\(\int \frac {\log (c+d x)}{x^3 (a+b x^4)} \, dx\) [298]

Optimal result

Integrand size = 19, antiderivative size = 537 \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=-\frac {d}{2 a c x}-\frac {d^2 \log (x)}{2 a c^2}+\frac {d^2 \log (c+d x)}{2 a c^2}-\frac {\log (c+d x)}{2 a x^2}-\frac {\sqrt {b} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac {\sqrt {b} \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}} \]

[Out]

-1/2*d/a/c/x-1/2*d^2*ln(x)/a/c^2+1/2*d^2*ln(d*x+c)/a/c^2-1/2*ln(d*x+c)/a/x^2+1/4*ln(d*((-a)^(1/4)-b^(1/4)*x)/(
b^(1/4)*c+(-a)^(1/4)*d))*ln(d*x+c)*b^(1/2)/(-a)^(3/2)+1/4*ln(-d*((-a)^(1/4)+b^(1/4)*x)/(b^(1/4)*c-(-a)^(1/4)*d
))*ln(d*x+c)*b^(1/2)/(-a)^(3/2)-1/4*ln(d*x+c)*ln(-d*(b^(1/4)*x+(-(-a)^(1/2))^(1/2))/(b^(1/4)*c-d*(-(-a)^(1/2))
^(1/2)))*b^(1/2)/(-a)^(3/2)-1/4*ln(d*x+c)*ln(d*(-b^(1/4)*x+(-(-a)^(1/2))^(1/2))/(b^(1/4)*c+d*(-(-a)^(1/2))^(1/
2)))*b^(1/2)/(-a)^(3/2)+1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-a)^(1/4)*d))*b^(1/2)/(-a)^(3/2)+1/4*polylog
(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+(-a)^(1/4)*d))*b^(1/2)/(-a)^(3/2)-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-d*(-(
-a)^(1/2))^(1/2)))*b^(1/2)/(-a)^(3/2)-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+d*(-(-a)^(1/2))^(1/2)))*b^(1/2)
/(-a)^(3/2)

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 537, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, Rules used = {281, 331, 211, 2463, 2442, 46, 266, 2441, 2440, 2438} \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac {d^2 \log (x)}{2 a c^2}+\frac {d^2 \log (c+d x)}{2 a c^2}-\frac {\log (c+d x)}{2 a x^2}-\frac {d}{2 a c x} \]

[In]

Int[Log[c + d*x]/(x^3*(a + b*x^4)),x]

[Out]

-1/2*d/(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]*Lo
g[(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*(S
qrt[-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 46

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] && Lt
Q[m + n + 2, 0])

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 266

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 281

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 331

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 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 2440

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 2441

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 2442

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 2463

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]

Rubi steps \begin{align*} \text {integral}& = \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} \text {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} \text {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} \text {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} \text {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] (verified)

Result contains complex when optimal does not.

Time = 0.13 (sec) , antiderivative size = 506, normalized size of antiderivative = 0.94 \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=-\frac {\log (c+d x)}{2 a x^2}-\frac {\sqrt {b} \log \left (\frac {d \left (i \sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac {\sqrt {b} \log \left (-\frac {d \left (i \sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}+\frac {\sqrt {b} \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 (-a)^{3/2}}-\frac {d \left (\frac {1}{c x}+\frac {d \log (x)}{c^2}-\frac {d \log (c+d x)}{c^2}\right )}{2 a}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}-\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}}+\frac {\sqrt {b} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 (-a)^{3/2}} \]

[In]

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

[Out]

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

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.66 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.29

method	result	size
derivativedivides	\(d^{2} \left (\frac {-\frac {\ln \left (-d x \right )}{2 c^{2}}-\frac {1}{2 c d x}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (-d x +c \right )}{2 c^{2} d^{2} x^{2}}}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{4 a}\right )\)	\(158\)
default	\(d^{2} \left (\frac {-\frac {\ln \left (-d x \right )}{2 c^{2}}-\frac {1}{2 c d x}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (-d x +c \right )}{2 c^{2} d^{2} x^{2}}}{a}-\frac {\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}}{4 a}\right )\)	\(158\)
risch	\(-\frac {d^{2} \ln \left (-d x \right )}{2 a \,c^{2}}-\frac {d}{2 a c x}+\frac {d^{2} \ln \left (d x +c \right )}{2 a \,c^{2}}-\frac {\ln \left (d x +c \right )}{2 a \,x^{2}}-\frac {d^{2} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{4 a}\)	\(162\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{3}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{3}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\int { \frac {\log \left (d x + c\right )}{{\left (b x^{4} + a\right )} x^{3}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log (c+d x)}{x^3 \left (a+b x^4\right )} \, dx=\int \frac {\ln \left (c+d\,x\right )}{x^3\,\left (b\,x^4+a\right )} \,d x \]

[In]

int(log(c + d*x)/(x^3*(a + b*x^4)),x)

[Out]

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