3.298 \(\int \frac{\log (c+d x)}{x^3 (a+b x^4)} \, dx\)

Optimal. Leaf size=537 \[ -\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{d}{2 a c x} \]

[Out]

-d/(2*a*c*x) - (d^2*Log[x])/(2*a*c^2) + (d^2*Log[c + d*x])/(2*a*c^2) - Log[c + d*x]/(2*a*x^2) - (Sqrt[b]*Log[(
d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[
(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(3/2)) - (Sqrt[b]*Log[-((d*(Sqr
t[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[-((d*
((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(
1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*(-a)^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^
(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*(-a)^(3/2)) + (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4
)*d)])/(4*(-a)^(3/2)) + (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.650977, antiderivative size = 537, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {275, 325, 205, 2416, 2395, 44, 260, 2394, 2393, 2391} \[ -\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{d}{2 a c x} \]

Antiderivative was successfully verified.

[In]

Int[Log[c + d*x]/(x^3*(a + b*x^4)),x]

[Out]

-d/(2*a*c*x) - (d^2*Log[x])/(2*a*c^2) + (d^2*Log[c + d*x])/(2*a*c^2) - Log[c + d*x]/(2*a*x^2) - (Sqrt[b]*Log[(
d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[
(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-a)^(3/2)) - (Sqrt[b]*Log[-((d*(Sqr
t[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[-((d*
((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(
1/4)*(c + d*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*(-a)^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^
(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*(-a)^(3/2)) + (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4
)*d)])/(4*(-a)^(3/2)) + (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(3/2))

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 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^3 \left (a+b x^4\right )} \, dx &=\int \left (\frac{\log (c+d x)}{a x^3}-\frac{b x \log (c+d x)}{a \left (a+b x^4\right )}\right ) \, dx\\ &=\frac{\int \frac{\log (c+d x)}{x^3} \, dx}{a}-\frac{b \int \frac{x \log (c+d x)}{a+b x^4} \, dx}{a}\\ &=-\frac{\log (c+d x)}{2 a x^2}-\frac{b \int \left (-\frac{\sqrt{b} x \log (c+d x)}{2 \sqrt{-a} \left (\sqrt{-a} \sqrt{b}-b x^2\right )}-\frac{\sqrt{b} x \log (c+d x)}{2 \sqrt{-a} \left (\sqrt{-a} \sqrt{b}+b x^2\right )}\right ) \, dx}{a}+\frac{d \int \frac{1}{x^2 (c+d x)} \, dx}{2 a}\\ &=-\frac{\log (c+d x)}{2 a x^2}-\frac{b^{3/2} \int \frac{x \log (c+d x)}{\sqrt{-a} \sqrt{b}-b x^2} \, dx}{2 (-a)^{3/2}}-\frac{b^{3/2} \int \frac{x \log (c+d x)}{\sqrt{-a} \sqrt{b}+b x^2} \, dx}{2 (-a)^{3/2}}+\frac{d \int \left (\frac{1}{c x^2}-\frac{d}{c^2 x}+\frac{d^2}{c^2 (c+d x)}\right ) \, dx}{2 a}\\ &=-\frac{d}{2 a c x}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{b^{3/2} \int \left (-\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}+\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 (-a)^{3/2}}-\frac{b^{3/2} \int \left (\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}-\frac{\log (c+d x)}{2 b^{3/4} \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}\right ) \, dx}{2 (-a)^{3/2}}\\ &=-\frac{d}{2 a c x}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}+\frac{b^{3/4} \int \frac{\log (c+d x)}{\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x} \, dx}{4 (-a)^{3/2}}-\frac{b^{3/4} \int \frac{\log (c+d x)}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 (-a)^{3/2}}-\frac{b^{3/4} \int \frac{\log (c+d x)}{\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x} \, dx}{4 (-a)^{3/2}}+\frac{b^{3/4} \int \frac{\log (c+d x)}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 (-a)^{3/2}}\\ &=-\frac{d}{2 a c x}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{\sqrt{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 (-a)^{3/2}}+\frac{\sqrt{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)^{3/2}}-\frac{\sqrt{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 (-a)^{3/2}}+\frac{\sqrt{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)^{3/2}}+\frac{\left (\sqrt{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 (-a)^{3/2}}-\frac{\left (\sqrt{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)^{3/2}}+\frac{\left (\sqrt{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 (-a)^{3/2}}-\frac{\left (\sqrt{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)^{3/2}}\\ &=-\frac{d}{2 a c x}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{\sqrt{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 (-a)^{3/2}}+\frac{\sqrt{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)^{3/2}}-\frac{\sqrt{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 (-a)^{3/2}}+\frac{\sqrt{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)^{3/2}}+\frac{\sqrt{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 (-a)^{3/2}}+\frac{\sqrt{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 (-a)^{3/2}}-\frac{\sqrt{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)^{3/2}}-\frac{\sqrt{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)^{3/2}}\\ &=-\frac{d}{2 a c x}-\frac{d^2 \log (x)}{2 a c^2}+\frac{d^2 \log (c+d x)}{2 a c^2}-\frac{\log (c+d x)}{2 a x^2}-\frac{\sqrt{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 (-a)^{3/2}}+\frac{\sqrt{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)^{3/2}}-\frac{\sqrt{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 (-a)^{3/2}}+\frac{\sqrt{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)^{3/2}}-\frac{\sqrt{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt{-\sqrt{-a}} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{Li}_2\left (\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.196353, size = 506, normalized size = 0.94 \[ \frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (-\sqrt [4]{b} x+i \sqrt [4]{-a}\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{b} x+i \sqrt [4]{-a}\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac{\sqrt{b} \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac{d \left (\frac{d \log (x)}{c^2}-\frac{d \log (c+d x)}{c^2}+\frac{1}{c x}\right )}{2 a}-\frac{\log (c+d x)}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-Log[c + d*x]/(2*a*x^2) - (Sqrt[b]*Log[(d*(I*(-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]*Log[c + d*
x])/(4*(-a)^(3/2)) + (Sqrt[b]*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*(-
a)^(3/2)) - (Sqrt[b]*Log[-((d*(I*(-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - I*(-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)
^(3/2)) + (Sqrt[b]*Log[-((d*((-a)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*(-a)^(3/2)
) - (d*(1/(c*x) + (d*Log[x])/c^2 - (d*Log[c + d*x])/c^2))/(2*a) + (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(
1/4)*c - (-a)^(1/4)*d)])/(4*(-a)^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)
])/(4*(-a)^(3/2)) - (Sqrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)])/(4*(-a)^(3/2)) + (S
qrt[b]*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*(-a)^(3/2))

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Maple [C]  time = 0.422, size = 161, normalized size = 0.3 \begin{align*} -{\frac{{d}^{2}}{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}}^{2}-2\,{\it \_R1}\,c+{c}^{2}} \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}^{2}\ln \left ( dx \right ) }{2\,{c}^{2}a}}-{\frac{d}{2\,acx}}+{\frac{{d}^{2}\ln \left ( dx+c \right ) }{2\,{c}^{2}a}}-{\frac{\ln \left ( dx+c \right ) }{2\,a{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*x+c)/x^3/(b*x^4+a),x)

[Out]

-1/4*d^2/a*sum(1/(_R1^2-2*_R1*c+c^2)*(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))-1/2*d^2/a/c^2*ln(d*x)-1/2*d/a/c/x+1/2*d^2*ln(d*x+c)/c^2/a-1
/2*ln(d*x+c)/a/x^2

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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^3/(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^{7} + a x^{3}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(log(d*x + c)/(b*x^7 + a*x^3), 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**3/(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^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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