3.311 \(\int \frac{x^3 (a+b \log (c (d+e x)^n))^2}{f+g x^2} \, dx\)

Optimal. Leaf size=499 \[ -\frac{b f n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}-\frac{b f n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac{b^2 f n^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^2}+\frac{b^2 f n^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{g^2}-\frac{b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g}-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}-\frac{f \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^2}-\frac{f \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^2}+\frac{2 a b d n x}{e g}+\frac{2 b^2 d n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g}+\frac{b^2 n^2 (d+e x)^2}{4 e^2 g}-\frac{2 b^2 d n^2 x}{e g} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.687801, antiderivative size = 499, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 12, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.414, Rules used = {2416, 2401, 2389, 2296, 2295, 2390, 2305, 2304, 2396, 2433, 2374, 6589} \[ -\frac{b f n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}-\frac{b f n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{g^2}+\frac{b^2 f n^2 \text{PolyLog}\left (3,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^2}+\frac{b^2 f n^2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{g^2}-\frac{b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g}-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}-\frac{f \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^2}-\frac{f \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 g^2}+\frac{2 a b d n x}{e g}+\frac{2 b^2 d n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g}+\frac{b^2 n^2 (d+e x)^2}{4 e^2 g}-\frac{2 b^2 d n^2 x}{e g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 2401

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

Rule 2389

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

Rule 2296

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2390

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

Rule 2305

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

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

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{x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx &=\int \left (\frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}-\frac{f x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g \left (f+g x^2\right )}\right ) \, dx\\ &=\frac{\int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{g}-\frac{f \int \frac{x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{g}\\ &=\frac{\int \left (-\frac{d \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}+\frac{(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e}\right ) \, dx}{g}-\frac{f \int \left (-\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt{g} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 \sqrt{g} \left (\sqrt{-f}+\sqrt{g} x\right )}\right ) \, dx}{g}\\ &=\frac{f \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}-\sqrt{g} x} \, dx}{2 g^{3/2}}-\frac{f \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt{-f}+\sqrt{g} x} \, dx}{2 g^{3/2}}+\frac{\int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{e g}-\frac{d \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{e g}\\ &=-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^2}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}+\frac{\operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2 g}-\frac{d \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2 g}+\frac{(b e f n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{d+e x} \, dx}{g^2}+\frac{(b e f n) \int \frac{\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{d+e x} \, dx}{g^2}\\ &=-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^2}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}+\frac{(b f n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{e \sqrt{-f}+d \sqrt{g}}{e}-\frac{\sqrt{g} x}{e}\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{g^2}+\frac{(b f n) \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (\frac{e \left (\frac{e \sqrt{-f}-d \sqrt{g}}{e}+\frac{\sqrt{g} x}{e}\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{g^2}-\frac{(b n) \operatorname{Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2 g}+\frac{(2 b d n) \operatorname{Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2 g}\\ &=\frac{2 a b d n x}{e g}+\frac{b^2 n^2 (d+e x)^2}{4 e^2 g}-\frac{b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g}-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^2}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}-\frac{b f n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^2}-\frac{b f n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^2}+\frac{\left (2 b^2 d n\right ) \operatorname{Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2 g}+\frac{\left (b^2 f n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (-\frac{\sqrt{g} x}{e \sqrt{-f}-d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{g^2}+\frac{\left (b^2 f n^2\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{\sqrt{g} x}{e \sqrt{-f}+d \sqrt{g}}\right )}{x} \, dx,x,d+e x\right )}{g^2}\\ &=\frac{2 a b d n x}{e g}-\frac{2 b^2 d n^2 x}{e g}+\frac{b^2 n^2 (d+e x)^2}{4 e^2 g}+\frac{2 b^2 d n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g}-\frac{b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g}-\frac{d (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g}+\frac{(d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{e \sqrt{-f}+d \sqrt{g}}\right )}{2 g^2}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right )}{2 g^2}-\frac{b f n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^2}-\frac{b f n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^2}+\frac{b^2 f n^2 \text{Li}_3\left (-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{g^2}+\frac{b^2 f n^2 \text{Li}_3\left (\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}+d \sqrt{g}}\right )}{g^2}\\ \end{align*}

Mathematica [C]  time = 0.465138, size = 637, normalized size = 1.28 \[ \frac{2 b n \left (-2 e^2 f \left (\text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )+\log (d+e x) \log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )\right )-2 e^2 f \left (\text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+i e \sqrt{f}}\right )+\log (d+e x) \log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+i e \sqrt{f}}\right )\right )-2 g \left (d^2-e^2 x^2\right ) \log (d+e x)+e g x (2 d-e x)\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )-b^2 n^2 \left (2 e^2 f \left (-2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )+2 \log (d+e x) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )+\log ^2(d+e x) \log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}-i e \sqrt{f}}\right )\right )+2 e^2 f \left (-2 \text{PolyLog}\left (3,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+i e \sqrt{f}}\right )+2 \log (d+e x) \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+i e \sqrt{f}}\right )+\log ^2(d+e x) \log \left (1-\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+i e \sqrt{f}}\right )\right )+g \left (2 \left (d^2-e^2 x^2\right ) \log ^2(d+e x)+\left (-6 d^2-4 d e x+2 e^2 x^2\right ) \log (d+e x)+e x (6 d-e x)\right )\right )-2 e^2 f \log \left (f+g x^2\right ) \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2+2 e^2 g x^2 \left (a+b \log \left (c (d+e x)^n\right )-b n \log (d+e x)\right )^2}{4 e^2 g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*e^2*g*x^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 - 2*e^2*f*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*
x)^n])^2*Log[f + g*x^2] + 2*b*n*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(e*g*x*(2*d - e*x) - 2*g*(d^2 -
e^2*x^2)*Log[d + e*x] - 2*e^2*f*(Log[d + e*x]*Log[1 - (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] + Poly
Log[2, (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])]) - 2*e^2*f*(Log[d + e*x]*Log[1 - (Sqrt[g]*(d + e*x))/
(I*e*Sqrt[f] + d*Sqrt[g])] + PolyLog[2, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])])) - b^2*n^2*(g*(e*x*(6*
d - e*x) + (-6*d^2 - 4*d*e*x + 2*e^2*x^2)*Log[d + e*x] + 2*(d^2 - e^2*x^2)*Log[d + e*x]^2) + 2*e^2*f*(Log[d +
e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])] + 2*Log[d + e*x]*PolyLog[2, (Sqrt[g]*(d + e*x
))/((-I)*e*Sqrt[f] + d*Sqrt[g])] - 2*PolyLog[3, (Sqrt[g]*(d + e*x))/((-I)*e*Sqrt[f] + d*Sqrt[g])]) + 2*e^2*f*(
Log[d + e*x]^2*Log[1 - (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] + 2*Log[d + e*x]*PolyLog[2, (Sqrt[g]*(d
+ e*x))/(I*e*Sqrt[f] + d*Sqrt[g])] - 2*PolyLog[3, (Sqrt[g]*(d + e*x))/(I*e*Sqrt[f] + d*Sqrt[g])])))/(4*e^2*g^2
)

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Maple [F]  time = 1.504, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{3} \left ( a+b\ln \left ( c \left ( ex+d \right ) ^{n} \right ) \right ) ^{2}}{g{x}^{2}+f}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, a^{2}{\left (\frac{x^{2}}{g} - \frac{f \log \left (g x^{2} + f\right )}{g^{2}}\right )} + \int \frac{b^{2} x^{3} \log \left ({\left (e x + d\right )}^{n}\right )^{2} + 2 \,{\left (b^{2} \log \left (c\right ) + a b\right )} x^{3} \log \left ({\left (e x + d\right )}^{n}\right ) +{\left (b^{2} \log \left (c\right )^{2} + 2 \, a b \log \left (c\right )\right )} x^{3}}{g x^{2} + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f),x, algorithm="maxima")

[Out]

1/2*a^2*(x^2/g - f*log(g*x^2 + f)/g^2) + integrate((b^2*x^3*log((e*x + d)^n)^2 + 2*(b^2*log(c) + a*b)*x^3*log(
(e*x + d)^n) + (b^2*log(c)^2 + 2*a*b*log(c))*x^3)/(g*x^2 + f), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} x^{3} \log \left ({\left (e x + d\right )}^{n} c\right )^{2} + 2 \, a b x^{3} \log \left ({\left (e x + d\right )}^{n} c\right ) + a^{2} x^{3}}{g x^{2} + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*x^3*log((e*x + d)^n*c)^2 + 2*a*b*x^3*log((e*x + d)^n*c) + a^2*x^3)/(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**3*(a+b*ln(c*(e*x+d)**n))**2/(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{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{2} x^{3}}{g x^{2} + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f),x, algorithm="giac")

[Out]

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