3.82 \(\int \frac{\log ^2(c (a+b x^2)^p)}{x^5} \, dx\)

Optimal. Leaf size=129 \[ \frac{b^2 p^2 \text{PolyLog}\left (2,\frac{a}{a+b x^2}\right )}{2 a^2}-\frac{b^2 p \log \left (1-\frac{a}{a+b x^2}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}+\frac{b^2 p^2 \log (x)}{a^2}-\frac{b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4} \]

[Out]

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

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Rubi [A]  time = 0.268723, antiderivative size = 147, normalized size of antiderivative = 1.14, number of steps used = 10, number of rules used = 10, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.556, Rules used = {2454, 2398, 2411, 2347, 2344, 2301, 2317, 2391, 2314, 31} \[ -\frac{b^2 p^2 \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right )}{2 a^2}+\frac{b^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2}-\frac{b^2 p \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}+\frac{b^2 p^2 \log (x)}{a^2}-\frac{b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]^2/x^5,x]

[Out]

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

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 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 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 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 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 ^2\left (c \left (a+b x^2\right )^p\right )}{x^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\log ^2\left (c (a+b x)^p\right )}{x^3} \, dx,x,x^2\right )\\ &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac{1}{2} (b p) \operatorname{Subst}\left (\int \frac{\log \left (c (a+b x)^p\right )}{x^2 (a+b x)} \, dx,x,x^2\right )\\ &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac{1}{2} p \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 )\\ &=-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac{p \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}-\frac{(b p) \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}\\ &=-\frac{b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac{(b p) \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^2}+\frac{\left (b^2 p\right ) \operatorname{Subst}\left (\int \frac{\log \left (c x^p\right )}{x} \, dx,x,a+b x^2\right )}{2 a^2}+\frac{\left (b p^2\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x}{b}} \, dx,x,a+b x^2\right )}{2 a^2}\\ &=\frac{b^2 p^2 \log (x)}{a^2}-\frac{b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac{b^2 p \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}+\frac{b^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}+\frac{\left (b^2 p^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{x}{a}\right )}{x} \, dx,x,a+b x^2\right )}{2 a^2}\\ &=\frac{b^2 p^2 \log (x)}{a^2}-\frac{b p \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2 x^2}-\frac{b^2 p \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 a^2}+\frac{b^2 \log ^2\left (c \left (a+b x^2\right )^p\right )}{4 a^2}-\frac{\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4}-\frac{b^2 p^2 \text{Li}_2\left (1+\frac{b x^2}{a}\right )}{2 a^2}\\ \end{align*}

Mathematica [A]  time = 0.0790633, size = 137, normalized size = 1.06 \[ \frac{\frac{b x^2 \left (-2 b p x^2 \left (p \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right )+\log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )\right )+b x^2 \log ^2\left (c \left (a+b x^2\right )^p\right )-2 a p \log \left (c \left (a+b x^2\right )^p\right )+2 b p^2 x^2 \left (2 \log (x)-\log \left (a+b x^2\right )\right )\right )}{a^2}-\log ^2\left (c \left (a+b x^2\right )^p\right )}{4 x^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.5, size = 1080, normalized size = 8.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*p^2*ln(x)/a^2+1/4*I*b^2*p/a^2*ln(b*x^2+a)*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)+b^2*p^2/a^2*ln(x)*ln((-b*x+
(-a*b)^(1/2))/(-a*b)^(1/2))+b^2*p^2/a^2*ln(x)*ln((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))+1/4*I/x^4*ln((b*x^2+a)^p)*Pi
*csgn(I*c*(b*x^2+a)^p)^3+1/2*I*b^2*p/a^2*ln(x)*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-1/4*I*b^
2*p/a^2*ln(b*x^2+a)*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)+1/4*I*b*p/a/x^2*Pi*csgn(I*(b*x^2+a)
^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)-1/4*b^2*p^2/a^2*ln(b*x^2+a)^2-1/2*b^2*p^2/a^2*ln(b*x^2+a)+b^2*p^2/a^2*dilo
g((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))+b^2*p^2/a^2*dilog((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))-1/2/x^4*ln((b*x^2+a)^p)
*ln(c)-1/4*I/x^4*ln((b*x^2+a)^p)*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-1/2*I*b^2*p/a^2*ln(x)*Pi*csgn(I*c*(b*x^2
+a)^p)^2*csgn(I*c)-1/2*I*b^2*p/a^2*ln(x)*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2+1/4*I*b^2*p/a^2*ln(b*x
^2+a)*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-1/4*I*b*p/a/x^2*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)-1/4*
I*b*p/a/x^2*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2-1/4*I/x^4*ln((b*x^2+a)^p)*Pi*csgn(I*(b*x^2+a)^p)*cs
gn(I*c*(b*x^2+a)^p)^2-1/4/x^4*ln((b*x^2+a)^p)^2+1/2*b^2*p/a^2*ln(b*x^2+a)*ln(c)-1/2*b*p/a/x^2*ln(c)-b^2*p/a^2*
ln(x)*ln(c)-1/16*(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^4+1/2*I*b^2*p/a
^2*ln(x)*Pi*csgn(I*c*(b*x^2+a)^p)^3+1/4*I/x^4*ln((b*x^2+a)^p)*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csg
n(I*c)+1/4*I*b*p/a/x^2*Pi*csgn(I*c*(b*x^2+a)^p)^3-1/4*I*b^2*p/a^2*ln(b*x^2+a)*Pi*csgn(I*c*(b*x^2+a)^p)^3-b^2*p
*ln((b*x^2+a)^p)/a^2*ln(x)+1/2*b^2*p*ln((b*x^2+a)^p)/a^2*ln(b*x^2+a)-1/2*b*p*ln((b*x^2+a)^p)/a/x^2

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Maxima [A]  time = 1.11725, size = 192, normalized size = 1.49 \begin{align*} -\frac{1}{4} \, b^{2} p^{2}{\left (\frac{\log \left (b x^{2} + a\right )^{2}}{a^{2}} - \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 )}}{a^{2}} + \frac{2 \, \log \left (b x^{2} + a\right )}{a^{2}} - \frac{4 \, \log \left (x\right )}{a^{2}}\right )} + \frac{1}{2} \, b p{\left (\frac{b \log \left (b x^{2} + a\right )}{a^{2}} - \frac{b \log \left (x^{2}\right )}{a^{2}} - \frac{1}{a x^{2}}\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}}{4 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{5}}, 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^5,x, algorithm="fricas")

[Out]

integral(log((b*x^2 + a)^p*c)^2/x^5, 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**5,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^{5}}\,{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^5,x, algorithm="giac")

[Out]

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