3.302 \(\int \frac{\log (c+d x)}{x^2 (a+b x^4)} \, dx\)

Optimal. Leaf size=536 \[ -\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}+\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}}+\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{5/4}}+\frac{\sqrt [4]{b} \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}+\frac{\sqrt [4]{b} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{5/4}}-\frac{\sqrt [4]{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt [4]{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}}+\frac{d \log (x)}{a c}-\frac{d \log (c+d x)}{a c}-\frac{\log (c+d x)}{a x} \]

[Out]

(d*Log[x])/(a*c) - (d*Log[c + d*x])/(a*c) - Log[c + d*x]/(a*x) + (b^(1/4)*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x)
)/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*(-Sqrt[-a])^(5/2)) + (b^(1/4)*Log[(d*((-a)^(1/4) - b^(1/4)
*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(5/4)) - (b^(1/4)*Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x
))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*(-Sqrt[-a])^(5/2)) - (b^(1/4)*Log[-((d*((-a)^(1/4) + b^(
1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(5/4)) - (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(
b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*(-Sqrt[-a])^(5/2)) + (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c +
Sqrt[-Sqrt[-a]]*d)])/(4*(-Sqrt[-a])^(5/2)) - (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d
)])/(4*(-a)^(5/4)) + (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(5/4))

________________________________________________________________________________________

Rubi [A]  time = 0.816492, antiderivative size = 536, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 16, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.842, Rules used = {325, 297, 1162, 617, 204, 1165, 628, 2416, 2395, 36, 29, 31, 2409, 2394, 2393, 2391} \[ -\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}+\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}}+\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{5/4}}+\frac{\sqrt [4]{b} \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}+\frac{\sqrt [4]{b} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{5/4}}-\frac{\sqrt [4]{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt [4]{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}}+\frac{d \log (x)}{a c}-\frac{d \log (c+d x)}{a c}-\frac{\log (c+d x)}{a x} \]

Antiderivative was successfully verified.

[In]

Int[Log[c + d*x]/(x^2*(a + b*x^4)),x]

[Out]

(d*Log[x])/(a*c) - (d*Log[c + d*x])/(a*c) - Log[c + d*x]/(a*x) + (b^(1/4)*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x)
)/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*(-Sqrt[-a])^(5/2)) + (b^(1/4)*Log[(d*((-a)^(1/4) - b^(1/4)
*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(5/4)) - (b^(1/4)*Log[-((d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x
))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*(-Sqrt[-a])^(5/2)) - (b^(1/4)*Log[-((d*((-a)^(1/4) + b^(
1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(5/4)) - (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(
b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*(-Sqrt[-a])^(5/2)) + (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c +
Sqrt[-Sqrt[-a]]*d)])/(4*(-Sqrt[-a])^(5/2)) - (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d
)])/(4*(-a)^(5/4)) + (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(5/4))

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 297

Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]},
Dist[1/(2*s), Int[(r + s*x^2)/(a + b*x^4), x], x] - Dist[1/(2*s), Int[(r - s*x^2)/(a + b*x^4), x], x]] /; Free
Q[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ,
b]]))

Rule 1162

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(2*d)/e, 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 617

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[(a*c)/b^2]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + (2*c*x)/b], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 1165

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-2*d)/e, 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2395

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[((f + g
*x)^(q + 1)*(a + b*Log[c*(d + e*x)^n]))/(g*(q + 1)), x] - Dist[(b*e*n)/(g*(q + 1)), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

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{\log (c+d x)}{x^2 \left (a+b x^4\right )} \, dx &=\int \left (\frac{\log (c+d x)}{a x^2}-\frac{b x^2 \log (c+d x)}{a \left (a+b x^4\right )}\right ) \, dx\\ &=\frac{\int \frac{\log (c+d x)}{x^2} \, dx}{a}-\frac{b \int \frac{x^2 \log (c+d x)}{a+b x^4} \, dx}{a}\\ &=-\frac{\log (c+d x)}{a x}-\frac{b \int \left (-\frac{\log (c+d x)}{2 \sqrt{b} \left (\sqrt{-a}-\sqrt{b} x^2\right )}+\frac{\log (c+d x)}{2 \sqrt{b} \left (\sqrt{-a}+\sqrt{b} x^2\right )}\right ) \, dx}{a}+\frac{d \int \frac{1}{x (c+d x)} \, dx}{a}\\ &=-\frac{\log (c+d x)}{a x}+\frac{\sqrt{b} \int \frac{\log (c+d x)}{\sqrt{-a}-\sqrt{b} x^2} \, dx}{2 a}-\frac{\sqrt{b} \int \frac{\log (c+d x)}{\sqrt{-a}+\sqrt{b} x^2} \, dx}{2 a}+\frac{d \int \frac{1}{x} \, dx}{a c}-\frac{d^2 \int \frac{1}{c+d x} \, dx}{a c}\\ &=\frac{d \log (x)}{a c}-\frac{d \log (c+d x)}{a c}-\frac{\log (c+d x)}{a x}-\frac{\sqrt{b} \int \left (\frac{\sqrt{-\sqrt{-a}} \log (c+d x)}{2 \sqrt{-a} \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}+\frac{\sqrt{-\sqrt{-a}} \log (c+d x)}{2 \sqrt{-a} \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 a}+\frac{\sqrt{b} \int \left (\frac{\log (c+d x)}{2 \sqrt [4]{-a} \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}+\frac{\log (c+d x)}{2 \sqrt [4]{-a} \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 a}\\ &=\frac{d \log (x)}{a c}-\frac{d \log (c+d x)}{a c}-\frac{\log (c+d x)}{a x}-\frac{\sqrt{b} \int \frac{\log (c+d x)}{\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x} \, dx}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt{b} \int \frac{\log (c+d x)}{\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x} \, dx}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt{b} \int \frac{\log (c+d x)}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 (-a)^{5/4}}-\frac{\sqrt{b} \int \frac{\log (c+d x)}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 (-a)^{5/4}}\\ &=\frac{d \log (x)}{a c}-\frac{d \log (c+d x)}{a c}-\frac{\log (c+d x)}{a x}+\frac{\sqrt [4]{b} \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt{-a}\right )^{5/2}}+\frac{\sqrt [4]{b} \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{5/4}}-\frac{\sqrt [4]{b} \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt [4]{b} \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{5/4}}-\frac{\left (\sqrt [4]{b} d\right ) \int \frac{\log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{c+d x} \, dx}{4 \left (-\sqrt{-a}\right )^{5/2}}+\frac{\left (\sqrt [4]{b} d\right ) \int \frac{\log \left (\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{c+d x} \, dx}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\left (\sqrt [4]{b} d\right ) \int \frac{\log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c+d x} \, dx}{4 (-a)^{5/4}}+\frac{\left (\sqrt [4]{b} d\right ) \int \frac{\log \left (\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{c+d x} \, dx}{4 (-a)^{5/4}}\\ &=\frac{d \log (x)}{a c}-\frac{d \log (c+d x)}{a c}-\frac{\log (c+d x)}{a x}+\frac{\sqrt [4]{b} \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt{-a}\right )^{5/2}}+\frac{\sqrt [4]{b} \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{5/4}}-\frac{\sqrt [4]{b} \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt [4]{b} \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{5/4}}+\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [4]{b} x}{-\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{x} \, dx,x,c+d x\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{x} \, dx,x,c+d x\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}+\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt [4]{b} x}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{x} \, dx,x,c+d x\right )}{4 (-a)^{5/4}}-\frac{\sqrt [4]{b} \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{x} \, dx,x,c+d x\right )}{4 (-a)^{5/4}}\\ &=\frac{d \log (x)}{a c}-\frac{d \log (c+d x)}{a c}-\frac{\log (c+d x)}{a x}+\frac{\sqrt [4]{b} \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt{-a}\right )^{5/2}}+\frac{\sqrt [4]{b} \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{5/4}}-\frac{\sqrt [4]{b} \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right ) \log (c+d x)}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt [4]{b} \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{5/4}}-\frac{\sqrt [4]{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}+\frac{\sqrt [4]{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{4 \left (-\sqrt{-a}\right )^{5/2}}-\frac{\sqrt [4]{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}}+\frac{\sqrt [4]{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{5/4}}\\ \end{align*}

Mathematica [A]  time = 0.601832, size = 525, normalized size = 0.98 \[ \frac{1}{4} \left (\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{\sqrt{-\sqrt{-a}} a}-\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{\sqrt{-\sqrt{-a}} a}+\frac{a \sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{(-a)^{9/4}}+\frac{\sqrt [4]{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{(-a)^{5/4}}-\frac{\sqrt [4]{b} \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{\sqrt{-\sqrt{-a}} a}+\frac{\sqrt [4]{b} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{(-a)^{5/4}}+\frac{\sqrt [4]{b} \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d-\sqrt [4]{b} c}\right )}{\sqrt{-\sqrt{-a}} a}+\frac{a \sqrt [4]{b} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d-\sqrt [4]{b} c}\right )}{(-a)^{9/4}}+\frac{4 d (\log (x)-\log (c+d x))}{a c}-\frac{4 \log (c+d x)}{a x}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*d*(Log[x] - Log[c + d*x]))/(a*c) - (4*Log[c + d*x])/(a*x) - (b^(1/4)*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))
/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(Sqrt[-Sqrt[-a]]*a) + (b^(1/4)*Log[(d*((-a)^(1/4) - b^(1/4)*x)
)/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(-a)^(5/4) + (b^(1/4)*Log[(d*(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(-(b^(
1/4)*c) + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(Sqrt[-Sqrt[-a]]*a) + (a*b^(1/4)*Log[(d*((-a)^(1/4) + b^(1/4)*x))/
(-(b^(1/4)*c) + (-a)^(1/4)*d)]*Log[c + d*x])/(-a)^(9/4) + (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c -
Sqrt[-Sqrt[-a]]*d)])/(Sqrt[-Sqrt[-a]]*a) - (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a
]]*d)])/(Sqrt[-Sqrt[-a]]*a) + (a*b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(-a)^(9/4
) + (b^(1/4)*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(-a)^(5/4))/4

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Maple [C]  time = 0.421, size = 136, normalized size = 0.3 \begin{align*} -{\frac{d}{4\,a}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{4}-4\,{{\it \_Z}}^{3}bc+6\,{{\it \_Z}}^{2}b{c}^{2}-4\,{\it \_Z}\,b{c}^{3}+a{d}^{4}+b{c}^{4} \right ) }{\frac{1}{{\it \_R1}-c} \left ( \ln \left ( dx+c \right ) \ln \left ({\frac{-dx+{\it \_R1}-c}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{-dx+{\it \_R1}-c}{{\it \_R1}}} \right ) \right ) }}+{\frac{d\ln \left ( dx \right ) }{ac}}-{\frac{d\ln \left ( dx+c \right ) }{ac}}-{\frac{\ln \left ( dx+c \right ) }{ax}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*x+c)/x^2/(b*x^4+a),x)

[Out]

-1/4*d/a*sum(1/(_R1-c)*(ln(d*x+c)*ln((-d*x+_R1-c)/_R1)+dilog((-d*x+_R1-c)/_R1)),_R1=RootOf(_Z^4*b-4*_Z^3*b*c+6
*_Z^2*b*c^2-4*_Z*b*c^3+a*d^4+b*c^4))+d/a/c*ln(d*x)-d*ln(d*x+c)/a/c-ln(d*x+c)/a/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(log(d*x+c)/x^2/(b*x^4+a),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{\log \left (d x + c\right )}{b x^{6} + a x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(ln(d*x+c)/x**2/(b*x**4+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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