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

Optimal. Leaf size=217 \[ \frac{6 b e^2 n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d^5}-\frac{3 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}-\frac{6 e^2 \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3 (d+e x)^2}+\frac{3 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac{a+b \log \left (c x^n\right )}{2 d^3 x^2}-\frac{b e^2 n}{2 d^4 (d+e x)}-\frac{b e^2 n \log (x)}{2 d^5}+\frac{7 b e^2 n \log (d+e x)}{2 d^5}+\frac{3 b e n}{d^4 x}-\frac{b n}{4 d^3 x^2} \]

[Out]

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

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Rubi [A]  time = 0.273624, antiderivative size = 239, normalized size of antiderivative = 1.1, number of steps used = 12, 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} \[ -\frac{6 b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^5}-\frac{3 e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^5 (d+e x)}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b d^5 n}+\frac{e^2 \left (a+b \log \left (c x^n\right )\right )}{2 d^3 (d+e x)^2}-\frac{6 e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^5}+\frac{3 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac{a+b \log \left (c x^n\right )}{2 d^3 x^2}-\frac{b e^2 n}{2 d^4 (d+e x)}-\frac{b e^2 n \log (x)}{2 d^5}+\frac{7 b e^2 n \log (d+e x)}{2 d^5}+\frac{3 b e n}{d^4 x}-\frac{b n}{4 d^3 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.369389, size = 227, normalized size = 1.05 \[ -\frac{24 b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )-\frac{2 d^2 e^2 \left (a+b \log \left (c x^n\right )\right )}{(d+e x)^2}+\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac{12 d e^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+24 e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{12 d e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{12 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+\frac{b d^2 n}{x^2}+12 b e^2 n (\log (x)-\log (d+e x))+\frac{2 b e^2 n (\log (x) (d+e x)-(d+e x) \log (d+e x)+d)}{d+e x}-\frac{12 b d e n}{x}}{4 d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.174, size = 1119, normalized size = 5.2 \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)^3,x)

[Out]

6*b*n/d^5*e^2*ln(e*x+d)*ln(-e*x/d)-3*I*b*Pi*csgn(I*c*x^n)^3/d^5*e^2*ln(x)-1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)
^2/d^3/x^2-1/2*a/d^3/x^2+1/4*I*b*Pi*csgn(I*c*x^n)^3/d^3/x^2+3*b*ln(x^n)/d^4*e^2/(e*x+d)-1/4*I*b*Pi*csgn(I*x^n)
*csgn(I*c*x^n)*csgn(I*c)*e^2/d^3/(e*x+d)^2-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^5*e^2*ln(x)-3/2*I*b*
Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^4*e/x-3/2*I*b*Pi*csgn(I*c*x^n)^3/d^4*e/x-1/4*I*b*Pi*csgn(I*c*x^n)^2*c
sgn(I*c)/d^3/x^2-3/2*I*b*Pi*csgn(I*c*x^n)^3/d^4*e^2/(e*x+d)-1/4*I*b*Pi*csgn(I*c*x^n)^3*e^2/d^3/(e*x+d)^2+3*I*b
*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^5*e^2*ln(e*x+d)-3/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^4*e
^2/(e*x+d)-3*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^5*e^2*ln(e*x+d)+1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e^2/d
^3/(e*x+d)^2-3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^5*e^2*ln(e*x+d)+1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*
c)/d^3/x^2+3*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^5*e^2*ln(x)+3/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^4*e^2/(e*x+
d)+3/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^4*e/x+1/4*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*e^2/d^3/(e*x+d)^2+3/2*I*b
*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^4*e^2/(e*x+d)+3/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^4*e/x+3*I*b*Pi*csgn(I
*x^n)*csgn(I*c*x^n)^2/d^5*e^2*ln(x)-1/2*b*ln(c)/d^3/x^2+3*I*b*Pi*csgn(I*c*x^n)^3/d^5*e^2*ln(e*x+d)-1/2*b*ln(x^
n)/d^3/x^2+3*a/d^4*e/x-6*a/d^5*e^2*ln(e*x+d)+6*a/d^5*e^2*ln(x)+3*a/d^4*e^2/(e*x+d)+1/2*a*e^2/d^3/(e*x+d)^2-7/2
*b*e^2*n*ln(x)/d^5+7/2*b*e^2*n*ln(e*x+d)/d^5+6*b*ln(c)/d^5*e^2*ln(x)-6*b*ln(c)/d^5*e^2*ln(e*x+d)+3*b*ln(c)/d^4
*e^2/(e*x+d)+1/2*b*ln(c)*e^2/d^3/(e*x+d)^2+3*b*ln(c)/d^4*e/x-3*b*n/d^5*e^2*ln(x)^2+6*b*n/d^5*e^2*dilog(-e*x/d)
+6*b*ln(x^n)/d^5*e^2*ln(x)+3*b*ln(x^n)/d^4*e/x+1/2*b*ln(x^n)*e^2/d^3/(e*x+d)^2-6*b*ln(x^n)/d^5*e^2*ln(e*x+d)-1
/4*b*n/d^3/x^2+3*b*e*n/d^4/x-1/2*b*e^2*n/d^4/(e*x+d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a{\left (\frac{12 \, e^{3} x^{3} + 18 \, d e^{2} x^{2} + 4 \, d^{2} e x - d^{3}}{d^{4} e^{2} x^{4} + 2 \, d^{5} e x^{3} + d^{6} x^{2}} - \frac{12 \, e^{2} \log \left (e x + d\right )}{d^{5}} + \frac{12 \, e^{2} \log \left (x\right )}{d^{5}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{3} x^{6} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{4} + d^{3} 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)^3,x, algorithm="maxima")

[Out]

1/2*a*((12*e^3*x^3 + 18*d*e^2*x^2 + 4*d^2*e*x - d^3)/(d^4*e^2*x^4 + 2*d^5*e*x^3 + d^6*x^2) - 12*e^2*log(e*x +
d)/d^5 + 12*e^2*log(x)/d^5) + b*integrate((log(c) + log(x^n))/(e^3*x^6 + 3*d*e^2*x^5 + 3*d^2*e*x^4 + d^3*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^{3} x^{6} + 3 \, d e^{2} x^{5} + 3 \, d^{2} e x^{4} + d^{3} 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)^3,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(e^3*x^6 + 3*d*e^2*x^5 + 3*d^2*e*x^4 + d^3*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)**3,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 )}^{3} 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)^3,x, algorithm="giac")

[Out]

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