3.90 \(\int \frac{\log ^2(c (a+b x^2)^p)}{x^8} \, dx\)

Optimal. Leaf size=338 \[ -\frac{4 i b^{7/2} p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{7 a^{7/2}}-\frac{4 b^3 p \log \left (c \left (a+b x^2\right )^p\right )}{7 a^3 x}+\frac{4 b^2 p \log \left (c \left (a+b x^2\right )^p\right )}{21 a^2 x^3}-\frac{4 b^{7/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{7 a^{7/2}}-\frac{8 b^2 p^2}{105 a^2 x^3}+\frac{64 b^3 p^2}{105 a^3 x}-\frac{4 i b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{7 a^{7/2}}+\frac{184 b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{105 a^{7/2}}-\frac{8 b^{7/2} p^2 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{7 a^{7/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{35 a x^5} \]

[Out]

(-8*b^2*p^2)/(105*a^2*x^3) + (64*b^3*p^2)/(105*a^3*x) + (184*b^(7/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(105*a^(
7/2)) - (((4*I)/7)*b^(7/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/a^(7/2) - (8*b^(7/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt
[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(7*a^(7/2)) - (4*b*p*Log[c*(a + b*x^2)^p])/(35*a*x^5) + (4*b^2*
p*Log[c*(a + b*x^2)^p])/(21*a^2*x^3) - (4*b^3*p*Log[c*(a + b*x^2)^p])/(7*a^3*x) - (4*b^(7/2)*p*ArcTan[(Sqrt[b]
*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(7*a^(7/2)) - Log[c*(a + b*x^2)^p]^2/(7*x^7) - (((4*I)/7)*b^(7/2)*p^2*PolyL
og[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/a^(7/2)

________________________________________________________________________________________

Rubi [A]  time = 0.376727, antiderivative size = 338, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 11, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.611, Rules used = {2457, 2476, 2455, 325, 205, 2470, 12, 4920, 4854, 2402, 2315} \[ -\frac{4 i b^{7/2} p^2 \text{PolyLog}\left (2,1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{7 a^{7/2}}-\frac{4 b^3 p \log \left (c \left (a+b x^2\right )^p\right )}{7 a^3 x}+\frac{4 b^2 p \log \left (c \left (a+b x^2\right )^p\right )}{21 a^2 x^3}-\frac{4 b^{7/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{7 a^{7/2}}-\frac{8 b^2 p^2}{105 a^2 x^3}+\frac{64 b^3 p^2}{105 a^3 x}-\frac{4 i b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{7 a^{7/2}}+\frac{184 b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{105 a^{7/2}}-\frac{8 b^{7/2} p^2 \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{7 a^{7/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{35 a x^5} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]^2/x^8,x]

[Out]

(-8*b^2*p^2)/(105*a^2*x^3) + (64*b^3*p^2)/(105*a^3*x) + (184*b^(7/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(105*a^(
7/2)) - (((4*I)/7)*b^(7/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt[a]]^2)/a^(7/2) - (8*b^(7/2)*p^2*ArcTan[(Sqrt[b]*x)/Sqrt
[a]]*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/(7*a^(7/2)) - (4*b*p*Log[c*(a + b*x^2)^p])/(35*a*x^5) + (4*b^2*
p*Log[c*(a + b*x^2)^p])/(21*a^2*x^3) - (4*b^3*p*Log[c*(a + b*x^2)^p])/(7*a^3*x) - (4*b^(7/2)*p*ArcTan[(Sqrt[b]
*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p])/(7*a^(7/2)) - Log[c*(a + b*x^2)^p]^2/(7*x^7) - (((4*I)/7)*b^(7/2)*p^2*PolyL
og[2, 1 - (2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)])/a^(7/2)

Rule 2457

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

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

Rule 2455

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

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4920

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

Rule 4854

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

Rule 2402

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

Rule 2315

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

Rubi steps

\begin{align*} \int \frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^8} \, dx &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}+\frac{1}{7} (4 b p) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{x^6 \left (a+b x^2\right )} \, dx\\ &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}+\frac{1}{7} (4 b p) \int \left (\frac{\log \left (c \left (a+b x^2\right )^p\right )}{a x^6}-\frac{b \log \left (c \left (a+b x^2\right )^p\right )}{a^2 x^4}+\frac{b^2 \log \left (c \left (a+b x^2\right )^p\right )}{a^3 x^2}-\frac{b^3 \log \left (c \left (a+b x^2\right )^p\right )}{a^3 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}+\frac{(4 b p) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{x^6} \, dx}{7 a}-\frac{\left (4 b^2 p\right ) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{x^4} \, dx}{7 a^2}+\frac{\left (4 b^3 p\right ) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{x^2} \, dx}{7 a^3}-\frac{\left (4 b^4 p\right ) \int \frac{\log \left (c \left (a+b x^2\right )^p\right )}{a+b x^2} \, dx}{7 a^3}\\ &=-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{35 a x^5}+\frac{4 b^2 p \log \left (c \left (a+b x^2\right )^p\right )}{21 a^2 x^3}-\frac{4 b^3 p \log \left (c \left (a+b x^2\right )^p\right )}{7 a^3 x}-\frac{4 b^{7/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{7 a^{7/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}+\frac{\left (8 b^2 p^2\right ) \int \frac{1}{x^4 \left (a+b x^2\right )} \, dx}{35 a}-\frac{\left (8 b^3 p^2\right ) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{21 a^2}+\frac{\left (8 b^4 p^2\right ) \int \frac{1}{a+b x^2} \, dx}{7 a^3}+\frac{\left (8 b^5 p^2\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \left (a+b x^2\right )} \, dx}{7 a^3}\\ &=-\frac{8 b^2 p^2}{105 a^2 x^3}+\frac{8 b^3 p^2}{21 a^3 x}+\frac{8 b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{7 a^{7/2}}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{35 a x^5}+\frac{4 b^2 p \log \left (c \left (a+b x^2\right )^p\right )}{21 a^2 x^3}-\frac{4 b^3 p \log \left (c \left (a+b x^2\right )^p\right )}{7 a^3 x}-\frac{4 b^{7/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{7 a^{7/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}-\frac{\left (8 b^3 p^2\right ) \int \frac{1}{x^2 \left (a+b x^2\right )} \, dx}{35 a^2}+\frac{\left (8 b^4 p^2\right ) \int \frac{1}{a+b x^2} \, dx}{21 a^3}+\frac{\left (8 b^{9/2} p^2\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a+b x^2} \, dx}{7 a^{7/2}}\\ &=-\frac{8 b^2 p^2}{105 a^2 x^3}+\frac{64 b^3 p^2}{105 a^3 x}+\frac{32 b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{21 a^{7/2}}-\frac{4 i b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{7 a^{7/2}}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{35 a x^5}+\frac{4 b^2 p \log \left (c \left (a+b x^2\right )^p\right )}{21 a^2 x^3}-\frac{4 b^3 p \log \left (c \left (a+b x^2\right )^p\right )}{7 a^3 x}-\frac{4 b^{7/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{7 a^{7/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}-\frac{\left (8 b^4 p^2\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{i-\frac{\sqrt{b} x}{\sqrt{a}}} \, dx}{7 a^4}+\frac{\left (8 b^4 p^2\right ) \int \frac{1}{a+b x^2} \, dx}{35 a^3}\\ &=-\frac{8 b^2 p^2}{105 a^2 x^3}+\frac{64 b^3 p^2}{105 a^3 x}+\frac{184 b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{105 a^{7/2}}-\frac{4 i b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{7 a^{7/2}}-\frac{8 b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{7 a^{7/2}}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{35 a x^5}+\frac{4 b^2 p \log \left (c \left (a+b x^2\right )^p\right )}{21 a^2 x^3}-\frac{4 b^3 p \log \left (c \left (a+b x^2\right )^p\right )}{7 a^3 x}-\frac{4 b^{7/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{7 a^{7/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}+\frac{\left (8 b^4 p^2\right ) \int \frac{\log \left (\frac{2}{1+\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{1+\frac{b x^2}{a}} \, dx}{7 a^4}\\ &=-\frac{8 b^2 p^2}{105 a^2 x^3}+\frac{64 b^3 p^2}{105 a^3 x}+\frac{184 b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{105 a^{7/2}}-\frac{4 i b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{7 a^{7/2}}-\frac{8 b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{7 a^{7/2}}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{35 a x^5}+\frac{4 b^2 p \log \left (c \left (a+b x^2\right )^p\right )}{21 a^2 x^3}-\frac{4 b^3 p \log \left (c \left (a+b x^2\right )^p\right )}{7 a^3 x}-\frac{4 b^{7/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{7 a^{7/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}-\frac{\left (8 i b^{7/2} p^2\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+\frac{i \sqrt{b} x}{\sqrt{a}}}\right )}{7 a^{7/2}}\\ &=-\frac{8 b^2 p^2}{105 a^2 x^3}+\frac{64 b^3 p^2}{105 a^3 x}+\frac{184 b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{105 a^{7/2}}-\frac{4 i b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )^2}{7 a^{7/2}}-\frac{8 b^{7/2} p^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{7 a^{7/2}}-\frac{4 b p \log \left (c \left (a+b x^2\right )^p\right )}{35 a x^5}+\frac{4 b^2 p \log \left (c \left (a+b x^2\right )^p\right )}{21 a^2 x^3}-\frac{4 b^3 p \log \left (c \left (a+b x^2\right )^p\right )}{7 a^3 x}-\frac{4 b^{7/2} p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{7 a^{7/2}}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}-\frac{4 i b^{7/2} p^2 \text{Li}_2\left (1-\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )}{7 a^{7/2}}\\ \end{align*}

Mathematica [C]  time = 0.223705, size = 334, normalized size = 0.99 \[ -\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{7 x^7}+\frac{4 b p \left (-15 i b^{5/2} p x^5 \left (\text{PolyLog}\left (2,\frac{\sqrt{b} x+i \sqrt{a}}{\sqrt{b} x-i \sqrt{a}}\right )+\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )-2 i \log \left (\frac{2 \sqrt{a}}{\sqrt{a}+i \sqrt{b} x}\right )\right )\right )+5 a^{3/2} b x^2 \log \left (c \left (a+b x^2\right )^p\right )-3 a^{5/2} \log \left (c \left (a+b x^2\right )^p\right )-2 a^{3/2} b p x^2 \, _2F_1\left (-\frac{3}{2},1;-\frac{1}{2};-\frac{b x^2}{a}\right )-15 \sqrt{a} b^2 x^4 \log \left (c \left (a+b x^2\right )^p\right )-15 b^{5/2} x^5 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )+10 \sqrt{a} b^2 p x^4 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{b x^2}{a}\right )+30 b^{5/2} p x^5 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right )}{105 a^{7/2} x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b*x^2)^p]^2/x^8,x]

[Out]

-Log[c*(a + b*x^2)^p]^2/(7*x^7) + (4*b*p*(30*b^(5/2)*p*x^5*ArcTan[(Sqrt[b]*x)/Sqrt[a]] - 2*a^(3/2)*b*p*x^2*Hyp
ergeometric2F1[-3/2, 1, -1/2, -((b*x^2)/a)] + 10*Sqrt[a]*b^2*p*x^4*Hypergeometric2F1[-1/2, 1, 1/2, -((b*x^2)/a
)] - 3*a^(5/2)*Log[c*(a + b*x^2)^p] + 5*a^(3/2)*b*x^2*Log[c*(a + b*x^2)^p] - 15*Sqrt[a]*b^2*x^4*Log[c*(a + b*x
^2)^p] - 15*b^(5/2)*x^5*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*Log[c*(a + b*x^2)^p] - (15*I)*b^(5/2)*p*x^5*(ArcTan[(Sqrt[
b]*x)/Sqrt[a]]*(ArcTan[(Sqrt[b]*x)/Sqrt[a]] - (2*I)*Log[(2*Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)]) + PolyLog[2, (I*
Sqrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)])))/(105*a^(7/2)*x^5)

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Maple [F]  time = 0.945, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{2}}{{x}^{8}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x^2+a)^p)^2/x^8,x)

[Out]

int(ln(c*(b*x^2+a)^p)^2/x^8,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)^2/x^8,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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 )^{2}}{x^{8}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)^2/x^8,x, algorithm="fricas")

[Out]

integral(log((b*x^2 + a)^p*c)^2/x^8, 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)**2/x**8,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 )^{2}}{x^{8}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x^2+a)^p)^2/x^8,x, algorithm="giac")

[Out]

integrate(log((b*x^2 + a)^p*c)^2/x^8, x)