3.343 \(\int \frac{x^4 \log (c (d+e x^2)^p)}{f+g x^2} \, dx\)

Optimal. Leaf size=667 \[ \frac{i f^{3/2} p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 g^{5/2}}+\frac{i f^{3/2} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 g^{5/2}}-\frac{i f^{3/2} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{5/2}}+\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}-\frac{f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}-\frac{2 d^{3/2} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2} g}-\frac{f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{g^{5/2}}-\frac{f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{g^{5/2}}-\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g^2}+\frac{2 d p x}{3 e g}+\frac{2 f^{3/2} p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{g^{5/2}}+\frac{2 f p x}{g^2}-\frac{2 p x^3}{9 g} \]

[Out]

(2*f*p*x)/g^2 + (2*d*p*x)/(3*e*g) - (2*p*x^3)/(9*g) - (2*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*g^2
) - (2*d^(3/2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)*g) + (2*f^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*
Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) - (f^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(-2*Sqrt[f]*Sqrt[g]*(S
qrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g^(5/2) - (f^(3/2)*p*
ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt
[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g^(5/2) - (f*x*Log[c*(d + e*x^2)^p])/g^2 + (x^3*Log[c*(d + e*x^2)^p])/(3*g) +
(f^(3/2)*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/g^(5/2) - (I*f^(3/2)*p*PolyLog[2, 1 - (2*Sqrt[f])/(
Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) + ((I/2)*f^(3/2)*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/
((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g^(5/2) + ((I/2)*f^(3/2)*p*PolyLog[2, 1 - (
2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g
^(5/2)

________________________________________________________________________________________

Rubi [A]  time = 0.717945, antiderivative size = 667, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 13, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.52, Rules used = {2476, 2448, 321, 205, 2455, 302, 2470, 12, 4928, 4856, 2402, 2315, 2447} \[ \frac{i f^{3/2} p \text{PolyLog}\left (2,1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 g^{5/2}}+\frac{i f^{3/2} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{2 g^{5/2}}-\frac{i f^{3/2} p \text{PolyLog}\left (2,1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{5/2}}+\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}-\frac{f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}-\frac{2 d^{3/2} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2} g}-\frac{f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (-\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{g^{5/2}}-\frac{f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (\sqrt{f}-i \sqrt{g} x\right ) \left (\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}\right )}\right )}{g^{5/2}}-\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g^2}+\frac{2 d p x}{3 e g}+\frac{2 f^{3/2} p \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right ) \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{g^{5/2}}+\frac{2 f p x}{g^2}-\frac{2 p x^3}{9 g} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*Log[c*(d + e*x^2)^p])/(f + g*x^2),x]

[Out]

(2*f*p*x)/g^2 + (2*d*p*x)/(3*e*g) - (2*p*x^3)/(9*g) - (2*Sqrt[d]*f*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[e]*g^2
) - (2*d^(3/2)*p*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(3*e^(3/2)*g) + (2*f^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*
Sqrt[f])/(Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) - (f^(3/2)*p*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(-2*Sqrt[f]*Sqrt[g]*(S
qrt[-d] - Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g^(5/2) - (f^(3/2)*p*
ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[(2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt
[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g^(5/2) - (f*x*Log[c*(d + e*x^2)^p])/g^2 + (x^3*Log[c*(d + e*x^2)^p])/(3*g) +
(f^(3/2)*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/g^(5/2) - (I*f^(3/2)*p*PolyLog[2, 1 - (2*Sqrt[f])/(
Sqrt[f] - I*Sqrt[g]*x)])/g^(5/2) + ((I/2)*f^(3/2)*p*PolyLog[2, 1 + (2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] - Sqrt[e]*x))/
((I*Sqrt[e]*Sqrt[f] - Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g^(5/2) + ((I/2)*f^(3/2)*p*PolyLog[2, 1 - (
2*Sqrt[f]*Sqrt[g]*(Sqrt[-d] + Sqrt[e]*x))/((I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])*(Sqrt[f] - I*Sqrt[g]*x))])/g
^(5/2)

Rule 2476

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m_.)*((f_) + (g_.)*(x_)^(s_))^(r_.),
x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c,
d, e, f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] && IntegerQ[s]

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 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 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 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 2470

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 4928

Int[(((a_.) + ArcTan[(c_.)*(x_)]*(b_.))*(x_)^(m_.))/((d_) + (e_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[a
+ b*ArcTan[c*x], x^m/(d + e*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] && IntegerQ[m] &&  !(EqQ[m, 1] && NeQ[a,
0])

Rule 4856

Int[((a_.) + ArcTan[(c_.)*(x_)]*(b_.))/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[((a + b*ArcTan[c*x])*Log[2/(1 -
I*c*x)])/e, x] + (Dist[(b*c)/e, Int[Log[2/(1 - I*c*x)]/(1 + c^2*x^2), x], x] - Dist[(b*c)/e, Int[Log[(2*c*(d
+ e*x))/((c*d + I*e)*(1 - I*c*x))]/(1 + c^2*x^2), x], x] + Simp[((a + b*ArcTan[c*x])*Log[(2*c*(d + e*x))/((c*d
+ I*e)*(1 - I*c*x))])/e, x]) /; FreeQ[{a, b, c, d, e}, x] && NeQ[c^2*d^2 + e^2, 0]

Rule 2402

Int[Log[(c_.)/((d_) + (e_.)*(x_))]/((f_) + (g_.)*(x_)^2), x_Symbol] :> -Dist[e/g, Subst[Int[Log[2*d*x]/(1 - 2*
d*x), x], x, 1/(d + e*x)], x] /; FreeQ[{c, d, e, f, g}, x] && EqQ[c, 2*d] && EqQ[e^2*f + d^2*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 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

\begin{align*} \int \frac{x^4 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx &=\int \left (-\frac{f \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^2 \log \left (c \left (d+e x^2\right )^p\right )}{g}+\frac{f^2 \log \left (c \left (d+e x^2\right )^p\right )}{g^2 \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac{f \int \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{g^2}+\frac{f^2 \int \frac{\log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx}{g^2}+\frac{\int x^2 \log \left (c \left (d+e x^2\right )^p\right ) \, dx}{g}\\ &=-\frac{f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}+\frac{(2 e f p) \int \frac{x^2}{d+e x^2} \, dx}{g^2}-\frac{\left (2 e f^2 p\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{f} \sqrt{g} \left (d+e x^2\right )} \, dx}{g^2}-\frac{(2 e p) \int \frac{x^4}{d+e x^2} \, dx}{3 g}\\ &=\frac{2 f p x}{g^2}-\frac{f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}-\frac{\left (2 e f^{3/2} p\right ) \int \frac{x \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{d+e x^2} \, dx}{g^{5/2}}-\frac{(2 d f p) \int \frac{1}{d+e x^2} \, dx}{g^2}-\frac{(2 e p) \int \left (-\frac{d}{e^2}+\frac{x^2}{e}+\frac{d^2}{e^2 \left (d+e x^2\right )}\right ) \, dx}{3 g}\\ &=\frac{2 f p x}{g^2}+\frac{2 d p x}{3 e g}-\frac{2 p x^3}{9 g}-\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g^2}-\frac{f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}-\frac{\left (2 e f^{3/2} p\right ) \int \left (-\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 \sqrt{e} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{2 \sqrt{e} \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{g^{5/2}}-\frac{\left (2 d^2 p\right ) \int \frac{1}{d+e x^2} \, dx}{3 e g}\\ &=\frac{2 f p x}{g^2}+\frac{2 d p x}{3 e g}-\frac{2 p x^3}{9 g}-\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g^2}-\frac{2 d^{3/2} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2} g}-\frac{f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}+\frac{\left (\sqrt{e} f^{3/2} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{-d}-\sqrt{e} x} \, dx}{g^{5/2}}-\frac{\left (\sqrt{e} f^{3/2} p\right ) \int \frac{\tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right )}{\sqrt{-d}+\sqrt{e} x} \, dx}{g^{5/2}}\\ &=\frac{2 f p x}{g^2}+\frac{2 d p x}{3 e g}-\frac{2 p x^3}{9 g}-\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g^2}-\frac{2 d^{3/2} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2} g}+\frac{2 f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{5/2}}-\frac{f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{5/2}}-\frac{f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{5/2}}-\frac{f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}-2 \frac{(f p) \int \frac{\log \left (\frac{2}{1-\frac{i \sqrt{g} x}{\sqrt{f}}}\right )}{1+\frac{g x^2}{f}} \, dx}{g^2}+\frac{(f p) \int \frac{\log \left (\frac{2 \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{f} \left (-i \sqrt{e}+\frac{\sqrt{-d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{g^2}+\frac{(f p) \int \frac{\log \left (\frac{2 \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{f} \left (i \sqrt{e}+\frac{\sqrt{-d} \sqrt{g}}{\sqrt{f}}\right ) \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right )}\right )}{1+\frac{g x^2}{f}} \, dx}{g^2}\\ &=\frac{2 f p x}{g^2}+\frac{2 d p x}{3 e g}-\frac{2 p x^3}{9 g}-\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g^2}-\frac{2 d^{3/2} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2} g}+\frac{2 f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{5/2}}-\frac{f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{5/2}}-\frac{f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{5/2}}-\frac{f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}+\frac{i f^{3/2} p \text{Li}_2\left (1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 g^{5/2}}+\frac{i f^{3/2} p \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 g^{5/2}}-2 \frac{\left (i f^{3/2} p\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1-\frac{i \sqrt{g} x}{\sqrt{f}}}\right )}{g^{5/2}}\\ &=\frac{2 f p x}{g^2}+\frac{2 d p x}{3 e g}-\frac{2 p x^3}{9 g}-\frac{2 \sqrt{d} f p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e} g^2}-\frac{2 d^{3/2} p \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{3 e^{3/2} g}+\frac{2 f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{5/2}}-\frac{f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{5/2}}-\frac{f^{3/2} p \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{g^{5/2}}-\frac{f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}-\frac{i f^{3/2} p \text{Li}_2\left (1-\frac{2 \sqrt{f}}{\sqrt{f}-i \sqrt{g} x}\right )}{g^{5/2}}+\frac{i f^{3/2} p \text{Li}_2\left (1+\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}-\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 g^{5/2}}+\frac{i f^{3/2} p \text{Li}_2\left (1-\frac{2 \sqrt{f} \sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\left (i \sqrt{e} \sqrt{f}+\sqrt{-d} \sqrt{g}\right ) \left (\sqrt{f}-i \sqrt{g} x\right )}\right )}{2 g^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.548642, size = 691, normalized size = 1.04 \[ -\frac{i f^{3/2} p \left (\text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-i \sqrt{-d} \sqrt{g}}\right )+\text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}-i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+i \sqrt{-d} \sqrt{g}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}+i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}-i \sqrt{-d} \sqrt{g}}\right )-\text{PolyLog}\left (2,\frac{\sqrt{e} \left (\sqrt{f}+i \sqrt{g} x\right )}{\sqrt{e} \sqrt{f}+i \sqrt{-d} \sqrt{g}}\right )+\log \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )+\log \left (1-\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}-i \sqrt{e} \sqrt{f}}\right )-\log \left (1+\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}-\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}-i \sqrt{e} \sqrt{f}}\right )-\log \left (1+\frac{i \sqrt{g} x}{\sqrt{f}}\right ) \log \left (\frac{\sqrt{g} \left (\sqrt{-d}+\sqrt{e} x\right )}{\sqrt{-d} \sqrt{g}+i \sqrt{e} \sqrt{f}}\right )\right )}{2 g^{5/2}}+\frac{f^{3/2} \tan ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{g^{5/2}}-\frac{f x \log \left (c \left (d+e x^2\right )^p\right )}{g^2}+\frac{x^3 \log \left (c \left (d+e x^2\right )^p\right )}{3 g}+\frac{2 d p \left (\sqrt{e} x-\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )\right )}{3 e^{3/2} g}+\frac{2 f p \left (x-\frac{\sqrt{d} \tan ^{-1}\left (\frac{\sqrt{e} x}{\sqrt{d}}\right )}{\sqrt{e}}\right )}{g^2}-\frac{2 p x^3}{9 g} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x^4*Log[c*(d + e*x^2)^p])/(f + g*x^2),x]

[Out]

(-2*p*x^3)/(9*g) + (2*d*p*(Sqrt[e]*x - Sqrt[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]]))/(3*e^(3/2)*g) + (2*f*p*(x - (Sqrt
[d]*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/Sqrt[e]))/g^2 - (f*x*Log[c*(d + e*x^2)^p])/g^2 + (x^3*Log[c*(d + e*x^2)^p])/(
3*g) + (f^(3/2)*ArcTan[(Sqrt[g]*x)/Sqrt[f]]*Log[c*(d + e*x^2)^p])/g^(5/2) - ((I/2)*f^(3/2)*p*(Log[(Sqrt[g]*(Sq
rt[-d] - Sqrt[e]*x))/(I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] + Log[(Sqrt[g]*(Sq
rt[-d] + Sqrt[e]*x))/((-I)*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 - (I*Sqrt[g]*x)/Sqrt[f]] - Log[(Sqrt[g]*
(Sqrt[-d] - Sqrt[e]*x))/((-I)*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 + (I*Sqrt[g]*x)/Sqrt[f]] - Log[(Sqrt[
g]*(Sqrt[-d] + Sqrt[e]*x))/(I*Sqrt[e]*Sqrt[f] + Sqrt[-d]*Sqrt[g])]*Log[1 + (I*Sqrt[g]*x)/Sqrt[f]] + PolyLog[2,
(Sqrt[e]*(Sqrt[f] - I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] - I*Sqrt[-d]*Sqrt[g])] + PolyLog[2, (Sqrt[e]*(Sqrt[f] - I*
Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt[g])] - PolyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sq
rt[f] - I*Sqrt[-d]*Sqrt[g])] - PolyLog[2, (Sqrt[e]*(Sqrt[f] + I*Sqrt[g]*x))/(Sqrt[e]*Sqrt[f] + I*Sqrt[-d]*Sqrt
[g])]))/g^(5/2)

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Maple [C]  time = 0.631, size = 1011, normalized size = 1.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*ln(c*(e*x^2+d)^p)/(g*x^2+f),x)

[Out]

1/3*(ln((e*x^2+d)^p)-p*ln(e*x^2+d))/g*x^3-p*f/g^2*x*ln(e*x^2+d)-1/6*I*Pi*csgn(I*c*(e*x^2+d)^p)^3/g*x^3+1/6*I*P
i*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2/g*x^3+1/6*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/g*x^3+p*Sum(1/2
*(ln(x-_alpha)*ln(e*x^2+d)-2*e*(1/2*ln(x-_alpha)*(ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1)-x+_alph
a)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1))+ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)-x+_a
lpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)))/e+1/2*(dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f
,index=1)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=1))+dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d
*g-e*f,index=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g+d*g-e*f,index=2)))/e))*f^2/g^3/_alpha,_alpha=RootOf(
_Z^2*g+f))-2/9*p*x^3/g+1/2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2*f^2/g^2/(f*g)^(1/2)*arctan(x*g/(f*
g)^(1/2))-2/3*p/g*d^2/e/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-2*p*f/g^2*d/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))-(l
n((e*x^2+d)^p)-p*ln(e*x^2+d))/g^2*f*x+2/3*d*p*x/e/g-ln(c)/g^2*f*x+ln(c)*f^2/g^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(
1/2))+1/3*p/g*x^3*ln(e*x^2+d)+1/3*ln(c)/g*x^3+2*f*p*x/g^2+1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^3/g^2*f*x+1/2*I*Pi*cs
gn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)/g^2*f*x+1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)*f^2/g^2/(
f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+(ln((e*x^2+d)^p)-p*ln(e*x^2+d))*f^2/g^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))
-1/2*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*csgn(I*c)*f^2/g^2/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-1/2*
I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)^2/g^2*f*x-1/6*I*Pi*csgn(I*(e*x^2+d)^p)*csgn(I*c*(e*x^2+d)^p)*cs
gn(I*c)/g*x^3-1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^2*csgn(I*c)/g^2*f*x-1/2*I*Pi*csgn(I*c*(e*x^2+d)^p)^3*f^2/g^2/(f*g
)^(1/2)*arctan(x*g/(f*g)^(1/2))

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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(x^4*log(c*(e*x^2+d)^p)/(g*x^2+f),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^{4} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="fricas")

[Out]

integral(x^4*log((e*x^2 + d)^p*c)/(g*x^2 + f), 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**4*ln(c*(e*x**2+d)**p)/(g*x**2+f),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="giac")

[Out]

integrate(x^4*log((e*x^2 + d)^p*c)/(g*x^2 + f), x)