3.4.10 \(\int \frac {x^5 (a+b \log (c (d+e x)^n))^2}{f+g x^2} \, dx\) [310]

Optimal. Leaf size=831 \[ -\frac {2 a b d f n x}{e g^2}+\frac {2 b^2 d f n^2 x}{e g^2}-\frac {2 b^2 d^3 n^2 x}{e^3 g}-\frac {b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac {3 b^2 d^2 n^2 (d+e x)^2}{4 e^4 g}-\frac {2 b^2 d n^2 (d+e x)^3}{9 e^4 g}+\frac {b^2 n^2 (d+e x)^4}{32 e^4 g}+\frac {b^2 d^4 n^2 \log ^2(d+e x)}{4 e^4 g}-\frac {2 b^2 d f n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}+\frac {2 b d^3 n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{e^4 g}+\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}-\frac {3 b d^2 n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^4 g}+\frac {2 b d n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right )}{3 e^4 g}-\frac {b n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 e^4 g}-\frac {b d^4 n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^4 g}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac {d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac {f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{2 g^3}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3}+\frac {b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}+\frac {b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^3}-\frac {b^2 f^2 n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b^2 f^2 n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^3} \]

[Out]

-2*a*b*d*f*n*x/e/g^2+2*b^2*d*f*n^2*x/e/g^2-2*b^2*d^3*n^2*x/e^3/g-1/4*b^2*f*n^2*(e*x+d)^2/e^2/g^2+3/4*b^2*d^2*n
^2*(e*x+d)^2/e^4/g-2/9*b^2*d*n^2*(e*x+d)^3/e^4/g+1/32*b^2*n^2*(e*x+d)^4/e^4/g+1/4*b^2*d^4*n^2*ln(e*x+d)^2/e^4/
g-2*b^2*d*f*n*(e*x+d)*ln(c*(e*x+d)^n)/e^2/g^2+2*b*d^3*n*(e*x+d)*(a+b*ln(c*(e*x+d)^n))/e^4/g+1/2*b*f*n*(e*x+d)^
2*(a+b*ln(c*(e*x+d)^n))/e^2/g^2-3/2*b*d^2*n*(e*x+d)^2*(a+b*ln(c*(e*x+d)^n))/e^4/g+2/3*b*d*n*(e*x+d)^3*(a+b*ln(
c*(e*x+d)^n))/e^4/g-1/8*b*n*(e*x+d)^4*(a+b*ln(c*(e*x+d)^n))/e^4/g-1/2*b*d^4*n*ln(e*x+d)*(a+b*ln(c*(e*x+d)^n))/
e^4/g+1/4*x^4*(a+b*ln(c*(e*x+d)^n))^2/g+d*f*(e*x+d)*(a+b*ln(c*(e*x+d)^n))^2/e^2/g^2-1/2*f*(e*x+d)^2*(a+b*ln(c*
(e*x+d)^n))^2/e^2/g^2+1/2*f^2*(a+b*ln(c*(e*x+d)^n))^2*ln(e*((-f)^(1/2)-x*g^(1/2))/(e*(-f)^(1/2)+d*g^(1/2)))/g^
3+1/2*f^2*(a+b*ln(c*(e*x+d)^n))^2*ln(e*((-f)^(1/2)+x*g^(1/2))/(e*(-f)^(1/2)-d*g^(1/2)))/g^3+b*f^2*n*(a+b*ln(c*
(e*x+d)^n))*polylog(2,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))/g^3+b*f^2*n*(a+b*ln(c*(e*x+d)^n))*polylog(2,(
e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))/g^3-b^2*f^2*n^2*polylog(3,-(e*x+d)*g^(1/2)/(e*(-f)^(1/2)-d*g^(1/2)))/
g^3-b^2*f^2*n^2*polylog(3,(e*x+d)*g^(1/2)/(e*(-f)^(1/2)+d*g^(1/2)))/g^3

________________________________________________________________________________________

Rubi [A]
time = 0.79, antiderivative size = 831, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 19, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.655, Rules used = {2463, 2448, 2436, 2333, 2332, 2437, 2342, 2341, 2445, 2458, 45, 2372, 12, 14, 2338, 2443, 2481, 2421, 6724} \begin {gather*} \frac {b^2 n^2 \log ^2(d+e x) d^4}{4 e^4 g}-\frac {b n \log (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^4}{2 e^4 g}-\frac {2 b^2 n^2 x d^3}{e^3 g}+\frac {2 b n (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right ) d^3}{e^4 g}+\frac {3 b^2 n^2 (d+e x)^2 d^2}{4 e^4 g}-\frac {3 b n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right ) d^2}{2 e^4 g}-\frac {2 b^2 n^2 (d+e x)^3 d}{9 e^4 g}+\frac {f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 d}{e^2 g^2}+\frac {2 b^2 f n^2 x d}{e g^2}-\frac {2 a b f n x d}{e g^2}-\frac {2 b^2 f n (d+e x) \log \left (c (d+e x)^n\right ) d}{e^2 g^2}+\frac {2 b n (d+e x)^3 \left (a+b \log \left (c (d+e x)^n\right )\right ) d}{3 e^4 g}+\frac {b^2 n^2 (d+e x)^4}{32 e^4 g}-\frac {b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac {f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}-\frac {b n (d+e x)^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{8 e^4 g}+\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{\sqrt {g} d+e \sqrt {-f}}\right )}{2 g^3}+\frac {f^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (\frac {e \left (\sqrt {g} x+\sqrt {-f}\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{2 g^3}+\frac {b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}+\frac {b f^2 n \left (a+b \log \left (c (d+e x)^n\right )\right ) \text {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{g^3}-\frac {b^2 f^2 n^2 \text {PolyLog}\left (3,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b^2 f^2 n^2 \text {PolyLog}\left (3,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{g^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(x^5*(a + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2),x]

[Out]

(-2*a*b*d*f*n*x)/(e*g^2) + (2*b^2*d*f*n^2*x)/(e*g^2) - (2*b^2*d^3*n^2*x)/(e^3*g) - (b^2*f*n^2*(d + e*x)^2)/(4*
e^2*g^2) + (3*b^2*d^2*n^2*(d + e*x)^2)/(4*e^4*g) - (2*b^2*d*n^2*(d + e*x)^3)/(9*e^4*g) + (b^2*n^2*(d + e*x)^4)
/(32*e^4*g) + (b^2*d^4*n^2*Log[d + e*x]^2)/(4*e^4*g) - (2*b^2*d*f*n*(d + e*x)*Log[c*(d + e*x)^n])/(e^2*g^2) +
(2*b*d^3*n*(d + e*x)*(a + b*Log[c*(d + e*x)^n]))/(e^4*g) + (b*f*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e
^2*g^2) - (3*b*d^2*n*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n]))/(2*e^4*g) + (2*b*d*n*(d + e*x)^3*(a + b*Log[c*(d
+ e*x)^n]))/(3*e^4*g) - (b*n*(d + e*x)^4*(a + b*Log[c*(d + e*x)^n]))/(8*e^4*g) - (b*d^4*n*Log[d + e*x]*(a + b*
Log[c*(d + e*x)^n]))/(2*e^4*g) + (x^4*(a + b*Log[c*(d + e*x)^n])^2)/(4*g) + (d*f*(d + e*x)*(a + b*Log[c*(d + e
*x)^n])^2)/(e^2*g^2) - (f*(d + e*x)^2*(a + b*Log[c*(d + e*x)^n])^2)/(2*e^2*g^2) + (f^2*(a + b*Log[c*(d + e*x)^
n])^2*Log[(e*(Sqrt[-f] - Sqrt[g]*x))/(e*Sqrt[-f] + d*Sqrt[g])])/(2*g^3) + (f^2*(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^3) + (b*f^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[
2, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d*Sqrt[g]))])/g^3 + (b*f^2*n*(a + b*Log[c*(d + e*x)^n])*PolyLog[2, (Sqr
t[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3 - (b^2*f^2*n^2*PolyLog[3, -((Sqrt[g]*(d + e*x))/(e*Sqrt[-f] - d
*Sqrt[g]))])/g^3 - (b^2*f^2*n^2*PolyLog[3, (Sqrt[g]*(d + e*x))/(e*Sqrt[-f] + d*Sqrt[g])])/g^3

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2332

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

Rule 2333

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 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2341

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 2342

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 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2421

Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> Simp
[(-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*L
og[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 2436

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 2437

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 2443

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 2445

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f
+ g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])^p/(g*(q + 1))), x] - Dist[b*e*n*(p/(g*(q + 1))), Int[(f + g*x)^(q
+ 1)*((a + b*Log[c*(d + e*x)^n])^(p - 1)/(d + e*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*
f - d*g, 0] && GtQ[p, 0] && NeQ[q, -1] && IntegersQ[2*p, 2*q] && ( !IGtQ[q, 0] || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2448

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 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[(g*(x/e))^q*((e*h - d*i)/e + i*(x/e))^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

Rule 2463

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 2481

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 6724

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^5 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx &=\int \left (-\frac {f x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2}+\frac {x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g}+\frac {f^2 x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{g^2 \left (f+g x^2\right )}\right ) \, dx\\ &=-\frac {f \int x \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{g^2}+\frac {f^2 \int \frac {x \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{f+g x^2} \, dx}{g^2}+\frac {\int x^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{g}\\ &=\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac {f \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^2}+\frac {f^2 \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^2}-\frac {(b e n) \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{d+e x} \, dx}{2 g}\\ &=\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}-\frac {f^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}-\sqrt {g} x} \, dx}{2 g^{5/2}}+\frac {f^2 \int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^2}{\sqrt {-f}+\sqrt {g} x} \, dx}{2 g^{5/2}}-\frac {f \int (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{e g^2}+\frac {(d f) \int \left (a+b \log \left (c (d+e x)^n\right )\right )^2 \, dx}{e g^2}-\frac {(b n) \text {Subst}\left (\int \frac {\left (-\frac {d}{e}+\frac {x}{e}\right )^4 \left (a+b \log \left (c x^n\right )\right )}{x} \, dx,x,d+e x\right )}{2 g}\\ &=\frac {b n \left (\frac {48 d^3 (d+e x)}{e^4}-\frac {36 d^2 (d+e x)^2}{e^4}+\frac {16 d (d+e x)^3}{e^4}-\frac {3 (d+e x)^4}{e^4}-\frac {12 d^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac {f^2 \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^3}+\frac {f^2 \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^3}-\frac {f \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2 g^2}+\frac {(d f) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^2 \, dx,x,d+e x\right )}{e^2 g^2}-\frac {\left (b e f^2 n\right ) \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^3}-\frac {\left (b e f^2 n\right ) \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^3}+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {x \left (-48 d^3+36 d^2 x-16 d x^2+3 x^3\right )+12 d^4 \log (x)}{12 e^4 x} \, dx,x,d+e x\right )}{2 g}\\ &=\frac {b n \left (\frac {48 d^3 (d+e x)}{e^4}-\frac {36 d^2 (d+e x)^2}{e^4}+\frac {16 d (d+e x)^3}{e^4}-\frac {3 (d+e x)^4}{e^4}-\frac {12 d^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac {d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac {f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}+\frac {f^2 \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^3}+\frac {f^2 \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^3}-\frac {\left (b f^2 n\right ) \text {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^3}-\frac {\left (b f^2 n\right ) \text {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^3}+\frac {(b f n) \text {Subst}\left (\int x \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2 g^2}-\frac {(2 b d f n) \text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right ) \, dx,x,d+e x\right )}{e^2 g^2}+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \frac {x \left (-48 d^3+36 d^2 x-16 d x^2+3 x^3\right )+12 d^4 \log (x)}{x} \, dx,x,d+e x\right )}{24 e^4 g}\\ &=-\frac {2 a b d f n x}{e g^2}-\frac {b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac {b n \left (\frac {48 d^3 (d+e x)}{e^4}-\frac {36 d^2 (d+e x)^2}{e^4}+\frac {16 d (d+e x)^3}{e^4}-\frac {3 (d+e x)^4}{e^4}-\frac {12 d^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac {d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac {f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}+\frac {f^2 \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^3}+\frac {f^2 \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^3}+\frac {b f^2 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^3}+\frac {b f^2 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^3}-\frac {\left (2 b^2 d f n\right ) \text {Subst}\left (\int \log \left (c x^n\right ) \, dx,x,d+e x\right )}{e^2 g^2}-\frac {\left (b^2 f^2 n^2\right ) \text {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^3}-\frac {\left (b^2 f^2 n^2\right ) \text {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^3}+\frac {\left (b^2 n^2\right ) \text {Subst}\left (\int \left (-48 d^3+36 d^2 x-16 d x^2+3 x^3+\frac {12 d^4 \log (x)}{x}\right ) \, dx,x,d+e x\right )}{24 e^4 g}\\ &=-\frac {2 a b d f n x}{e g^2}+\frac {2 b^2 d f n^2 x}{e g^2}-\frac {2 b^2 d^3 n^2 x}{e^3 g}-\frac {b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac {3 b^2 d^2 n^2 (d+e x)^2}{4 e^4 g}-\frac {2 b^2 d n^2 (d+e x)^3}{9 e^4 g}+\frac {b^2 n^2 (d+e x)^4}{32 e^4 g}-\frac {2 b^2 d f n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}+\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac {b n \left (\frac {48 d^3 (d+e x)}{e^4}-\frac {36 d^2 (d+e x)^2}{e^4}+\frac {16 d (d+e x)^3}{e^4}-\frac {3 (d+e x)^4}{e^4}-\frac {12 d^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac {d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac {f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}+\frac {f^2 \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^3}+\frac {f^2 \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^3}+\frac {b f^2 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^3}+\frac {b f^2 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^3}-\frac {b^2 f^2 n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b^2 f^2 n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^3}+\frac {\left (b^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,d+e x\right )}{2 e^4 g}\\ &=-\frac {2 a b d f n x}{e g^2}+\frac {2 b^2 d f n^2 x}{e g^2}-\frac {2 b^2 d^3 n^2 x}{e^3 g}-\frac {b^2 f n^2 (d+e x)^2}{4 e^2 g^2}+\frac {3 b^2 d^2 n^2 (d+e x)^2}{4 e^4 g}-\frac {2 b^2 d n^2 (d+e x)^3}{9 e^4 g}+\frac {b^2 n^2 (d+e x)^4}{32 e^4 g}+\frac {b^2 d^4 n^2 \log ^2(d+e x)}{4 e^4 g}-\frac {2 b^2 d f n (d+e x) \log \left (c (d+e x)^n\right )}{e^2 g^2}+\frac {b f n (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{2 e^2 g^2}+\frac {b n \left (\frac {48 d^3 (d+e x)}{e^4}-\frac {36 d^2 (d+e x)^2}{e^4}+\frac {16 d (d+e x)^3}{e^4}-\frac {3 (d+e x)^4}{e^4}-\frac {12 d^4 \log (d+e x)}{e^4}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{24 g}+\frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{4 g}+\frac {d f (d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{e^2 g^2}-\frac {f (d+e x)^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{2 e^2 g^2}+\frac {f^2 \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^3}+\frac {f^2 \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^3}+\frac {b f^2 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^3}+\frac {b f^2 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^3}-\frac {b^2 f^2 n^2 \text {Li}_3\left (-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{g^3}-\frac {b^2 f^2 n^2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{g^3}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.60, size = 862, normalized size = 1.04 \begin {gather*} \frac {-144 e^4 f g x^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+72 e^4 g^2 x^4 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2+144 e^4 f^2 \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )^2 \log \left (f+g x^2\right )-12 b n \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right ) \left (12 e^2 f g \left (e x (2 d-e x)-2 \left (d^2-e^2 x^2\right ) \log (d+e x)\right )+g^2 \left (e x \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+12 \left (d^4-e^4 x^4\right ) \log (d+e x)\right )-24 e^4 f^2 \left (\log (d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+\text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )-24 e^4 f^2 \left (\log (d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+\text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )\right )+b^2 n^2 \left (-72 e^2 f g \left (e x (-6 d+e x)+\left (6 d^2+4 d e x-2 e^2 x^2\right ) \log (d+e x)-2 \left (d^2-e^2 x^2\right ) \log ^2(d+e x)\right )-g^2 \left (e x \left (300 d^3-78 d^2 e x+28 d e^2 x^2-9 e^3 x^3\right )-12 \left (25 d^4+12 d^3 e x-6 d^2 e^2 x^2+4 d e^3 x^3-3 e^4 x^4\right ) \log (d+e x)+72 \left (d^4-e^4 x^4\right ) \log ^2(d+e x)\right )+144 e^4 f^2 \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{-i e \sqrt {f}+d \sqrt {g}}\right )\right )+144 e^4 f^2 \left (\log ^2(d+e x) \log \left (1-\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )+2 \log (d+e x) \text {Li}_2\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )-2 \text {Li}_3\left (\frac {\sqrt {g} (d+e x)}{i e \sqrt {f}+d \sqrt {g}}\right )\right )\right )}{288 e^4 g^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(x^5*(a + b*Log[c*(d + e*x)^n])^2)/(f + g*x^2),x]

[Out]

(-144*e^4*f*g*x^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2 + 72*e^4*g^2*x^4*(a - b*n*Log[d + e*x] + b*L
og[c*(d + e*x)^n])^2 + 144*e^4*f^2*(a - b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])^2*Log[f + g*x^2] - 12*b*n*(a
- b*n*Log[d + e*x] + b*Log[c*(d + e*x)^n])*(12*e^2*f*g*(e*x*(2*d - e*x) - 2*(d^2 - e^2*x^2)*Log[d + e*x]) + g^
2*(e*x*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 12*(d^4 - e^4*x^4)*Log[d + e*x]) - 24*e^4*f^2*(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])]) - 24*e^4*f^2*(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*(-72*e^2*f*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) - g^2*(e*x*(300*d^3 - 78*d^2*e*x + 28*d*e^2*x
^2 - 9*e^3*x^3) - 12*(25*d^4 + 12*d^3*e*x - 6*d^2*e^2*x^2 + 4*d*e^3*x^3 - 3*e^4*x^4)*Log[d + e*x] + 72*(d^4 -
e^4*x^4)*Log[d + e*x]^2) + 144*e^4*f^2*(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])]) + 144*e^4*f^2*(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])])))/(288*e^4*g^3)

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {x^{5} \left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )^{2}}{g \,x^{2}+f}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f),x)

[Out]

int(x^5*(a+b*ln(c*(e*x+d)^n))^2/(g*x^2+f),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f),x, algorithm="maxima")

[Out]

1/4*a^2*(2*f^2*log(g*x^2 + f)/g^3 + (g*x^4 - 2*f*x^2)/g^2) + integrate((b^2*x^5*log((x*e + d)^n)^2 + 2*(b^2*lo
g(c) + a*b)*x^5*log((x*e + d)^n) + (b^2*log(c)^2 + 2*a*b*log(c))*x^5)/(g*x^2 + f), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f),x, algorithm="fricas")

[Out]

integral((b^2*x^5*log((x*e + d)^n*c)^2 + 2*a*b*x^5*log((x*e + d)^n*c) + a^2*x^5)/(g*x^2 + f), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**5*(a+b*ln(c*(e*x+d)**n))**2/(g*x**2+f),x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^5*(a+b*log(c*(e*x+d)^n))^2/(g*x^2+f),x, algorithm="giac")

[Out]

integrate((b*log((x*e + d)^n*c) + a)^2*x^5/(g*x^2 + f), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^5\,{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^2}{g\,x^2+f} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^5*(a + b*log(c*(d + e*x)^n))^2)/(f + g*x^2),x)

[Out]

int((x^5*(a + b*log(c*(d + e*x)^n))^2)/(f + g*x^2), x)

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