3.3.40 \(\int \frac {a+b \log (c x^n)}{x^4 (d+e x^2)^3} \, dx\) [240]

Optimal. Leaf size=260 \[ -\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {35 i b e^{3/2} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 i b e^{3/2} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}} \]

[Out]

-35/72*b*n/d^3/x^3+35/8*b*e*n/d^4/x+1/4*(a+b*ln(c*x^n))/d/x^3/(e*x^2+d)^2+1/8*(7*a-b*n+7*b*ln(c*x^n))/d^2/x^3/
(e*x^2+d)+1/24*(-35*a+12*b*n-35*b*ln(c*x^n))/d^3/x^3+1/8*e*(35*a-12*b*n+35*b*ln(c*x^n))/d^4/x+1/8*e^(3/2)*arct
an(x*e^(1/2)/d^(1/2))*(35*a-12*b*n+35*b*ln(c*x^n))/d^(9/2)-35/16*I*b*e^(3/2)*n*polylog(2,-I*x*e^(1/2)/d^(1/2))
/d^(9/2)+35/16*I*b*e^(3/2)*n*polylog(2,I*x*e^(1/2)/d^(1/2))/d^(9/2)

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Rubi [A]
time = 0.32, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2385, 2380, 2341, 211, 2361, 12, 4940, 2438} \begin {gather*} -\frac {35 i b e^{3/2} n \text {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 i b e^{3/2} n \text {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {e^{3/2} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{8 d^{9/2}}+\frac {e \left (35 a+35 b \log \left (c x^n\right )-12 b n\right )}{8 d^4 x}-\frac {35 a+35 b \log \left (c x^n\right )-12 b n}{24 d^3 x^3}+\frac {7 a+7 b \log \left (c x^n\right )-b n}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {35 b e n}{8 d^4 x}-\frac {35 b n}{72 d^3 x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-35*b*n)/(72*d^3*x^3) + (35*b*e*n)/(8*d^4*x) + (a + b*Log[c*x^n])/(4*d*x^3*(d + e*x^2)^2) + (7*a - b*n + 7*b*
Log[c*x^n])/(8*d^2*x^3*(d + e*x^2)) - (35*a - 12*b*n + 35*b*Log[c*x^n])/(24*d^3*x^3) + (e*(35*a - 12*b*n + 35*
b*Log[c*x^n]))/(8*d^4*x) + (e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(35*a - 12*b*n + 35*b*Log[c*x^n]))/(8*d^(9/2))
- (((35*I)/16)*b*e^(3/2)*n*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]])/d^(9/2) + (((35*I)/16)*b*e^(3/2)*n*PolyLog[2
, (I*Sqrt[e]*x)/Sqrt[d]])/d^(9/2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

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 2361

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> With[{u = IntHide[1/(d + e*x^2),
x]}, Simp[u*(a + b*Log[c*x^n]), x] - Dist[b*n, Int[u/x, x], x]] /; FreeQ[{a, b, c, d, e, n}, x]

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]

Rule 4940

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Dist[I*(b/2), Int[Log[1 - I*c*x
]/x, x], x] - Dist[I*(b/2), Int[Log[1 + I*c*x]/x, x], x]) /; FreeQ[{a, b, c}, x]

Rubi steps

\begin {align*} \int \frac {a+b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^3} \, dx &=\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}-\frac {\int \frac {-7 a+b n-7 b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {\int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{x^4 \left (d+e x^2\right )} \, dx}{8 d^2}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {\int \left (\frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{d x^4}-\frac {e \left (-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )\right )}{d^2 x^2}+\frac {e^2 \left (-7 b n-5 (-7 a+b n)+35 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^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}+\frac {\int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{x^4} \, dx}{8 d^3}-\frac {e \int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{x^2} \, dx}{8 d^4}+\frac {e^2 \int \frac {-7 b n-5 (-7 a+b n)+35 b \log \left (c x^n\right )}{d+e x^2} \, dx}{8 d^4}\\ &=-\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {\left (35 b e^2 n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{8 d^4}\\ &=-\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {\left (35 b e^{3/2} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{8 d^{9/2}}\\ &=-\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {\left (35 i b e^{3/2} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{9/2}}+\frac {\left (35 i b e^{3/2} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{9/2}}\\ &=-\frac {35 b n}{72 d^3 x^3}+\frac {35 b e n}{8 d^4 x}+\frac {a+b \log \left (c x^n\right )}{4 d x^3 \left (d+e x^2\right )^2}+\frac {7 a-b n+7 b \log \left (c x^n\right )}{8 d^2 x^3 \left (d+e x^2\right )}-\frac {35 a-12 b n+35 b \log \left (c x^n\right )}{24 d^3 x^3}+\frac {e \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^4 x}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (35 a-12 b n+35 b \log \left (c x^n\right )\right )}{8 d^{9/2}}-\frac {35 i b e^{3/2} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}+\frac {35 i b e^{3/2} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{9/2}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(584\) vs. \(2(260)=520\).
time = 1.09, size = 584, normalized size = 2.25 \begin {gather*} \frac {1}{144} \left (-\frac {16 b n}{d^3 x^3}+\frac {432 b e n}{d^4 x}-\frac {48 \left (a+b \log \left (c x^n\right )\right )}{d^3 x^3}+\frac {432 e \left (a+b \log \left (c x^n\right )\right )}{d^4 x}-\frac {9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}-\frac {99 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^4 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {9 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}+\frac {99 e^{3/2} \left (a+b \log \left (c x^n\right )\right )}{d^4 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {99 b e^{3/2} n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{9/2}}-\frac {99 b e^{3/2} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{9/2}}-\frac {9 b e^{3/2} n \left (\frac {1}{\sqrt {-d} \left (\sqrt {-d}+\sqrt {e} x\right )}-\frac {\log (x)}{d}+\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right )}{d}\right )}{(-d)^{7/2}}-\frac {315 e^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{9/2}}+\frac {9 b e^{3/2} n \left (\frac {1}{\sqrt {-d} \left (\sqrt {-d}-\sqrt {e} x\right )}-\frac {\log (x)}{d}+\frac {\log \left ((-d)^{3/2}+d \sqrt {e} x\right )}{d}\right )}{(-d)^{7/2}}+\frac {315 e^{3/2} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{9/2}}+\frac {315 b e^{3/2} n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{9/2}}-\frac {315 b e^{3/2} n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{9/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-16*b*n)/(d^3*x^3) + (432*b*e*n)/(d^4*x) - (48*(a + b*Log[c*x^n]))/(d^3*x^3) + (432*e*(a + b*Log[c*x^n]))/(d
^4*x) - (9*e^(3/2)*(a + b*Log[c*x^n]))/((-d)^(7/2)*(Sqrt[-d] - Sqrt[e]*x)^2) - (99*e^(3/2)*(a + b*Log[c*x^n]))
/(d^4*(Sqrt[-d] - Sqrt[e]*x)) + (9*e^(3/2)*(a + b*Log[c*x^n]))/((-d)^(7/2)*(Sqrt[-d] + Sqrt[e]*x)^2) + (99*e^(
3/2)*(a + b*Log[c*x^n]))/(d^4*(Sqrt[-d] + Sqrt[e]*x)) + (99*b*e^(3/2)*n*(Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/
(-d)^(9/2) - (99*b*e^(3/2)*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]))/(-d)^(9/2) - (9*b*e^(3/2)*n*(1/(Sqrt[-d]*(S
qrt[-d] + Sqrt[e]*x)) - Log[x]/d + Log[Sqrt[-d] + Sqrt[e]*x]/d))/(-d)^(7/2) - (315*e^(3/2)*(a + b*Log[c*x^n])*
Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(9/2) + (9*b*e^(3/2)*n*(1/(Sqrt[-d]*(Sqrt[-d] - Sqrt[e]*x)) - Log[x]/d + L
og[(-d)^(3/2) + d*Sqrt[e]*x]/d))/(-d)^(7/2) + (315*e^(3/2)*(a + b*Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)
])/(-d)^(9/2) + (315*b*e^(3/2)*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(9/2) - (315*b*e^(3/2)*n*PolyLog[2, (d
*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(9/2))/144

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.15, size = 1729, normalized size = 6.65

method	result	size
risch	\(\text {Expression too large to display}\)	\(1729\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*x^n))/x^4/(e*x^2+d)^3,x,method=_RETURNVERBOSE)

[Out]

-1/9*b*n/d^3/x^3-3/2*I*b*Pi*csgn(I*c*x^n)^3/d^4*e/x-3/16*b*n*e^4/d^4*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((e*x+(-
e*d)^(1/2))/(-e*d)^(1/2))*x^4-11/16*I*b*Pi*csgn(I*c*x^n)^3*e^3/d^4/(e*x^2+d)^2*x^3-35/16*b*n*e^2/d^4/(-e*d)^(1
/2)*dilog((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))-3/2*b*n*e^2/d^4/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))+35/16*b*n*e^2/d
^4/(-e*d)^(1/2)*dilog((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))-1/8*b*n*e^2/d^4*x/(e*x^2+d)+11/8*a*e^3/d^4/(e*x^2+d)^2
*x^3+13/8*a*e^2/d^3/(e*x^2+d)^2*x+35/8*a*e^2/d^4/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))-13/16*I*b*Pi*csgn(I*c)*cs
gn(I*x^n)*csgn(I*c*x^n)*e^2/d^3/(e*x^2+d)^2*x-35/16*I*b*Pi*csgn(I*c*x^n)^3*e^2/d^4/(e*d)^(1/2)*arctan(x*e/(e*d
)^(1/2))+3/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^4*e/x-1/3*b*ln(c)/d^3/x^3-13/16*I*b*Pi*csgn(I*c*x^n)^3*e^2/d^3
/(e*x^2+d)^2*x-1/3*a/d^3/x^3+11/8*b*e^3/d^4/(e*x^2+d)^2*x^3*ln(x^n)+11/16*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2*e^3
/d^4/(e*x^2+d)^2*x^3+3*a/d^4*e/x+11/8*b*ln(c)*e^3/d^4/(e*x^2+d)^2*x^3+13/8*b*ln(c)*e^2/d^3/(e*x^2+d)^2*x+35/8*
b*ln(c)*e^2/d^4/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))+3*b*ln(c)/d^4*e/x-b*n*e^3/d^4*ln(x)/(e*x^2+d)^2*x^3-b*n*e^
2/d^3*ln(x)/(e*x^2+d)^2*x+3/2*b*n*e^2/d^4/(-e*d)^(1/2)*ln(x)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))-3/2*b*n*e^2/
d^4/(-e*d)^(1/2)*ln(x)*ln((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))+b*n*e^2/d^4*ln(x)*x/(e*x^2+d)-35/8*b*e^2/d^4/(e*d)^
(1/2)*arctan(x*e/(e*d)^(1/2))*n*ln(x)-1/6*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^3/x^3+1/6*I*b*Pi*csgn(I*c)*csgn(I
*x^n)*csgn(I*c*x^n)/d^3/x^3+1/2*b*n*e^2/d^3*ln(x)/(e*x^2+d)/(-e*d)^(1/2)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))+
13/8*b*e^2/d^3/(e*x^2+d)^2*x*ln(x^n)+35/8*b*e^2/d^4/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))*ln(x^n)+3/16*b*n*e^4/d
^4*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))*x^4+3/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n
)^2/d^4*e/x+1/6*I*b*Pi*csgn(I*c*x^n)^3/d^3/x^3-1/2*b*n*e^2/d^3*ln(x)/(e*x^2+d)/(-e*d)^(1/2)*ln((e*x+(-e*d)^(1/
2))/(-e*d)^(1/2))+3/16*b*n*e^2/d^2*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))-3/16*b*
n*e^2/d^2*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))-1/3*b/d^3/x^3*ln(x^n)-3/8*b*n*e^3
/d^3*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))*x^2+1/2*b*n*e^3/d^4*ln(x)/(e*x^2+d)/(-
e*d)^(1/2)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))*x^2+3/8*b*n*e^3/d^3*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((-e*x+(-
e*d)^(1/2))/(-e*d)^(1/2))*x^2-1/2*b*n*e^3/d^4*ln(x)/(e*x^2+d)/(-e*d)^(1/2)*ln((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))
*x^2+3*b*e*n/d^4/x+13/16*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e^2/d^3/(e*x^2+d)^2*x+35/16*I*b*Pi*csgn(I*x^n)*csg
n(I*c*x^n)^2*e^2/d^4/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))-1/6*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3/x^3-3/2*I*
b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^4*e/x-35/16*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*e^2/d^4/(e*d
)^(1/2)*arctan(x*e/(e*d)^(1/2))-11/16*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)*e^3/d^4/(e*x^2+d)^2*x^3+3*b*l
n(x^n)/d^4*e/x+35/16*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2*e^2/d^4/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))+11/16*I*b*Pi
*csgn(I*x^n)*csgn(I*c*x^n)^2*e^3/d^4/(e*x^2+d)^2*x^3+13/16*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2*e^2/d^3/(e*x^2+d)^
2*x

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^4/(e*x^2+d)^3,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^4/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

integral((b*log(c*x^n) + a)/(x^10*e^3 + 3*d*x^8*e^2 + 3*d^2*x^6*e + d^3*x^4), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*x**n))/x**4/(e*x**2+d)**3,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*x^n))/x^4/(e*x^2+d)^3,x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)/((x^2*e + d)^3*x^4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,x^n\right )}{x^4\,{\left (e\,x^2+d\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*log(c*x^n))/(x^4*(d + e*x^2)^3),x)

[Out]

int((a + b*log(c*x^n))/(x^4*(d + e*x^2)^3), x)

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