\(\int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx\) [296]

Optimal result

Integrand size = 19, antiderivative size = 530 \[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\frac {c x}{2 b d}-\frac {x^2}{4 b}-\frac {c^2 \log (c+d x)}{2 b d^2}+\frac {x^2 \log (c+d x)}{2 b}-\frac {\sqrt {-a} \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^{3/2}}+\frac {\sqrt {-a} \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^{3/2}}-\frac {\sqrt {-a} \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^{3/2}}+\frac {\sqrt {-a} \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^{3/2}}-\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}-\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 530, 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, 327, 211, 2463, 2442, 45, 266, 2441, 2440, 2438} \[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=-\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}-\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/2}}-\frac {\sqrt {-a} \log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^{3/2}}-\frac {\sqrt {-a} \log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}-\frac {c^2 \log (c+d x)}{2 b d^2}+\frac {x^2 \log (c+d x)}{2 b}+\frac {c x}{2 b d}-\frac {x^2}{4 b} \]

[In]

Int[(x^5*Log[c + d*x])/(a + b*x^4),x]

[Out]

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

Rule 45

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 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 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 {x \log (c+d x)}{b}-\frac {a x \log (c+d x)}{b \left (a+b x^4\right )}\right ) \, dx \\ & = \frac {\int x \log (c+d x) \, dx}{b}-\frac {a \int \frac {x \log (c+d x)}{a+b x^4} \, dx}{b} \\ & = \frac {x^2 \log (c+d x)}{2 b}-\frac {a \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}{b}-\frac {d \int \frac {x^2}{c+d x} \, dx}{2 b} \\ & = \frac {x^2 \log (c+d x)}{2 b}-\frac {\sqrt {-a} \int \frac {x \log (c+d x)}{\sqrt {-a} \sqrt {b}-b x^2} \, dx}{2 \sqrt {b}}-\frac {\sqrt {-a} \int \frac {x \log (c+d x)}{\sqrt {-a} \sqrt {b}+b x^2} \, dx}{2 \sqrt {b}}-\frac {d \int \left (-\frac {c}{d^2}+\frac {x}{d}+\frac {c^2}{d^2 (c+d x)}\right ) \, dx}{2 b} \\ & = \frac {c x}{2 b d}-\frac {x^2}{4 b}-\frac {c^2 \log (c+d x)}{2 b d^2}+\frac {x^2 \log (c+d x)}{2 b}-\frac {\sqrt {-a} \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 \sqrt {b}}-\frac {\sqrt {-a} \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 \sqrt {b}} \\ & = \frac {c x}{2 b d}-\frac {x^2}{4 b}-\frac {c^2 \log (c+d x)}{2 b d^2}+\frac {x^2 \log (c+d x)}{2 b}+\frac {\sqrt {-a} \int \frac {\log (c+d x)}{\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x} \, dx}{4 b^{5/4}}-\frac {\sqrt {-a} \int \frac {\log (c+d x)}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 b^{5/4}}-\frac {\sqrt {-a} \int \frac {\log (c+d x)}{\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x} \, dx}{4 b^{5/4}}+\frac {\sqrt {-a} \int \frac {\log (c+d x)}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 b^{5/4}} \\ & = \frac {c x}{2 b d}-\frac {x^2}{4 b}-\frac {c^2 \log (c+d x)}{2 b d^2}+\frac {x^2 \log (c+d x)}{2 b}-\frac {\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^{3/2}}+\frac {\sqrt {-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^{3/2}}-\frac {\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^{3/2}}+\frac {\sqrt {-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^{3/2}}+\frac {\left (\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^{3/2}}-\frac {\left (\sqrt {-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^{3/2}}+\frac {\left (\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^{3/2}}-\frac {\left (\sqrt {-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^{3/2}} \\ & = \frac {c x}{2 b d}-\frac {x^2}{4 b}-\frac {c^2 \log (c+d x)}{2 b d^2}+\frac {x^2 \log (c+d x)}{2 b}-\frac {\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^{3/2}}+\frac {\sqrt {-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^{3/2}}-\frac {\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^{3/2}}+\frac {\sqrt {-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^{3/2}}+\frac {\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^{3/2}}+\frac {\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^{3/2}}-\frac {\sqrt {-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^{3/2}}-\frac {\sqrt {-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^{3/2}} \\ & = \frac {c x}{2 b d}-\frac {x^2}{4 b}-\frac {c^2 \log (c+d x)}{2 b d^2}+\frac {x^2 \log (c+d x)}{2 b}-\frac {\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^{3/2}}+\frac {\sqrt {-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^{3/2}}-\frac {\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^{3/2}}+\frac {\sqrt {-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^{3/2}}-\frac {\sqrt {-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}-\frac {\sqrt {-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^{3/2}}+\frac {\sqrt {-a} \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/2}} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.16 (sec) , antiderivative size = 484, normalized size of antiderivative = 0.91 \[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\frac {2 \sqrt {b} c d x-\sqrt {b} d^2 x^2-2 \sqrt {b} c^2 \log (c+d x)+2 \sqrt {b} d^2 x^2 \log (c+d x)+\sqrt {-a} d^2 \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt {-a} d^2 \log \left (\frac {d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)-\sqrt {-a} d^2 \log \left (\frac {d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{-i \sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+\sqrt {-a} d^2 \log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{-\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)+\sqrt {-a} d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )-\sqrt {-a} d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )-\sqrt {-a} d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+\sqrt {-a} d^2 \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^{3/2} d^2} \]

[In]

Integrate[(x^5*Log[c + d*x])/(a + b*x^4),x]

[Out]

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

Maple [C] (verified)

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

Time = 0.64 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.31

method	result	size
derivativedivides	\(\frac {-\frac {d^{4} \left (-\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}+\frac {\left (d x +c \right )^{2}}{4}+c \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )\right )}{b}-\frac {a \,d^{8} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{4 b^{2}}}{d^{6}}\)	\(163\)
default	\(\frac {-\frac {d^{4} \left (-\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}+\frac {\left (d x +c \right )^{2}}{4}+c \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )\right )}{b}-\frac {a \,d^{8} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{4 b^{2}}}{d^{6}}\)	\(163\)
risch	\(\frac {x^{2} \ln \left (d x +c \right )}{2 b}-\frac {c^{2} \ln \left (d x +c \right )}{2 b \,d^{2}}-\frac {x^{2}}{4 b}+\frac {c x}{2 d b}+\frac {3 c^{2}}{4 d^{2} b}-\frac {d^{2} a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\frac {\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )}{\textit {\_R1}^{2}-2 \textit {\_R1} c +c^{2}}\right )}{4 b^{2}}\)	\(164\)

[In]

int(x^5*ln(d*x+c)/(b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

1/d^6*(-d^4/b*(-1/2*(d*x+c)^2*ln(d*x+c)+1/4*(d*x+c)^2+c*((d*x+c)*ln(d*x+c)-d*x-c))-1/4*a*d^8/b^2*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]

\[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{5} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]

[In]

integrate(x^5*log(d*x+c)/(b*x^4+a),x, algorithm="fricas")

[Out]

integral(x^5*log(d*x + c)/(b*x^4 + a), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\text {Timed out} \]

[In]

integrate(x**5*ln(d*x+c)/(b*x**4+a),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{5} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]

[In]

integrate(x^5*log(d*x+c)/(b*x^4+a),x, algorithm="maxima")

[Out]

integrate(x^5*log(d*x + c)/(b*x^4 + a), x)

Giac [F]

\[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{5} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]

[In]

integrate(x^5*log(d*x+c)/(b*x^4+a),x, algorithm="giac")

[Out]

integrate(x^5*log(d*x + c)/(b*x^4 + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^5 \log (c+d x)}{a+b x^4} \, dx=\int \frac {x^5\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \]

[In]

int((x^5*log(c + d*x))/(a + b*x^4),x)

[Out]

int((x^5*log(c + d*x))/(a + b*x^4), x)