\(\int \frac {x^2 (a+b \log (c (d+e x)^n))}{(f+g x^2)^2} \, dx\) [272]

Optimal result

Integrand size = 27, antiderivative size = 491 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {b e n \log (d+e x)}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {b e n \log (d+e x)}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 \left (e \sqrt {-f}+d \sqrt {g}\right ) g^{3/2}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 \left (e \sqrt {-f}-d \sqrt {g}\right ) g^{3/2}}-\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.60 (sec) , antiderivative size = 491, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {294, 211, 2463, 2456, 2442, 36, 31, 2441, 2440, 2438} \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}-\sqrt {g} x\right )}-\frac {a+b \log \left (c (d+e x)^n\right )}{4 g^{3/2} \left (\sqrt {-f}+\sqrt {g} x\right )}+\frac {\log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{d \sqrt {g}+e \sqrt {-f}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 \sqrt {-f} g^{3/2}}-\frac {\log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{4 \sqrt {-f} g^{3/2}}-\frac {b n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{\sqrt {g} d+e \sqrt {-f}}\right )}{4 \sqrt {-f} g^{3/2}}+\frac {b e n \log (d+e x)}{4 g^{3/2} \left (e \sqrt {-f}-d \sqrt {g}\right )}-\frac {b e n \log (d+e x)}{4 g^{3/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}+\frac {b e n \log \left (\sqrt {-f}-\sqrt {g} x\right )}{4 g^{3/2} \left (d \sqrt {g}+e \sqrt {-f}\right )}-\frac {b e n \log \left (\sqrt {-f}+\sqrt {g} x\right )}{4 g^{3/2} \left (e \sqrt {-f}-d \sqrt {g}\right )} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 382, normalized size of antiderivative = 0.78 \[ \int \frac {x^2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{\left (f+g x^2\right )^2} \, dx=\frac {\frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}-\sqrt {g} x}-\frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {-f}+\sqrt {g} x}+\frac {b e n \left (-\log (d+e x)+\log \left (\sqrt {-f}-\sqrt {g} x\right )\right )}{e \sqrt {-f}+d \sqrt {g}}+\frac {\left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}-\sqrt {g} x\right )}{e \sqrt {-f}+d \sqrt {g}}\right )}{\sqrt {-f}}+\frac {b e n \left (\log (d+e x)-\log \left (\sqrt {-f}+\sqrt {g} x\right )\right )}{e \sqrt {-f}-d \sqrt {g}}+\frac {f \left (a+b \log \left (c (d+e x)^n\right )\right ) \log \left (\frac {e \left (\sqrt {-f}+\sqrt {g} x\right )}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {b f n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}-d \sqrt {g}}\right )}{(-f)^{3/2}}+\frac {b n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (d+e x)}{e \sqrt {-f}+d \sqrt {g}}\right )}{\sqrt {-f}}}{4 g^{3/2}} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

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

Time = 1.31 (sec) , antiderivative size = 1521, normalized size of antiderivative = 3.10

method	result	size
risch	\(\text {Expression too large to display}\)	\(1521\)

[In]

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

[Out]

1/2*b*e^2/(e^2*g*x^2+e^2*f)/g*x*n*ln(e*x+d)-1/2*b*e^2/(e^2*g*x^2+e^2*f)/g*x*ln((e*x+d)^n)-1/2*b/g/(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*b/g/(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*n/g/(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*
f/g/(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))-1/4*b*e^2*n*g*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
*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+1/4*b*e^2*n*g*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*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^2-1/2*b*e^3*n*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^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^2-1/4*b*e^4*n*f^2/g*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^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+1/4*b*e^4*n*f^2/g*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*ln(e*x+d)/(d^2*g+e^2*f
)/(e^2*g*x^2+e^2*f)*x*d^2-1/2*b*e^4*n*f/g*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*f/g*ln(e*x+d
)/(d^2*g+e^2*f)/(e^2*g*x^2+e^2*f)*d+1/4*b*n/g/(-f*g)^(1/2)*dilog((e*(-f*g)^(1/2)-g*(e*x+d)+d*g)/(e*(-f*g)^(1/2
)+d*g))-1/4*b*n/g/(-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*n/g*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*n/g*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*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)*csgn(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)*(-1/2/g*x/(g*x^2+f)+1/2/g/(f*g)^(1/2)*arctan(g*x/(f*g)^(1/2)))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

integrate(x^2*(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^2/(g*x^2 + f)^2, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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