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

Optimal. Leaf size=498 \[ -\frac{a \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^2}-\frac{a \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac{a \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac{a \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac{a \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac{a \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac{a \log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^2}-\frac{a \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac{c^2 x^2}{8 b d^2}+\frac{c^3 x}{4 b d^3}-\frac{c^4 \log (c+d x)}{4 b d^4}+\frac{c x^3}{12 b d}+\frac{x^4 \log (c+d x)}{4 b}-\frac{x^4}{16 b} \]

[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)

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Rubi [A]  time = 0.810366, antiderivative size = 498, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 8, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.421, Rules used = {266, 43, 2416, 2395, 260, 2394, 2393, 2391} \[ -\frac{a \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^2}-\frac{a \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac{a \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac{a \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac{a \log (c+d x) \log \left (\frac{d \left (\sqrt{-\sqrt{-a}}-\sqrt [4]{b} x\right )}{\sqrt{-\sqrt{-a}} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac{a \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac{a \log (c+d x) \log \left (-\frac{d \left (\sqrt{-\sqrt{-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt{-\sqrt{-a}} d}\right )}{4 b^2}-\frac{a \log (c+d x) \log \left (-\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac{c^2 x^2}{8 b d^2}+\frac{c^3 x}{4 b d^3}-\frac{c^4 \log (c+d x)}{4 b d^4}+\frac{c x^3}{12 b d}+\frac{x^4 \log (c+d x)}{4 b}-\frac{x^4}{16 b} \]

Antiderivative was successfully verified.

[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 266

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 43

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 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 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{x^7 \log (c+d x)}{a+b x^4} \, dx &=\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 \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 b^2}+\frac{a \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 b^2}+\frac{a \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 b^2}+\frac{a \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 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]  time = 0.321199, size = 446, normalized size = 0.9 \[ -\frac{12 a d^4 \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )+12 a d^4 \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )+12 a d^4 \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+12 a d^4 \text{PolyLog}\left (2,\frac{\sqrt [4]{b} (c+d x)}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )+12 a d^4 \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )+12 a d^4 \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+i \sqrt [4]{b} c}\right )+12 a d^4 \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{\sqrt [4]{-a} d-i \sqrt [4]{b} c}\right )+12 a d^4 \log (c+d x) \log \left (\frac{d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d-\sqrt [4]{b} c}\right )+6 b c^2 d^2 x^2-12 b c^3 d x+12 b c^4 \log (c+d x)-4 b c d^3 x^3-12 b d^4 x^4 \log (c+d x)+3 b d^4 x^4}{48 b^2 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(-12*b*c^3*d*x + 6*b*c^2*d^2*x^2 - 4*b*c*d^3*x^3 + 3*b*d^4*x^4 + 12*b*c^4*Log[c + d*x] - 12*b*d^4*x^4*Log[c +
d*x] + 12*a*d^4*Log[(d*((-a)^(1/4) - b^(1/4)*x))/(b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x] + 12*a*d^4*Log[(d*(
(-a)^(1/4) - I*b^(1/4)*x))/(I*b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x] + 12*a*d^4*Log[(d*((-a)^(1/4) + I*b^(1/4
)*x))/((-I)*b^(1/4)*c + (-a)^(1/4)*d)]*Log[c + d*x] + 12*a*d^4*Log[(d*((-a)^(1/4) + b^(1/4)*x))/(-(b^(1/4)*c)
+ (-a)^(1/4)*d)]*Log[c + d*x] + 12*a*d^4*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - (-a)^(1/4)*d)] + 12*a*d^4
*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c - I*(-a)^(1/4)*d)] + 12*a*d^4*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/
4)*c + I*(-a)^(1/4)*d)] + 12*a*d^4*PolyLog[2, (b^(1/4)*(c + d*x))/(b^(1/4)*c + (-a)^(1/4)*d)])/(48*b^2*d^4)

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Maple [C]  time = 0.385, size = 175, normalized size = 0.4 \begin{align*}{\frac{{x}^{4}\ln \left ( dx+c \right ) }{4\,b}}-{\frac{{c}^{4}\ln \left ( dx+c \right ) }{4\,b{d}^{4}}}-{\frac{{x}^{4}}{16\,b}}+{\frac{c{x}^{3}}{12\,bd}}-{\frac{{c}^{2}{x}^{2}}{8\,b{d}^{2}}}+{\frac{{c}^{3}x}{4\,b{d}^{3}}}+{\frac{25\,{c}^{4}}{48\,b{d}^{4}}}-{\frac{a}{4\,{b}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{4}-4\,bc{{\it \_Z}}^{3}+6\,{c}^{2}b{{\it \_Z}}^{2}-4\,b{c}^{3}{\it \_Z}+a{d}^{4}+b{c}^{4} \right ) }\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{7} \log \left (d x + c\right )}{b x^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{7} \log \left (d x + c\right )}{b x^{4} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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(x**7*ln(d*x+c)/(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{x^{7} \log \left (d x + c\right )}{b x^{4} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)