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

Optimal. Leaf size=421 \[ \frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )}{e^4}-\frac{2 d^3 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e^4}-\frac{2 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{3 a^{3/2} e}-\frac{d^3 \log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^4}+\frac{d^2 x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e^2}+\frac{x^3 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{3 e}+\frac{d^3 p \log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{-a} d-\sqrt{b} e}\right )}{e^4}+\frac{2 \sqrt{b} d^2 p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} e^3}-\frac{b d p \log \left (a x^2+b\right )}{2 a e^2}+\frac{2 b p x}{3 a e}-\frac{2 d^3 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^4} \]

[Out]

(2*b*p*x)/(3*a*e) + (2*Sqrt[b]*d^2*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(Sqrt[a]*e^3) - (2*b^(3/2)*p*ArcTan[(Sqrt[a]
*x)/Sqrt[b]])/(3*a^(3/2)*e) + (d^2*x*Log[c*(a + b/x^2)^p])/e^3 - (d*x^2*Log[c*(a + b/x^2)^p])/(2*e^2) + (x^3*L
og[c*(a + b/x^2)^p])/(3*e) - (d^3*Log[c*(a + b/x^2)^p]*Log[d + e*x])/e^4 - (2*d^3*p*Log[-((e*x)/d)]*Log[d + e*
x])/e^4 + (d^3*p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*
(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/e^4 - (b*d*p*Log[b + a*x^2])/(2*a*e^2) + (d^3
*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/e^4 + (d^3*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sq
rt[-a]*d + Sqrt[b]*e)])/e^4 - (2*d^3*p*PolyLog[2, 1 + (e*x)/d])/e^4

________________________________________________________________________________________

Rubi [A]  time = 0.585927, antiderivative size = 421, normalized size of antiderivative = 1., number of steps used = 25, number of rules used = 14, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.609, Rules used = {2466, 2448, 263, 205, 2455, 260, 193, 321, 2462, 2416, 2394, 2315, 2393, 2391} \[ \frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )}{e^4}+\frac{d^3 p \text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )}{e^4}-\frac{2 d^3 p \text{PolyLog}\left (2,\frac{e x}{d}+1\right )}{e^4}-\frac{2 b^{3/2} p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{3 a^{3/2} e}-\frac{d^3 \log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^4}+\frac{d^2 x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{e^3}-\frac{d x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{2 e^2}+\frac{x^3 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )}{3 e}+\frac{d^3 p \log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )}{e^4}+\frac{d^3 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{-a} d-\sqrt{b} e}\right )}{e^4}+\frac{2 \sqrt{b} d^2 p \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{\sqrt{a} e^3}-\frac{b d p \log \left (a x^2+b\right )}{2 a e^2}+\frac{2 b p x}{3 a e}-\frac{2 d^3 p \log \left (-\frac{e x}{d}\right ) \log (d+e x)}{e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*p*x)/(3*a*e) + (2*Sqrt[b]*d^2*p*ArcTan[(Sqrt[a]*x)/Sqrt[b]])/(Sqrt[a]*e^3) - (2*b^(3/2)*p*ArcTan[(Sqrt[a]
*x)/Sqrt[b]])/(3*a^(3/2)*e) + (d^2*x*Log[c*(a + b/x^2)^p])/e^3 - (d*x^2*Log[c*(a + b/x^2)^p])/(2*e^2) + (x^3*L
og[c*(a + b/x^2)^p])/(3*e) - (d^3*Log[c*(a + b/x^2)^p]*Log[d + e*x])/e^4 - (2*d^3*p*Log[-((e*x)/d)]*Log[d + e*
x])/e^4 + (d^3*p*Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e*
(Sqrt[b] + Sqrt[-a]*x))/(Sqrt[-a]*d - Sqrt[b]*e))]*Log[d + e*x])/e^4 - (b*d*p*Log[b + a*x^2])/(2*a*e^2) + (d^3
*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - Sqrt[b]*e)])/e^4 + (d^3*p*PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sq
rt[-a]*d + Sqrt[b]*e)])/e^4 - (2*d^3*p*PolyLog[2, 1 + (e*x)/d])/e^4

Rule 2466

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

Rule 2448

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)], x_Symbol] :> Simp[x*Log[c*(d + e*x^n)^p], x] - Dist[e*n*p, Int[
x^n/(d + e*x^n), x], x] /; FreeQ[{c, d, e, n, p}, x]

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

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 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 260

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

Rule 193

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && LtQ[n, 0]
&& IntegerQ[p]

Rule 321

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

Rule 2462

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

Rule 2416

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

Rule 2394

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

Rule 2393

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

Rubi steps

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

Mathematica [C]  time = 0.41833, size = 375, normalized size = 0.89 \[ -\frac{6 d^3 p \left (-\text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d-\sqrt{b} e}\right )-\text{PolyLog}\left (2,\frac{\sqrt{-a} (d+e x)}{\sqrt{-a} d+\sqrt{b} e}\right )+2 \text{PolyLog}\left (2,\frac{e x}{d}+1\right )-\log (d+e x) \log \left (\frac{e \left (\sqrt{b}-\sqrt{-a} x\right )}{\sqrt{-a} d+\sqrt{b} e}\right )-\log (d+e x) \log \left (\frac{e \left (\sqrt{-a} x+\sqrt{b}\right )}{\sqrt{b} e-\sqrt{-a} d}\right )+2 \log \left (-\frac{e x}{d}\right ) \log (d+e x)\right )+6 d^3 \log (d+e x) \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )-6 d^2 e x \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+3 d e^2 x^2 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )-2 e^3 x^3 \log \left (c \left (a+\frac{b}{x^2}\right )^p\right )+\frac{12 \sqrt{b} d^2 e p \tan ^{-1}\left (\frac{\sqrt{b}}{\sqrt{a} x}\right )}{\sqrt{a}}+\frac{3 b d e^2 p \left (\log \left (a+\frac{b}{x^2}\right )+2 \log (x)\right )}{a}-\frac{4 b e^3 p x \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\frac{b}{a x^2}\right )}{a}}{6 e^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((12*Sqrt[b]*d^2*e*p*ArcTan[Sqrt[b]/(Sqrt[a]*x)])/Sqrt[a] - (4*b*e^3*p*x*Hypergeometric2F1[-1/2, 1, 1/2, -(b/
(a*x^2))])/a - 6*d^2*e*x*Log[c*(a + b/x^2)^p] + 3*d*e^2*x^2*Log[c*(a + b/x^2)^p] - 2*e^3*x^3*Log[c*(a + b/x^2)
^p] + (3*b*d*e^2*p*(Log[a + b/x^2] + 2*Log[x]))/a + 6*d^3*Log[c*(a + b/x^2)^p]*Log[d + e*x] + 6*d^3*p*(2*Log[-
((e*x)/d)]*Log[d + e*x] - Log[(e*(Sqrt[b] - Sqrt[-a]*x))/(Sqrt[-a]*d + Sqrt[b]*e)]*Log[d + e*x] - Log[(e*(Sqrt
[b] + Sqrt[-a]*x))/(-(Sqrt[-a]*d) + Sqrt[b]*e)]*Log[d + e*x] - PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d - S
qrt[b]*e)] - PolyLog[2, (Sqrt[-a]*(d + e*x))/(Sqrt[-a]*d + Sqrt[b]*e)] + 2*PolyLog[2, 1 + (e*x)/d]))/(6*e^4)

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Maple [F]  time = 0.773, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3}}{ex+d}\ln \left ( c \left ( a+{\frac{b}{{x}^{2}}} \right ) ^{p} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(x^3*ln(c*(a+b/x^2)^p)/(e*x+d),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(x^3*log(c*(a+b/x^2)^p)/(e*x+d),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{x^{3} \log \left (c \left (\frac{a x^{2} + b}{x^{2}}\right )^{p}\right )}{e x + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [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(x^3*log(c*(a+b/x^2)^p)/(e*x+d),x, algorithm="giac")

[Out]

Timed out