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

Optimal. Leaf size=134 \[ \frac{i b \sqrt{e} n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2}}-\frac{i b \sqrt{e} n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2}}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac{a+b \log \left (c x^n\right )}{d x}-\frac{b n}{d x} \]

[Out]

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

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Rubi [A]  time = 0.17457, antiderivative size = 134, normalized size of antiderivative = 1., 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 = {325, 205, 2351, 2304, 2324, 12, 4848, 2391} \[ \frac{i b \sqrt{e} n \text{PolyLog}\left (2,-\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2}}-\frac{i b \sqrt{e} n \text{PolyLog}\left (2,\frac{i \sqrt{e} x}{\sqrt{d}}\right )}{2 d^{3/2}}-\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{d^{3/2}}-\frac{a+b \log \left (c x^n\right )}{d x}-\frac{b n}{d x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

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 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 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]

Rubi steps

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

Mathematica [A]  time = 0.139715, size = 173, normalized size = 1.29 \[ \frac{d \left (b d \sqrt{e} n x \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )-b d \sqrt{e} n x \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )-d \sqrt{e} x \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )+d \sqrt{e} x \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )+2 d \sqrt{-d} \left (a+b \log \left (c x^n\right )\right )-2 b (-d)^{3/2} n\right )}{2 (-d)^{7/2} x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.268, size = 531, normalized size = 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^2/(e*x^2+d),x)

[Out]

b*e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*n*ln(x)-b*e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))*ln(x^n)-b*ln(x^n)/
d/x-1/2*b*n*e/d/(-d*e)^(1/2)*ln(x)*ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/2*b*n*e/d/(-d*e)^(1/2)*ln(x)*ln((e*x
+(-d*e)^(1/2))/(-d*e)^(1/2))-1/2*b*n*e/d/(-d*e)^(1/2)*dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+1/2*b*n*e/d/(-d*
e)^(1/2)*dilog((e*x+(-d*e)^(1/2))/(-d*e)^(1/2))-b*n/d/x-1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)*e/d/(d*e)^(1/2)*a
rctan(x*e/(d*e)^(1/2))+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)/d/x-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)
^2/d/x-1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)^2*e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/2*I*b*Pi*csgn(I*c*x^n)
^3*e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))+1/2*I*b*Pi*csgn(I*x^n)*csgn(I*c*x^n)*csgn(I*c)*e/d/(d*e)^(1/2)*arct
an(x*e/(d*e)^(1/2))+1/2*I*b*Pi*csgn(I*c*x^n)^3/d/x-1/2*I*b*Pi*csgn(I*c*x^n)^2*csgn(I*c)/d/x-b*ln(c)*e/d/(d*e)^
(1/2)*arctan(x*e/(d*e)^(1/2))-b*ln(c)/d/x-a*e/d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-a/d/x

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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 )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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