3.97 \(\int \frac{\log ^3(c (a+b x^2)^p)}{x^7} \, dx\)

Optimal. Leaf size=352 \[ -\frac{b^3 p^2 \text{PolyLog}\left (2,\frac{a}{a+b x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^3}+\frac{b^3 p^3 \text{PolyLog}\left (2,\frac{a}{a+b x^2}\right )}{2 a^3}-\frac{b^3 p^3 \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right )}{a^3}-\frac{b^3 p^3 \text{PolyLog}\left (3,\frac{a}{a+b x^2}\right )}{a^3}-\frac{b^3 p^2 \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^3}-\frac{b^3 p^2 \log \left (1-\frac{a}{a+b x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3}-\frac{b^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}+\frac{b^3 p \log \left (1-\frac{a}{a+b x^2}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3}+\frac{b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}+\frac{b^3 p^3 \log (x)}{a^3}-\frac{b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.744987, antiderivative size = 331, normalized size of antiderivative = 0.94, number of steps used = 22, number of rules used = 16, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.889, Rules used = {2454, 2398, 2411, 2347, 2344, 2302, 30, 2317, 2374, 6589, 2318, 2391, 2319, 2301, 2314, 31} \[ \frac{b^3 p^2 \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^3}-\frac{3 b^3 p^3 \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right )}{2 a^3}-\frac{b^3 p^3 \text{PolyLog}\left (3,\frac{b x^2}{a}+1\right )}{a^3}-\frac{3 b^3 p^2 \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3}-\frac{b^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac{b^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 a^3}+\frac{b^3 p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^3}+\frac{b^3 p \log \left (-\frac{b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3}+\frac{b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}+\frac{b^3 p^3 \log (x)}{a^3}-\frac{b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]^3/x^7,x]

[Out]

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

Rule 2454

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 2398

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 2411

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 2347

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 2344

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

Rule 2302

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2317

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 2374

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 6589

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]

Rule 2318

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 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 2319

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2314

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

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

\begin{align*} \int \frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log ^3\left (c (a+b x)^p\right )}{x^4} \, dx,x,x^2\right )\\ &=-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{1}{2} (b p) \operatorname{Subst}\left (\int \frac{\log ^2\left (c (a+b x)^p\right )}{x^3 (a+b x)} \, dx,x,x^2\right )\\ &=-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{1}{2} p \operatorname{Subst}\left (\int \frac{\log ^2\left (c x^p\right )}{x \left (-\frac{a}{b}+\frac{x}{b}\right )^3} \, dx,x,a+b x^2\right )\\ &=-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{p \operatorname{Subst}\left (\int \frac{\log ^2\left (c x^p\right )}{\left (-\frac{a}{b}+\frac{x}{b}\right )^3} \, dx,x,a+b x^2\right )}{2 a}-\frac{(b p) \operatorname{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 )}{2 a}\\ &=-\frac{b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{(b p) \operatorname{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 )}{2 a^2}+\frac{\left (b^2 p\right ) \operatorname{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 )}{2 a^2}+\frac{\left (b p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{x \left (-\frac{a}{b}+\frac{x}{b}\right )^2} \, dx,x,a+b x^2\right )}{2 a}\\ &=-\frac{b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}+\frac{b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac{\left (b^2 p\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (c x^p\right )}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x^2\right )}{2 a^3}-\frac{\left (b^3 p\right ) \operatorname{Subst}\left (\int \frac{\log ^2\left (c x^p\right )}{x} \, dx,x,a+b x^2\right )}{2 a^3}+\frac{\left (b p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{\left (-\frac{a}{b}+\frac{x}{b}\right )^2} \, dx,x,a+b x^2\right )}{2 a^2}-\frac{\left (b^2 p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x^2\right )}{a^3}-\frac{\left (b^2 p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{x \left (-\frac{a}{b}+\frac{x}{b}\right )} \, dx,x,a+b x^2\right )}{2 a^2}\\ &=-\frac{b^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac{b^3 p^2 \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a^3}-\frac{b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}+\frac{b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}+\frac{b^3 p \log \left (-\frac{b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{b^3 \operatorname{Subst}\left (\int x^2 \, dx,x,\log \left (c \left (a+b x^2\right )^p\right )\right )}{2 a^3}-\frac{\left (b^2 p^2\right ) \operatorname{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^3}+\frac{\left (b^3 p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{x} \, dx,x,a+b x^2\right )}{2 a^3}-\frac{\left (b^3 p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right ) \log \left (1-\frac{x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{a^3}+\frac{\left (b^2 p^3\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x^2\right )}{2 a^3}+\frac{\left (b^3 p^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{a^3}\\ &=\frac{b^3 p^3 \log (x)}{a^3}-\frac{b^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac{3 b^3 p^2 \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3}+\frac{b^3 p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^3}-\frac{b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}+\frac{b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}+\frac{b^3 p \log \left (-\frac{b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3}-\frac{b^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 a^3}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{b^3 p^3 \text{Li}_2\left (1+\frac{b x^2}{a}\right )}{a^3}+\frac{b^3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text{Li}_2\left (1+\frac{b x^2}{a}\right )}{a^3}+\frac{\left (b^3 p^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{2 a^3}-\frac{\left (b^3 p^3\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{a^3}\\ &=\frac{b^3 p^3 \log (x)}{a^3}-\frac{b^2 p^2 \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}-\frac{3 b^3 p^2 \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^3}+\frac{b^3 p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^3}-\frac{b p \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a x^4}+\frac{b^2 p \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3 x^2}+\frac{b^3 p \log \left (-\frac{b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a^3}-\frac{b^3 \log ^3\left (c \left (a+b x^2\right )^p\right )}{6 a^3}-\frac{\log ^3\left (c \left (a+b x^2\right )^p\right )}{6 x^6}-\frac{3 b^3 p^3 \text{Li}_2\left (1+\frac{b x^2}{a}\right )}{2 a^3}+\frac{b^3 p^2 \log \left (c \left (a+b x^2\right )^p\right ) \text{Li}_2\left (1+\frac{b x^2}{a}\right )}{a^3}-\frac{b^3 p^3 \text{Li}_3\left (1+\frac{b x^2}{a}\right )}{a^3}\\ \end{align*}

Mathematica [A]  time = 0.393036, size = 571, normalized size = 1.62 \[ -\frac{6 b^3 p^2 x^6 \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right ) \left (3 p-2 \log \left (c \left (a+b x^2\right )^p\right )\right )+12 b^3 p^3 x^6 \text{PolyLog}\left (3,\frac{b x^2}{a}+1\right )+3 a^2 b p x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )+2 a^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-6 b^3 p^2 x^6 \log ^2\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+36 b^3 p^2 x^6 \log (x) \log \left (c \left (a+b x^2\right )^p\right )-18 b^3 p^2 x^6 \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+24 b^3 p^2 x^6 \log (x) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-12 b^3 p^2 x^6 \log \left (-\frac{b x^2}{a}\right ) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+6 a b^2 p^2 x^4 \log \left (c \left (a+b x^2\right )^p\right )-12 b^3 p x^6 \log (x) \log ^2\left (c \left (a+b x^2\right )^p\right )+6 b^3 p x^6 \log \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )-6 a b^2 p x^4 \log ^2\left (c \left (a+b x^2\right )^p\right )+2 b^3 p^3 x^6 \log ^3\left (a+b x^2\right )+9 b^3 p^3 x^6 \log ^2\left (a+b x^2\right )-12 b^3 p^3 x^6 \log (x) \log ^2\left (a+b x^2\right )+6 b^3 p^3 x^6 \log \left (-\frac{b x^2}{a}\right ) \log ^2\left (a+b x^2\right )-6 b^3 p^3 x^6 \log \left (-\frac{b x^2}{a}\right )+6 b^3 p^3 x^6 \log \left (a+b x^2\right )-36 b^3 p^3 x^6 \log (x) \log \left (a+b x^2\right )+18 b^3 p^3 x^6 \log \left (-\frac{b x^2}{a}\right ) \log \left (a+b x^2\right )}{12 a^3 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p]^3/x^7,x]

[Out]

-(-6*b^3*p^3*x^6*Log[-((b*x^2)/a)] + 6*b^3*p^3*x^6*Log[a + b*x^2] - 36*b^3*p^3*x^6*Log[x]*Log[a + b*x^2] + 18*
b^3*p^3*x^6*Log[-((b*x^2)/a)]*Log[a + b*x^2] + 9*b^3*p^3*x^6*Log[a + b*x^2]^2 - 12*b^3*p^3*x^6*Log[x]*Log[a +
b*x^2]^2 + 6*b^3*p^3*x^6*Log[-((b*x^2)/a)]*Log[a + b*x^2]^2 + 2*b^3*p^3*x^6*Log[a + b*x^2]^3 + 6*a*b^2*p^2*x^4
*Log[c*(a + b*x^2)^p] + 36*b^3*p^2*x^6*Log[x]*Log[c*(a + b*x^2)^p] - 18*b^3*p^2*x^6*Log[a + b*x^2]*Log[c*(a +
b*x^2)^p] + 24*b^3*p^2*x^6*Log[x]*Log[a + b*x^2]*Log[c*(a + b*x^2)^p] - 12*b^3*p^2*x^6*Log[-((b*x^2)/a)]*Log[a
+ b*x^2]*Log[c*(a + b*x^2)^p] - 6*b^3*p^2*x^6*Log[a + b*x^2]^2*Log[c*(a + b*x^2)^p] + 3*a^2*b*p*x^2*Log[c*(a
+ b*x^2)^p]^2 - 6*a*b^2*p*x^4*Log[c*(a + b*x^2)^p]^2 - 12*b^3*p*x^6*Log[x]*Log[c*(a + b*x^2)^p]^2 + 6*b^3*p*x^
6*Log[a + b*x^2]*Log[c*(a + b*x^2)^p]^2 + 2*a^3*Log[c*(a + b*x^2)^p]^3 + 6*b^3*p^2*x^6*(3*p - 2*Log[c*(a + b*x
^2)^p])*PolyLog[2, 1 + (b*x^2)/a] + 12*b^3*p^3*x^6*PolyLog[3, 1 + (b*x^2)/a])/(12*a^3*x^6)

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Maple [F]  time = 1.332, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}}{{x}^{7}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^2+a)^p)^3/x^7,x)

[Out]

int(ln(c*(b*x^2+a)^p)^3/x^7,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{\log \left ({\left (b x^{2} + a\right )}^{p}\right )^{3}}{6 \, x^{6}} + \int \frac{b x^{2} \log \left (c\right )^{3} + a \log \left (c\right )^{3} +{\left (b{\left (p + 3 \, \log \left (c\right )\right )} x^{2} + 3 \, a \log \left (c\right )\right )} \log \left ({\left (b x^{2} + a\right )}^{p}\right )^{2} + 3 \,{\left (b x^{2} \log \left (c\right )^{2} + a \log \left (c\right )^{2}\right )} \log \left ({\left (b x^{2} + a\right )}^{p}\right )}{b x^{9} + a x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)^3/x^7,x, algorithm="maxima")

[Out]

-1/6*log((b*x^2 + a)^p)^3/x^6 + integrate((b*x^2*log(c)^3 + a*log(c)^3 + (b*(p + 3*log(c))*x^2 + 3*a*log(c))*l
og((b*x^2 + a)^p)^2 + 3*(b*x^2*log(c)^2 + a*log(c)^2)*log((b*x^2 + a)^p))/(b*x^9 + a*x^7), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{7}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)^3/x^7,x, algorithm="fricas")

[Out]

integral(log((b*x^2 + a)^p*c)^3/x^7, 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(ln(c*(b*x**2+a)**p)**3/x**7,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3}}{x^{7}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)^3/x^7,x, algorithm="giac")

[Out]

integrate(log((b*x^2 + a)^p*c)^3/x^7, x)