3.3.39 \(\int \frac {a+b \log (c x^n)}{x^2 (d+e x^2)^3} \, dx\) [239]

Optimal. Leaf size=219 \[ -\frac {15 b n}{8 d^3 x}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}-\frac {15 a-8 b n+15 b \log \left (c x^n\right )}{8 d^3 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a-8 b n+15 b \log \left (c x^n\right )\right )}{8 d^{7/2}}+\frac {15 i b \sqrt {e} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}}-\frac {15 i b \sqrt {e} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}} \]

[Out]

-15/8*b*n/d^3/x+1/4*(a+b*ln(c*x^n))/d/x/(e*x^2+d)^2+1/8*(5*a-b*n+5*b*ln(c*x^n))/d^2/x/(e*x^2+d)+1/8*(-15*a+8*b
*n-15*b*ln(c*x^n))/d^3/x-1/8*arctan(x*e^(1/2)/d^(1/2))*(15*a-8*b*n+15*b*ln(c*x^n))*e^(1/2)/d^(7/2)+15/16*I*b*n
*polylog(2,-I*x*e^(1/2)/d^(1/2))*e^(1/2)/d^(7/2)-15/16*I*b*n*polylog(2,I*x*e^(1/2)/d^(1/2))*e^(1/2)/d^(7/2)

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Rubi [A]
time = 0.23, antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {15 i b \sqrt {e} n \text {PolyLog}\left (2,-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}}-\frac {15 i b \sqrt {e} n \text {PolyLog}\left (2,\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}}-\frac {\sqrt {e} \text {ArcTan}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a+15 b \log \left (c x^n\right )-8 b n\right )}{8 d^{7/2}}-\frac {15 a+15 b \log \left (c x^n\right )-8 b n}{8 d^3 x}+\frac {5 a+5 b \log \left (c x^n\right )-b n}{8 d^2 x \left (d+e x^2\right )}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}-\frac {15 b n}{8 d^3 x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*b*n)/(8*d^3*x) + (a + b*Log[c*x^n])/(4*d*x*(d + e*x^2)^2) + (5*a - b*n + 5*b*Log[c*x^n])/(8*d^2*x*(d + e*
x^2)) - (15*a - 8*b*n + 15*b*Log[c*x^n])/(8*d^3*x) - (Sqrt[e]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]*(15*a - 8*b*n + 15*b
*Log[c*x^n]))/(8*d^(7/2)) + (((15*I)/16)*b*Sqrt[e]*n*PolyLog[2, ((-I)*Sqrt[e]*x)/Sqrt[d]])/d^(7/2) - (((15*I)/
16)*b*Sqrt[e]*n*PolyLog[2, (I*Sqrt[e]*x)/Sqrt[d]])/d^(7/2)

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^2 \left (d+e x^2\right )^3} \, dx &=\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}-\frac {\int \frac {-5 a+b n-5 b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )^2} \, dx}{4 d}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}+\frac {\int \frac {-5 b n-3 (-5 a+b n)+15 b \log \left (c x^n\right )}{x^2 \left (d+e x^2\right )} \, dx}{8 d^2}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}+\frac {\int \left (\frac {-5 b n-3 (-5 a+b n)+15 b \log \left (c x^n\right )}{d x^2}-\frac {e \left (-5 b n-3 (-5 a+b n)+15 b \log \left (c x^n\right )\right )}{d \left (d+e x^2\right )}\right ) \, dx}{8 d^2}\\ &=\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}+\frac {\int \frac {-5 b n-3 (-5 a+b n)+15 b \log \left (c x^n\right )}{x^2} \, dx}{8 d^3}-\frac {e \int \frac {-5 b n-3 (-5 a+b n)+15 b \log \left (c x^n\right )}{d+e x^2} \, dx}{8 d^3}\\ &=-\frac {15 b n}{8 d^3 x}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}-\frac {15 a-8 b n+15 b \log \left (c x^n\right )}{8 d^3 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a-8 b n+15 b \log \left (c x^n\right )\right )}{8 d^{7/2}}+\frac {(15 b e n) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \sqrt {e} x} \, dx}{8 d^3}\\ &=-\frac {15 b n}{8 d^3 x}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}-\frac {15 a-8 b n+15 b \log \left (c x^n\right )}{8 d^3 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a-8 b n+15 b \log \left (c x^n\right )\right )}{8 d^{7/2}}+\frac {\left (15 b \sqrt {e} n\right ) \int \frac {\tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{8 d^{7/2}}\\ &=-\frac {15 b n}{8 d^3 x}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}-\frac {15 a-8 b n+15 b \log \left (c x^n\right )}{8 d^3 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a-8 b n+15 b \log \left (c x^n\right )\right )}{8 d^{7/2}}+\frac {\left (15 i b \sqrt {e} n\right ) \int \frac {\log \left (1-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{7/2}}-\frac {\left (15 i b \sqrt {e} n\right ) \int \frac {\log \left (1+\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{x} \, dx}{16 d^{7/2}}\\ &=-\frac {15 b n}{8 d^3 x}+\frac {a+b \log \left (c x^n\right )}{4 d x \left (d+e x^2\right )^2}+\frac {5 a-b n+5 b \log \left (c x^n\right )}{8 d^2 x \left (d+e x^2\right )}-\frac {15 a-8 b n+15 b \log \left (c x^n\right )}{8 d^3 x}-\frac {\sqrt {e} \tan ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) \left (15 a-8 b n+15 b \log \left (c x^n\right )\right )}{8 d^{7/2}}+\frac {15 i b \sqrt {e} n \text {Li}_2\left (-\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}}-\frac {15 i b \sqrt {e} n \text {Li}_2\left (\frac {i \sqrt {e} x}{\sqrt {d}}\right )}{16 d^{7/2}}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(552\) vs. \(2(219)=438\).
time = 1.01, size = 552, normalized size = 2.52 \begin {gather*} \frac {1}{16} \left (-\frac {16 b n}{d^3 x}-\frac {16 \left (a+b \log \left (c x^n\right )\right )}{d^3 x}+\frac {d \sqrt {e} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{7/2} \left (\sqrt {-d}-\sqrt {e} x\right )^2}+\frac {7 \sqrt {e} \left (a+b \log \left (c x^n\right )\right )}{d^3 \left (\sqrt {-d}-\sqrt {e} x\right )}+\frac {\sqrt {e} \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2} \left (\sqrt {-d}+\sqrt {e} x\right )^2}-\frac {7 \sqrt {e} \left (a+b \log \left (c x^n\right )\right )}{d^3 \left (\sqrt {-d}+\sqrt {e} x\right )}+\frac {7 b \sqrt {e} n \left (\log (x)-\log \left (\sqrt {-d}-\sqrt {e} x\right )\right )}{(-d)^{7/2}}-\frac {7 b \sqrt {e} n \left (\log (x)-\log \left (\sqrt {-d}+\sqrt {e} x\right )\right )}{(-d)^{7/2}}+\frac {b d \sqrt {e} 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 {15 \sqrt {e} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{7/2}}+\frac {b \sqrt {e} 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)^{5/2}}+\frac {15 \sqrt {e} \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{7/2}}+\frac {15 b \sqrt {e} n \text {Li}_2\left (\frac {\sqrt {e} x}{\sqrt {-d}}\right )}{(-d)^{7/2}}-\frac {15 b \sqrt {e} n \text {Li}_2\left (\frac {d \sqrt {e} x}{(-d)^{3/2}}\right )}{(-d)^{7/2}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-16*b*n)/(d^3*x) - (16*(a + b*Log[c*x^n]))/(d^3*x) + (d*Sqrt[e]*(a + b*Log[c*x^n]))/((-d)^(7/2)*(Sqrt[-d] -
Sqrt[e]*x)^2) + (7*Sqrt[e]*(a + b*Log[c*x^n]))/(d^3*(Sqrt[-d] - Sqrt[e]*x)) + (Sqrt[e]*(a + b*Log[c*x^n]))/((-
d)^(5/2)*(Sqrt[-d] + Sqrt[e]*x)^2) - (7*Sqrt[e]*(a + b*Log[c*x^n]))/(d^3*(Sqrt[-d] + Sqrt[e]*x)) + (7*b*Sqrt[e
]*n*(Log[x] - Log[Sqrt[-d] - Sqrt[e]*x]))/(-d)^(7/2) - (7*b*Sqrt[e]*n*(Log[x] - Log[Sqrt[-d] + Sqrt[e]*x]))/(-
d)^(7/2) + (b*d*Sqrt[e]*n*(1/(Sqrt[-d]*(Sqrt[-d] + Sqrt[e]*x)) - Log[x]/d + Log[Sqrt[-d] + Sqrt[e]*x]/d))/(-d)
^(7/2) - (15*Sqrt[e]*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(7/2) + (b*Sqrt[e]*n*(1/(Sqrt[-d]*
(Sqrt[-d] - Sqrt[e]*x)) - Log[x]/d + Log[(-d)^(3/2) + d*Sqrt[e]*x]/d))/(-d)^(5/2) + (15*Sqrt[e]*(a + b*Log[c*x
^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(7/2) + (15*b*Sqrt[e]*n*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(7
/2) - (15*b*Sqrt[e]*n*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(7/2))/16

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.15, size = 1518, normalized size = 6.93

method	result	size
risch	\(\text {Expression too large to display}\)	\(1518\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-15/8*b/d^3*e/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))*ln(x^n)+7/16*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^3*
e^2/(e*x^2+d)^2*x^3+b*n/d^3*e/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3/x
-1/2*b*n*e/d^3*ln(x)*x/(e*x^2+d)-1/2*b*n/d^3*e/(-e*d)^(1/2)*ln(x)*ln((-e*x+(-e*d)^(1/2))/(-e*d)^(1/2))+1/2*b*n
*e^2/d^3*ln(x)/(e*x^2+d)^2*x^3+15/8*b/d^3*e/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))*n*ln(x)+1/2*b*n/d^3*e/(-e*d)^(
1/2)*ln(x)*ln((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))-9/8*b*ln(c)/d^2*e/(e*x^2+d)^2*x-15/8*b*ln(c)/d^3*e/(e*d)^(1/2)*
arctan(x*e/(e*d)^(1/2))-7/8*b*ln(c)/d^3*e^2/(e*x^2+d)^2*x^3+1/2*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^3
/x+15/16*I*b*Pi*csgn(I*c*x^n)^3/d^3*e/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))+1/2*b*n*e/d^2*ln(x)/(e*x^2+d)^2*x-7/
8*a/d^3*e^2/(e*x^2+d)^2*x^3-9/8*a/d^2*e/(e*x^2+d)^2*x-15/8*a/d^3*e/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))+1/2*I*b
*Pi*csgn(I*c*x^n)^3/d^3/x-7/8*b/d^3*e^2/(e*x^2+d)^2*x^3*ln(x^n)-a/d^3/x-15/16*b*n*e/d^3/(-e*d)^(1/2)*dilog((-e
*x+(-e*d)^(1/2))/(-e*d)^(1/2))-9/8*b/d^2*e/(e*x^2+d)^2*x*ln(x^n)+1/4*b*n*e/d^2*ln(x)/(e*x^2+d)/(-e*d)^(1/2)*ln
((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))-b*ln(c)/d^3/x+7/16*I*b*Pi*csgn(I*c*x^n)^3/d^3*e^2/(e*x^2+d)^2*x^3+9/16*I*b*P
i*csgn(I*c*x^n)^3/d^2*e/(e*x^2+d)^2*x+15/16*b*n*e/d^3/(-e*d)^(1/2)*dilog((e*x+(-e*d)^(1/2))/(-e*d)^(1/2))-1/4*
b*n*e/d^2*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/d*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/d*ln(x)/(e*x^2+d)^2/(-e*d)^(1/2)*ln((e*x+(-e*d)^(1/2
))/(-e*d)^(1/2))+1/8*b*n*e/d^3*x/(e*x^2+d)-b*ln(x^n)/d^3/x-1/2*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^3/x-7/16*I*b
*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3*e^2/(e*x^2+d)^2*x^3-9/16*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^2*e/(e*x^2+d
)^2*x+3/16*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^4-3/8*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))*x^2-1/4*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))*x^2+1/4*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))*x^2+15/16*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^3*e/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/
2))-7/16*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^3*e^2/(e*x^2+d)^2*x^3-15/16*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2/d^3
*e/(e*d)^(1/2)*arctan(x*e/(e*d)^(1/2))-15/16*I*b*Pi*csgn(I*c)*csgn(I*c*x^n)^2/d^3*e/(e*d)^(1/2)*arctan(x*e/(e*
d)^(1/2))+9/16*I*b*Pi*csgn(I*c)*csgn(I*x^n)*csgn(I*c*x^n)/d^2*e/(e*x^2+d)^2*x-b*n/d^3/x-9/16*I*b*Pi*csgn(I*c)*
csgn(I*c*x^n)^2/d^2*e/(e*x^2+d)^2*x+3/8*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))*x^2-3/16*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^4

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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^2/(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^2/(e*x^2+d)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \log {\left (c x^{n} \right )}}{x^{2} \left (d + e x^{2}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2/(e*x^2+d)^3,x, algorithm="giac")

[Out]

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

[Out]

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

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