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

Optimal. Leaf size=162 \[ -\frac{3 b e n \text{PolyLog}\left (2,-\frac{d}{e x^2}\right )}{4 d^4}+\frac{6 a+6 b \log \left (c x^n\right )-b n}{8 d^2 x^2 \left (d+e x^2\right )}+\frac{e \log \left (\frac{d}{e x^2}+1\right ) \left (12 a+12 b \log \left (c x^n\right )-5 b n\right )}{8 d^4}-\frac{12 a+12 b \log \left (c x^n\right )-5 b n}{8 d^3 x^2}+\frac{a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}-\frac{3 b n}{4 d^3 x^2} \]

[Out]

(-3*b*n)/(4*d^3*x^2) + (a + b*Log[c*x^n])/(4*d*x^2*(d + e*x^2)^2) + (6*a - b*n + 6*b*Log[c*x^n])/(8*d^2*x^2*(d
+ e*x^2)) - (12*a - 5*b*n + 12*b*Log[c*x^n])/(8*d^3*x^2) + (e*Log[1 + d/(e*x^2)]*(12*a - 5*b*n + 12*b*Log[c*x
^n]))/(8*d^4) - (3*b*e*n*PolyLog[2, -(d/(e*x^2))])/(4*d^4)

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Rubi [A]  time = 0.390189, antiderivative size = 195, normalized size of antiderivative = 1.2, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.348, Rules used = {2340, 266, 44, 2351, 2304, 2301, 2337, 2391} \[ \frac{3 b e n \text{PolyLog}\left (2,-\frac{e x^2}{d}\right )}{4 d^4}-\frac{e \left (12 a+12 b \log \left (c x^n\right )-5 b n\right )^2}{96 b d^4 n}+\frac{e \log \left (\frac{e x^2}{d}+1\right ) \left (12 a+12 b \log \left (c x^n\right )-5 b n\right )}{8 d^4}+\frac{6 a+6 b \log \left (c x^n\right )-b n}{8 d^2 x^2 \left (d+e x^2\right )}-\frac{12 a+12 b \log \left (c x^n\right )-5 b n}{8 d^3 x^2}+\frac{a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}-\frac{3 b n}{4 d^3 x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-3*b*n)/(4*d^3*x^2) + (a + b*Log[c*x^n])/(4*d*x^2*(d + e*x^2)^2) + (6*a - b*n + 6*b*Log[c*x^n])/(8*d^2*x^2*(d
+ e*x^2)) - (12*a - 5*b*n + 12*b*Log[c*x^n])/(8*d^3*x^2) - (e*(12*a - 5*b*n + 12*b*Log[c*x^n])^2)/(96*b*d^4*n
) + (e*(12*a - 5*b*n + 12*b*Log[c*x^n])*Log[1 + (e*x^2)/d])/(8*d^4) + (3*b*e*n*PolyLog[2, -((e*x^2)/d)])/(4*d^
4)

Rule 2340

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.), x_Symbol] :> -Simp[(
(f*x)^(m + 1)*(d + e*x^2)^(q + 1)*(a + b*Log[c*x^n]))/(2*d*f*(q + 1)), x] + Dist[1/(2*d*(q + 1)), Int[(f*x)^m*
(d + e*x^2)^(q + 1)*(a*(m + 2*q + 3) + b*n + b*(m + 2*q + 3)*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m
, n}, x] && ILtQ[q, -1] && ILtQ[m, 0]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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 2337

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.))/((d_) + (e_.)*(x_)^(r_)), x_Symbol] :> Si
mp[(f^m*Log[1 + (e*x^r)/d]*(a + b*Log[c*x^n])^p)/(e*r), x] - Dist[(b*f^m*n*p)/(e*r), Int[(Log[1 + (e*x^r)/d]*(
a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, r}, x] && EqQ[m, r - 1] && IGtQ[p, 0] &
& (IntegerQ[m] || GtQ[f, 0]) && NeQ[r, n]

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 \left (d+e x^2\right )^3} \, dx &=\frac{a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}-\frac{\int \frac{-6 a+b n-6 b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac{a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac{6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}+\frac{\int \frac{-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )} \, dx}{8 d^2}\\ &=\frac{a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac{6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}+\frac{\int \left (\frac{-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )}{d x^3}-\frac{e \left (-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )\right )}{d^2 x}+\frac{e^2 x \left (-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )\right )}{d^2 \left (d+e x^2\right )}\right ) \, dx}{8 d^2}\\ &=\frac{a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac{6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}+\frac{\int \frac{-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )}{x^3} \, dx}{8 d^3}-\frac{e \int \frac{-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )}{x} \, dx}{8 d^4}+\frac{e^2 \int \frac{x \left (-6 b n-4 (-6 a+b n)+24 b \log \left (c x^n\right )\right )}{d+e x^2} \, dx}{8 d^4}\\ &=-\frac{3 b n}{4 d^3 x^2}+\frac{a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac{6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}-\frac{12 a-5 b n+12 b \log \left (c x^n\right )}{8 d^3 x^2}-\frac{e \left (12 a-5 b n+12 b \log \left (c x^n\right )\right )^2}{96 b d^4 n}+\frac{e \left (12 a-5 b n+12 b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^2}{d}\right )}{8 d^4}-\frac{(3 b e n) \int \frac{\log \left (1+\frac{e x^2}{d}\right )}{x} \, dx}{2 d^4}\\ &=-\frac{3 b n}{4 d^3 x^2}+\frac{a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac{6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}-\frac{12 a-5 b n+12 b \log \left (c x^n\right )}{8 d^3 x^2}-\frac{e \left (12 a-5 b n+12 b \log \left (c x^n\right )\right )^2}{96 b d^4 n}+\frac{e \left (12 a-5 b n+12 b \log \left (c x^n\right )\right ) \log \left (1+\frac{e x^2}{d}\right )}{8 d^4}+\frac{3 b e n \text{Li}_2\left (-\frac{e x^2}{d}\right )}{4 d^4}\\ \end{align*}

Mathematica [C]  time = 1.10452, size = 507, normalized size = 3.13 \[ \frac{b n \left (24 e \left (\text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )+\log (x) \log \left (1+\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )+24 e \left (\text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )+\log (x) \log \left (1-\frac{i \sqrt{e} x}{\sqrt{d}}\right )\right )+\frac{9 e^{3/2} x \log (x)}{\sqrt{e} x-i \sqrt{d}}+\frac{9 i e \left (\sqrt{e} x+i \sqrt{d}\right ) \log \left (\sqrt{e} x+i \sqrt{d}\right )-9 i e^{3/2} x \log (x)}{\sqrt{d}-i \sqrt{e} x}+e \left (\frac{d}{d+i \sqrt{d} \sqrt{e} x}-\frac{d \log (x)}{\left (\sqrt{d}+i \sqrt{e} x\right )^2}-\log \left (-\sqrt{e} x+i \sqrt{d}\right )+\log (x)\right )-9 e \log \left (-\sqrt{e} x+i \sqrt{d}\right )+e \left (\frac{d}{d-i \sqrt{d} \sqrt{e} x}-\frac{d \log (x)}{\left (\sqrt{d}-i \sqrt{e} x\right )^2}-\log \left (\sqrt{e} x+i \sqrt{d}\right )+\log (x)\right )-\frac{4 d (2 \log (x)+1)}{x^2}-24 e \log ^2(x)\right )-\frac{4 d^2 e \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{\left (d+e x^2\right )^2}-\frac{16 d e \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{d+e x^2}+24 e \log \left (d+e x^2\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-\frac{8 d \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{x^2}-48 e \log (x) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{16 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-8*d*(a - b*n*Log[x] + b*Log[c*x^n]))/x^2 - (4*d^2*e*(a - b*n*Log[x] + b*Log[c*x^n]))/(d + e*x^2)^2 - (16*d*
e*(a - b*n*Log[x] + b*Log[c*x^n]))/(d + e*x^2) - 48*e*Log[x]*(a - b*n*Log[x] + b*Log[c*x^n]) + 24*e*(a - b*n*L
og[x] + b*Log[c*x^n])*Log[d + e*x^2] + b*n*((9*e^(3/2)*x*Log[x])/((-I)*Sqrt[d] + Sqrt[e]*x) - 24*e*Log[x]^2 -
(4*d*(1 + 2*Log[x]))/x^2 + e*(d/(d + I*Sqrt[d]*Sqrt[e]*x) + Log[x] - (d*Log[x])/(Sqrt[d] + I*Sqrt[e]*x)^2 - Lo
g[I*Sqrt[d] - Sqrt[e]*x]) - 9*e*Log[I*Sqrt[d] - Sqrt[e]*x] + e*(d/(d - I*Sqrt[d]*Sqrt[e]*x) + Log[x] - (d*Log[
x])/(Sqrt[d] - I*Sqrt[e]*x)^2 - Log[I*Sqrt[d] + Sqrt[e]*x]) + ((-9*I)*e^(3/2)*x*Log[x] + (9*I)*e*(I*Sqrt[d] +
Sqrt[e]*x)*Log[I*Sqrt[d] + Sqrt[e]*x])/(Sqrt[d] - I*Sqrt[e]*x) + 24*e*(Log[x]*Log[1 + (I*Sqrt[e]*x)/Sqrt[d]] +
PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]]) + 24*e*(Log[x]*Log[1 - (I*Sqrt[e]*x)/Sqrt[d]] + PolyLog[2, (I*Sqrt[e]*x
)/Sqrt[d]])))/(16*d^4)

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Maple [C]  time = 0.181, size = 1030, normalized size = 6.4 \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^2+d)^3,x)

[Out]

3/2*b*n/d^4*e*ln(x)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+3/2*b*n/d^4*e*ln(x)*ln((e*x+(-d*e)^(1/2))/(-d*e)^(1/2
))-3/2*b*n/d^4*e*ln(x)*ln(e*x^2+d)+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*e/d^3/(e*x^2+d)+1/8*I*b*Pi*c
sgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*e/d^2/(e*x^2+d)^2-1/2*a/d^3/x^2-a*e/d^3/(e*x^2+d)+3/2*a*e/d^4*ln(e*x^2+d)-1
/4*a*e/d^2/(e*x^2+d)^2+1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^3/x^2+3/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*
x^n)*csgn(I*c)/d^4*e*ln(x)+3/2*b*ln(x^n)*e/d^4*ln(e*x^2+d)-1/4*b*ln(x^n)*e/d^2/(e*x^2+d)^2-b*ln(x^n)*e/d^3/(e*
x^2+d)+3/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^4*e*ln(e*x^2+d)-3*a/d^4*e*ln(x)-3*b*ln(c)/d^4*e*ln(x)-b*ln(c)*
e/d^3/(e*x^2+d)+3/2*b*ln(c)/d^4*e*ln(e*x^2+d)-1/4*b*ln(c)*e/d^2/(e*x^2+d)^2+3/2*b*n/d^4*e*dilog((-e*x+(-d*e)^(
1/2))/(-d*e)^(1/2))+3/2*b*n/d^4*e*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-5/8*b*n/d^4*e*ln(e*x^2+d)+1/8*b*n*e/d
^3/(e*x^2+d)-3/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^4*e*ln(x)-1/8*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*e/d^2/(e*
x^2+d)^2-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e/d^3/(e*x^2+d)-1/2*b*ln(x^n)/d^3/x^2+3/4*I*b*Pi*csgn(I*c*x^n)
^2*csgn(I*c)/d^4*e*ln(e*x^2+d)-1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*e/d^3/(e*x^2+d)-1/8*I*b*Pi*csgn(I*x^n)*csg
n(I*c*x^n)^2*e/d^2/(e*x^2+d)^2+1/2*I*b*Pi*csgn(I*c*x^n)^3*e/d^3/(e*x^2+d)-3/4*I*b*Pi*csgn(I*c*x^n)^3/d^4*e*ln(
e*x^2+d)+1/8*I*b*Pi*csgn(I*c*x^n)^3*e/d^2/(e*x^2+d)^2-3/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d^4*e*ln(x)+1/4*I*b
*Pi*csgn(I*c*x^n)^3/d^3/x^2-1/2*b*ln(c)/d^3/x^2+3/2*b*n/d^4*e*ln(x)^2-3*b*ln(x^n)/d^4*e*ln(x)-1/4*I*b*Pi*csgn(
I*c*x^n)^2*csgn(I*c)/d^3/x^2-1/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3/x^2+3/2*I*b*Pi*csgn(I*c*x^n)^3/d^4*e*l
n(x)-3/4*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d^4*e*ln(e*x^2+d)+5/4*b*e*n*ln(x)/d^4-1/4*b*n/d^3/x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{4} \, a{\left (\frac{6 \, e^{2} x^{4} + 9 \, d e x^{2} + 2 \, d^{2}}{d^{3} e^{2} x^{6} + 2 \, d^{4} e x^{4} + d^{5} x^{2}} - \frac{6 \, e \log \left (e x^{2} + d\right )}{d^{4}} + \frac{12 \, e \log \left (x\right )}{d^{4}}\right )} + b \int \frac{\log \left (c\right ) + \log \left (x^{n}\right )}{e^{3} x^{9} + 3 \, d e^{2} x^{7} + 3 \, d^{2} e x^{5} + 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^2+d)^3,x, algorithm="maxima")

[Out]

-1/4*a*((6*e^2*x^4 + 9*d*e*x^2 + 2*d^2)/(d^3*e^2*x^6 + 2*d^4*e*x^4 + d^5*x^2) - 6*e*log(e*x^2 + d)/d^4 + 12*e*
log(x)/d^4) + b*integrate((log(c) + log(x^n))/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + 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^{9} + 3 \, d e^{2} x^{7} + 3 \, d^{2} e x^{5} + 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^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(e^3*x^9 + 3*d*e^2*x^7 + 3*d^2*e*x^5 + 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**2+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^{2} + 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^2+d)^3,x, algorithm="giac")

[Out]

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