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

Optimal. Leaf size=371 \[ \frac{e^2 p \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right )}{2 d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )}{d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{-a} e+\sqrt{b} d}\right )}{d^3}+\frac{e^2 \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac{e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac{e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac{e^2 p \log (d+e x) \log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{-a} e+\sqrt{b} d}\right )}{d^3}+\frac{e^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{b} d-\sqrt{-a} e}\right )}{d^3}-\frac{2 \sqrt{b} e p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} d^2}-\frac{b p \log \left (a+b x^2\right )}{2 a d}+\frac{b p \log (x)}{a d} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.394818, antiderivative size = 371, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 15, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.652, Rules used = {2466, 2454, 2395, 36, 29, 31, 2455, 205, 2394, 2315, 2462, 260, 2416, 2393, 2391} \[ \frac{e^2 p \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right )}{2 d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )}{d^3}+\frac{e^2 p \text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{-a} e+\sqrt{b} d}\right )}{d^3}+\frac{e^2 \log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 d^3}-\frac{e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{d^3}+\frac{e \log \left (c \left (a+b x^2\right )^p\right )}{d^2 x}-\frac{\log \left (c \left (a+b x^2\right )^p\right )}{2 d x^2}+\frac{e^2 p \log (d+e x) \log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{-a} e+\sqrt{b} d}\right )}{d^3}+\frac{e^2 p \log (d+e x) \log \left (-\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{b} d-\sqrt{-a} e}\right )}{d^3}-\frac{2 \sqrt{b} e p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} d^2}-\frac{b p \log \left (a+b x^2\right )}{2 a d}+\frac{b p \log (x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 2395

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

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

Mathematica [A]  time = 0.211551, size = 320, normalized size = 0.86 \[ \frac{e^2 \left (p \text{PolyLog}\left (2,\frac{b x^2}{a}+1\right )+\log \left (-\frac{b x^2}{a}\right ) \log \left (c \left (a+b x^2\right )^p\right )\right )+2 e^2 p \left (\text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{b} d-\sqrt{-a} e}\right )+\text{PolyLog}\left (2,\frac{\sqrt{b} (d+e x)}{\sqrt{-a} e+\sqrt{b} d}\right )+\log (d+e x) \left (\log \left (\frac{e \left (\sqrt{-a}-\sqrt{b} x\right )}{\sqrt{-a} e+\sqrt{b} d}\right )+\log \left (\frac{e \left (\sqrt{-a}+\sqrt{b} x\right )}{\sqrt{-a} e-\sqrt{b} d}\right )\right )\right )-\frac{d^2 \log \left (c \left (a+b x^2\right )^p\right )}{x^2}-2 e^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+\frac{2 d e \log \left (c \left (a+b x^2\right )^p\right )}{x}+\frac{b d^2 p \left (2 \log (x)-\log \left (a+b x^2\right )\right )}{a}-\frac{4 \sqrt{b} d e p \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a}}}{2 d^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.358, size = 1071, normalized size = 2.9 \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)/x^3/(e*x+d),x)

[Out]

-1/2*b*p*ln(b*x^2+a)/a/d-2*b*p/d^2*e/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))+1/4*I*Pi*csgn(I*c*(b*x^2+a)^p)^3/d/x^
2-1/4*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)^2/d/x^2+1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p
)*csgn(I*c)*e^2/d^3*ln(e*x+d)-1/4*I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)/d/x^2+ln((b*x^2+a)^p)*e/d^2/x-ln((b*x
^2+a)^p)*e^2/d^3*ln(e*x+d)+ln((b*x^2+a)^p)*e^2/d^3*ln(x)-p*e^2/d^3*dilog((-b*x+(-a*b)^(1/2))/(-a*b)^(1/2))-p*e
^2/d^3*dilog((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))+p*e^2/d^3*dilog((e*(-a*b)^(1/2)-b*(e*x+d)+b*d)/(e*(-a*b)^(1/2)+b
*d))+p*e^2/d^3*dilog((e*(-a*b)^(1/2)+b*(e*x+d)-b*d)/(e*(-a*b)^(1/2)-b*d))-1/2*ln((b*x^2+a)^p)/d/x^2-ln(c)*e^2/
d^3*ln(e*x+d)+ln(c)*e^2/d^3*ln(x)+ln(c)*e/d^2/x-p*e^2/d^3*ln(x)*ln((b*x+(-a*b)^(1/2))/(-a*b)^(1/2))+p*e^2/d^3*
ln(e*x+d)*ln((e*(-a*b)^(1/2)-b*(e*x+d)+b*d)/(e*(-a*b)^(1/2)+b*d))+p*e^2/d^3*ln(e*x+d)*ln((e*(-a*b)^(1/2)+b*(e*
x+d)-b*d)/(e*(-a*b)^(1/2)-b*d))+b*p*ln(x)/a/d+1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^3*e^2/d^3*ln(e*x+d)-1/2*I*Pi*csgn
(I*c*(b*x^2+a)^p)^3*e/d^2/x-1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)*e^2/d^3*ln(x)-1/2*ln(
c)/d/x^2-1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)*e/d^2/x-p*e^2/d^3*ln(x)*ln((-b*x+(-a*b)^
(1/2))/(-a*b)^(1/2))-1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^3*e^2/d^3*ln(x)+1/2*I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x
^2+a)^p)^2*e/d^2/x-1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)*e^2/d^3*ln(e*x+d)+1/2*I*Pi*csgn(I*(b*x^2+a)^p)*c
sgn(I*c*(b*x^2+a)^p)^2*e^2/d^3*ln(x)+1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)*e/d^2/x-1/2*I*Pi*csgn(I*(b*x^2
+a)^p)*csgn(I*c*(b*x^2+a)^p)^2*e^2/d^3*ln(e*x+d)+1/2*I*Pi*csgn(I*c*(b*x^2+a)^p)^2*csgn(I*c)*e^2/d^3*ln(x)+1/4*
I*Pi*csgn(I*(b*x^2+a)^p)*csgn(I*c*(b*x^2+a)^p)*csgn(I*c)/d/x^2

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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)/x^3/(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{\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x^{4} + d 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)/x^3/(e*x+d),x, algorithm="fricas")

[Out]

integral(log((b*x^2 + a)^p*c)/(e*x^4 + d*x^3), 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)/x**3/(e*x+d),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 )}{{\left (e x + d\right )} 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)/x^3/(e*x+d),x, algorithm="giac")

[Out]

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