\(\int \frac {a+b \log (c x^n)}{x^2 (d+e x)^4} \, dx\) [60]

Optimal result

Integrand size = 21, antiderivative size = 211 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^4} \, dx=-\frac {b n}{d^4 x}+\frac {b e n}{6 d^3 (d+e x)^2}+\frac {4 b e n}{3 d^4 (d+e x)}+\frac {4 b e n \log (x)}{3 d^5}-\frac {a+b \log \left (c x^n\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac {4 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {13 b e n \log (d+e x)}{3 d^5}-\frac {4 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^5} \]

[Out]

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

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {46, 2393, 2341, 2356, 2351, 31, 2379, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^4} \, dx=\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac {4 e \log \left (\frac {d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {a+b \log \left (c x^n\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^3}-\frac {4 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x}\right )}{d^5}+\frac {4 b e n \log (x)}{3 d^5}-\frac {13 b e n \log (d+e x)}{3 d^5}+\frac {4 b e n}{3 d^4 (d+e x)}-\frac {b n}{d^4 x}+\frac {b e n}{6 d^3 (d+e x)^2} \]

[In]

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

[Out]

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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 2351

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

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && 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 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}& = \int \left (\frac {a+b \log \left (c x^n\right )}{d^4 x^2}+\frac {e^2 \left (a+b \log \left (c x^n\right )\right )}{d^2 (d+e x)^4}+\frac {2 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^3}+\frac {3 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^2}-\frac {4 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x (d+e x)}\right ) \, dx \\ & = \frac {\int \frac {a+b \log \left (c x^n\right )}{x^2} \, dx}{d^4}-\frac {(4 e) \int \frac {a+b \log \left (c x^n\right )}{x (d+e x)} \, dx}{d^4}+\frac {\left (3 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^4}+\frac {\left (2 e^2\right ) \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^3}+\frac {e^2 \int \frac {a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^2} \\ & = -\frac {b n}{d^4 x}-\frac {a+b \log \left (c x^n\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac {4 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {(4 b e n) \int \frac {\log \left (1+\frac {d}{e x}\right )}{x} \, dx}{d^5}+\frac {(b e n) \int \frac {1}{x (d+e x)^2} \, dx}{d^3}+\frac {(b e n) \int \frac {1}{x (d+e x)^3} \, dx}{3 d^2}-\frac {\left (3 b e^2 n\right ) \int \frac {1}{d+e x} \, dx}{d^5} \\ & = -\frac {b n}{d^4 x}-\frac {a+b \log \left (c x^n\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac {4 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {3 b e n \log (d+e x)}{d^5}-\frac {4 b e n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^5}+\frac {(b e n) \int \left (\frac {1}{d^2 x}-\frac {e}{d (d+e x)^2}-\frac {e}{d^2 (d+e x)}\right ) \, dx}{d^3}+\frac {(b e n) \int \left (\frac {1}{d^3 x}-\frac {e}{d (d+e x)^3}-\frac {e}{d^2 (d+e x)^2}-\frac {e}{d^3 (d+e x)}\right ) \, dx}{3 d^2} \\ & = -\frac {b n}{d^4 x}+\frac {b e n}{6 d^3 (d+e x)^2}+\frac {4 b e n}{3 d^4 (d+e x)}+\frac {4 b e n \log (x)}{3 d^5}-\frac {a+b \log \left (c x^n\right )}{d^4 x}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{3 d^2 (d+e x)^3}-\frac {e \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^2}+\frac {3 e^2 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac {4 e \log \left (1+\frac {d}{e x}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}-\frac {13 b e n \log (d+e x)}{3 d^5}-\frac {4 b e n \text {Li}_2\left (-\frac {d}{e x}\right )}{d^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.09 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^4} \, dx=\frac {-\frac {6 b d n}{x}-\frac {6 d \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {2 d^3 e \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}-\frac {6 d^2 e \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac {18 d e \left (a+b \log \left (c x^n\right )\right )}{d+e x}-\frac {12 e \left (a+b \log \left (c x^n\right )\right )^2}{b n}+b e n \left (\frac {d (3 d+2 e x)}{(d+e x)^2}+2 \log (x)-2 \log (d+e x)\right )+18 b e n (\log (x)-\log (d+e x))+6 b e n \left (\frac {d}{d+e x}+\log (x)-\log (d+e x)\right )+24 e \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {e x}{d}\right )+24 b e n \operatorname {PolyLog}\left (2,-\frac {e x}{d}\right )}{6 d^5} \]

[In]

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

[Out]

((-6*b*d*n)/x - (6*d*(a + b*Log[c*x^n]))/x - (2*d^3*e*(a + b*Log[c*x^n]))/(d + e*x)^3 - (6*d^2*e*(a + b*Log[c*
x^n]))/(d + e*x)^2 - (18*d*e*(a + b*Log[c*x^n]))/(d + e*x) - (12*e*(a + b*Log[c*x^n])^2)/(b*n) + b*e*n*((d*(3*
d + 2*e*x))/(d + e*x)^2 + 2*Log[x] - 2*Log[d + e*x]) + 18*b*e*n*(Log[x] - Log[d + e*x]) + 6*b*e*n*(d/(d + e*x)
+ Log[x] - Log[d + e*x]) + 24*e*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] + 24*b*e*n*PolyLog[2, -((e*x)/d)])/(6*d^5
)

Maple [C] (warning: unable to verify)

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

Time = 0.75 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.75

method	result	size
risch	\(-\frac {b \ln \left (x^{n}\right ) e}{3 d^{2} \left (e x +d \right )^{3}}+\frac {4 b \ln \left (x^{n}\right ) e \ln \left (e x +d \right )}{d^{5}}-\frac {3 b \ln \left (x^{n}\right ) e}{d^{4} \left (e x +d \right )}-\frac {b \ln \left (x^{n}\right ) e}{d^{3} \left (e x +d \right )^{2}}-\frac {b \ln \left (x^{n}\right )}{d^{4} x}-\frac {4 b \ln \left (x^{n}\right ) e \ln \left (x \right )}{d^{5}}+\frac {2 b n e \ln \left (x \right )^{2}}{d^{5}}-\frac {4 b n e \ln \left (e x +d \right ) \ln \left (-\frac {e x}{d}\right )}{d^{5}}-\frac {4 b n e \operatorname {dilog}\left (-\frac {e x}{d}\right )}{d^{5}}+\frac {4 b e n}{3 d^{4} \left (e x +d \right )}-\frac {13 b e n \ln \left (e x +d \right )}{3 d^{5}}+\frac {b e n}{6 d^{3} \left (e x +d \right )^{2}}-\frac {b n}{d^{4} x}+\frac {13 b e n \ln \left (x \right )}{3 d^{5}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (-\frac {e}{3 d^{2} \left (e x +d \right )^{3}}+\frac {4 e \ln \left (e x +d \right )}{d^{5}}-\frac {3 e}{d^{4} \left (e x +d \right )}-\frac {e}{d^{3} \left (e x +d \right )^{2}}-\frac {1}{d^{4} x}-\frac {4 e \ln \left (x \right )}{d^{5}}\right )\)	\(370\)

[In]

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

[Out]

-1/3*b*ln(x^n)/d^2/(e*x+d)^3*e+4*b*ln(x^n)/d^5*e*ln(e*x+d)-3*b*ln(x^n)/d^4*e/(e*x+d)-b*ln(x^n)/d^3/(e*x+d)^2*e
-b*ln(x^n)/d^4/x-4*b*ln(x^n)/d^5*e*ln(x)+2*b*n/d^5*e*ln(x)^2-4*b*n/d^5*e*ln(e*x+d)*ln(-e*x/d)-4*b*n/d^5*e*dilo
g(-e*x/d)+4/3*b*e*n/d^4/(e*x+d)-13/3*b*e*n*ln(e*x+d)/d^5+1/6*b*e*n/d^3/(e*x+d)^2-b*n/d^4/x+13/3*b*e*n*ln(x)/d^
5+(-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)*(-1/3/d^2/(e*x+d)^3*e+4/d^5*e*ln(e*x+d)-3/d^4*e/(e*x+d)
-1/d^3/(e*x+d)^2*e-1/d^4/x-4/d^5*e*ln(x))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 57.44 (sec) , antiderivative size = 614, normalized size of antiderivative = 2.91 \[ \int \frac {a+b \log \left (c x^n\right )}{x^2 (d+e x)^4} \, dx=\text {Too large to display} \]

[In]

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

[Out]

a*e**2*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/d**2 + 2*a*e**2*Piecewise((x/d**3, Eq(e, 0
)), (-1/(2*e*(d + e*x)**2), True))/d**3 + 3*a*e**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/d*
*4 - a/(d**4*x) + 4*a*e**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/d**5 - 4*a*e*log(x)/d**5 - b*e**
2*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))/d**2 + b*e**2*Piecewise(
(x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))*log(c*x**n)/d**2 - 2*b*e**2*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))/d**3 + 2*b*e**2*Piecewise((
x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))*log(c*x**n)/d**3 - 3*b*e**2*n*Piecewise((x/d**2, Eq(e, 0)),
(-log(x)/(d*e) + log(d/e + x)/(d*e), True))/d**4 + 3*b*e**2*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x),
True))*log(c*x**n)/d**4 - b*n/(d**4*x) - b*log(c*x**n)/(d**4*x) - 4*b*e**2*n*Piecewise((x/d, Eq(e, 0)), (Piece
wise((-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_pola
r(I*pi)/d), True))/e, True))/d**5 + 4*b*e**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))*log(c*x**n)/d*
*5 + 2*b*e*n*log(x)**2/d**5 - 4*b*e*log(x)*log(c*x**n)/d**5

Maxima [F]

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

[In]

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

[Out]

-1/3*a*((12*e^3*x^3 + 30*d*e^2*x^2 + 22*d^2*e*x + 3*d^3)/(d^4*e^3*x^4 + 3*d^5*e^2*x^3 + 3*d^6*e*x^2 + d^7*x) -
12*e*log(e*x + d)/d^5 + 12*e*log(x)/d^5) + b*integrate((log(c) + log(x^n))/(e^4*x^6 + 4*d*e^3*x^5 + 6*d^2*e^2
*x^4 + 4*d^3*e*x^3 + d^4*x^2), x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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