\(\int \frac {x^4 (a+b \log (c x^n))}{(d+e x)^4} \, dx\) [54]

Optimal result

Integrand size = 21, antiderivative size = 183 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=-\frac {4 b n x}{e^4}+\frac {(12 a+13 b n) x}{3 e^4}+\frac {4 b x \log \left (c x^n\right )}{e^4}-\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^3 \left (4 a+b n+4 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^2 \left (12 a+7 b n+12 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}-\frac {d \left (12 a+13 b n+12 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 e^5}-\frac {4 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^5} \]

[Out]

-4*b*n*x/e^4+1/3*(13*b*n+12*a)*x/e^4+4*b*x*ln(c*x^n)/e^4-1/3*x^4*(a+b*ln(c*x^n))/e/(e*x+d)^3-1/6*x^3*(4*a+b*n+
4*b*ln(c*x^n))/e^2/(e*x+d)^2-1/6*x^2*(12*a+7*b*n+12*b*ln(c*x^n))/e^3/(e*x+d)-1/3*d*(12*a+13*b*n+12*b*ln(c*x^n)
)*ln(1+e*x/d)/e^5-4*b*d*n*polylog(2,-e*x/d)/e^5

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2384, 45, 2393, 2332, 2354, 2438} \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=-\frac {d \log \left (\frac {e x}{d}+1\right ) \left (12 a+12 b \log \left (c x^n\right )+13 b n\right )}{3 e^5}-\frac {x^2 \left (12 a+12 b \log \left (c x^n\right )+7 b n\right )}{6 e^3 (d+e x)}-\frac {x^3 \left (4 a+4 b \log \left (c x^n\right )+b n\right )}{6 e^2 (d+e x)^2}-\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}+\frac {x (12 a+13 b n)}{3 e^4}+\frac {4 b x \log \left (c x^n\right )}{e^4}-\frac {4 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{e^5}-\frac {4 b n x}{e^4} \]

[In]

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

[Out]

(-4*b*n*x)/e^4 + ((12*a + 13*b*n)*x)/(3*e^4) + (4*b*x*Log[c*x^n])/e^4 - (x^4*(a + b*Log[c*x^n]))/(3*e*(d + e*x
)^3) - (x^3*(4*a + b*n + 4*b*Log[c*x^n]))/(6*e^2*(d + e*x)^2) - (x^2*(12*a + 7*b*n + 12*b*Log[c*x^n]))/(6*e^3*
(d + e*x)) - (d*(12*a + 13*b*n + 12*b*Log[c*x^n])*Log[1 + (e*x)/d])/(3*e^5) - (4*b*d*n*PolyLog[2, -((e*x)/d)])
/e^5

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 2332

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

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 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 2393

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

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^4 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}+\frac {\int \frac {x^3 \left (4 a+b n+4 b \log \left (c x^n\right )\right )}{(d+e x)^3} \, dx}{3 e} \\ & = -\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^3 \left (4 a+b n+4 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}+\frac {\int \frac {x^2 \left (4 b n+3 (4 a+b n)+12 b \log \left (c x^n\right )\right )}{(d+e x)^2} \, dx}{6 e^2} \\ & = -\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^3 \left (4 a+b n+4 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^2 \left (12 a+7 b n+12 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}+\frac {\int \frac {x \left (12 b n+2 (4 b n+3 (4 a+b n))+24 b \log \left (c x^n\right )\right )}{d+e x} \, dx}{6 e^3} \\ & = -\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^3 \left (4 a+b n+4 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^2 \left (12 a+7 b n+12 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}+\frac {\int \left (\frac {12 b n+2 (4 b n+3 (4 a+b n))+24 b \log \left (c x^n\right )}{e}-\frac {d \left (12 b n+2 (4 b n+3 (4 a+b n))+24 b \log \left (c x^n\right )\right )}{e (d+e x)}\right ) \, dx}{6 e^3} \\ & = -\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^3 \left (4 a+b n+4 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^2 \left (12 a+7 b n+12 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}+\frac {\int \left (12 b n+2 (4 b n+3 (4 a+b n))+24 b \log \left (c x^n\right )\right ) \, dx}{6 e^4}-\frac {d \int \frac {12 b n+2 (4 b n+3 (4 a+b n))+24 b \log \left (c x^n\right )}{d+e x} \, dx}{6 e^4} \\ & = \frac {(12 a+13 b n) x}{3 e^4}-\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^3 \left (4 a+b n+4 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^2 \left (12 a+7 b n+12 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}-\frac {d \left (12 a+13 b n+12 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 e^5}+\frac {(4 b) \int \log \left (c x^n\right ) \, dx}{e^4}+\frac {(4 b d n) \int \frac {\log \left (1+\frac {e x}{d}\right )}{x} \, dx}{e^5} \\ & = -\frac {4 b n x}{e^4}+\frac {(12 a+13 b n) x}{3 e^4}+\frac {4 b x \log \left (c x^n\right )}{e^4}-\frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}-\frac {x^3 \left (4 a+b n+4 b \log \left (c x^n\right )\right )}{6 e^2 (d+e x)^2}-\frac {x^2 \left (12 a+7 b n+12 b \log \left (c x^n\right )\right )}{6 e^3 (d+e x)}-\frac {d \left (12 a+13 b n+12 b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )}{3 e^5}-\frac {4 b d n \text {Li}_2\left (-\frac {e x}{d}\right )}{e^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.13 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\frac {6 a e x-6 b e n x+6 b e x \log \left (c x^n\right )-\frac {2 d^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}+\frac {12 d^3 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {36 d^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+b d n \left (\frac {d (3 d+2 e x)}{(d+e x)^2}+2 \log (x)-2 \log (d+e x)\right )+36 b d n (\log (x)-\log (d+e x))-12 b d n \left (\frac {d}{d+e x}+\log (x)-\log (d+e x)\right )-24 d \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )-24 b d n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{6 e^5} \]

[In]

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

[Out]

(6*a*e*x - 6*b*e*n*x + 6*b*e*x*Log[c*x^n] - (2*d^4*(a + b*Log[c*x^n]))/(d + e*x)^3 + (12*d^3*(a + b*Log[c*x^n]
))/(d + e*x)^2 - (36*d^2*(a + b*Log[c*x^n]))/(d + e*x) + b*d*n*((d*(3*d + 2*e*x))/(d + e*x)^2 + 2*Log[x] - 2*L
og[d + e*x]) + 36*b*d*n*(Log[x] - Log[d + e*x]) - 12*b*d*n*(d/(d + e*x) + Log[x] - Log[d + e*x]) - 24*d*(a + b
*Log[c*x^n])*Log[1 + (e*x)/d] - 24*b*d*n*PolyLog[2, -((e*x)/d)])/(6*e^5)

Maple [C] (warning: unable to verify)

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

Time = 0.53 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.94

method	result	size
risch	\(\frac {b \ln \left (x^{n}\right ) x}{e^{4}}-\frac {b \ln \left (x^{n}\right ) d^{4}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {4 b \ln \left (x^{n}\right ) d \ln \left (e x +d \right )}{e^{5}}-\frac {6 b \ln \left (x^{n}\right ) d^{2}}{e^{5} \left (e x +d \right )}+\frac {2 b \ln \left (x^{n}\right ) d^{3}}{e^{5} \left (e x +d \right )^{2}}-\frac {b n x}{e^{4}}-\frac {b n d}{e^{5}}+\frac {b n \,d^{3}}{6 e^{5} \left (e x +d \right )^{2}}-\frac {13 b n d \ln \left (e x +d \right )}{3 e^{5}}-\frac {5 b n \,d^{2}}{3 e^{5} \left (e x +d \right )}+\frac {13 b n d \ln \left (e x \right )}{3 e^{5}}+\frac {4 b n d \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{e^{5}}+\frac {4 b n d \operatorname {dilog}\left (-\frac {e x}{d}\right )}{e^{5}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {x}{e^{4}}-\frac {d^{4}}{3 e^{5} \left (e x +d \right )^{3}}-\frac {4 d \ln \left (e x +d \right )}{e^{5}}-\frac {6 d^{2}}{e^{5} \left (e x +d \right )}+\frac {2 d^{3}}{e^{5} \left (e x +d \right )^{2}}\right )\)	\(355\)

[In]

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

[Out]

b*ln(x^n)/e^4*x-1/3*b*ln(x^n)/e^5*d^4/(e*x+d)^3-4*b*ln(x^n)/e^5*d*ln(e*x+d)-6*b*ln(x^n)/e^5*d^2/(e*x+d)+2*b*ln
(x^n)/e^5*d^3/(e*x+d)^2-b*n*x/e^4-b*n/e^5*d+1/6*b*n/e^5*d^3/(e*x+d)^2-13/3*b*n/e^5*d*ln(e*x+d)-5/3*b*n/e^5*d^2
/(e*x+d)+13/3*b*n/e^5*d*ln(e*x)+4*b*n/e^5*d*ln(e*x+d)*ln(-e*x/d)+4*b*n/e^5*d*dilog(-e*x/d)+(-1/2*I*b*Pi*csgn(I
*c)*csgn(I*x^n)*csgn(I*c*x^n)+1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2-1/2*
I*b*Pi*csgn(I*c*x^n)^3+b*ln(c)+a)*(x/e^4-1/3/e^5*d^4/(e*x+d)^3-4/e^5*d*ln(e*x+d)-6/e^5*d^2/(e*x+d)+2/e^5*d^3/(
e*x+d)^2)

Fricas [F]

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

[In]

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

[Out]

integral((b*x^4*log(c*x^n) + a*x^4)/(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 [A] (verification not implemented)

Time = 31.90 (sec) , antiderivative size = 563, normalized size of antiderivative = 3.08 \[ \int \frac {x^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

a*d**4*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/e**4 - 4*a*d**3*Piecewise((x/d**3, Eq(e, 0
)), (-1/(2*e*(d + e*x)**2), True))/e**4 + 6*a*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/e*
*4 - 4*a*d*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**4 + a*x/e**4 - b*d**4*n*Piecewise((x/d**4, Eq
(e, 0)), (-3*d/(6*d**4*e + 12*d**3*e**2*x + 6*d**2*e**3*x**2) - 2*e*x/(6*d**4*e + 12*d**3*e**2*x + 6*d**2*e**3
*x**2) - log(x)/(3*d**3*e) + log(d/e + x)/(3*d**3*e), True))/e**4 + b*d**4*Piecewise((x/d**4, Eq(e, 0)), (-1/(
3*e*(d + e*x)**3), True))*log(c*x**n)/e**4 + 4*b*d**3*n*Piecewise((x/d**3, Eq(e, 0)), (-1/(2*d**2*e + 2*d*e**2
*x) - log(x)/(2*d**2*e) + log(d/e + x)/(2*d**2*e), True))/e**4 - 4*b*d**3*Piecewise((x/d**3, Eq(e, 0)), (-1/(2
*e*(d + e*x)**2), True))*log(c*x**n)/e**4 - 6*b*d**2*n*Piecewise((x/d**2, Eq(e, 0)), (-log(x)/(d*e) + log(d/e
+ x)/(d*e), True))/e**4 + 6*b*d**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))*log(c*x**n)/e**4 +
4*b*d*n*Piecewise((x/d, Eq(e, 0)), (Piecewise((-polylog(2, e*x*exp_polar(I*pi)/d), (Abs(x) < 1) & (1/Abs(x) <
1)), (log(d)*log(x) - polylog(2, e*x*exp_polar(I*pi)/d), Abs(x) < 1), (-log(d)*log(1/x) - polylog(2, e*x*exp_
polar(I*pi)/d), 1/Abs(x) < 1), (-meijerg(((), (1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0
, 0)), x)*log(d) - polylog(2, e*x*exp_polar(I*pi)/d), True))/e, True))/e**4 - 4*b*d*Piecewise((x/d, Eq(e, 0)),
(log(d + e*x)/e, True))*log(c*x**n)/e**4 - b*n*x/e**4 + b*x*log(c*x**n)/e**4

Maxima [F]

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

[In]

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

[Out]

-1/3*a*((18*d^2*e^2*x^2 + 30*d^3*e*x + 13*d^4)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) - 3*x/e^4 + 12*
d*log(e*x + d)/e^5) + b*integrate((x^4*log(c) + x^4*log(x^n))/(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)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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