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

Optimal. Leaf size=210 \[ -\frac{3 i b n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{5/2} \sqrt{e}}+\frac{3 i b n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{5/2} \sqrt{e}}+\frac{3 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{5/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac{b n x}{8 d^2 \left (d+e x^2\right )}-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \sqrt{e}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.145805, antiderivative size = 210, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.35, Rules used = {2323, 205, 2324, 12, 4848, 2391, 199} \[ -\frac{3 i b n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{5/2} \sqrt{e}}+\frac{3 i b n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{16 d^{5/2} \sqrt{e}}+\frac{3 x \left (a+b \log \left (c x^n\right )\right )}{8 d^2 \left (d+e x^2\right )}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{8 d^{5/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )}{4 d \left (d+e x^2\right )^2}-\frac{b n x}{8 d^2 \left (d+e x^2\right )}-\frac{b n \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{2 d^{5/2} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2323

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

Rule 205

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

Rule 2324

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 12

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

Rule 4848

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]

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]

Rule 199

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[(x*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Dist[(n*(p +
1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (In
tegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[p]
)

Rubi steps

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

Mathematica [B]  time = 0.989545, size = 544, normalized size = 2.59 \[ \frac{1}{16} \left (\frac{3 b n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{5/2} \sqrt{e}}-\frac{3 b n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{5/2} \sqrt{e}}+\frac{3 \left (a+b \log \left (c x^n\right )\right )}{d^2 e x+(-d)^{5/2} \sqrt{e}}+\frac{3 \left (a+b \log \left (c x^n\right )\right )}{d^2 e x+(-d)^{3/2} d \sqrt{e}}-\frac{3 \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2} \sqrt{e}}+\frac{3 \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2} \sqrt{e}}+\frac{d \left (a+b \log \left (c x^n\right )\right )}{(-d)^{5/2} \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )^2}+\frac{a+b \log \left (c x^n\right )}{(-d)^{3/2} \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )^2}-\frac{b n \left (\log (x) \left (d-\sqrt{-d} \sqrt{e} x\right )+\left (\sqrt{-d} \sqrt{e} x-d\right ) \log \left (\sqrt{-d}+\sqrt{e} x\right )+d\right )}{d^3 \left (\sqrt{-d} \sqrt{e}+e x\right )}-\frac{b n \left (\log (x) \left (\sqrt{-d} \sqrt{e} x+d\right )-\left (\sqrt{-d} \sqrt{e} x+d\right ) \log \left (d \sqrt{e} x+(-d)^{3/2}\right )+d\right )}{d^3 e x+(-d)^{7/2} \sqrt{e}}+\frac{3 b n \left (\log (x)-\log \left (\sqrt{-d}-\sqrt{e} x\right )\right )}{(-d)^{5/2} \sqrt{e}}-\frac{3 b n \left (\log (x)-\log \left (\sqrt{-d}+\sqrt{e} x\right )\right )}{(-d)^{5/2} \sqrt{e}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.306, size = 1047, normalized size = 5. \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))/(e*x^2+d)^3,x)

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

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^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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))/(e*x**2+d)**3,x)

[Out]

Timed out

________________________________________________________________________________________

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}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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