3.53 \(\int \frac{x^5 (a+b \log (c x^n))}{(d+e x)^4} \, dx\)

Optimal. Leaf size=229 \[ \frac{10 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^6}+\frac{d^2 \log \left (\frac{e x}{d}+1\right ) \left (60 a+60 b \log \left (c x^n\right )+47 b n\right )}{6 e^6}-\frac{x^4 \left (5 a+5 b \log \left (c x^n\right )+b n\right )}{6 e^2 (d+e x)^2}-\frac{x^3 \left (20 a+20 b \log \left (c x^n\right )+9 b n\right )}{6 e^3 (d+e x)}-\frac{x^5 \left (a+b \log \left (c x^n\right )\right )}{3 e (d+e x)^3}+\frac{x^2 \left (60 a+60 b \log \left (c x^n\right )+47 b n\right )}{12 e^4}-\frac{d x (60 a+47 b n)}{6 e^5}-\frac{10 b d x \log \left (c x^n\right )}{e^5}+\frac{10 b d n x}{e^5}-\frac{5 b n x^2}{2 e^4} \]

[Out]

(10*b*d*n*x)/e^5 - (d*(60*a + 47*b*n)*x)/(6*e^5) - (5*b*n*x^2)/(2*e^4) - (10*b*d*x*Log[c*x^n])/e^5 - (x^5*(a +
b*Log[c*x^n]))/(3*e*(d + e*x)^3) - (x^4*(5*a + b*n + 5*b*Log[c*x^n]))/(6*e^2*(d + e*x)^2) - (x^3*(20*a + 9*b*
n + 20*b*Log[c*x^n]))/(6*e^3*(d + e*x)) + (x^2*(60*a + 47*b*n + 60*b*Log[c*x^n]))/(12*e^4) + (d^2*(60*a + 47*b
*n + 60*b*Log[c*x^n])*Log[1 + (e*x)/d])/(6*e^6) + (10*b*d^2*n*PolyLog[2, -((e*x)/d)])/e^6

________________________________________________________________________________________

Rubi [A]  time = 0.319522, antiderivative size = 260, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 10, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.476, Rules used = {43, 2351, 2295, 2304, 2319, 44, 2314, 31, 2317, 2391} \[ \frac{10 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{e^6}+\frac{d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac{5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac{10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac{10 d^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{e^6}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}-\frac{4 a d x}{e^5}-\frac{4 b d x \log \left (c x^n\right )}{e^5}-\frac{b d^4 n}{6 e^6 (d+e x)^2}+\frac{13 b d^3 n}{6 e^6 (d+e x)}+\frac{13 b d^2 n \log (x)}{6 e^6}+\frac{47 b d^2 n \log (d+e x)}{6 e^6}+\frac{4 b d n x}{e^5}-\frac{b n x^2}{4 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*Log[c*x^n]))/(d + e*x)^4,x]

[Out]

(-4*a*d*x)/e^5 + (4*b*d*n*x)/e^5 - (b*n*x^2)/(4*e^4) - (b*d^4*n)/(6*e^6*(d + e*x)^2) + (13*b*d^3*n)/(6*e^6*(d
+ e*x)) + (13*b*d^2*n*Log[x])/(6*e^6) - (4*b*d*x*Log[c*x^n])/e^5 + (x^2*(a + b*Log[c*x^n]))/(2*e^4) + (d^5*(a
+ b*Log[c*x^n]))/(3*e^6*(d + e*x)^3) - (5*d^4*(a + b*Log[c*x^n]))/(2*e^6*(d + e*x)^2) - (10*d^2*x*(a + b*Log[c
*x^n]))/(e^5*(d + e*x)) + (47*b*d^2*n*Log[d + e*x])/(6*e^6) + (10*d^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d])/e^6
+ (10*b*d^2*n*PolyLog[2, -((e*x)/d)])/e^6

Rule 43

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 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 2295

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

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 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 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 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{x^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^4} \, dx &=\int \left (-\frac{4 d \left (a+b \log \left (c x^n\right )\right )}{e^5}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{e^4}-\frac{d^5 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^4}+\frac{5 d^4 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^3}-\frac{10 d^3 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)^2}+\frac{10 d^2 \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}\right ) \, dx\\ &=-\frac{(4 d) \int \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^5}+\frac{\left (10 d^2\right ) \int \frac{a+b \log \left (c x^n\right )}{d+e x} \, dx}{e^5}-\frac{\left (10 d^3\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^2} \, dx}{e^5}+\frac{\left (5 d^4\right ) \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^3} \, dx}{e^5}-\frac{d^5 \int \frac{a+b \log \left (c x^n\right )}{(d+e x)^4} \, dx}{e^5}+\frac{\int x \left (a+b \log \left (c x^n\right )\right ) \, dx}{e^4}\\ &=-\frac{4 a d x}{e^5}-\frac{b n x^2}{4 e^4}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac{d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac{5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac{10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac{10 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^6}-\frac{(4 b d) \int \log \left (c x^n\right ) \, dx}{e^5}-\frac{\left (10 b d^2 n\right ) \int \frac{\log \left (1+\frac{e x}{d}\right )}{x} \, dx}{e^6}+\frac{\left (5 b d^4 n\right ) \int \frac{1}{x (d+e x)^2} \, dx}{2 e^6}-\frac{\left (b d^5 n\right ) \int \frac{1}{x (d+e x)^3} \, dx}{3 e^6}+\frac{\left (10 b d^2 n\right ) \int \frac{1}{d+e x} \, dx}{e^5}\\ &=-\frac{4 a d x}{e^5}+\frac{4 b d n x}{e^5}-\frac{b n x^2}{4 e^4}-\frac{4 b d x \log \left (c x^n\right )}{e^5}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac{d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac{5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac{10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac{10 b d^2 n \log (d+e x)}{e^6}+\frac{10 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^6}+\frac{10 b d^2 n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^6}+\frac{\left (5 b d^4 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 e^6}-\frac{\left (b d^5 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 e^6}\\ &=-\frac{4 a d x}{e^5}+\frac{4 b d n x}{e^5}-\frac{b n x^2}{4 e^4}-\frac{b d^4 n}{6 e^6 (d+e x)^2}+\frac{13 b d^3 n}{6 e^6 (d+e x)}+\frac{13 b d^2 n \log (x)}{6 e^6}-\frac{4 b d x \log \left (c x^n\right )}{e^5}+\frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{2 e^4}+\frac{d^5 \left (a+b \log \left (c x^n\right )\right )}{3 e^6 (d+e x)^3}-\frac{5 d^4 \left (a+b \log \left (c x^n\right )\right )}{2 e^6 (d+e x)^2}-\frac{10 d^2 x \left (a+b \log \left (c x^n\right )\right )}{e^5 (d+e x)}+\frac{47 b d^2 n \log (d+e x)}{6 e^6}+\frac{10 d^2 \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x}{d}\right )}{e^6}+\frac{10 b d^2 n \text{Li}_2\left (-\frac{e x}{d}\right )}{e^6}\\ \end{align*}

Mathematica [A]  time = 0.297395, size = 249, normalized size = 1.09 \[ \frac{120 b d^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{4 d^5 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^3}-\frac{30 d^4 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac{120 d^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+120 d^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )+6 e^2 x^2 \left (a+b \log \left (c x^n\right )\right )-48 a d e x-48 b d e x \log \left (c x^n\right )-2 b d^2 n \left (\frac{d (3 d+2 e x)}{(d+e x)^2}-2 \log (d+e x)+2 \log (x)\right )-120 b d^2 n (\log (x)-\log (d+e x))+30 b d^2 n \left (\frac{d}{d+e x}-\log (d+e x)+\log (x)\right )+48 b d e n x-3 b e^2 n x^2}{12 e^6} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(a + b*Log[c*x^n]))/(d + e*x)^4,x]

[Out]

(-48*a*d*e*x + 48*b*d*e*n*x - 3*b*e^2*n*x^2 - 48*b*d*e*x*Log[c*x^n] + 6*e^2*x^2*(a + b*Log[c*x^n]) + (4*d^5*(a
+ b*Log[c*x^n]))/(d + e*x)^3 - (30*d^4*(a + b*Log[c*x^n]))/(d + e*x)^2 + (120*d^3*(a + b*Log[c*x^n]))/(d + e*
x) - 2*b*d^2*n*((d*(3*d + 2*e*x))/(d + e*x)^2 + 2*Log[x] - 2*Log[d + e*x]) - 120*b*d^2*n*(Log[x] - Log[d + e*x
]) + 30*b*d^2*n*(d/(d + e*x) + Log[x] - Log[d + e*x]) + 120*d^2*(a + b*Log[c*x^n])*Log[1 + (e*x)/d] + 120*b*d^
2*n*PolyLog[2, -((e*x)/d)])/(12*e^6)

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Maple [C]  time = 0.204, size = 1153, 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(x^5*(a+b*ln(c*x^n))/(e*x+d)^4,x)

[Out]

5/4*I*b*Pi*csgn(I*c*x^n)^3/e^6*d^4/(e*x+d)^2-1/6*I*b*Pi*csgn(I*c*x^n)^3*d^5/e^6/(e*x+d)^3-10*b*n/e^6*d^2*ln(e*
x+d)*ln(-e*x/d)+1/2*a/e^4*x^2-5/2*b*ln(c)/e^6*d^4/(e*x+d)^2+1/3*b*ln(c)*d^5/e^6/(e*x+d)^3+10*b*ln(c)/e^6*d^3/(
e*x+d)-4*b*ln(c)/e^5*d*x+10*b*ln(c)/e^6*d^2*ln(e*x+d)-1/4*I*b*Pi*csgn(I*c*x^n)^3/e^4*x^2-10*b*n/e^6*d^2*dilog(
-e*x/d)+1/2*b*ln(x^n)/e^4*x^2-1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*d^5/e^6/(e*x+d)^3-47/6*b*n/e^6*d^
2*ln(e*x)+47/6*b*n/e^6*d^2*ln(e*x+d)+13/6*b*n/e^6*d^3/(e*x+d)-1/6*b*n/e^6*d^4/(e*x+d)^2-5*I*b*Pi*csgn(I*c*x^n)
^3/e^6*d^2*ln(e*x+d)+2*I*b*Pi*csgn(I*c*x^n)^3/e^5*d*x-5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^6*d^3/(e*
x+d)+2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^5*d*x+5/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^6*d
^4/(e*x+d)^2-5*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/e^6*d^2*ln(e*x+d)-5/2*a/e^6*d^4/(e*x+d)^2+5*I*b*Pi*c
sgn(I*x^n)*csgn(I*c*x^n)^2/e^6*d^2*ln(e*x+d)+5*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^6*d^3/(e*x+d)+5*I*b*Pi*csgn(
I*x^n)*csgn(I*c*x^n)^2/e^6*d^3/(e*x+d)+5*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^6*d^2*ln(e*x+d)-5/4*I*b*Pi*csgn(I*
c*x^n)^2*csgn(I*c)/e^6*d^4/(e*x+d)^2-2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^5*d*x-1/4*I*b*Pi*csgn(I*x^n)*csgn(I*
c*x^n)*csgn(I*c)/e^4*x^2+1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*d^5/e^6/(e*x+d)^3-2*I*b*Pi*csgn(I*x^n)*csgn(I*
c*x^n)^2/e^5*d*x-5/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^6*d^4/(e*x+d)^2+1/6*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)
*d^5/e^6/(e*x+d)^3-5*I*b*Pi*csgn(I*c*x^n)^3/e^6*d^3/(e*x+d)+1/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/e^4*x^2+1/4*I
*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/e^4*x^2+1/3*b*ln(x^n)*d^5/e^6/(e*x+d)^3+10*b*ln(x^n)/e^6*d^3/(e*x+d)+17/4*b*
n/e^6*d^2-4*b*ln(x^n)/e^5*d*x-5/2*b*ln(x^n)/e^6*d^4/(e*x+d)^2+1/3*a*d^5/e^6/(e*x+d)^3+10*a/e^6*d^3/(e*x+d)-4*a
/e^5*d*x+10*a/e^6*d^2*ln(e*x+d)+1/2*b*ln(c)/e^4*x^2+10*b*ln(x^n)/e^6*d^2*ln(e*x+d)-1/4*b*n*x^2/e^4+4*b*d*n*x/e
^5

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{6} \, a{\left (\frac{60 \, d^{3} e^{2} x^{2} + 105 \, d^{4} e x + 47 \, d^{5}}{e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}} + \frac{60 \, d^{2} \log \left (e x + d\right )}{e^{6}} + \frac{3 \,{\left (e x^{2} - 8 \, d x\right )}}{e^{5}}\right )} + b \int \frac{x^{5} \log \left (c\right ) + x^{5} \log \left (x^{n}\right )}{e^{4} x^{4} + 4 \, d e^{3} x^{3} + 6 \, d^{2} e^{2} x^{2} + 4 \, d^{3} e x + d^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*a*((60*d^3*e^2*x^2 + 105*d^4*e*x + 47*d^5)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6) + 60*d^2*log(e*
x + d)/e^6 + 3*(e*x^2 - 8*d*x)/e^5) + b*integrate((x^5*log(c) + x^5*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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="fricas")

[Out]

integral((b*x^5*log(c*x^n) + a*x^5)/(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)

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Sympy [A]  time = 127.016, size = 598, normalized size = 2.61 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*ln(c*x**n))/(e*x+d)**4,x)

[Out]

-a*d**5*Piecewise((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))/e**5 + 5*a*d**4*Piecewise((x/d**3, Eq(e,
0)), (-1/(2*e*(d + e*x)**2), True))/e**5 - 10*a*d**3*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2*x), True))/
e**5 + 10*a*d**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))/e**5 - 4*a*d*x/e**5 + a*x**2/(2*e**4) + b*
d**5*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**5 - b*d**5*Piecewi
se((x/d**4, Eq(e, 0)), (-1/(3*e*(d + e*x)**3), True))*log(c*x**n)/e**5 - 5*b*d**4*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**5 + 5*b*d**4*Piecewis
e((x/d**3, Eq(e, 0)), (-1/(2*e*(d + e*x)**2), True))*log(c*x**n)/e**5 + 10*b*d**3*n*Piecewise((x/d**2, Eq(e, 0
)), (-log(x)/(d*e) + log(d/e + x)/(d*e), True))/e**5 - 10*b*d**3*Piecewise((x/d**2, Eq(e, 0)), (-1/(d*e + e**2
*x), True))*log(c*x**n)/e**5 - 10*b*d**2*n*Piecewise((x/d, Eq(e, 0)), (Piecewise((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), (-me
ijerg(((), (1, 1)), ((0, 0), ()), x)*log(d) + meijerg(((1, 1), ()), ((), (0, 0)), x)*log(d) - polylog(2, e*x*e
xp_polar(I*pi)/d), True))/e, True))/e**5 + 10*b*d**2*Piecewise((x/d, Eq(e, 0)), (log(d + e*x)/e, True))*log(c*
x**n)/e**5 + 4*b*d*n*x/e**5 - 4*b*d*x*log(c*x**n)/e**5 - b*n*x**2/(4*e**4) + b*x**2*log(c*x**n)/(2*e**4)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c x^{n}\right ) + a\right )} x^{5}}{{\left (e x + d\right )}^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*x^n))/(e*x+d)^4,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^5/(e*x + d)^4, x)