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

Optimal. Leaf size=498 \[ -\frac {a \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (\frac {d \left (\sqrt {-\sqrt {-a}}-\sqrt [4]{b} x\right )}{\sqrt {-\sqrt {-a}} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (-\frac {d \left (\sqrt {-\sqrt {-a}}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt {-\sqrt {-a}} d}\right )}{4 b^2}-\frac {a \log (c+d x) \log \left (-\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )}{4 b^2}-\frac {c^4 \log (c+d x)}{4 b d^4}+\frac {c^3 x}{4 b d^3}-\frac {c^2 x^2}{8 b d^2}+\frac {x^4 \log (c+d x)}{4 b}+\frac {c x^3}{12 b d}-\frac {x^4}{16 b} \]

[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

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Rubi [A]  time = 0.81, antiderivative size = 498, normalized size of antiderivative = 1.00, 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 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 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 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 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]

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 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 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 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]

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*}

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Mathematica [C]  time = 0.34, size = 446, normalized size = 0.90 \[ -\frac {12 a d^4 \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-\sqrt [4]{-a} d}\right )+12 a d^4 \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c-i \sqrt [4]{-a} d}\right )+12 a d^4 \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+i \sqrt [4]{-a} d}\right )+12 a d^4 \text {Li}_2\left (\frac {\sqrt [4]{b} (c+d x)}{\sqrt [4]{b} c+\sqrt [4]{-a} d}\right )+12 a d^4 \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+\sqrt [4]{b} c}\right )+12 a d^4 \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}-i \sqrt [4]{b} x\right )}{\sqrt [4]{-a} d+i \sqrt [4]{b} c}\right )+12 a d^4 \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}+i \sqrt [4]{b} x\right )}{\sqrt [4]{-a} d-i \sqrt [4]{b} c}\right )+12 a d^4 \log (c+d x) \log \left (\frac {d \left (\sqrt [4]{-a}+\sqrt [4]{b} x\right )}{\sqrt [4]{-a} d-\sqrt [4]{b} c}\right )+12 b c^4 \log (c+d x)-12 b c^3 d x+6 b c^2 d^2 x^2-12 b d^4 x^4 \log (c+d x)-4 b c d^3 x^3+3 b d^4 x^4}{48 b^2 d^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/48*(-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*Lo
g[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)])/(b^2*d^4)

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fricas [F]  time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {x^{7} \log \left (d x + c\right )}{b x^{4} + a}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{7} \log \left (d x + c\right )}{b x^{4} + a}\,{d x} \]

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)

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maple [C]  time = 0.26, size = 175, normalized size = 0.35 \[ \frac {x^{4} \ln \left (d x +c \right )}{4 b}-\frac {x^{4}}{16 b}+\frac {c \,x^{3}}{12 b d}-\frac {c^{2} x^{2}}{8 b \,d^{2}}-\frac {a \left (\ln \left (\frac {-d x +\RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )-c}{\RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}\right ) \ln \left (d x +c \right )+\dilog \left (\frac {-d x +\RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )-c}{\RootOf \left (\textit {\_Z}^{4} b -4 b c \,\textit {\_Z}^{3}+6 b \,c^{2} \textit {\_Z}^{2}-4 b \,c^{3} \textit {\_Z} +a \,d^{4}+b \,c^{4}\right )}\right )\right )}{4 b^{2}}-\frac {c^{4} \ln \left (d x +c \right )}{4 b \,d^{4}}+\frac {c^{3} x}{4 b \,d^{3}}+\frac {25 c^{4}}{48 b \,d^{4}} \]

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*c^3*x/b/d^3+25/48/
d^4/b*c^4-1/4/b^2*sum(ln((-d*x+_R1-c)/_R1)*ln(d*x+c)+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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{7} \log \left (d x + c\right )}{b x^{4} + a}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^7\,\ln \left (c+d\,x\right )}{b\,x^4+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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