3.61 \(\int \frac{a+b \log (c x^n)}{x^3 (d+e x)^4} \, dx\)

Optimal. Leaf size=263 \[ \frac{10 b e^2 n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d^6}-\frac{6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}-\frac{10 e^2 \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^6}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}-\frac{a+b \log \left (c x^n\right )}{2 d^4 x^2}-\frac{11 b e^2 n}{6 d^5 (d+e x)}-\frac{b e^2 n}{6 d^4 (d+e x)^2}-\frac{11 b e^2 n \log (x)}{6 d^6}+\frac{47 b e^2 n \log (d+e x)}{6 d^6}+\frac{4 b e n}{d^5 x}-\frac{b n}{4 d^4 x^2} \]

[Out]

-(b*n)/(4*d^4*x^2) + (4*b*e*n)/(d^5*x) - (b*e^2*n)/(6*d^4*(d + e*x)^2) - (11*b*e^2*n)/(6*d^5*(d + e*x)) - (11*
b*e^2*n*Log[x])/(6*d^6) - (a + b*Log[c*x^n])/(2*d^4*x^2) + (4*e*(a + b*Log[c*x^n]))/(d^5*x) + (e^2*(a + b*Log[
c*x^n]))/(3*d^3*(d + e*x)^3) + (3*e^2*(a + b*Log[c*x^n]))/(2*d^4*(d + e*x)^2) - (6*e^3*x*(a + b*Log[c*x^n]))/(
d^6*(d + e*x)) - (10*e^2*Log[1 + d/(e*x)]*(a + b*Log[c*x^n]))/d^6 + (47*b*e^2*n*Log[d + e*x])/(6*d^6) + (10*b*
e^2*n*PolyLog[2, -(d/(e*x))])/d^6

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Rubi [A]  time = 0.346497, antiderivative size = 285, normalized size of antiderivative = 1.08, number of steps used = 15, number of rules used = 9, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.429, Rules used = {44, 2351, 2304, 2301, 2319, 2314, 31, 2317, 2391} \[ -\frac{10 b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^6}-\frac{6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}+\frac{5 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^6 n}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}-\frac{10 e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^6}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}-\frac{a+b \log \left (c x^n\right )}{2 d^4 x^2}-\frac{11 b e^2 n}{6 d^5 (d+e x)}-\frac{b e^2 n}{6 d^4 (d+e x)^2}-\frac{11 b e^2 n \log (x)}{6 d^6}+\frac{47 b e^2 n \log (d+e x)}{6 d^6}+\frac{4 b e n}{d^5 x}-\frac{b n}{4 d^4 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x^3*(d + e*x)^4),x]

[Out]

-(b*n)/(4*d^4*x^2) + (4*b*e*n)/(d^5*x) - (b*e^2*n)/(6*d^4*(d + e*x)^2) - (11*b*e^2*n)/(6*d^5*(d + e*x)) - (11*
b*e^2*n*Log[x])/(6*d^6) - (a + b*Log[c*x^n])/(2*d^4*x^2) + (4*e*(a + b*Log[c*x^n]))/(d^5*x) + (e^2*(a + b*Log[
c*x^n]))/(3*d^3*(d + e*x)^3) + (3*e^2*(a + b*Log[c*x^n]))/(2*d^4*(d + e*x)^2) - (6*e^3*x*(a + b*Log[c*x^n]))/(
d^6*(d + e*x)) + (5*e^2*(a + b*Log[c*x^n])^2)/(b*d^6*n) + (47*b*e^2*n*Log[d + e*x])/(6*d^6) - (10*e^2*(a + b*L
og[c*x^n])*Log[1 + (e*x)/d])/d^6 - (10*b*e^2*n*PolyLog[2, -((e*x)/d)])/d^6

Rule 44

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] && L
tQ[m + n + 2, 0])

Rule 2351

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 2304

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 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2319

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 2314

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 31

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

Rule 2317

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 2391

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*} \int \frac{a+b \log \left (c x^n\right )}{x^3 (d+e x)^4} \, dx &=\int \left (\frac{a+b \log \left (c x^n\right )}{d^4 x^3}-\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x^2}+\frac{10 e^2 \left (a+b \log \left (c x^n\right )\right )}{d^6 x}-\frac{e^3 \left (a+b \log \left (c x^n\right )\right )}{d^3 (d+e x)^4}-\frac{3 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)^3}-\frac{6 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)^2}-\frac{10 e^3 \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}\right ) \, dx\\ &=\frac{\int \frac{a+b \log \left (c x^n\right )}{x^3} \, dx}{d^4}-\frac{(4 e) \int \frac{a+b \log \left (c x^n\right )}{x^2} \, dx}{d^5}+\frac{\left (10 e^2\right ) \int \frac{a+b \log \left (c x^n\right )}{x} \, dx}{d^6}-\frac{\left (10 e^3\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{d^6}-\frac{\left (6 e^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{d^5}-\frac{\left (3 e^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{d^4}-\frac{e^3 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{d^3}\\ &=-\frac{b n}{4 d^4 x^2}+\frac{4 b e n}{d^5 x}-\frac{a+b \log \left (c x^n\right )}{2 d^4 x^2}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac{6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}+\frac{5 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^6 n}-\frac{10 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^6}+\frac{\left (10 b e^2 n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{d^6}-\frac{\left (3 b e^2 n\right ) \int \frac{1}{x (d+e x)^2} \, dx}{2 d^4}-\frac{\left (b e^2 n\right ) \int \frac{1}{x (d+e x)^3} \, dx}{3 d^3}+\frac{\left (6 b e^3 n\right ) \int \frac{1}{d+e x} \, dx}{d^6}\\ &=-\frac{b n}{4 d^4 x^2}+\frac{4 b e n}{d^5 x}-\frac{a+b \log \left (c x^n\right )}{2 d^4 x^2}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac{6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}+\frac{5 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^6 n}+\frac{6 b e^2 n \log (d+e x)}{d^6}-\frac{10 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^6}-\frac{10 b e^2 n \text{Li}_2\left (-\frac{e x}{d}\right )}{d^6}-\frac{\left (3 b e^2 n\right ) \int \left (\frac{1}{d^2 x}-\frac{e}{d (d+e x)^2}-\frac{e}{d^2 (d+e x)}\right ) \, dx}{2 d^4}-\frac{\left (b e^2 n\right ) \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^3}\\ &=-\frac{b n}{4 d^4 x^2}+\frac{4 b e n}{d^5 x}-\frac{b e^2 n}{6 d^4 (d+e x)^2}-\frac{11 b e^2 n}{6 d^5 (d+e x)}-\frac{11 b e^2 n \log (x)}{6 d^6}-\frac{a+b \log \left (c x^n\right )}{2 d^4 x^2}+\frac{4 e \left (a+b \log \left (c x^n\right )\right )}{d^5 x}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{3 d^3 (d+e x)^3}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^4 (d+e x)^2}-\frac{6 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^6 (d+e x)}+\frac{5 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^6 n}+\frac{47 b e^2 n \log (d+e x)}{6 d^6}-\frac{10 e^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{d^6}-\frac{10 b e^2 n \text{Li}_2\left (-\frac{e x}{d}\right )}{d^6}\\ \end{align*}

Mathematica [A]  time = 0.339154, size = 276, normalized size = 1.05 \[ \frac{-120 b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{4 d^3 e^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}+\frac{18 d^2 e^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}-\frac{6 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}+\frac{72 d e^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}-120 e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+\frac{48 d e \left (a+b \log \left (c x^n\right )\right )}{x}+\frac{60 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}-\frac{3 b d^2 n}{x^2}-\frac{2 b d e^2 n (3 d+2 e x)}{(d+e x)^2}-\frac{18 b d e^2 n}{d+e x}-72 b e^2 n (\log (x)-\log (d+e x))+22 b e^2 n \log (d+e x)+\frac{48 b d e n}{x}-22 b e^2 n \log (x)}{12 d^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x)^4),x]

[Out]

((-3*b*d^2*n)/x^2 + (48*b*d*e*n)/x - (18*b*d*e^2*n)/(d + e*x) - (2*b*d*e^2*n*(3*d + 2*e*x))/(d + e*x)^2 - 22*b
*e^2*n*Log[x] - (6*d^2*(a + b*Log[c*x^n]))/x^2 + (48*d*e*(a + b*Log[c*x^n]))/x + (4*d^3*e^2*(a + b*Log[c*x^n])
)/(d + e*x)^3 + (18*d^2*e^2*(a + b*Log[c*x^n]))/(d + e*x)^2 + (72*d*e^2*(a + b*Log[c*x^n]))/(d + e*x) + (60*e^
2*(a + b*Log[c*x^n])^2)/(b*n) - 72*b*e^2*n*(Log[x] - Log[d + e*x]) + 22*b*e^2*n*Log[d + e*x] - 120*e^2*(a + b*
Log[c*x^n])*Log[1 + (e*x)/d] - 120*b*e^2*n*PolyLog[2, -((e*x)/d)])/(12*d^6)

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Maple [C]  time = 0.177, size = 1324, normalized size = 5. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x^3/(e*x+d)^4,x)

[Out]

10*b*n/d^6*e^2*ln(e*x+d)*ln(-e*x/d)-10*b*ln(c)/d^6*e^2*ln(e*x+d)+6*b*ln(c)*e^2/d^5/(e*x+d)+1/4*I*b*Pi*csgn(I*c
*x^n)^3/d^4/x^2+5*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^6*e^2*ln(x)+1/6*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*e^2/d^3/
(e*x+d)^3+3/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^4*e^2/(e*x+d)^2+2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^5*e/x+
5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^6*e^2*ln(x)+3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*e^2/d^5/(e*x+d)+1/4*I*b*
Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^4/x^2+3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e^2/d^5/(e*x+d)+3/4*I*b*Pi
*csgn(I*x^n)*csgn(I*c*x^n)^2/d^4*e^2/(e*x+d)^2+1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e^2/d^3/(e*x+d)^3-5*I*b*
Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^6*e^2*ln(e*x+d)+2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^5*e/x-5*I*b*Pi*csgn(I*c*
x^n)^2*csgn(I*c)/d^6*e^2*ln(e*x+d)-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*e^2/d^5/(e*x+d)+5*I*b*Pi*csgn(
I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^6*e^2*ln(e*x+d)-1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*e^2/d^3/(e*x+d
)^3-3/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^4*e^2/(e*x+d)^2-5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I
*c)/d^6*e^2*ln(x)-2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^5*e/x-5*I*b*Pi*csgn(I*c*x^n)^3/d^6*e^2*ln(x)-
1/6*I*b*Pi*csgn(I*c*x^n)^3*e^2/d^3/(e*x+d)^3-2*I*b*Pi*csgn(I*c*x^n)^3/d^5*e/x-1/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(
I*c)/d^4/x^2-1/2*a/d^4/x^2-1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^4/x^2+5*I*b*Pi*csgn(I*c*x^n)^3/d^6*e^2*ln(
e*x+d)-3*I*b*Pi*csgn(I*c*x^n)^3*e^2/d^5/(e*x+d)-3/4*I*b*Pi*csgn(I*c*x^n)^3/d^4*e^2/(e*x+d)^2+3/2*b*ln(c)/d^4*e
^2/(e*x+d)^2+1/3*b*ln(c)*e^2/d^3/(e*x+d)^3+4*b*ln(c)/d^5*e/x+10*b*ln(c)/d^6*e^2*ln(x)-1/2*b*ln(x^n)/d^4/x^2+4*
a/d^5*e/x-10*a/d^6*e^2*ln(e*x+d)+10*a/d^6*e^2*ln(x)+6*a*e^2/d^5/(e*x+d)-5*b*n/d^6*e^2*ln(x)^2+10*b*n/d^6*e^2*d
ilog(-e*x/d)+3/2*a/d^4*e^2/(e*x+d)^2+1/3*a*e^2/d^3/(e*x+d)^3-1/2*b*ln(c)/d^4/x^2-10*b*ln(x^n)/d^6*e^2*ln(e*x+d
)-47/6*b*e^2*n*ln(x)/d^6+47/6*b*e^2*n*ln(e*x+d)/d^6+6*b*ln(x^n)*e^2/d^5/(e*x+d)+3/2*b*ln(x^n)/d^4*e^2/(e*x+d)^
2+1/3*b*ln(x^n)*e^2/d^3/(e*x+d)^3+10*b*ln(x^n)/d^6*e^2*ln(x)+4*b*ln(x^n)/d^5*e/x-1/4*b*n/d^4/x^2+4*b*e*n/d^5/x
-1/6*b*e^2*n/d^4/(e*x+d)^2-11/6*b*e^2*n/d^5/(e*x+d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \, a{\left (\frac{60 \, e^{4} x^{4} + 150 \, d e^{3} x^{3} + 110 \, d^{2} e^{2} x^{2} + 15 \, d^{3} e x - 3 \, d^{4}}{d^{5} e^{3} x^{5} + 3 \, d^{6} e^{2} x^{4} + 3 \, d^{7} e x^{3} + d^{8} x^{2}} - \frac{60 \, e^{2} \log \left (e x + d\right )}{d^{6}} + \frac{60 \, e^{2} \log \left (x\right )}{d^{6}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{4} x^{7} + 4 \, d e^{3} x^{6} + 6 \, d^{2} e^{2} x^{5} + 4 \, d^{3} e x^{4} + d^{4} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*a*((60*e^4*x^4 + 150*d*e^3*x^3 + 110*d^2*e^2*x^2 + 15*d^3*e*x - 3*d^4)/(d^5*e^3*x^5 + 3*d^6*e^2*x^4 + 3*d^
7*e*x^3 + d^8*x^2) - 60*e^2*log(e*x + d)/d^6 + 60*e^2*log(x)/d^6) + b*integrate((log(c) + log(x^n))/(e^4*x^7 +
4*d*e^3*x^6 + 6*d^2*e^2*x^5 + 4*d^3*e*x^4 + d^4*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \log \left (c x^{n}\right ) + a}{e^{4} x^{7} + 4 \, d e^{3} x^{6} + 6 \, d^{2} e^{2} x^{5} + 4 \, d^{3} e x^{4} + d^{4} x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(e*x+d)^4,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(e^4*x^7 + 4*d*e^3*x^6 + 6*d^2*e^2*x^5 + 4*d^3*e*x^4 + d^4*x^3), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**3/(e*x+d)**4,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \log \left (c x^{n}\right ) + a}{{\left (e x + d\right )}^{4} x^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^3/(e*x+d)^4,x, algorithm="giac")

[Out]

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