3.313 \(\int \frac{x^2 (a+b \log (c x^n))}{\sqrt{d-e x} \sqrt{d+e x}} \, dx\)

Optimal. Leaf size=406 \[ \frac{i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}\left (\frac{e x}{d}\right )}\right )}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b n x \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )^2}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{e x}{d}\right )}\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.612182, antiderivative size = 406, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 12, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2342, 321, 216, 2350, 12, 14, 195, 4625, 3717, 2190, 2279, 2391} \[ \frac{i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{PolyLog}\left (2,e^{2 i \sin ^{-1}\left (\frac{e x}{d}\right )}\right )}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b n x \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )^2}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{e x}{d}\right )}\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(q_)*((d2_) + (e2_.)*(x_))^(q_), x_
Symbol] :> Dist[((d1 + e1*x)^q*(d2 + e2*x)^q)/(1 + (e1*e2*x^2)/(d1*d2))^q, Int[x^m*(1 + (e1*e2*x^2)/(d1*d2))^q
*(a + b*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[d2*e1 + d1*e2, 0] && IntegerQ[m]
&& IntegerQ[q - 1/2]

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 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 2350

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = IntHide[(f*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] /; ((EqQ[r, 1] || EqQ[r, 2]) && IntegerQ[m] && IntegerQ[q - 1/2]) || InverseFunctionFreeQ[u, x]] /
; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && IntegerQ[2*q] && ((IntegerQ[m] && IntegerQ[r]) || IGtQ[q, 0])

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 4625

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Tan[x], x], x, ArcSin[c*
x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 3717

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Simp[(I*(c + d*x)^(m + 1))/(d*
(m + 1)), x] - Dist[2*I, Int[((c + d*x)^m*E^(2*I*k*Pi)*E^(2*I*(e + f*x)))/(1 + E^(2*I*k*Pi)*E^(2*I*(e + f*x)))
, x], x] /; FreeQ[{c, d, e, f}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 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^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}} \, dx &=\frac{\sqrt{1-\frac{e^2 x^2}{d^2}} \int \frac{x^2 \left (a+b \log \left (c x^n\right )\right )}{\sqrt{1-\frac{e^2 x^2}{d^2}}} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{d^2 \left (-e x \sqrt{\frac{d^2-e^2 x^2}{d^2}}+d \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{2 e^3 x} \, dx}{\sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b d^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{-e x \sqrt{\frac{d^2-e^2 x^2}{d^2}}+d \sin ^{-1}\left (\frac{e x}{d}\right )}{x} \, dx}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b d^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \left (-e \sqrt{1-\frac{e^2 x^2}{d^2}}+\frac{d \sin ^{-1}\left (\frac{e x}{d}\right )}{x}\right ) \, dx}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}\\ &=-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{\sin ^{-1}\left (\frac{e x}{d}\right )}{x} \, dx}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b d^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \sqrt{1-\frac{e^2 x^2}{d^2}} \, dx}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{b n x \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int x \cot (x) \, dx,x,\sin ^{-1}\left (\frac{e x}{d}\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b d^2 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \int \frac{1}{\sqrt{1-\frac{e^2 x^2}{d^2}}} \, dx}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{b n x \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )^2}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{e^{2 i x} x}{1-e^{2 i x}} \, dx,x,\sin ^{-1}\left (\frac{e x}{d}\right )\right )}{e^3 \sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{b n x \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )^2}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{e x}{d}\right )}\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{\left (b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}\left (\frac{e x}{d}\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{b n x \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )^2}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{e x}{d}\right )}\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{\left (i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}}\right ) \operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 i \sin ^{-1}\left (\frac{e x}{d}\right )}\right )}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}\\ &=\frac{b n x \left (d^2-e^2 x^2\right )}{4 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )^2}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \log \left (1-e^{2 i \sin ^{-1}\left (\frac{e x}{d}\right )}\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}-\frac{x \left (d^2-e^2 x^2\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^2 \sqrt{d-e x} \sqrt{d+e x}}+\frac{d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 e^3 \sqrt{d-e x} \sqrt{d+e x}}+\frac{i b d^3 n \sqrt{1-\frac{e^2 x^2}{d^2}} \text{Li}_2\left (e^{2 i \sin ^{-1}\left (\frac{e x}{d}\right )}\right )}{4 e^3 \sqrt{d-e x} \sqrt{d+e x}}\\ \end{align*}

Mathematica [A]  time = 2.72016, size = 316, normalized size = 0.78 \[ \frac{\frac{b n \left (\frac{e^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \left (-\text{PolyLog}\left (2,e^{-2 \sinh ^{-1}\left (x \sqrt{-\frac{e^2}{d^2}}\right )}\right )-2 \log (x) \log \left (\sqrt{1-\frac{e^2 x^2}{d^2}}+x \sqrt{-\frac{e^2}{d^2}}\right )+\sinh ^{-1}\left (x \sqrt{-\frac{e^2}{d^2}}\right )^2+2 \sinh ^{-1}\left (x \sqrt{-\frac{e^2}{d^2}}\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (x \sqrt{-\frac{e^2}{d^2}}\right )}\right )\right )}{\left (-\frac{e^2}{d^2}\right )^{3/2}}+e x (2 \log (x)-1) \left (e^2 x^2-d^2\right )+d^3 \sqrt{1-\frac{e^2 x^2}{d^2}} \sin ^{-1}\left (\frac{e x}{d}\right )\right )}{\sqrt{d-e x} \sqrt{d+e x}}+2 d^2 \tan ^{-1}\left (\frac{e x}{\sqrt{d-e x} \sqrt{d+e x}}\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-2 e x \sqrt{d-e x} \sqrt{d+e x} \left (a+b \log \left (c x^n\right )-b n \log (x)\right )}{4 e^3} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(-2*e*x*Sqrt[d - e*x]*Sqrt[d + e*x]*(a - b*n*Log[x] + b*Log[c*x^n]) + 2*d^2*ArcTan[(e*x)/(Sqrt[d - e*x]*Sqrt[d
+ e*x])]*(a - b*n*Log[x] + b*Log[c*x^n]) + (b*n*(d^3*Sqrt[1 - (e^2*x^2)/d^2]*ArcSin[(e*x)/d] + e*x*(-d^2 + e^
2*x^2)*(-1 + 2*Log[x]) + (e^3*Sqrt[1 - (e^2*x^2)/d^2]*(ArcSinh[Sqrt[-(e^2/d^2)]*x]^2 + 2*ArcSinh[Sqrt[-(e^2/d^
2)]*x]*Log[1 - E^(-2*ArcSinh[Sqrt[-(e^2/d^2)]*x])] - 2*Log[x]*Log[Sqrt[-(e^2/d^2)]*x + Sqrt[1 - (e^2*x^2)/d^2]
] - PolyLog[2, E^(-2*ArcSinh[Sqrt[-(e^2/d^2)]*x])]))/(-(e^2/d^2))^(3/2)))/(Sqrt[d - e*x]*Sqrt[d + e*x]))/(4*e^
3)

________________________________________________________________________________________

Maple [F]  time = 0.643, size = 0, normalized size = 0. \begin{align*} \int{{x}^{2} \left ( a+b\ln \left ( c{x}^{n} \right ) \right ){\frac{1}{\sqrt{-ex+d}}}{\frac{1}{\sqrt{ex+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a+b*ln(c*x^n))/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

[Out]

int(x^2*(a+b*ln(c*x^n))/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

integral(-(sqrt(e*x + d)*sqrt(-e*x + d)*b*x^2*log(c*x^n) + sqrt(e*x + d)*sqrt(-e*x + d)*a*x^2)/(e^2*x^2 - d^2)
, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(a+b*ln(c*x**n))/(-e*x+d)**(1/2)/(e*x+d)**(1/2),x)

[Out]

Integral(x**2*(a + b*log(c*x**n))/(sqrt(d - e*x)*sqrt(d + e*x)), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(a+b*log(c*x^n))/(-e*x+d)^(1/2)/(e*x+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^2/(sqrt(e*x + d)*sqrt(-e*x + d)), x)