\(\int \frac {x (a+b \log (c x^n))^2}{d+e x} \, dx\) [94]

Optimal result

Integrand size = 21, antiderivative size = 130 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=-\frac {2 a b n x}{e}+\frac {2 b^2 n^2 x}{e}-\frac {2 b^2 n x \log \left (c x^n\right )}{e}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )}{e^2}-\frac {2 b d n \left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^2}+\frac {2 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2395, 2333, 2332, 2354, 2421, 6724} \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=-\frac {2 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{e^2}-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{e^2}+\frac {x \left (a+b \log \left (c x^n\right )\right )^2}{e}-\frac {2 a b n x}{e}-\frac {2 b^2 n x \log \left (c x^n\right )}{e}+\frac {2 b^2 d n^2 \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )}{e^2}+\frac {2 b^2 n^2 x}{e} \]

[In]

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

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2354

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

Rule 2395

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.79 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{d+e x} \, dx=\frac {e x \left (a+b \log \left (c x^n\right )\right )^2-2 b e n x \left (a-b n+b \log \left (c x^n\right )\right )-d \left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac {e x}{d}\right )-2 b d n \left (\left (a+b \log \left (c x^n\right )\right ) \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )-b n \operatorname {PolyLog}\left (3,-\frac {e x}{d}\right )\right )}{e^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.43 (sec) , antiderivative size = 528, normalized size of antiderivative = 4.06

method	result	size
risch	\(\frac {b^{2} \ln \left (x^{n}\right )^{2} x}{e}-\frac {b^{2} \ln \left (x^{n}\right )^{2} d \ln \left (e x +d \right )}{e^{2}}-\frac {2 b^{2} n \ln \left (x^{n}\right ) x}{e}+\frac {2 b^{2} n^{2} x}{e}-\frac {2 b^{2} d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right ) \ln \left (x \right ) n^{2}}{e^{2}}+\frac {2 b^{2} n d \ln \left (x^{n}\right ) \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{2}}-\frac {2 b^{2} d \operatorname {dilog}\left (-\frac {e x}{d}\right ) \ln \left (x \right ) n^{2}}{e^{2}}+\frac {2 b^{2} n d \ln \left (x^{n}\right ) \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{2}}+\frac {b^{2} d \,n^{2} \ln \left (e x +d \right ) \ln \left (x \right )^{2}}{e^{2}}-\frac {b^{2} d \,n^{2} \ln \left (x \right )^{2} \ln \left (1+\frac {e x}{d}\right )}{e^{2}}-\frac {2 b^{2} d \,n^{2} \ln \left (x \right ) \operatorname {Li}_{2}\left (-\frac {e x}{d}\right )}{e^{2}}+\frac {2 b^{2} d \,n^{2} \operatorname {Li}_{3}\left (-\frac {e x}{d}\right )}{e^{2}}+\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 ) x}{e}-\frac {\ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{2}}-n \left (\frac {e x +d}{e^{2}}-\frac {d \left (\operatorname {dilog}\left (-\frac {e x}{d}\right )+\ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )\right )}{e^{2}}\right )\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 {x}{e}-\frac {d \ln \left (e x +d \right )}{e^{2}}\right )}{4}\)	\(528\)

[In]

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

[Out]

b^2*ln(x^n)^2/e*x-b^2*ln(x^n)^2*d/e^2*ln(e*x+d)-2*b^2*n*ln(x^n)/e*x+2*b^2*n^2*x/e-2*b^2*d/e^2*ln(e*x+d)*ln(-e*
x/d)*ln(x)*n^2+2*b^2*n*d/e^2*ln(x^n)*ln(e*x+d)*ln(-e*x/d)-2*b^2*d/e^2*dilog(-e*x/d)*ln(x)*n^2+2*b^2*n*d/e^2*ln
(x^n)*dilog(-e*x/d)+b^2*d/e^2*n^2*ln(e*x+d)*ln(x)^2-b^2*d/e^2*n^2*ln(x)^2*ln(1+e*x/d)-2*b^2*d/e^2*n^2*ln(x)*po
lylog(2,-e*x/d)+2*b^2*d*n^2*polylog(3,-e*x/d)/e^2+(-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*x-ln(x
^n)*d/e^2*ln(e*x+d)-n*((e*x+d)/e^2-d/e^2*(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*(x/e-d/e^2*ln(e*x+d))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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