}+\frac {e^{2} \ln \left (x \right )^{2}}{d^{3}}-\frac {2 e^{2} \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{d^{3}}\right )}{2}\right )+\frac {{\left (-i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )+i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}+i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}-i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}+2 b \ln \left (c \right )+2 a \right )}^{2} \left (-\frac {e^{2} \ln \left (e x +d \right )}{d^{3}}-\frac {1}{2 d \,x^{2}}+\frac {e^{2} \ln \left (x \right )}{d^{3}}+\frac {e}{d^{2} x}\right )}{4}\)	\(731\)

[In]

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

[Out]

-b^2*ln(x^n)^2*e^2/d^3*ln(e*x+d)-1/2*b^2*ln(x^n)^2/d/x^2+b^2*ln(x^n)^2*e^2/d^3*ln(x)+b^2*ln(x^n)^2*e/d^2/x+2*b
^2*n*ln(x^n)*e/d^2/x-1/2*b^2*n*ln(x^n)/d/x^2+2*b^2*e*n^2/d^2/x-1/4*b^2*n^2/d/x^2-b^2*n*e^2/d^3*ln(x^n)*ln(x)^2
+1/3*b^2*e^2/d^3*ln(x)^3*n^2-2*b^2*e^2/d^3*ln(x)*ln(e*x+d)*ln(-e*x/d)*n^2-2*b^2*e^2/d^3*ln(x)*dilog(-e*x/d)*n^
2+2*b^2*n*e^2/d^3*ln(x^n)*ln(e*x+d)*ln(-e*x/d)+2*b^2*n*e^2/d^3*ln(x^n)*dilog(-e*x/d)+b^2*e^2/d^3*n^2*ln(e*x+d)
*ln(x)^2-b^2*e^2/d^3*n^2*ln(x)^2*ln(1+e*x/d)-2*b^2*e^2/d^3*n^2*ln(x)*polylog(2,-e*x/d)+2*b^2*e^2/d^3*n^2*polyl
og(3,-e*x/d)+(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn(I*x^n)*
csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)*b*(-ln(x^n)*e^2/d^3*ln(e*x+d)-1/2*ln(x^n)/d/x^2+ln(x^n)*
e^2/d^3*ln(x)+ln(x^n)*e/d^2/x-1/2*n*(1/d^2*(-2*e/x+1/2*d/x^2)+e^2/d^3*ln(x)^2-2*e^2/d^3*(dilog(-e*x/d)+ln(e*x+
d)*ln(-e*x/d))))+1/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+I*b*Pi*csgn
(I*x^n)*csgn(I*c*x^n)^2-I*b*Pi*csgn(I*c*x^n)^3+2*b*ln(c)+2*a)^2*(-e^2/d^3*ln(e*x+d)-1/2/d/x^2+e^2/d^3*ln(x)+e/
d^2/x)

Fricas [F]

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

[In]

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

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e*x^4 + d*x^3), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b \log \left (c x^n\right )\right )^2}{x^3 (d+e x)} \, dx=\int \frac {{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x^3\,\left (d+e\,x\right )} \,d x \]

[In]

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

[Out]

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