\(\int \frac {\log ^3(c (a+b x^2)^p)}{x^5} \, dx\) [96]

Optimal result

Integrand size = 18, antiderivative size = 219 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\frac {3 b^2 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}-\frac {3 b p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {3 b^2 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{4 a^2}+\frac {3 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \operatorname {PolyLog}\left (2,\frac {a}{a+b x^2}\right )}{2 a^2}+\frac {3 b^2 p^3 \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{2 a^2}+\frac {3 b^2 p^3 \operatorname {PolyLog}\left (3,\frac {a}{a+b x^2}\right )}{2 a^2} \]

[Out]

3/2*b^2*p^2*ln(-b*x^2/a)*ln(c*(b*x^2+a)^p)/a^2-3/4*b*p*(b*x^2+a)*ln(c*(b*x^2+a)^p)^2/a^2/x^2-1/4*ln(c*(b*x^2+a
)^p)^3/x^4-3/4*b^2*p*ln(c*(b*x^2+a)^p)^2*ln(1-a/(b*x^2+a))/a^2+3/2*b^2*p^2*ln(c*(b*x^2+a)^p)*polylog(2,a/(b*x^
2+a))/a^2+3/2*b^2*p^3*polylog(2,1+b*x^2/a)/a^2+3/2*b^2*p^3*polylog(3,a/(b*x^2+a))/a^2

Rubi [A] (verified)

Time = 0.25 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {2504, 2445, 2458, 2389, 2379, 2421, 6724, 2355, 2354, 2438} \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\frac {3 b^2 p^2 \operatorname {PolyLog}\left (2,\frac {a}{b x^2+a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}+\frac {3 b^2 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}-\frac {3 b^2 p \log \left (1-\frac {a}{a+b x^2}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2}+\frac {3 b^2 p^3 \operatorname {PolyLog}\left (2,\frac {b x^2}{a}+1\right )}{2 a^2}+\frac {3 b^2 p^3 \operatorname {PolyLog}\left (3,\frac {a}{b x^2+a}\right )}{2 a^2}-\frac {3 b p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4} \]

[In]

Int[Log[c*(a + b*x^2)^p]^3/x^5,x]

[Out]

(3*b^2*p^2*Log[-((b*x^2)/a)]*Log[c*(a + b*x^2)^p])/(2*a^2) - (3*b*p*(a + b*x^2)*Log[c*(a + b*x^2)^p]^2)/(4*a^2
*x^2) - Log[c*(a + b*x^2)^p]^3/(4*x^4) - (3*b^2*p*Log[c*(a + b*x^2)^p]^2*Log[1 - a/(a + b*x^2)])/(4*a^2) + (3*
b^2*p^2*Log[c*(a + b*x^2)^p]*PolyLog[2, a/(a + b*x^2)])/(2*a^2) + (3*b^2*p^3*PolyLog[2, 1 + (b*x^2)/a])/(2*a^2
) + (3*b^2*p^3*PolyLog[3, a/(a + b*x^2)])/(2*a^2)

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2355

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Symbol] :> Simp[x*((a + b*Log[c*x^n])
^p/(d*(d + e*x))), x] - Dist[b*n*(p/d), Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d,
e, n, p}, x] && GtQ[p, 0]

Rule 2379

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r_.))), x_Symbol] :> Simp[(-Log[1 +
d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)), x] + Dist[b*n*(p/(d*r)), Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^
(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c*x^n])^p/m), x] + Dist[b*n*(p/m), Int[PolyLog[2, (-d)*f*x^m]*((a + b*L
og[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]

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 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2504

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[I
nt[x^(Simplify[(m + 1)/n] - 1)*(a + b*Log[c*(d + e*x)^p])^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p,
q}, x] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0] || IGtQ[q, 0]) &&  !(EqQ[q, 1] && ILtQ[n, 0] &&
IGtQ[m, 0])

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} \text {Subst}\left (\int \frac {\log ^3\left (c (a+b x)^p\right )}{x^3} \, dx,x,x^2\right ) \\ & = -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {1}{4} (3 b p) \text {Subst}\left (\int \frac {\log ^2\left (c (a+b x)^p\right )}{x^2 (a+b x)} \, dx,x,x^2\right ) \\ & = -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {1}{4} (3 p) \text {Subst}\left (\int \frac {\log ^2\left (c x^p\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x^2\right ) \\ & = -\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac {(3 p) \text {Subst}\left (\int \frac {\log ^2\left (c x^p\right )}{\left (-\frac {a}{b}+\frac {x}{b}\right )^2} \, dx,x,a+b x^2\right )}{4 a}-\frac {(3 b p) \text {Subst}\left (\int \frac {\log ^2\left (c x^p\right )}{x \left (-\frac {a}{b}+\frac {x}{b}\right )} \, dx,x,a+b x^2\right )}{4 a} \\ & = -\frac {3 b p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {3 b^2 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{4 a^2}+\frac {\left (3 b p^2\right ) \text {Subst}\left (\int \frac {\log \left (c x^p\right )}{-\frac {a}{b}+\frac {x}{b}} \, dx,x,a+b x^2\right )}{2 a^2}+\frac {\left (3 b^2 p^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {a}{x}\right ) \log \left (c x^p\right )}{x} \, dx,x,a+b x^2\right )}{2 a^2} \\ & = \frac {3 b^2 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}-\frac {3 b p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {3 b^2 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{4 a^2}+\frac {3 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (\frac {a}{a+b x^2}\right )}{2 a^2}-\frac {\left (3 b^2 p^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{2 a^2}-\frac {\left (3 b^2 p^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {a}{x}\right )}{x} \, dx,x,a+b x^2\right )}{2 a^2} \\ & = \frac {3 b^2 p^2 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}-\frac {3 b p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2 x^2}-\frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac {3 b^2 p \log ^2\left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{4 a^2}+\frac {3 b^2 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text {Li}_2\left (\frac {a}{a+b x^2}\right )}{2 a^2}+\frac {3 b^2 p^3 \text {Li}_2\left (1+\frac {b x^2}{a}\right )}{2 a^2}+\frac {3 b^2 p^3 \text {Li}_3\left (\frac {a}{a+b x^2}\right )}{2 a^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(477\) vs. \(2(219)=438\).

Time = 0.28 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.18 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=-\frac {3 b p \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{4 a x^2}-\frac {3 b^2 p \log (x) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{2 a^2}+\frac {3 b^2 p \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{4 a^2}-\frac {3 p \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{4 x^4}-\frac {\left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^3}{4 x^4}+3 p^2 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (-\frac {\log ^2\left (a+b x^2\right )}{4 x^4}+\frac {b \left (4 b x^2 \log (x)+\log \left (a+b x^2\right ) \left (-2 \left (a+b x^2\right )-2 b x^2 \log \left (-\frac {b x^2}{a}\right )+b x^2 \log \left (a+b x^2\right )\right )-2 b x^2 \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )\right )}{4 a^2 x^2}\right )+\frac {b^2 p^3 \left (\frac {\left (a+b x^2\right ) \left (a \left (3-2 \log \left (a+b x^2\right )\right )+\left (a+b x^2\right ) \left (-3+\log \left (a+b x^2\right )\right )\right ) \log ^2\left (a+b x^2\right )}{b^2 x^4}-3 \left (-2+\log \left (a+b x^2\right )\right ) \log \left (a+b x^2\right ) \log \left (1-\frac {a+b x^2}{a}\right )-6 \left (-1+\log \left (a+b x^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {a+b x^2}{a}\right )+6 \operatorname {PolyLog}\left (3,\frac {a+b x^2}{a}\right )\right )}{4 a^2} \]

[In]

Integrate[Log[c*(a + b*x^2)^p]^3/x^5,x]

[Out]

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

Maple [F]

\[\int \frac {{\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{x^{5}}d x\]

[In]

int(ln(c*(b*x^2+a)^p)^3/x^5,x)

[Out]

int(ln(c*(b*x^2+a)^p)^3/x^5,x)

Fricas [F]

\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{5}} \,d x } \]

[In]

integrate(log(c*(b*x^2+a)^p)^3/x^5,x, algorithm="fricas")

[Out]

integral(log((b*x^2 + a)^p*c)^3/x^5, x)

Sympy [F]

\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}}{x^{5}}\, dx \]

[In]

integrate(ln(c*(b*x**2+a)**p)**3/x**5,x)

[Out]

Integral(log(c*(a + b*x**2)**p)**3/x**5, x)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.23 \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=-\frac {1}{4} \, {\left (\frac {3 \, {\left (\log \left (b x^{2} + a\right )^{2} \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + 2 \, {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right ) \log \left (b x^{2} + a\right ) - 2 \, {\rm Li}_{3}(\frac {b x^{2} + a}{a})\right )} b p^{2}}{a^{2}} - \frac {6 \, {\left (p^{2} - p \log \left (c\right )\right )} {\left (\log \left (b x^{2} + a\right ) \log \left (-\frac {b x^{2} + a}{a} + 1\right ) + {\rm Li}_2\left (\frac {b x^{2} + a}{a}\right )\right )} b}{a^{2}} - \frac {6 \, {\left (2 \, p \log \left (c\right ) - \log \left (c\right )^{2}\right )} b \log \left (x\right )}{a^{2}} - \frac {b p^{2} x^{2} \log \left (b x^{2} + a\right )^{3} - 3 \, {\left ({\left (p^{2} - p \log \left (c\right )\right )} b x^{2} + a p^{2}\right )} \log \left (b x^{2} + a\right )^{2} - 3 \, a \log \left (c\right )^{2} - 3 \, {\left ({\left (2 \, p \log \left (c\right ) - \log \left (c\right )^{2}\right )} b x^{2} + 2 \, a p \log \left (c\right )\right )} \log \left (b x^{2} + a\right )}{a^{2} x^{2}}\right )} b p - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{4 \, x^{4}} \]

[In]

integrate(log(c*(b*x^2+a)^p)^3/x^5,x, algorithm="maxima")

[Out]

-1/4*(3*(log(b*x^2 + a)^2*log(-(b*x^2 + a)/a + 1) + 2*dilog((b*x^2 + a)/a)*log(b*x^2 + a) - 2*polylog(3, (b*x^
2 + a)/a))*b*p^2/a^2 - 6*(p^2 - p*log(c))*(log(b*x^2 + a)*log(-(b*x^2 + a)/a + 1) + dilog((b*x^2 + a)/a))*b/a^
2 - 6*(2*p*log(c) - log(c)^2)*b*log(x)/a^2 - (b*p^2*x^2*log(b*x^2 + a)^3 - 3*((p^2 - p*log(c))*b*x^2 + a*p^2)*
log(b*x^2 + a)^2 - 3*a*log(c)^2 - 3*((2*p*log(c) - log(c)^2)*b*x^2 + 2*a*p*log(c))*log(b*x^2 + a))/(a^2*x^2))*
b*p - 1/4*log((b*x^2 + a)^p*c)^3/x^4

Giac [F]

\[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{5}} \,d x } \]

[In]

integrate(log(c*(b*x^2+a)^p)^3/x^5,x, algorithm="giac")

[Out]

integrate(log((b*x^2 + a)^p*c)^3/x^5, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3}{x^5} \,d x \]

[In]

int(log(c*(a + b*x^2)^p)^3/x^5,x)

[Out]

int(log(c*(a + b*x^2)^p)^3/x^5, x)