\(\int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx\) [293]

Optimal result

Integrand size = 19, antiderivative size = 498 \[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {x^4 \log (c+d x)}{4 b}-\frac {a \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^2} \]

[Out]

1/4*c^3*x/b/d^3-1/8*c^2*x^2/b/d^2+1/12*c*x^3/b/d-1/16*x^4/b-1/4*c^4*ln(d*x+c)/b/d^4+1/4*x^4*ln(d*x+c)/b-1/4*a*
ln(d*((-a)^(1/4)-b^(1/4)*x)/(b^(1/4)*c+(-a)^(1/4)*d))*ln(d*x+c)/b^2-1/4*a*ln(-d*((-a)^(1/4)+b^(1/4)*x)/(b^(1/4
)*c-(-a)^(1/4)*d))*ln(d*x+c)/b^2-1/4*a*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^2-1/4*a*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^2-1
/4*a*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-(-a)^(1/4)*d))/b^2-1/4*a*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c+(-a)^(
1/4)*d))/b^2-1/4*a*polylog(2,b^(1/4)*(d*x+c)/(b^(1/4)*c-d*(-(-a)^(1/2))^(1/2)))/b^2-1/4*a*polylog(2,b^(1/4)*(d
*x+c)/(b^(1/4)*c+d*(-(-a)^(1/2))^(1/2)))/b^2

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 498, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {272, 45, 2463, 2442, 266, 2441, 2440, 2438} \[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {x^4 \log (c+d x)}{4 b}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b} \]

[In]

Int[(x^7*Log[c + d*x])/(a + b*x^4),x]

[Out]

(c^3*x)/(4*b*d^3) - (c^2*x^2)/(8*b*d^2) + (c*x^3)/(12*b*d) - x^4/(16*b) - (c^4*Log[c + d*x])/(4*b*d^4) + (x^4*
Log[c + d*x])/(4*b) - (a*Log[(d*(Sqrt[-Sqrt[-a]] - b^(1/4)*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)]*Log[c + d*x])/
(4*b^2) - (a*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b^2) - (a*Log[-((d*
(Sqrt[-Sqrt[-a]] + b^(1/4)*x))/(b^(1/4)*c - Sqrt[-Sqrt[-a]]*d))]*Log[c + d*x])/(4*b^2) - (a*Log[-((d*((-a)^(1/
4) + b^(1/4)*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/
4)*c - Sqrt[-Sqrt[-a]]*d)])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + Sqrt[-Sqrt[-a]]*d)])/(4*b
^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x
))/(b^(1/4)*c + (-a)^(1/4)*d)])/(4*b^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 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 272

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

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^3 \log (c+d x)}{b}-\frac {a x^3 \log (c+d x)}{b \left (a+b x^4\right )}\right ) \, dx \\ & = \frac {\int x^3 \log (c+d x) \, dx}{b}-\frac {a \int \frac {x^3 \log (c+d x)}{a+b x^4} \, dx}{b} \\ & = \frac {x^4 \log (c+d x)}{4 b}-\frac {a \int \left (\frac {x \log (c+d x)}{2 \left (-\sqrt {-a} \sqrt {b}+b x^2\right )}+\frac {x \log (c+d x)}{2 \left (\sqrt {-a} \sqrt {b}+b x^2\right )}\right ) \, dx}{b}-\frac {d \int \frac {x^4}{c+d x} \, dx}{4 b} \\ & = \frac {x^4 \log (c+d x)}{4 b}-\frac {a \int \frac {x \log (c+d x)}{-\sqrt {-a} \sqrt {b}+b x^2} \, dx}{2 b}-\frac {a \int \frac {x \log (c+d x)}{\sqrt {-a} \sqrt {b}+b x^2} \, dx}{2 b}-\frac {d \int \left (-\frac {c^3}{d^4}+\frac {c^2 x}{d^3}-\frac {c x^2}{d^2}+\frac {x^3}{d}+\frac {c^4}{d^4 (c+d x)}\right ) \, dx}{4 b} \\ & = \frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {x^4 \log (c+d x)}{4 b}-\frac {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 b}-\frac {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 b} \\ & = \frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {x^4 \log (c+d x)}{4 b}+\frac {a \int \frac {\log (c+d x)}{\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x} \, dx}{4 b^{7/4}}+\frac {a \int \frac {\log (c+d x)}{\sqrt [4]{-a}-\sqrt [4]{b} x} \, dx}{4 b^{7/4}}-\frac {a \int \frac {\log (c+d x)}{\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x} \, dx}{4 b^{7/4}}-\frac {a \int \frac {\log (c+d x)}{\sqrt [4]{-a}+\sqrt [4]{b} x} \, dx}{4 b^{7/4}} \\ & = \frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {x^4 \log (c+d x)}{4 b}-\frac {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^2}-\frac {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^2}-\frac {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^2}-\frac {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^2}+\frac {(a d) \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^2}+\frac {(a d) \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^2}+\frac {(a d) \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^2}+\frac {(a d) \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^2} \\ & = \frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {x^4 \log (c+d x)}{4 b}-\frac {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^2}-\frac {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^2}-\frac {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^2}-\frac {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^2}+\frac {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^2}+\frac {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^2}+\frac {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^2}+\frac {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^2} \\ & = \frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {x^4 \log (c+d x)}{4 b}-\frac {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^2}-\frac {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^2}-\frac {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^2}-\frac {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^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^2} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.15 (sec) , antiderivative size = 480, normalized size of antiderivative = 0.96 \[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {x^4 \log (c+d x)}{4 b}-\frac {a \log \left (\frac {d \left (i \sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (-\frac {d \left (i \sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right ) \log (c+d x)}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^2} \]

[In]

Integrate[(x^7*Log[c + d*x])/(a + b*x^4),x]

[Out]

(c^3*x)/(4*b*d^3) - (c^2*x^2)/(8*b*d^2) + (c*x^3)/(12*b*d) - x^4/(16*b) - (c^4*Log[c + d*x])/(4*b*d^4) + (x^4*
Log[c + d*x])/(4*b) - (a*Log[(d*(I*(-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)]*Log[c + d*x])/(4*b^2
) - (a*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x])/(4*b^2) - (a*Log[-((d*(I*(-a
)^(1/4) + b^(1/4)*x))/(b^(1/4)*c - I*(-a)^(1/4)*d))]*Log[c + d*x])/(4*b^2) - (a*Log[-((d*((-a)^(1/4) + b^(1/4)
*x))/(b^(1/4)*c - (-a)^(1/4)*d))]*Log[c + d*x])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^
(1/4)*d)])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)])/(4*b^2) - (a*PolyLog[2,
(b^(1/4)*(c + d*x))/(b^(1/4)*c + I*(-a)^(1/4)*d)])/(4*b^2) - (a*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-
a)^(1/4)*d)])/(4*b^2)

Maple [C] (verified)

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

Time = 0.75 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.35

method	result	size
risch	\(-\frac {c^{4} \ln \left (d x +c \right )}{4 b \,d^{4}}+\frac {c^{3} x}{4 b \,d^{3}}+\frac {25 c^{4}}{48 d^{4} b}-\frac {c^{2} x^{2}}{8 b \,d^{2}}+\frac {c \,x^{3}}{12 b d}+\frac {x^{4} \ln \left (d x +c \right )}{4 b}-\frac {x^{4}}{16 b}-\frac {a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{4 b^{2}}\)	\(175\)
parts	\(\frac {x^{4} \ln \left (d x +c \right )}{4 b}-\frac {\ln \left (d x +c \right ) a \ln \left (b \,x^{4}+a \right )}{4 b^{2}}-\frac {d \left (\frac {\frac {\frac {1}{4} d^{3} x^{4}-\frac {1}{3} c \,x^{3} d^{2}+\frac {1}{2} d \,x^{2} c^{2}-x \,c^{3}}{d^{4}}+\frac {c^{4} \ln \left (d x +c \right )}{d^{5}}}{b}-\frac {a \ln \left (d x +c \right ) \ln \left (b \,x^{4}+a \right )}{b^{2} d}+\frac {a \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{b^{2} d}\right )}{4}\)	\(205\)
derivativedivides	\(\frac {-\frac {d^{4} \left (c^{3} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )-3 c^{2} \left (\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}\right )+3 c \left (\frac {\left (d x +c \right )^{3} \ln \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{3}}{9}\right )-\frac {\left (d x +c \right )^{4} \ln \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{4}}{16}\right )}{b}-\frac {a \,d^{8} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{4 b^{2}}}{d^{8}}\)	\(209\)
default	\(\frac {-\frac {d^{4} \left (c^{3} \left (\left (d x +c \right ) \ln \left (d x +c \right )-d x -c \right )-3 c^{2} \left (\frac {\left (d x +c \right )^{2} \ln \left (d x +c \right )}{2}-\frac {\left (d x +c \right )^{2}}{4}\right )+3 c \left (\frac {\left (d x +c \right )^{3} \ln \left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{3}}{9}\right )-\frac {\left (d x +c \right )^{4} \ln \left (d x +c \right )}{4}+\frac {\left (d x +c \right )^{4}}{16}\right )}{b}-\frac {a \,d^{8} \left (\munderset {\textit {\_R1} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}-4 c b \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\operatorname {dilog}\left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{4 b^{2}}}{d^{8}}\)	\(209\)

[In]

int(x^7*ln(d*x+c)/(b*x^4+a),x,method=_RETURNVERBOSE)

[Out]

-1/4*c^4*ln(d*x+c)/b/d^4+1/4*c^3*x/b/d^3+25/48/d^4/b*c^4-1/8*c^2*x^2/b/d^2+1/12*c*x^3/b/d+1/4*x^4*ln(d*x+c)/b-
1/16*x^4/b-1/4*a/b^2*sum(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]

\[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{7} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]

[In]

integrate(x^7*log(d*x+c)/(b*x^4+a),x, algorithm="fricas")

[Out]

integral(x^7*log(d*x + c)/(b*x^4 + a), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\text {Timed out} \]

[In]

integrate(x**7*ln(d*x+c)/(b*x**4+a),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{7} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]

[In]

integrate(x^7*log(d*x+c)/(b*x^4+a),x, algorithm="maxima")

[Out]

integrate(x^7*log(d*x + c)/(b*x^4 + a), x)

Giac [F]

\[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\int { \frac {x^{7} \log \left (d x + c\right )}{b x^{4} + a} \,d x } \]

[In]

integrate(x^7*log(d*x+c)/(b*x^4+a),x, algorithm="giac")

[Out]

integrate(x^7*log(d*x + c)/(b*x^4 + a), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^7 \log (c+d x)}{a+b x^4} \, dx=\int \frac {x^7\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \]

[In]

int((x^7*log(c + d*x))/(a + b*x^4),x)

[Out]

int((x^7*log(c + d*x))/(a + b*x^4), x)