3.3.86 \(\int \frac {\log (c+d x)}{x^4 (a+b x^3)} \, dx\) [286]

Optimal. Leaf size=414 \[ -\frac {d}{6 a c x^2}+\frac {d^2}{3 a c^2 x}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac {b \text {Li}_2\left (1+\frac {d x}{c}\right )}{a^2} \]

[Out]

-1/6*d/a/c/x^2+1/3*d^2/a/c^2/x+1/3*d^3*ln(x)/a/c^3-1/3*d^3*ln(d*x+c)/a/c^3-1/3*ln(d*x+c)/a/x^3-b*ln(-d*x/c)*ln
(d*x+c)/a^2+1/3*b*ln(-d*(a^(1/3)+b^(1/3)*x)/(b^(1/3)*c-a^(1/3)*d))*ln(d*x+c)/a^2+1/3*b*ln(-d*((-1)^(2/3)*a^(1/
3)+b^(1/3)*x)/(b^(1/3)*c-(-1)^(2/3)*a^(1/3)*d))*ln(d*x+c)/a^2+1/3*b*ln((-1)^(1/3)*d*(a^(1/3)+(-1)^(2/3)*b^(1/3
)*x)/(b^(1/3)*c+(-1)^(1/3)*a^(1/3)*d))*ln(d*x+c)/a^2+1/3*b*polylog(2,b^(1/3)*(d*x+c)/(b^(1/3)*c-a^(1/3)*d))/a^
2+1/3*b*polylog(2,b^(1/3)*(d*x+c)/(b^(1/3)*c+(-1)^(1/3)*a^(1/3)*d))/a^2+1/3*b*polylog(2,b^(1/3)*(d*x+c)/(b^(1/
3)*c-(-1)^(2/3)*a^(1/3)*d))/a^2-b*polylog(2,1+d*x/c)/a^2

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Rubi [A]
time = 0.36, antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 9, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.474, Rules used = {272, 46, 2463, 2442, 2441, 2352, 266, 2440, 2438} \begin {gather*} \frac {b \text {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a^2}+\frac {b \text {PolyLog}\left (2,\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac {b \text {PolyLog}\left (2,\frac {d x}{c}+1\right )}{a^2}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log (c+d x) \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \log (c+d x) \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \log (c+d x) \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 a^2}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}+\frac {d^2}{3 a c^2 x}-\frac {\log (c+d x)}{3 a x^3}-\frac {d}{6 a c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c + d*x]/(x^4*(a + b*x^3)),x]

[Out]

-1/6*d/(a*c*x^2) + d^2/(3*a*c^2*x) + (d^3*Log[x])/(3*a*c^3) - (d^3*Log[c + d*x])/(3*a*c^3) - Log[c + d*x]/(3*a
*x^3) - (b*Log[-((d*x)/c)]*Log[c + d*x])/a^2 + (b*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Lo
g[c + d*x])/(3*a^2) + (b*Log[-((d*((-1)^(2/3)*a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d))]*Log[c
+ d*x])/(3*a^2) + (b*Log[((-1)^(1/3)*d*(a^(1/3) + (-1)^(2/3)*b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*
Log[c + d*x])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, (b
^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)])/(3*a^2) + (b*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c
- (-1)^(2/3)*a^(1/3)*d)])/(3*a^2) - (b*PolyLog[2, 1 + (d*x)/c])/a^2

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[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 2352

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(-e^(-1))*PolyLog[2, 1 - c*x], x] /; FreeQ[{c, d, e
}, x] && EqQ[e + c*d, 0]

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*} \int \frac {\log (c+d x)}{x^4 \left (a+b x^3\right )} \, dx &=\int \left (\frac {\log (c+d x)}{a x^4}-\frac {b \log (c+d x)}{a^2 x}+\frac {b^2 x^2 \log (c+d x)}{a^2 \left (a+b x^3\right )}\right ) \, dx\\ &=\frac {\int \frac {\log (c+d x)}{x^4} \, dx}{a}-\frac {b \int \frac {\log (c+d x)}{x} \, dx}{a^2}+\frac {b^2 \int \frac {x^2 \log (c+d x)}{a+b x^3} \, dx}{a^2}\\ &=-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b^2 \int \left (\frac {\log (c+d x)}{3 b^{2/3} \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\log (c+d x)}{3 b^{2/3} \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}+\frac {\log (c+d x)}{3 b^{2/3} \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}\right ) \, dx}{a^2}+\frac {d \int \frac {1}{x^3 (c+d x)} \, dx}{3 a}+\frac {(b d) \int \frac {\log \left (-\frac {d x}{c}\right )}{c+d x} \, dx}{a^2}\\ &=-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}-\frac {b \text {Li}_2\left (1+\frac {d x}{c}\right )}{a^2}+\frac {b^{4/3} \int \frac {\log (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac {b^{4/3} \int \frac {\log (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac {b^{4/3} \int \frac {\log (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^2}+\frac {d \int \left (\frac {1}{c x^3}-\frac {d}{c^2 x^2}+\frac {d^2}{c^3 x}-\frac {d^3}{c^3 (c+d x)}\right ) \, dx}{3 a}\\ &=-\frac {d}{6 a c x^2}+\frac {d^2}{3 a c^2 x}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}-\frac {b \text {Li}_2\left (1+\frac {d x}{c}\right )}{a^2}-\frac {(b d) \int \frac {\log \left (\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 a^2}-\frac {(b d) \int \frac {\log \left (\frac {d \left (-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c-\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 a^2}-\frac {(b d) \int \frac {\log \left (\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{-\sqrt [3]{b} c+(-1)^{2/3} \sqrt [3]{a} d}\right )}{c+d x} \, dx}{3 a^2}\\ &=-\frac {d}{6 a c x^2}+\frac {d^2}{3 a c^2 x}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}-\frac {b \text {Li}_2\left (1+\frac {d x}{c}\right )}{a^2}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} c+\sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^2}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} c-\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^2}-\frac {b \text {Subst}\left (\int \frac {\log \left (1+\frac {\sqrt [3]{b} x}{-\sqrt [3]{b} c+(-1)^{2/3} \sqrt [3]{a} d}\right )}{x} \, dx,x,c+d x\right )}{3 a^2}\\ &=-\frac {d}{6 a c x^2}+\frac {d^2}{3 a c^2 x}+\frac {d^3 \log (x)}{3 a c^3}-\frac {d^3 \log (c+d x)}{3 a c^3}-\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (-\frac {d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2}-\frac {b \text {Li}_2\left (1+\frac {d x}{c}\right )}{a^2}\\ \end {align*}

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Mathematica [A]
time = 0.11, size = 405, normalized size = 0.98 \begin {gather*} -\frac {\log (c+d x)}{3 a x^3}-\frac {b \log \left (-\frac {d x}{c}\right ) \log (c+d x)}{a^2}+\frac {b \log \left (-\frac {d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (-\frac {(-1)^{2/3} d \left (\sqrt [3]{a}-\sqrt [3]{-1} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}+\frac {b \log \left (\frac {\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right ) \log (c+d x)}{3 a^2}-\frac {d \left (\frac {1}{c x^2}-\frac {2 d}{c^2 x}-\frac {2 d^2 \log (x)}{c^3}+\frac {2 d^2 \log (c+d x)}{c^3}\right )}{6 a}-\frac {b \text {Li}_2\left (\frac {c+d x}{c}\right )}{a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 a^2}+\frac {b \text {Li}_2\left (\frac {\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c + d*x]/(x^4*(a + b*x^3)),x]

[Out]

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

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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.77, size = 199, normalized size = 0.48

method	result	size
risch	\(-\frac {d}{6 a c \,x^{2}}+\frac {d^{2}}{3 a \,c^{2} x}+\frac {d^{3} \ln \left (-d x \right )}{3 a \,c^{3}}-\frac {d^{3} \ln \left (d x +c \right )}{3 a \,c^{3}}-\frac {\ln \left (d x +c \right )}{3 x^{3} a}+\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 a^{2}}-\frac {b \ln \left (-\frac {x d}{c}\right ) \ln \left (d x +c \right )}{a^{2}}-\frac {b \dilog \left (-\frac {x d}{c}\right )}{a^{2}}\)	\(186\)
derivativedivides	\(d^{3} \left (\frac {-\frac {1}{6 c \,d^{2} x^{2}}+\frac {1}{3 c^{2} d x}+\frac {\ln \left (-d x \right )}{3 c^{3}}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (3 c^{2}-3 c \left (d x +c \right )+\left (d x +c \right )^{2}\right )}{3 c^{3} d^{3} x^{3}}}{a}+\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 a^{2} d^{3}}-\frac {\left (\dilog \left (-\frac {x d}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {x d}{c}\right )\right ) b}{a^{2} d^{3}}\right )\)	\(199\)
default	\(d^{3} \left (\frac {-\frac {1}{6 c \,d^{2} x^{2}}+\frac {1}{3 c^{2} d x}+\frac {\ln \left (-d x \right )}{3 c^{3}}-\frac {\ln \left (d x +c \right ) \left (d x +c \right ) \left (3 c^{2}-3 c \left (d x +c \right )+\left (d x +c \right )^{2}\right )}{3 c^{3} d^{3} x^{3}}}{a}+\frac {b \left (\munderset {\textit {\_R1} =\RootOf \left (b \,\textit {\_Z}^{3}-3 c b \,\textit {\_Z}^{2}+3 b \,c^{2} \textit {\_Z} +a \,d^{3}-b \,c^{3}\right )}{\sum }\left (\ln \left (d x +c \right ) \ln \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )+\dilog \left (\frac {-d x +\textit {\_R1} -c}{\textit {\_R1}}\right )\right )\right )}{3 a^{2} d^{3}}-\frac {\left (\dilog \left (-\frac {x d}{c}\right )+\ln \left (d x +c \right ) \ln \left (-\frac {x d}{c}\right )\right ) b}{a^{2} d^{3}}\right )\)	\(199\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(d*x+c)/x^4/(b*x^3+a),x,method=_RETURNVERBOSE)

[Out]

d^3*((-1/6/c/d^2/x^2+1/3/c^2/d/x+1/3/c^3*ln(-d*x)-1/3*ln(d*x+c)*(d*x+c)*(3*c^2-3*c*(d*x+c)+(d*x+c)^2)/c^3/d^3/
x^3)/a+1/3*b*sum(ln(d*x+c)*ln((-d*x+_R1-c)/_R1)+dilog((-d*x+_R1-c)/_R1),_R1=RootOf(_Z^3*b-3*_Z^2*b*c+3*_Z*b*c^
2+a*d^3-b*c^3))/a^2/d^3-(dilog(-x*d/c)+ln(d*x+c)*ln(-x*d/c))/a^2/d^3*b)

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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(log(d*x+c)/x^4/(b*x^3+a),x, algorithm="maxima")

[Out]

integrate(log(d*x + c)/((b*x^3 + a)*x^4), x)

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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(log(d*x+c)/x^4/(b*x^3+a),x, algorithm="fricas")

[Out]

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

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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(ln(d*x+c)/x**4/(b*x**3+a),x)

[Out]

Timed out

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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(log(d*x+c)/x^4/(b*x^3+a),x, algorithm="giac")

[Out]

integrate(log(d*x + c)/((b*x^3 + a)*x^4), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\ln \left (c+d\,x\right )}{x^4\,\left (b\,x^3+a\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c + d*x)/(x^4*(a + b*x^3)),x)

[Out]

int(log(c + d*x)/(x^4*(a + b*x^3)), x)

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