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

Optimal. Leaf size=119 \[ \frac{3 b p^2 \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac{3 b p^3 \text{PolyLog}\left (3,\frac{b x^2}{a}+1\right )}{a}+\frac{3 b p \log \left (-\frac{b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac{\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2} \]

[Out]

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

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Rubi [A]  time = 0.145445, antiderivative size = 119, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2454, 2397, 2396, 2433, 2374, 6589} \[ \frac{3 b p^2 \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )}{a}-\frac{3 b p^3 \text{PolyLog}\left (3,\frac{b x^2}{a}+1\right )}{a}+\frac{3 b p \log \left (-\frac{b x^2}{a}\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )}{2 a}-\frac{\left (a+b x^2\right ) \log ^3\left (c \left (a+b x^2\right )^p\right )}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x^2)^p]^3/x^3,x]

[Out]

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

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 2397

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

Rule 2396

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

Rule 2433

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 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]

Rubi steps

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

Mathematica [B]  time = 0.287264, size = 302, normalized size = 2.54 \[ -\frac{-6 b p^2 x^2 \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right ) \log \left (c \left (a+b x^2\right )^p\right )+6 b p^3 x^2 \text{PolyLog}\left (3,\frac{b x^2}{a}+1\right )-3 b p^2 x^2 \log ^2\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )+12 b p^2 x^2 \log (x) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-6 b p^2 x^2 \log \left (-\frac{b x^2}{a}\right ) \log \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )-6 b p x^2 \log (x) \log ^2\left (c \left (a+b x^2\right )^p\right )+3 b p x^2 \log \left (a+b x^2\right ) \log ^2\left (c \left (a+b x^2\right )^p\right )+a \log ^3\left (c \left (a+b x^2\right )^p\right )+b p^3 x^2 \log ^3\left (a+b x^2\right )-6 b p^3 x^2 \log (x) \log ^2\left (a+b x^2\right )+3 b p^3 x^2 \log \left (-\frac{b x^2}{a}\right ) \log ^2\left (a+b x^2\right )}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 1.296, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( \ln \left ( c \left ( b{x}^{2}+a \right ) ^{p} \right ) \right ) ^{3}}{{x}^{3}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(ln(c*(b*x^2+a)^p)^3/x^3,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}}{2 \, x^{2}} + \int \frac{b x^{2} \log \left (c\right )^{3} + a \log \left (c\right )^{3} + 3 \,{\left (b{\left (p + \log \left (c\right )\right )} x^{2} + 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^{5} + a x^{3}}\,{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^3,x, algorithm="maxima")

[Out]

-1/2*log((b*x^2 + a)^p)^3/x^2 + integrate((b*x^2*log(c)^3 + a*log(c)^3 + 3*(b*(p + log(c))*x^2 + a*log(c))*log
((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^5 + a*x^3), 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^{3}}, 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^3,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{3}}\,{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^3,x, algorithm="giac")

[Out]

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