\(\int \frac {x^4 (a+b \log (c (d+e x)^n))}{(f+g x^2)^2} \, dx\) [271]

Optimal result

Integrand size = 27, antiderivative size = 534 \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {a x}{g^2}-\frac {b n x}{g^2}-\frac {b e f n \log (d+e x)}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{5/2}}+\frac {b e f n \log (d+e x)}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{5/2}}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt {-f}+\sqrt {g} x\right )}-\frac {b e f n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{5/2}}+\frac {3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 g^{5/2}}+\frac {b e f n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{5/2}}-\frac {3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 g^{5/2}}-\frac {3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 g^{5/2}}+\frac {3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 g^{5/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 534, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {294, 327, 211, 2463, 2436, 2332, 2456, 2442, 36, 31, 2441, 2440, 2438} \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt {-f}-\sqrt {g} x\right )}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {3 \sqrt {-f} \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2}}-\frac {3 \sqrt {-f} \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 g^{5/2}}+\frac {a x}{g^2}+\frac {b (d+e x) \log \left (c (d+e x)^n\right )}{e g^2}-\frac {3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 g^{5/2}}+\frac {3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 g^{5/2}}-\frac {b e f n \log (d+e x)}{4 g^{5/2} \left (e \sqrt {-f}-d \sqrt {g}\right )}+\frac {b e f n \log (d+e x)}{4 g^{5/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}-\frac {b e f n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 g^{5/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}+\frac {b e f n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 g^{5/2} \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {b n x}{g^2} \]

[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 31

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

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 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 294

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 327

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 2332

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

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 2438

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]

Rule 2440

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 2441

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 2442

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 2456

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 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]

Rubi steps \begin{align*} \text {integral}& = \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 \text {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 ) \text {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 ) \text {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 ) \text {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 ) \text {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] (verified)

Time = 0.47 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.81 \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {4 a \sqrt {g} x-4 b \sqrt {g} n x+\frac {4 b \sqrt {g} (d+e x) \log \left (c (d+e x)^n\right )}{e}-\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {-f}-\sqrt {g} x}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {-f}+\sqrt {g} x}+\frac {b e f n \left (\log (d+e x)-\log \left (\sqrt {-f}-\sqrt {g} x\right )\right )}{e \sqrt {-f}+d \sqrt {g}}+3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )+\frac {b e f n \left (\log (d+e x)-\log \left (\sqrt {-f}+\sqrt {g} x\right )\right )}{-e \sqrt {-f}+d \sqrt {g}}-3 \sqrt {-f} \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )-3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )+3 b \sqrt {-f} n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 g^{5/2}} \]

[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))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 1.32 (sec) , antiderivative size = 1619, normalized size of antiderivative = 3.03

method	result	size
risch	\(\text {Expression too large to display}\)	\(1619\)

[In]

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

[Out]

-b*d*n/e/g^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*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))*x^2*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))*x^2*d^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^2-3/4*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))+3/4*b*n/g^2*f/(-f*g)^(1/2)*di
log((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)+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+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))+1/2*b*e^2/g^2*f*x/(e^2*g*x^2+e^2*f)*ln((e*x+d)^n)+b*ln
((e*x+d)^n)/g^2*x+b/e/g^2*d*ln((e*x+d)^n)-1/4*b*e*n/g^2*f/(d^2*g+e^2*f)*d*ln(g*(e*x+d)^2-2*(e*x+d)*d*g+d^2*g+f
*e^2)-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*b*e^2/g^
2*f*x/(e^2*g*x^2+e^2*f)*n*ln(e*x+d)+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+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+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)+(-1/2*I*b*Pi*csgn(I*c)*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)+1/2*I*b*Pi*csgn(I*c)*cs
gn(I*c*(e*x+d)^n)^2+1/2*I*b*Pi*csgn(I*(e*x+d)^n)*csgn(I*c*(e*x+d)^n)^2-1/2*I*b*Pi*csgn(I*c*(e*x+d)^n)^3+b*ln(c
)+a)*(x/g^2-1/g^2*f*(-1/2*x/(g*x^2+f)+3/2/(f*g)^(1/2)*arctan(g*x/(f*g)^(1/2))))+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)*x^2*d+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)*l
n((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))+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)*x*d^2-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))-b*n*x/g^2

Fricas [F]

\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{4}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{4}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )} x^{4}}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^4\,\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]

[In]

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

[Out]

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