\(\int \frac {x (a+b \log (c x^n))^2}{(d+e x)^4} \, dx\) [116]

Optimal result

Integrand size = 21, antiderivative size = 210 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\frac {b^2 n^2}{3 d e^2 (d+e x)}-\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 e^2 (d+e x)^2}+\frac {b n \left (a+b \log \left (c x^n\right )\right )}{3 d e^2 (d+e x)}+\frac {\left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 e^2}+\frac {d \left (a+b \log \left (c x^n\right )\right )^2}{3 e^2 (d+e x)^3}-\frac {\left (a+b \log \left (c x^n\right )\right )^2}{2 e^2 (d+e x)^2}-\frac {b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^2 e^2}-\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d^2 e^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2383, 2381, 2384, 2354, 2438, 2373, 45} \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=-\frac {b n \log \left (\frac {e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )+b n\right )}{3 d^2 e^2}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 (d+e x)^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}-\frac {b^2 n^2 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{3 d^2 e^2}+\frac {b^2 n^2 \log (d+e x)}{3 d^2 e^2}+\frac {b^2 n^2}{3 d e^2 (d+e x)} \]

[In]

Int[(x*(a + b*Log[c*x^n])^2)/(d + e*x)^4,x]

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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 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 2381

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

Rule 2383

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

Rule 2384

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {\int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^3} \, dx}{3 d}-\frac {(2 b n) \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx}{3 d} \\ & = -\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 (d+e x)^2}-\frac {(b n) \int \frac {x \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx}{3 d^2}+\frac {\left (b^2 n^2\right ) \int \frac {x}{(d+e x)^2} \, dx}{3 d^2} \\ & = -\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 (d+e x)^2}-\frac {(b n) \int \frac {a+b n+b \log \left (c x^n\right )}{d+e x} \, dx}{3 d^2 e}+\frac {\left (b^2 n^2\right ) \int \left (-\frac {d}{e (d+e x)^2}+\frac {1}{e (d+e x)}\right ) \, dx}{3 d^2} \\ & = \frac {b^2 n^2}{3 d e^2 (d+e x)}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 (d+e x)^2}+\frac {b^2 n^2 \log (d+e x)}{3 d^2 e^2}-\frac {b n \left (a+b n+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^2 e^2}+\frac {\left (b^2 n^2\right ) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{3 d^2 e^2} \\ & = \frac {b^2 n^2}{3 d e^2 (d+e x)}-\frac {b n x^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^2}+\frac {b n x \left (a+b \log \left (c x^n\right )\right )}{3 d^2 e (d+e x)}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{3 d (d+e x)^3}+\frac {x^2 \left (a+b \log \left (c x^n\right )\right )^2}{6 d^2 (d+e x)^2}+\frac {b^2 n^2 \log (d+e x)}{3 d^2 e^2}-\frac {b n \left (a+b n+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 d^2 e^2}-\frac {b^2 n^2 \text {Li}_2\left (-\frac {e x}{d}\right )}{3 d^2 e^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.34 \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\frac {2 b^2 d^3 n^2+2 a b d^2 e n x+4 b^2 d^2 e n^2 x+3 a^2 d e^2 x^2+2 a b d e^2 n x^2+2 b^2 d e^2 n^2 x^2+a^2 e^3 x^3+b^2 e^2 x^2 (3 d+e x) \log ^2\left (c x^n\right )-2 a b d^3 n \log \left (1+\frac {e x}{d}\right )-6 a b d^2 e n x \log \left (1+\frac {e x}{d}\right )-6 a b d e^2 n x^2 \log \left (1+\frac {e x}{d}\right )-2 a b e^3 n x^3 \log \left (1+\frac {e x}{d}\right )-2 b \log \left (c x^n\right ) \left (-e x (b d n (d+e x)+a e x (3 d+e x))+b n (d+e x)^3 \log \left (1+\frac {e x}{d}\right )\right )-2 b^2 n^2 (d+e x)^3 \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{6 d^2 e^2 (d+e x)^3} \]

[In]

Integrate[(x*(a + b*Log[c*x^n])^2)/(d + e*x)^4,x]

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 0.56 (sec) , antiderivative size = 508, normalized size of antiderivative = 2.42

method	result	size
risch	\(-\frac {b^{2} \ln \left (x^{n}\right )^{2}}{2 e^{2} \left (e x +d \right )^{2}}+\frac {b^{2} \ln \left (x^{n}\right )^{2} d}{3 e^{2} \left (e x +d \right )^{3}}-\frac {b^{2} n \ln \left (x^{n}\right )}{3 e^{2} \left (e x +d \right )^{2}}-\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (e x +d \right )}{3 e^{2} d^{2}}+\frac {b^{2} n \ln \left (x^{n}\right )}{3 e^{2} d \left (e x +d \right )}+\frac {b^{2} n \ln \left (x^{n}\right ) \ln \left (x \right )}{3 e^{2} d^{2}}-\frac {b^{2} n^{2} \ln \left (x \right )^{2}}{6 e^{2} d^{2}}+\frac {b^{2} n^{2}}{3 d \,e^{2} \left (e x +d \right )}+\frac {b^{2} n^{2} \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{3 e^{2} d^{2}}+\frac {b^{2} n^{2} \operatorname {dilog}\left (-\frac {e x}{d}\right )}{3 e^{2} d^{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 )}{2 e^{2} \left (e x +d \right )^{2}}+\frac {\ln \left (x^{n}\right ) d}{3 e^{2} \left (e x +d \right )^{3}}-\frac {n \left (\frac {\ln \left (e x +d \right )}{d^{2}}-\frac {1}{d \left (e x +d \right )}+\frac {1}{\left (e x +d \right )^{2}}-\frac {\ln \left (x \right )}{d^{2}}\right )}{6 e^{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 {1}{2 e^{2} \left (e x +d \right )^{2}}+\frac {d}{3 e^{2} \left (e x +d \right )^{3}}\right )}{4}\)	\(508\)

[In]

int(x*(a+b*ln(c*x^n))^2/(e*x+d)^4,x,method=_RETURNVERBOSE)

[Out]

-1/2*b^2*ln(x^n)^2/e^2/(e*x+d)^2+1/3*b^2*ln(x^n)^2/e^2*d/(e*x+d)^3-1/3*b^2*n*ln(x^n)/e^2/(e*x+d)^2-1/3*b^2*n*l
n(x^n)/e^2/d^2*ln(e*x+d)+1/3*b^2*n*ln(x^n)/e^2/d/(e*x+d)+1/3*b^2*n*ln(x^n)/e^2/d^2*ln(x)-1/6*b^2*n^2/e^2/d^2*l
n(x)^2+1/3*b^2*n^2/d/e^2/(e*x+d)+1/3*b^2*n^2/e^2/d^2*ln(e*x+d)*ln(-e*x/d)+1/3*b^2*n^2/e^2/d^2*dilog(-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*(-1/2*ln(x^n)/e^2/(e*x+d)^2+1/3*ln(x^n)/e^2*d/(e*x+d)^3-1/6*n/e^2*(1/
d^2*ln(e*x+d)-1/d/(e*x+d)+1/(e*x+d)^2-1/d^2*ln(x)))+1/4*(-I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)+I*b*Pi*cs
gn(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*(-1/2/e^2/(
e*x+d)^2+1/3/e^2*d/(e*x+d)^3)

Fricas [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate(x*(a+b*log(c*x^n))^2/(e*x+d)^4,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^4*x^4 + 4*d*e^3*x^3 + 6*d^2*e^2*x^2 + 4*d^3*e*x
+ d^4), x)

Sympy [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int \frac {x \left (a + b \log {\left (c x^{n} \right )}\right )^{2}}{\left (d + e x\right )^{4}}\, dx \]

[In]

integrate(x*(a+b*ln(c*x**n))**2/(e*x+d)**4,x)

[Out]

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

Maxima [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate(x*(a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/3*a*b*n*(x/(d*e^3*x^2 + 2*d^2*e^2*x + d^3*e) - log(e*x + d)/(d^2*e^2) + log(x)/(d^2*e^2)) - 1/6*((3*e*x + d)
*log(x^n)^2/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2) - 6*integrate(1/3*(3*e^2*x^2*log(c)^2 + (4*d*e*n*x
+ d^2*n + 3*(e^2*n + 2*e^2*log(c))*x^2)*log(x^n))/(e^6*x^5 + 4*d*e^5*x^4 + 6*d^2*e^4*x^3 + 4*d^3*e^3*x^2 + d^
4*e^2*x), x))*b^2 - 1/3*(3*e*x + d)*a*b*log(c*x^n)/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2) - 1/6*(3*e*
x + d)*a^2/(e^5*x^3 + 3*d*e^4*x^2 + 3*d^2*e^3*x + d^3*e^2)

Giac [F]

\[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} x}{{\left (e x + d\right )}^{4}} \,d x } \]

[In]

integrate(x*(a+b*log(c*x^n))^2/(e*x+d)^4,x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b \log \left (c x^n\right )\right )^2}{(d+e x)^4} \, dx=\int \frac {x\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{{\left (d+e\,x\right )}^4} \,d x \]

[In]

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

[Out]

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