\(\int \frac {a+b \log (c x^n)}{x^3 (d+e x^2)^3} \, dx\) [235]

Optimal result

Integrand size = 23, antiderivative size = 162 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^3} \, dx=-\frac {3 b n}{4 d^3 x^2}+\frac {a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}+\frac {6 a-b n+6 b \log \left (c x^n\right )}{8 d^2 x^2 \left (d+e x^2\right )}-\frac {12 a-5 b n+12 b \log \left (c x^n\right )}{8 d^3 x^2}+\frac {e \log \left (1+\frac {d}{e x^2}\right ) \left (12 a-5 b n+12 b \log \left (c x^n\right )\right )}{8 d^4}-\frac {3 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{4 d^4} \]

[Out]

-3/4*b*n/d^3/x^2+1/4*(a+b*ln(c*x^n))/d/x^2/(e*x^2+d)^2+1/8*(6*a-b*n+6*b*ln(c*x^n))/d^2/x^2/(e*x^2+d)+1/8*(-12*
a+5*b*n-12*b*ln(c*x^n))/d^3/x^2+1/8*e*ln(1+d/e/x^2)*(12*a-5*b*n+12*b*ln(c*x^n))/d^4-3/4*b*e*n*polylog(2,-d/e/x
^2)/d^4

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2385, 2380, 2341, 2379, 2438} \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {e \log \left (\frac {d}{e x^2}+1\right ) \left (12 a+12 b \log \left (c x^n\right )-5 b n\right )}{8 d^4}-\frac {12 a+12 b \log \left (c x^n\right )-5 b n}{8 d^3 x^2}+\frac {6 a+6 b \log \left (c x^n\right )-b n}{8 d^2 x^2 \left (d+e x^2\right )}+\frac {a+b \log \left (c x^n\right )}{4 d x^2 \left (d+e x^2\right )^2}-\frac {3 b e n \operatorname {PolyLog}\left (2,-\frac {d}{e x^2}\right )}{4 d^4}-\frac {3 b n}{4 d^3 x^2} \]

[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*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)

Rule 2341

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 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2380

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

Rule 2385

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 2438

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.73 (sec) , antiderivative size = 507, normalized size of antiderivative = 3.13 \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^3} \, dx=\frac {-\frac {8 d \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{x^2}-\frac {4 d^2 e \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{\left (d+e x^2\right )^2}-\frac {16 d e \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{d+e x^2}-48 e \log (x) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+24 e \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \log \left (d+e x^2\right )+b n \left (\frac {9 e^{3/2} x \log (x)}{-i \sqrt {d}+\sqrt {e} x}-24 e \log ^2(x)-\frac {4 d (1+2 \log (x))}{x^2}+e \left (\frac {d}{d+i \sqrt {d} \sqrt {e} x}+\log (x)-\frac {d \log (x)}{\left (\sqrt {d}+i \sqrt {e} x\right )^2}-\log \left (i \sqrt {d}-\sqrt {e} x\right )\right )-9 e \log \left (i \sqrt {d}-\sqrt {e} x\right )+e \left (\frac {d}{d-i \sqrt {d} \sqrt {e} x}+\log (x)-\frac {d \log (x)}{\left (\sqrt {d}-i \sqrt {e} x\right )^2}-\log \left (i \sqrt {d}+\sqrt {e} x\right )\right )+\frac {-9 i e^{3/2} x \log (x)+9 i e \left (i \sqrt {d}+\sqrt {e} x\right ) \log \left (i \sqrt {d}+\sqrt {e} x\right )}{\sqrt {d}-i \sqrt {e} x}+24 e \left (\log (x) \log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )+24 e \left (\log (x) \log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )+\operatorname {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )\right )\right )}{16 d^4} \]

[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)

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.18 (sec) , antiderivative size = 440, normalized size of antiderivative = 2.72

method	result	size
risch	\(-\frac {b \ln \left (x^{n}\right ) e}{4 d^{2} \left (e \,x^{2}+d \right )^{2}}+\frac {3 b \ln \left (x^{n}\right ) e \ln \left (e \,x^{2}+d \right )}{2 d^{4}}-\frac {b \ln \left (x^{n}\right ) e}{d^{3} \left (e \,x^{2}+d \right )}-\frac {b \ln \left (x^{n}\right )}{2 d^{3} x^{2}}-\frac {3 b \ln \left (x^{n}\right ) e \ln \left (x \right )}{d^{4}}+\frac {3 b n e \ln \left (x \right )^{2}}{2 d^{4}}-\frac {3 b n e \ln \left (x \right ) \ln \left (e \,x^{2}+d \right )}{2 d^{4}}+\frac {3 b n e \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{4}}+\frac {3 b n e \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{4}}+\frac {3 b n e \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{4}}+\frac {3 b n e \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{2 d^{4}}-\frac {5 b n e \ln \left (e \,x^{2}+d \right )}{8 d^{4}}+\frac {b n e}{8 d^{3} \left (e \,x^{2}+d \right )}-\frac {b n}{4 d^{3} x^{2}}+\frac {5 b e n \ln \left (x \right )}{4 d^{4}}+\left (-\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}+\frac {i b \pi \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i c \,x^{n}\right )^{2}}{2}-\frac {i b \pi \operatorname {csgn}\left (i c \,x^{n}\right )^{3}}{2}+b \ln \left (c \right )+a \right ) \left (\frac {e^{2} \left (-\frac {d^{2}}{2 e \left (e \,x^{2}+d \right )^{2}}+\frac {3 \ln \left (e \,x^{2}+d \right )}{e}-\frac {2 d}{e \left (e \,x^{2}+d \right )}\right )}{2 d^{4}}-\frac {1}{2 d^{3} x^{2}}-\frac {3 e \ln \left (x \right )}{d^{4}}\right )\)	\(440\)

[In]

int((a+b*ln(c*x^n))/x^3/(e*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^3} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]

[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)

Giac [F]

\[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^3} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{{\left (e x^{2} + d\right )}^{3} x^{3}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x^3 \left (d+e x^2\right )^3} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x^3\,{\left (e\,x^2+d\right )}^3} \,d x \]

[In]

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

[Out]

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