3.3.99 \(\int \frac {x^4 \log (c+d x)}{a+b x^4} \, dx\) [299]

Optimal. Leaf size=521 \[ -\frac {x}{b}+\frac {(c+d x) \log (c+d x)}{b d}+\frac {\sqrt {-\sqrt {-a}} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^{5/4}}-\frac {\sqrt {-\sqrt {-a}} \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^{5/4}}-\frac {\sqrt {-\sqrt {-a}} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{5/4}} \]

[Out]

-x/b+(d*x+c)*ln(d*x+c)/b/d+1/4*(-a)^(1/4)*ln(d*((-a)^(1/4)-b^(1/4)*x)/(b^(1/4)*c+(-a)^(1/4)*d))*ln(d*x+c)/b^(5
/4)-1/4*(-a)^(1/4)*ln(-d*((-a)^(1/4)+b^(1/4)*x)/(b^(1/4)*c-(-a)^(1/4)*d))*ln(d*x+c)/b^(5/4)-1/4*(-a)^(1/4)*pol
ylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-a)^(1/4)*d))/b^(5/4)+1/4*(-a)^(1/4)*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+(
-a)^(1/4)*d))/b^(5/4)-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)))*(
-(-a)^(1/2))^(1/2)/b^(5/4)+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
)))*(-(-a)^(1/2))^(1/2)/b^(5/4)-1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-d*(-(-a)^(1/2))^(1/2)))*(-(-a)^(1/2))
^(1/2)/b^(5/4)+1/4*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+d*(-(-a)^(1/2))^(1/2)))*(-(-a)^(1/2))^(1/2)/b^(5/4)

________________________________________________________________________________________

Rubi [A]
time = 0.67, antiderivative size = 521, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 14, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {327, 217, 1179, 642, 1176, 631, 210, 2463, 2436, 2332, 2456, 2441, 2440, 2438} \begin {gather*} -\frac {\sqrt {-\sqrt {-a}} \text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 b^{5/4}}-\frac {\sqrt [4]{-a} \text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{5/4}}+\frac {\sqrt [4]{-a} \text {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{5/4}}+\frac {\sqrt {-\sqrt {-a}} \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 b^{5/4}}+\frac {\sqrt [4]{-a} \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 b^{5/4}}-\frac {\sqrt {-\sqrt {-a}} \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 b^{5/4}}-\frac {\sqrt [4]{-a} \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 b^{5/4}}+\frac {(c+d x) \log (c+d x)}{b d}-\frac {x}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^4*Log[c + d*x])/(a + b*x^4),x]

[Out]

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

Rule 210

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

Rule 217

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

Rule 327

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

Rule 631

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 642

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 1176

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 1179

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 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2436

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, 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 2456

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

________________________________________________________________________________________

Mathematica [C] Result contains complex when optimal does not.
time = 0.18, size = 458, normalized size = 0.88 \begin {gather*} \frac {-4 \sqrt [4]{b} d x+4 \sqrt [4]{b} c \log (c+d x)+4 \sqrt [4]{b} d x \log (c+d x)+\sqrt [4]{-a} d \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)-i \sqrt [4]{-a} d \log \left (\frac {d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+i \sqrt [4]{-a} d \log \left (\frac {d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{-i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt [4]{-a} d \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)-\sqrt [4]{-a} d \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-i \sqrt [4]{-a} d \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )+i \sqrt [4]{-a} d \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+\sqrt [4]{-a} d \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{5/4} d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.54, size = 145, normalized size = 0.28

method	result	size
derivativedivides	\(\frac {\frac {\left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right ) d^{4}}{b}+\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 \textit {\_R1}^{2} c -3 \textit {\_R1} \,c^{2}+c^{3}}\right ) a \,d^{8}}{4 b^{2}}}{d^{5}}\)	\(145\)
default	\(\frac {\frac {\left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right ) d^{4}}{b}+\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 \textit {\_R1}^{2} c -3 \textit {\_R1} \,c^{2}+c^{3}}\right ) a \,d^{8}}{4 b^{2}}}{d^{5}}\)	\(145\)
risch	\(\frac {\ln \left (d x +c \right ) x}{b}+\frac {\ln \left (d x +c \right ) c}{d b}-\frac {x}{b}-\frac {c}{b d}+\frac {d^{3} \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{-\textit {\_R1}^{3}+3 \textit {\_R1}^{2} c -3 \textit {\_R1} \,c^{2}+c^{3}}\right ) a}{4 b^{2}}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d^5*(((d*x+c)*ln(d*x+c)-d*x-c)*d^4/b+1/4/b^2*sum(1/(-_R1^3+3*_R1^2*c-3*_R1*c^2+c^3)*(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))*a*d^8)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^4\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^4*log(c + d*x))/(a + b*x^4),x)

[Out]

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

________________________________________________________________________________________