\(\int \frac {\log ^2(c (a+b x^2)^p)}{x^7} \, dx\) [83]

Optimal result

Integrand size = 18, antiderivative size = 193 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=-\frac {b^2 p^2}{6 a^2 x^2}-\frac {b^3 p^2 \log (x)}{a^3}+\frac {b^3 p^2 \log \left (a+b x^2\right )}{6 a^3}-\frac {b p \log \left (c \left (a+b x^2\right )^p\right )}{6 a x^4}+\frac {b^2 p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 a^3 x^2}-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{6 x^6}+\frac {b^3 p \log \left (c \left (a+b x^2\right )^p\right ) \log \left (1-\frac {a}{a+b x^2}\right )}{3 a^3}-\frac {b^3 p^2 \operatorname {PolyLog}\left (2,\frac {a}{a+b x^2}\right )}{3 a^3} \] Output:

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

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.98 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=-\frac {a b^2 p^2 x^4+6 b^3 p^2 x^6 \log (x)-3 b^3 p^2 x^6 \log \left (a+b x^2\right )+a^2 b p x^2 \log \left (c \left (a+b x^2\right )^p\right )-2 a b^2 p x^4 \log \left (c \left (a+b x^2\right )^p\right )-2 b^3 p x^6 \log \left (-\frac {b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )+a^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+b^3 x^6 \log ^2\left (c \left (a+b x^2\right )^p\right )-2 b^3 p^2 x^6 \operatorname {PolyLog}\left (2,1+\frac {b x^2}{a}\right )}{6 a^3 x^6} \] Input:

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

Output:

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

Rubi [A] (warning: unable to verify)

Time = 1.41 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {2904, 2845, 2858, 25, 27, 2789, 2756, 54, 2009, 2789, 2751, 16, 2779, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx\)

\(\Big \downarrow \) 2904

\(\displaystyle \frac {1}{2} \int \frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{x^8}dx^2\)

\(\Big \downarrow \) 2845

\(\displaystyle \frac {1}{2} \left (\frac {2}{3} b p \int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{x^6 \left (b x^2+a\right )}dx^2-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2858

\(\displaystyle \frac {1}{2} \left (\frac {2}{3} p \int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{x^8}d\left (b x^2+a\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} p \int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{x^8}d\left (b x^2+a\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} b^3 p \int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^3 x^8}d\left (b x^2+a\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} b^3 p \left (\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^3 x^6}d\left (b x^2+a\right )}{a}+\frac {\int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^2 x^6}d\left (b x^2+a\right )}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2756

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} b^3 p \left (\frac {\int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^2 x^6}d\left (b x^2+a\right )}{a}+\frac {\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-\frac {1}{2} p \int \frac {1}{b^2 x^6}d\left (b x^2+a\right )}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 54

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} b^3 p \left (\frac {\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-\frac {1}{2} p \int \left (\frac {1}{b^2 x^4 a}-\frac {1}{b x^2 a^2}+\frac {1}{x^2 a^2}\right )d\left (b x^2+a\right )}{a}+\frac {\int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^2 x^6}d\left (b x^2+a\right )}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} b^3 p \left (\frac {\int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^2 x^6}d\left (b x^2+a\right )}{a}+\frac {\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-\frac {1}{2} p \left (-\frac {\log \left (-b x^2\right )}{a^2}+\frac {\log \left (a+b x^2\right )}{a^2}-\frac {1}{a b x^2}\right )}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2789

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} b^3 p \left (\frac {\frac {\int \frac {\log \left (c \left (b x^2+a\right )^p\right )}{b^2 x^4}d\left (b x^2+a\right )}{a}+\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}}{a}+\frac {\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-\frac {1}{2} p \left (-\frac {\log \left (-b x^2\right )}{a^2}+\frac {\log \left (a+b x^2\right )}{a^2}-\frac {1}{a b x^2}\right )}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2751

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} b^3 p \left (\frac {\frac {-\frac {p \int -\frac {1}{b x^2}d\left (b x^2+a\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}+\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}}{a}+\frac {\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-\frac {1}{2} p \left (-\frac {\log \left (-b x^2\right )}{a^2}+\frac {\log \left (a+b x^2\right )}{a^2}-\frac {1}{a b x^2}\right )}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} b^3 p \left (\frac {\frac {\int -\frac {\log \left (c \left (b x^2+a\right )^p\right )}{b x^4}d\left (b x^2+a\right )}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}}{a}+\frac {\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-\frac {1}{2} p \left (-\frac {\log \left (-b x^2\right )}{a^2}+\frac {\log \left (a+b x^2\right )}{a^2}-\frac {1}{a b x^2}\right )}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2779

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} b^3 p \left (\frac {\frac {\frac {p \int \frac {\log \left (1-\frac {a}{x^2}\right )}{x^2}d\left (b x^2+a\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}}{a}+\frac {\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-\frac {1}{2} p \left (-\frac {\log \left (-b x^2\right )}{a^2}+\frac {\log \left (a+b x^2\right )}{a^2}-\frac {1}{a b x^2}\right )}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {1}{2} \left (-\frac {2}{3} b^3 p \left (\frac {\frac {\log \left (c \left (a+b x^2\right )^p\right )}{2 b^2 x^4}-\frac {1}{2} p \left (-\frac {\log \left (-b x^2\right )}{a^2}+\frac {\log \left (a+b x^2\right )}{a^2}-\frac {1}{a b x^2}\right )}{a}+\frac {\frac {\frac {p \operatorname {PolyLog}\left (2,\frac {a}{x^2}\right )}{a}-\frac {\log \left (1-\frac {a}{x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}}{a}+\frac {\frac {p \log \left (-b x^2\right )}{a}-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a b x^2}}{a}}{a}\right )-\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{3 x^6}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 16

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

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 54

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E 
xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && 
 ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2751

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] - Simp[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 2756

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] 
- Simp[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] || (IntegersQ[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] & 
& NeQ[q, 1]))
 

rule 2779

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] + Simp[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 2789

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

rule 2838

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 2845

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] - Simp[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] && In 
tegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))
 

rule 2858

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_ 
.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))^(r_.), x_Symbol] :> Simp[1/e   Subst[In 
t[(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 2904

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*L 
og[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])
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.81 (sec) , antiderivative size = 607, normalized size of antiderivative = 3.15

method	result	size
risch	\(-\frac {{\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}^{2}}{6 x^{6}}-\frac {p \,b^{3} \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (b \,x^{2}+a \right )}{3 a^{3}}-\frac {p b \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{6 a \,x^{4}}+\frac {2 p \,b^{3} \ln \left (\left (b \,x^{2}+a \right )^{p}\right ) \ln \left (x \right )}{3 a^{3}}+\frac {p \,b^{2} \ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{3 a^{2} x^{2}}+\frac {b^{3} p^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{3}}-\frac {b^{2} p^{2}}{6 a^{2} x^{2}}-\frac {b^{3} p^{2} \ln \left (x \right )}{a^{3}}-\frac {2 p^{2} b^{3} \ln \left (x \right ) \ln \left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{3 a^{3}}-\frac {2 p^{2} b^{3} \ln \left (x \right ) \ln \left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{3 a^{3}}-\frac {2 p^{2} b^{3} \operatorname {dilog}\left (\frac {-b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{3 a^{3}}-\frac {2 p^{2} b^{3} \operatorname {dilog}\left (\frac {b x +\sqrt {-a b}}{\sqrt {-a b}}\right )}{3 a^{3}}+\frac {p^{2} b^{3} \ln \left (b \,x^{2}+a \right )^{2}}{6 a^{3}}+\left (i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right ) \left (-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{6 x^{6}}+\frac {p b \left (-\frac {b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{3}}-\frac {1}{4 a \,x^{4}}+\frac {b^{2} \ln \left (x \right )}{a^{3}}+\frac {b}{2 a^{2} x^{2}}\right )}{3}\right )-\frac {{\left (i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}-i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}+i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )+2 \ln \left (c \right )\right )}^{2}}{24 x^{6}}\)	\(607\)

Input:

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

Output:

-1/6*ln((b*x^2+a)^p)^2/x^6-1/3*p*b^3*ln((b*x^2+a)^p)/a^3*ln(b*x^2+a)-1/6*p 
*b*ln((b*x^2+a)^p)/a/x^4+2/3*p*b^3*ln((b*x^2+a)^p)/a^3*ln(x)+1/3*p*b^2*ln( 
(b*x^2+a)^p)/a^2/x^2+1/2*b^3*p^2*ln(b*x^2+a)/a^3-1/6*b^2*p^2/a^2/x^2-b^3*p 
^2*ln(x)/a^3-2/3*p^2*b^3/a^3*ln(x)*ln((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))-2/ 
3*p^2*b^3/a^3*ln(x)*ln((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))-2/3*p^2*b^3/a^3*di 
log((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))-2/3*p^2*b^3/a^3*dilog((b*x+(-a*b)^(1 
/2))/(-a*b)^(1/2))+1/6*p^2*b^3/a^3*ln(b*x^2+a)^2+(I*Pi*csgn(I*(b*x^2+a)^p) 
*csgn(I*c*(b*x^2+a)^p)^2-I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*cs 
gn(I*c)-I*Pi*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c 
)+2*ln(c))*(-1/6/x^6*ln((b*x^2+a)^p)+1/3*p*b*(-1/2*b^2/a^3*ln(b*x^2+a)-1/4 
/a/x^4+b^2/a^3*ln(x)+1/2*b/a^2/x^2))-1/24*(I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I 
*c*(b*x^2+a)^p)^2-I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c) 
-I*Pi*csgn(I*c*(b*x^2+a)^p)^3+I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+2*ln( 
c))^2/x^6
 

Fricas [F]

\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{7}} \,d x } \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int \frac {\log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{2}}{x^{7}}\, dx \] Input:

integrate(ln(c*(b*x**2+a)**p)**2/x**7,x)
 

Output:

Integral(log(c*(a + b*x**2)**p)**2/x**7, x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.90 \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=-\frac {1}{6} \, b^{2} p^{2} {\left (\frac {2 \, {\left (2 \, \log \left (\frac {b x^{2}}{a} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {b x^{2}}{a}\right )\right )} b}{a^{3}} - \frac {3 \, b \log \left (b x^{2} + a\right )}{a^{3}} - \frac {b x^{2} \log \left (b x^{2} + a\right )^{2} - 6 \, b x^{2} \log \left (x\right ) - a}{a^{3} x^{2}}\right )} - \frac {1}{6} \, b p {\left (\frac {2 \, b^{2} \log \left (b x^{2} + a\right )}{a^{3}} - \frac {2 \, b^{2} \log \left (x^{2}\right )}{a^{3}} - \frac {2 \, b x^{2} - a}{a^{2} x^{4}}\right )} \log \left ({\left (b x^{2} + a\right )}^{p} c\right ) - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{6 \, x^{6}} \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int { \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{2}}{x^{7}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\int \frac {{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^2}{x^7} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{x^7} \, dx=\frac {4 \left (\int \frac {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}{b \,x^{3}+a x}d x \right ) a \,b^{3} p \,x^{6}-{\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{2} a^{3}-\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) a^{2} b p \,x^{2}+2 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) a \,b^{2} p \,x^{4}+3 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b^{3} p \,x^{6}-6 \,\mathrm {log}\left (x \right ) b^{3} p^{2} x^{6}-a \,b^{2} p^{2} x^{4}}{6 a^{3} x^{6}} \] Input:

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

Output:

(4*int(log((a + b*x**2)**p*c)/(a*x + b*x**3),x)*a*b**3*p*x**6 - log((a + b 
*x**2)**p*c)**2*a**3 - log((a + b*x**2)**p*c)*a**2*b*p*x**2 + 2*log((a + b 
*x**2)**p*c)*a*b**2*p*x**4 + 3*log((a + b*x**2)**p*c)*b**3*p*x**6 - 6*log( 
x)*b**3*p**2*x**6 - a*b**2*p**2*x**4)/(6*a**3*x**6)