3.271 \(\int \frac{x^4 (a+b \log (c (d+e x)^n))}{(f+g x^2)^2} \, dx\)

Optimal. Leaf size=534 \[ -\frac{3 b \sqrt{-f} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{4 g^{5/2}}+\frac{3 b \sqrt{-f} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{4 g^{5/2}}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{3 \sqrt{-f} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2}}-\frac{3 \sqrt{-f} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2}}+\frac{a x}{g^2}+\frac{b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac{b e f n \log (d+e x)}{4 g^{5/2} \left (e \sqrt{-f}-d \sqrt{g}\right )}+\frac{b e f n \log (d+e x)}{4 g^{5/2} \left (d \sqrt{g}+e \sqrt{-f}\right )}-\frac{b e f n \log \left (\sqrt{-f}-\sqrt{g} x\right )}{4 g^{5/2} \left (d \sqrt{g}+e \sqrt{-f}\right )}+\frac{b e f n \log \left (\sqrt{-f}+\sqrt{g} x\right )}{4 g^{5/2} \left (e \sqrt{-f}-d \sqrt{g}\right )}-\frac{b n x}{g^2} \]

[Out]

(a*x)/g^2 - (b*n*x)/g^2 - (b*e*f*n*Log[d + e*x])/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) + (b*e*f*n*Log[d + e*x])
/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (b*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) - (f*(a + b*Log[c*(d + e*x)^n
]))/(4*g^(5/2)*(Sqrt[-f] - Sqrt[g]*x)) + (f*(a + b*Log[c*(d + e*x)^n]))/(4*g^(5/2)*(Sqrt[-f] + Sqrt[g]*x)) - (
b*e*f*n*Log[Sqrt[-f] - Sqrt[g]*x])/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n
])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2)) + (b*e*f*n*Log[Sqrt[-f] + Sqrt[g]*x])
/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) - (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/
(e*Sqrt[-f] - d*Sqrt[g])])/(4*g^(5/2)) - (3*b*Sqrt[-f]*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt
[g]))])/(4*g^(5/2)) + (3*b*Sqrt[-f]*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.929954, antiderivative size = 534, normalized size of antiderivative = 1., number of steps used = 31, number of rules used = 13, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.482, Rules used = {288, 321, 205, 2416, 2389, 2295, 2409, 2395, 36, 31, 2394, 2393, 2391} \[ -\frac{3 b \sqrt{-f} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )}{4 g^{5/2}}+\frac{3 b \sqrt{-f} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )}{4 g^{5/2}}-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt{-f}-\sqrt{g} x\right )}+\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt{-f}+\sqrt{g} x\right )}+\frac{3 \sqrt{-f} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2}}-\frac{3 \sqrt{-f} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2}}+\frac{a x}{g^2}+\frac{b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac{b e f n \log (d+e x)}{4 g^{5/2} \left (e \sqrt{-f}-d \sqrt{g}\right )}+\frac{b e f n \log (d+e x)}{4 g^{5/2} \left (d \sqrt{g}+e \sqrt{-f}\right )}-\frac{b e f n \log \left (\sqrt{-f}-\sqrt{g} x\right )}{4 g^{5/2} \left (d \sqrt{g}+e \sqrt{-f}\right )}+\frac{b e f n \log \left (\sqrt{-f}+\sqrt{g} x\right )}{4 g^{5/2} \left (e \sqrt{-f}-d \sqrt{g}\right )}-\frac{b n x}{g^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*x)/g^2 - (b*n*x)/g^2 - (b*e*f*n*Log[d + e*x])/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) + (b*e*f*n*Log[d + e*x])
/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (b*(d + e*x)*Log[c*(d + e*x)^n])/(e*g^2) - (f*(a + b*Log[c*(d + e*x)^n
]))/(4*g^(5/2)*(Sqrt[-f] - Sqrt[g]*x)) + (f*(a + b*Log[c*(d + e*x)^n]))/(4*g^(5/2)*(Sqrt[-f] + Sqrt[g]*x)) - (
b*e*f*n*Log[Sqrt[-f] - Sqrt[g]*x])/(4*(e*Sqrt[-f] + d*Sqrt[g])*g^(5/2)) + (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n
])*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2)) + (b*e*f*n*Log[Sqrt[-f] + Sqrt[g]*x])
/(4*(e*Sqrt[-f] - d*Sqrt[g])*g^(5/2)) - (3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/
(e*Sqrt[-f] - d*Sqrt[g])])/(4*g^(5/2)) - (3*b*Sqrt[-f]*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt
[g]))])/(4*g^(5/2)) + (3*b*Sqrt[-f]*n*PolyLog[2, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2))

Rule 288

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*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, 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 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 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 2295

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

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

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 31

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

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

Mathematica [A]  time = 0.955826, size = 434, normalized size = 0.81 \[ \frac{-3 b \sqrt{-f} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (d+e x)}{e \sqrt{-f}-d \sqrt{g}}\right )+3 b \sqrt{-f} n \text{PolyLog}\left (2,\frac{\sqrt{g} (d+e x)}{d \sqrt{g}+e \sqrt{-f}}\right )-\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{-f}-\sqrt{g} x}+\frac{f \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt{-f}+\sqrt{g} x}+3 \sqrt{-f} \log \left (\frac{e \left (\sqrt{-f}-\sqrt{g} x\right )}{d \sqrt{g}+e \sqrt{-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )-3 \sqrt{-f} \log \left (\frac{e \left (\sqrt{-f}+\sqrt{g} x\right )}{e \sqrt{-f}-d \sqrt{g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )+4 a \sqrt{g} x+\frac{4 b \sqrt{g} (d+e x) \log \left (c (d+e x)^n\right )}{e}+\frac{b e f n \left (\log (d+e x)-\log \left (\sqrt{-f}-\sqrt{g} x\right )\right )}{d \sqrt{g}+e \sqrt{-f}}-\frac{b e f n \left (\log (d+e x)-\log \left (\sqrt{-f}+\sqrt{g} x\right )\right )}{e \sqrt{-f}-d \sqrt{g}}-4 b \sqrt{g} n x}{4 g^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*a*Sqrt[g]*x - 4*b*Sqrt[g]*n*x + (4*b*Sqrt[g]*(d + e*x)*Log[c*(d + e*x)^n])/e - (f*(a + b*Log[c*(d + e*x)^n]
))/(Sqrt[-f] - Sqrt[g]*x) + (f*(a + b*Log[c*(d + e*x)^n]))/(Sqrt[-f] + Sqrt[g]*x) + (b*e*f*n*(Log[d + e*x] - L
og[Sqrt[-f] - Sqrt[g]*x]))/(e*Sqrt[-f] + d*Sqrt[g]) + 3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] -
Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])] - (b*e*f*n*(Log[d + e*x] - Log[Sqrt[-f] + Sqrt[g]*x]))/(e*Sqrt[-f] - d*
Sqrt[g]) - 3*Sqrt[-f]*(a + b*Log[c*(d + e*x)^n])*Log[(e*(Sqrt[-f] + Sqrt[g]*x))/(e*Sqrt[-f] - d*Sqrt[g])] - 3*
b*Sqrt[-f]*n*PolyLog[2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))] + 3*b*Sqrt[-f]*n*PolyLog[2, (Sqrt[g]*
(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(4*g^(5/2))

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Maple [C]  time = 0.549, size = 2021, normalized size = 3.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/g^2-b*n*x/g^2-1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g^2*x+3/2*b/g^2*f/(f*g)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g
)/e/(f*g)^(1/2))*n*ln(e*x+d)-b*n/g^2*f*ln(e*x+d)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2
)+d*g))-3/2*b/g^2*f/(f*g)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(f*g)^(1/2))*ln((e*x+d)^n)+b*ln(c)/g^2*x+3/4*
I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g^2*f/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-1/4*I*b*Pi*csgn(I*c*(e*x+d)^n)^3/g^2*f*
x/(g*x^2+f)+1/2*a/g^2*f*x/(g*x^2+f)-3/2*a/g^2*f/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))-b/e*n/g^2*d-1/4*b*e*n/g^2*
f/(d^2*g+e^2*f)*d*ln(g*(e*x+d)^2-2*d*g*(e*x+d)+d^2*g+f*e^2)-1/2*b*e^2/g^2*f/(e^2*g*x^2+e^2*f)*x*n*ln(e*x+d)-1/
2*b*e^2*n/g^2*f^2/(d^2*g+e^2*f)/(f*g)^(1/2)*arctan(1/2*(2*g*(e*x+d)-2*d*g)/e/(f*g)^(1/2))+1/2*I*b*Pi*csgn(I*c)
*csgn(I*c*(e*x+d)^n)^2/g^2*x+b/e/g^2*d*ln((e*x+d)^n)+b*ln((e*x+d)^n)/g^2*x-1/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)
^n)*csgn(I*c*(e*x+d)^n)/g^2*x+1/2*b*e^2/g^2*f/(e^2*g*x^2+e^2*f)*x*ln((e*x+d)^n)-3/4*b*n/g^2*f/(-f*g)^(1/2)*dil
og((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))+3/4*b*n/g^2*f/(-f*g)^(1/2)*dilog((e*(-f*g)^(1/2)+g*(e*
x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+1/2*b*ln(c)/g^2*f*x/(g*x^2+f)-3/2*b*ln(c)/g^2*f/(f*g)^(1/2)*arctan(x*g/(f*g)^(
1/2))+1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2/g^2*x+b*n/g^2*f*ln(e*x+d)/(-f*g)^(1/2)*ln((e*(-f*g)^(
1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))-3/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/g^2*f/(f*g)^(1/2)*arctan(
x*g/(f*g)^(1/2))-1/4*b*e^4*n/g*f^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g
*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))*x^2+1/4*b*e^4*n/g*f^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/
2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))*x^2+1/4*b*e^2*n/g*f^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*
g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2)+d*g))*d^2-1/4*b*e^2*n/g*f^2*ln(e*x
+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))*d^2+1
/4*b*e^2*n*f*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g
)^(1/2)+d*g))*d^2*x^2-1/4*b*e^2*n*f*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+
g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))*d^2*x^2+1/2*b*e^4*n/g^2*f^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)*x+1
/2*b*e^3*n/g^2*f^2*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)*d-3/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n
)^2/g^2*f/(f*g)^(1/2)*arctan(x*g/(f*g)^(1/2))+1/2*b*e^2*n/g*f*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)*d^2*x+
1/4*I*b*Pi*csgn(I*c)*csgn(I*c*(e*x+d)^n)^2/g^2*f*x/(g*x^2+f)+1/4*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^
2/g^2*f*x/(g*x^2+f)+1/2*b*e^3*n/g*f*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)*d*x^2-1/4*b*e^4*n/g^2*f^3*ln(e*x
+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)+g*(e*x+d)-d*g)/(e*(-f*g)^(1/2)-d*g))+1/4*b
*e^4*n/g^2*f^3*ln(e*x+d)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)/(-f*g)^(1/2)*ln((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f
*g)^(1/2)+d*g))+3/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/g^2*f/(f*g)^(1/2)*arctan(x*g/(f*g)^
(1/2))-1/4*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)/g^2*f*x/(g*x^2+f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*x^4*log((e*x + d)^n*c) + a*x^4)/(g^2*x^4 + 2*f*g*x^2 + f^2), 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*(a+b*ln(c*(e*x+d)**n))/(g*x**2+f)**2,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 )} x^{4}}{{\left (g x^{2} + f\right )}^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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