3.283 \(\int \frac{x^5 \log (c+d x)}{a+b x^3} \, dx\)

Optimal. Leaf size=371 \[ -\frac{a \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac{a \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 b^2}-\frac{a \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^2}-\frac{a \log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac{a \log (c+d x) \log \left (-\frac{d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^2}-\frac{a \log (c+d x) \log \left (\frac{\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 b^2}-\frac{c^2 x}{3 b d^2}+\frac{c^3 \log (c+d x)}{3 b d^3}+\frac{c x^2}{6 b d}+\frac{x^3 \log (c+d x)}{3 b}-\frac{x^3}{9 b} \]

[Out]

-(c^2*x)/(3*b*d^2) + (c*x^2)/(6*b*d) - x^3/(9*b) + (c^3*Log[c + d*x])/(3*b*d^3) + (x^3*Log[c + d*x])/(3*b) - (
a*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b^2) - (a*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*b^2) - (a*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*b^2) - (a*PolyLog[2, (b^(1/3)*(
c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*b^2) - (a*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3
)*d)])/(3*b^2) - (a*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(3*b^2)

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Rubi [A]  time = 0.595824, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ -\frac{a \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac{a \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 b^2}-\frac{a \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^2}-\frac{a \log (c+d x) \log \left (-\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac{a \log (c+d x) \log \left (-\frac{d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^2}-\frac{a \log (c+d x) \log \left (\frac{\sqrt [3]{-1} d \left (\sqrt [3]{a}+(-1)^{2/3} \sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )}{3 b^2}-\frac{c^2 x}{3 b d^2}+\frac{c^3 \log (c+d x)}{3 b d^3}+\frac{c x^2}{6 b d}+\frac{x^3 \log (c+d x)}{3 b}-\frac{x^3}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^5*Log[c + d*x])/(a + b*x^3),x]

[Out]

-(c^2*x)/(3*b*d^2) + (c*x^2)/(6*b*d) - x^3/(9*b) + (c^3*Log[c + d*x])/(3*b*d^3) + (x^3*Log[c + d*x])/(3*b) - (
a*Log[-((d*(a^(1/3) + b^(1/3)*x))/(b^(1/3)*c - a^(1/3)*d))]*Log[c + d*x])/(3*b^2) - (a*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*b^2) - (a*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*b^2) - (a*PolyLog[2, (b^(1/3)*(
c + d*x))/(b^(1/3)*c - a^(1/3)*d)])/(3*b^2) - (a*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3
)*d)])/(3*b^2) - (a*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(3*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^5 \log (c+d x)}{a+b x^3} \, dx &=\int \left (\frac{x^2 \log (c+d x)}{b}-\frac{a x^2 \log (c+d x)}{b \left (a+b x^3\right )}\right ) \, dx\\ &=\frac{\int x^2 \log (c+d x) \, dx}{b}-\frac{a \int \frac{x^2 \log (c+d x)}{a+b x^3} \, dx}{b}\\ &=\frac{x^3 \log (c+d x)}{3 b}-\frac{a \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}{b}-\frac{d \int \frac{x^3}{c+d x} \, dx}{3 b}\\ &=\frac{x^3 \log (c+d x)}{3 b}-\frac{a \int \frac{\log (c+d x)}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{5/3}}-\frac{a \int \frac{\log (c+d x)}{-\sqrt [3]{-1} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{5/3}}-\frac{a \int \frac{\log (c+d x)}{(-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 b^{5/3}}-\frac{d \int \left (\frac{c^2}{d^3}-\frac{c x}{d^2}+\frac{x^2}{d}-\frac{c^3}{d^3 (c+d x)}\right ) \, dx}{3 b}\\ &=-\frac{c^2 x}{3 b d^2}+\frac{c x^2}{6 b d}-\frac{x^3}{9 b}+\frac{c^3 \log (c+d x)}{3 b d^3}+\frac{x^3 \log (c+d x)}{3 b}-\frac{a \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 b^2}-\frac{a \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 b^2}-\frac{a \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 b^2}+\frac{(a 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 b^2}+\frac{(a 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 b^2}+\frac{(a 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 b^2}\\ &=-\frac{c^2 x}{3 b d^2}+\frac{c x^2}{6 b d}-\frac{x^3}{9 b}+\frac{c^3 \log (c+d x)}{3 b d^3}+\frac{x^3 \log (c+d x)}{3 b}-\frac{a \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 b^2}-\frac{a \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 b^2}-\frac{a \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 b^2}+\frac{a \operatorname{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 b^2}+\frac{a \operatorname{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 b^2}+\frac{a \operatorname{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 b^2}\\ &=-\frac{c^2 x}{3 b d^2}+\frac{c x^2}{6 b d}-\frac{x^3}{9 b}+\frac{c^3 \log (c+d x)}{3 b d^3}+\frac{x^3 \log (c+d x)}{3 b}-\frac{a \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 b^2}-\frac{a \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 b^2}-\frac{a \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 b^2}-\frac{a \text{Li}_2\left (\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )}{3 b^2}-\frac{a \text{Li}_2\left (\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c+\sqrt [3]{-1} \sqrt [3]{a} d}\right )}{3 b^2}-\frac{a \text{Li}_2\left (\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )}{3 b^2}\\ \end{align*}

Mathematica [A]  time = 0.327861, size = 345, normalized size = 0.93 \[ -\frac{6 a d^3 \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-\sqrt [3]{a} d}\right )+6 a d^3 \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )+6 a d^3 \text{PolyLog}\left (2,\frac{\sqrt [3]{b} (c+d x)}{\sqrt [3]{b} c-(-1)^{2/3} \sqrt [3]{a} d}\right )+6 a d^3 \log (c+d x) \log \left (\frac{d \left (\sqrt [3]{-1} \sqrt [3]{a}-\sqrt [3]{b} x\right )}{\sqrt [3]{-1} \sqrt [3]{a} d+\sqrt [3]{b} c}\right )+6 a d^3 \log (c+d x) \log \left (\frac{d \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{a} d-\sqrt [3]{b} c}\right )+6 a d^3 \log (c+d x) \log \left (\frac{d \left ((-1)^{2/3} \sqrt [3]{a}+\sqrt [3]{b} x\right )}{(-1)^{2/3} \sqrt [3]{a} d-\sqrt [3]{b} c}\right )+6 b c^2 d x-6 b c^3 \log (c+d x)-3 b c d^2 x^2-6 b d^3 x^3 \log (c+d x)+2 b d^3 x^3}{18 b^2 d^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^5*Log[c + d*x])/(a + b*x^3),x]

[Out]

-(6*b*c^2*d*x - 3*b*c*d^2*x^2 + 2*b*d^3*x^3 - 6*b*c^3*Log[c + d*x] - 6*b*d^3*x^3*Log[c + d*x] + 6*a*d^3*Log[(d
*((-1)^(1/3)*a^(1/3) - b^(1/3)*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)]*Log[c + d*x] + 6*a*d^3*Log[(d*(a^(1/3)
+ b^(1/3)*x))/(-(b^(1/3)*c) + a^(1/3)*d)]*Log[c + d*x] + 6*a*d^3*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] + 6*a*d^3*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c - a^(1/3)*
d)] + 6*a*d^3*PolyLog[2, (b^(1/3)*(c + d*x))/(b^(1/3)*c + (-1)^(1/3)*a^(1/3)*d)] + 6*a*d^3*PolyLog[2, (b^(1/3)
*(c + d*x))/(b^(1/3)*c - (-1)^(2/3)*a^(1/3)*d)])/(18*b^2*d^3)

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Maple [C]  time = 0.39, size = 153, normalized size = 0.4 \begin{align*}{\frac{{x}^{3}\ln \left ( dx+c \right ) }{3\,b}}+{\frac{{c}^{3}\ln \left ( dx+c \right ) }{3\,b{d}^{3}}}-{\frac{{x}^{3}}{9\,b}}+{\frac{c{x}^{2}}{6\,bd}}-{\frac{{c}^{2}x}{3\,b{d}^{2}}}-{\frac{11\,{c}^{3}}{18\,b{d}^{3}}}-{\frac{a}{3\,{b}^{2}}\sum _{{\it \_R1}={\it RootOf} \left ( b{{\it \_Z}}^{3}-3\,bc{{\it \_Z}}^{2}+3\,{c}^{2}b{\it \_Z}+a{d}^{3}-b{c}^{3} \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^5*ln(d*x+c)/(b*x^3+a),x)

[Out]

1/3*x^3*ln(d*x+c)/b+1/3*c^3*ln(d*x+c)/b/d^3-1/9*x^3/b+1/6*c*x^2/b/d-1/3/b/d^2*c^2*x-11/18/b/d^3*c^3-1/3*a/b^2*
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
))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \log \left (d x + c\right )}{b x^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*log(d*x+c)/(b*x^3+a),x, algorithm="maxima")

[Out]

integrate(x^5*log(d*x + c)/(b*x^3 + a), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{x^{5} \log \left (d x + c\right )}{b x^{3} + a}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*log(d*x+c)/(b*x^3+a),x, algorithm="fricas")

[Out]

integral(x^5*log(d*x + c)/(b*x^3 + 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**5*ln(d*x+c)/(b*x**3+a),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5} \log \left (d x + c\right )}{b x^{3} + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*log(d*x+c)/(b*x^3+a),x, algorithm="giac")

[Out]

integrate(x^5*log(d*x + c)/(b*x^3 + a), x)