3.45 \(\int \frac{a+b \log (c x^n)}{x^3 (d+e x)^2} \, dx\)

Optimal. Leaf size=154 \[ \frac{3 b e^2 n \text{PolyLog}\left (2,-\frac{d}{e x}\right )}{d^4}-\frac{e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}-\frac{3 e^2 \log \left (\frac{d}{e x}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac{a+b \log \left (c x^n\right )}{2 d^2 x^2}+\frac{b e^2 n \log (d+e x)}{d^4}+\frac{2 b e n}{d^3 x}-\frac{b n}{4 d^2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.213345, antiderivative size = 178, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {44, 2351, 2304, 2301, 2314, 31, 2317, 2391} \[ -\frac{3 b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )}{d^4}-\frac{e^3 x \left (a+b \log \left (c x^n\right )\right )}{d^4 (d+e x)}+\frac{3 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{2 b d^4 n}-\frac{3 e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{d^4}+\frac{2 e \left (a+b \log \left (c x^n\right )\right )}{d^3 x}-\frac{a+b \log \left (c x^n\right )}{2 d^2 x^2}+\frac{b e^2 n \log (d+e x)}{d^4}+\frac{2 b e n}{d^3 x}-\frac{b n}{4 d^2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*x^n])/(x^3*(d + e*x)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.218132, size = 165, normalized size = 1.07 \[ -\frac{12 b e^2 n \text{PolyLog}\left (2,-\frac{e x}{d}\right )+\frac{2 d^2 \left (a+b \log \left (c x^n\right )\right )}{x^2}-\frac{4 d e^2 \left (a+b \log \left (c x^n\right )\right )}{d+e x}+12 e^2 \log \left (\frac{e x}{d}+1\right ) \left (a+b \log \left (c x^n\right )\right )-\frac{8 d e \left (a+b \log \left (c x^n\right )\right )}{x}-\frac{6 e^2 \left (a+b \log \left (c x^n\right )\right )^2}{b n}+\frac{b d^2 n}{x^2}+4 b e^2 n (\log (x)-\log (d+e x))-\frac{8 b d e n}{x}}{4 d^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*x^n])/(x^3*(d + e*x)^2),x]

[Out]

-((b*d^2*n)/x^2 - (8*b*d*e*n)/x + (2*d^2*(a + b*Log[c*x^n]))/x^2 - (8*d*e*(a + b*Log[c*x^n]))/x - (4*d*e^2*(a
+ b*Log[c*x^n]))/(d + e*x) - (6*e^2*(a + b*Log[c*x^n])^2)/(b*n) + 4*b*e^2*n*(Log[x] - Log[d + e*x]) + 12*e^2*(
a + b*Log[c*x^n])*Log[1 + (e*x)/d] + 12*b*e^2*n*PolyLog[2, -((e*x)/d)])/(4*d^4)

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Maple [C]  time = 0.168, size = 910, normalized size = 5.9 \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)^2,x)

[Out]

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

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

[Out]

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

[Out]

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

[Out]

integrate((b*log(c*x^n) + a)/((e*x + d)^2*x^3), x)