3.113 \(\int \frac {(f+g x) (a+b \sin ^{-1}(c x))^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=935 \[ -\frac {i b^2 d (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {i b^2 d (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 d (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 d (e f-d g) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 (e f-d g) \log (d+e x) c^2}{e^2 \left (c^2 d^2-e^2\right )}+\frac {b^2 (e f-d g) \sqrt {1-c^2 x^2} \sin ^{-1}(c x) c}{e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b \left (2 e^2 g-c^2 d (e f+d g)\right ) \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) c}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {2 i b^2 g \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 g \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 g \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 g \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {a b (e f-d g) \sqrt {1-c^2 x^2} c}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}+\frac {a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)} \]

[Out]

a*b*g^2*arcsin(c*x)/e^2/(-d*g+e*f)+1/2*b^2*g^2*arcsin(c*x)^2/e^2/(-d*g+e*f)-1/2*(g*x+f)^2*(a+b*arcsin(c*x))^2/
(-d*g+e*f)/(e*x+d)^2-a*b*c*(2*e^2*g-c^2*d*(d*g+e*f))*arctan((c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2)
)/e^2/(c^2*d^2-e^2)^(3/2)-b^2*c^2*(-d*g+e*f)*ln(e*x+d)/e^2/(c^2*d^2-e^2)-I*b^2*c^3*d*(-d*g+e*f)*arcsin(c*x)*ln
(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^2/(c^2*d^2-e^2)^(3/2)+I*b^2*c^3*d*(-d*g+e*f)*ar
csin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^2/(c^2*d^2-e^2)^(3/2)-b^2*c^3*d*(-d
*g+e*f)*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^2/(c^2*d^2-e^2)^(3/2)+b^2*c^3*d*
(-d*g+e*f)*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^2/(c^2*d^2-e^2)^(3/2)-2*I*b^2
*c*g*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^2/(c^2*d^2-e^2)^(1/2)+2*I*b^
2*c*g*arcsin(c*x)*ln(1-I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^2/(c^2*d^2-e^2)^(1/2)-2*b^2
*c*g*polylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d-(c^2*d^2-e^2)^(1/2)))/e^2/(c^2*d^2-e^2)^(1/2)+2*b^2*c*g*pol
ylog(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^2/(c^2*d^2-e^2)^(1/2)+a*b*c*(-d*g+e*f)*(-c^
2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)/(e*x+d)+b^2*c*(-d*g+e*f)*arcsin(c*x)*(-c^2*x^2+1)^(1/2)/e/(c^2*d^2-e^2)/(e*x+d)

________________________________________________________________________________________

Rubi [A]  time = 2.93, antiderivative size = 935, normalized size of antiderivative = 1.00, number of steps used = 33, number of rules used = 20, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.870, Rules used = {37, 4755, 12, 1651, 844, 216, 725, 204, 4799, 4797, 4641, 4773, 3324, 3323, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac {i b^2 d (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {i b^2 d (e f-d g) \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 d (e f-d g) \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b^2 d (e f-d g) \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c^3}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {b^2 (e f-d g) \log (d+e x) c^2}{e^2 \left (c^2 d^2-e^2\right )}+\frac {b^2 (e f-d g) \sqrt {1-c^2 x^2} \sin ^{-1}(c x) c}{e \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {a b \left (2 e^2 g-c^2 d (e f+d g)\right ) \tan ^{-1}\left (\frac {d x c^2+e}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right ) c}{e^2 \left (c^2 d^2-e^2\right )^{3/2}}-\frac {2 i b^2 g \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 i b^2 g \sin ^{-1}(c x) \log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c}{e^2 \sqrt {c^2 d^2-e^2}}-\frac {2 b^2 g \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right ) c}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {2 b^2 g \text {PolyLog}\left (2,\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right ) c}{e^2 \sqrt {c^2 d^2-e^2}}+\frac {a b (e f-d g) \sqrt {1-c^2 x^2} c}{e \left (c^2 d^2-e^2\right ) (d+e x)}+\frac {b^2 g^2 \sin ^{-1}(c x)^2}{2 e^2 (e f-d g)}-\frac {(f+g x)^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 (e f-d g) (d+e x)^2}+\frac {a b g^2 \sin ^{-1}(c x)}{e^2 (e f-d g)} \]

Antiderivative was successfully verified.

[In]

Int[((f + g*x)*(a + b*ArcSin[c*x])^2)/(d + e*x)^3,x]

[Out]

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

Rule 12

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

Rule 31

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

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

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 725

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 844

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1651

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, d
+ e*x, x], R = PolynomialRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + c*x^2)^(p + 1))/((m + 1
)*(c*d^2 + a*e^2)), x] + Dist[1/((m + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*ExpandToSum[(m
+ 1)*(c*d^2 + a*e^2)*Q + c*d*R*(m + 1) - c*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e, p}, x] && Po
lyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[m, -1]

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 2264

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.)*(F_)^(v_)), x_Symbol] :> With[{q =
Rt[b^2 - 4*a*c, 2]}, Dist[(2*c)/q, Int[((f + g*x)^m*F^u)/(b - q + 2*c*F^u), x], x] - Dist[(2*c)/q, Int[((f +
g*x)^m*F^u)/(b + q + 2*c*F^u), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[v, 2*u] && LinearQ[u, x] && NeQ[
b^2 - 4*a*c, 0] && 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]

Rule 2668

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 3323

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[2, Int[((c + d*x)^m*E
^(I*(e + f*x)))/(I*b + 2*a*E^(I*(e + f*x)) - I*b*E^(2*I*(e + f*x))), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&
NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 3324

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*(c + d*x)^m*Cos[
e + f*x])/(f*(a^2 - b^2)*(a + b*Sin[e + f*x])), x] + (Dist[a/(a^2 - b^2), Int[(c + d*x)^m/(a + b*Sin[e + f*x])
, x], x] - Dist[(b*d*m)/(f*(a^2 - b^2)), Int[((c + d*x)^(m - 1)*Cos[e + f*x])/(a + b*Sin[e + f*x]), x], x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 4755

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(p_.), x_Symbol] :>
With[{u = IntHide[(f + g*x)^p*(d + e*x)^m, x]}, Dist[(a + b*ArcSin[c*x])^n, u, x] - Dist[b*c*n, Int[SimplifyIn
tegrand[(u*(a + b*ArcSin[c*x])^(n - 1))/Sqrt[1 - c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && I
GtQ[n, 0] && IGtQ[p, 0] && ILtQ[m, 0] && LtQ[m + p + 1, 0]

Rule 4773

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol]
:> Dist[1/(c^(m + 1)*Sqrt[d]), Subst[Int[(a + b*x)^n*(c*f + g*Sin[x])^m, x], x, ArcSin[c*x]], x] /; FreeQ[{a,
b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[m] && GtQ[d, 0] && (GtQ[m, 0] || IGtQ[n, 0])

Rule 4797

Int[ArcSin[(c_.)*(x_)]^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> With[{u = ExpandIntegrand[(d + e*
x^2)^p*ArcSin[c*x]^n, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{c, d, e}, x] && RationalFunctionQ[RFx, x] && I
GtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rule 4799

Int[(ArcSin[(c_.)*(x_)]*(b_.) + (a_))^(n_.)*(RFx_)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegran
d[(d + e*x^2)^p, RFx*(a + b*ArcSin[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && RationalFunctionQ[RFx, x] &
& IGtQ[n, 0] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2]

Rubi steps

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

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Mathematica [A]  time = 1.70, size = 574, normalized size = 0.61 \[ \frac {\frac {2 b c (e f-d g) \left (-i c^2 d (d+e x) \left (\left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )-\log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )\right )-i b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+i b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )+e \sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2} \left (a+b \sin ^{-1}(c x)\right )-b c \sqrt {c^2 d^2-e^2} (d+e x) \log (d+e x)\right )}{\left (c^2 d^2-e^2\right )^{3/2} (d+e x)}+\frac {4 b c g \left (-i \left (a+b \sin ^{-1}(c x)\right ) \left (\log \left (1+\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}-c d}\right )-\log \left (1-\frac {i e e^{i \sin ^{-1}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )\right )-b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )+b \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{\sqrt {c^2 d^2-e^2}}-\frac {(e f-d g) \left (a+b \sin ^{-1}(c x)\right )^2}{(d+e x)^2}-\frac {2 g \left (a+b \sin ^{-1}(c x)\right )^2}{d+e x}}{2 e^2} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((f + g*x)*(a + b*ArcSin[c*x])^2)/(d + e*x)^3,x]

[Out]

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

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fricas [F]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{2} g x + a^{2} f + {\left (b^{2} g x + b^{2} f\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b g x + a b f\right )} \arcsin \left (c x\right )}{e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arcsin(c*x))^2/(e*x+d)^3,x, algorithm="fricas")

[Out]

integral((a^2*g*x + a^2*f + (b^2*g*x + b^2*f)*arcsin(c*x)^2 + 2*(a*b*g*x + a*b*f)*arcsin(c*x))/(e^3*x^3 + 3*d*
e^2*x^2 + 3*d^2*e*x + d^3), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (e x + d\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arcsin(c*x))^2/(e*x+d)^3,x, algorithm="giac")

[Out]

integrate((g*x + f)*(b*arcsin(c*x) + a)^2/(e*x + d)^3, x)

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maple [B]  time = 1.76, size = 3096, normalized size = 3.31 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*x+f)*(a+b*arcsin(c*x))^2/(e*x+d)^3,x)

[Out]

c^3*b^2/e*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*d*f*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^
2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-c^2*a*b/e^2/(c^2*d^2-e^2)/(c*x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x
+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*d*g-c^3*b^2*arcsin(c*x)/(c^2*d^2-e^2)/(c*e*x+c*d)^2*(-c^2*x^2+1)^(1/2)*x*d*g-
c^4*b^2*arcsin(c*x)^2/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*x*d^2*g+c^3*b^2/e^2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*g
*d^2*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-c^
3*b^2/e*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*d*f*arcsin(c*x)*ln((-I*d*c-(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2
+e^2)^(1/2))/(-I*d*c+(-c^2*d^2+e^2)^(1/2)))-c^3*b^2/e^2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*g*d^2*arcsin(c*x)
*ln((-I*d*c-(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(-I*d*c+(-c^2*d^2+e^2)^(1/2)))+I*c^4*b^2*arcsin
(c*x)/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^2*d^3*g+c^3*b^2*arcsin(c*x)/(c^2*d^2-e^2)/(c*e*x+c*d)^2*e*(-c^2*x^2+1)^(1/
2)*x*f+I*c^4*b^2*arcsin(c*x)/(c^2*d^2-e^2)/(c*e*x+c*d)^2*x^2*d*g+I*c^3*b^2/e^2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e
^2)^2*g*d^2*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+I*c^
3*b^2/e*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*d*f*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2
))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))+c^3*a*b/e^3*d^2/(c^2*d^2-e^2)/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)
/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(
1/2))/(c*x+d*c/e))*g-c^3*a*b/e^2*d/(c^2*d^2-e^2)/(-(c^2*d^2-e^2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(
c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c
/e))*f-I*c^4*b^2*arcsin(c*x)/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*d^2*f-2*I*c^4*b^2*arcsin(c*x)/(c^2*d^2-e^2)/(c*e*x+
c*d)^2*x*d*f-I*c^4*b^2*arcsin(c*x)/(c^2*d^2-e^2)/(c*e*x+c*d)^2*e*x^2*f-I*c^3*b^2/e*(-c^2*d^2+e^2)^(1/2)/(c^2*d
^2-e^2)^2*d*f*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-I*
c^3*b^2/e^2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*g*d^2*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2
)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+1/2*c^2*b^2*arcsin(c*x)^2/(c^2*d^2-e^2)/(c*e*x+c*d)^2*d*g+2*c*b^2*(-c^2
*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*g*arcsin(c*x)*ln((-I*d*c-(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(-
I*d*c+(-c^2*d^2+e^2)^(1/2)))-2*c*b^2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*g*arcsin(c*x)*ln((I*d*c+(I*c*x+(-c^2
*x^2+1)^(1/2))*e+(-c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))+1/2*c^2*b^2*arcsin(c*x)^2/(c^2*d^2-e^2)/(
c*e*x+c*d)^2*e*f+2*I*c*b^2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*g*dilog((I*d*c+(I*c*x+(-c^2*x^2+1)^(1/2))*e+(-
c^2*d^2+e^2)^(1/2))/(I*d*c+(-c^2*d^2+e^2)^(1/2)))-2*I*c*b^2*(-c^2*d^2+e^2)^(1/2)/(c^2*d^2-e^2)^2*g*dilog((I*d*
c+(I*c*x+(-c^2*x^2+1)^(1/2))*e-(-c^2*d^2+e^2)^(1/2))/(I*d*c-(-c^2*d^2+e^2)^(1/2)))-2*c*a*b/e^3*g/(-(c^2*d^2-e^
2)/e^2)^(1/2)*ln((-2*(c^2*d^2-e^2)/e^2+2*d*c/e*(c*x+d*c/e)+2*(-(c^2*d^2-e^2)/e^2)^(1/2)*(-(c*x+d*c/e)^2+2*d*c/
e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2))/(c*x+d*c/e))-2*c^2*b^2/e^2/(c^2*d^2-e^2)*d*g*ln(I*c*x+(-c^2*x^2+1)^(1/
2))+c^2*b^2/e^2/(c^2*d^2-e^2)*d*g*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2*e+2*I*d*c*(I*c*x+(-c^2*x^2+1)^(1/2))-e)-2*c*
a*b*arcsin(c*x)*g/e^2/(c*e*x+c*d)-c^2*a*b*arcsin(c*x)/e/(c*e*x+c*d)^2*f+1/2*c^2*a^2/e^2/(c*e*x+c*d)^2*d*g-c^2*
b^2/e/(c^2*d^2-e^2)*f*ln((I*c*x+(-c^2*x^2+1)^(1/2))^2*e+2*I*d*c*(I*c*x+(-c^2*x^2+1)^(1/2))-e)+2*c^2*b^2/e/(c^2
*d^2-e^2)*f*ln(I*c*x+(-c^2*x^2+1)^(1/2))-c*a^2*g/e^2/(c*e*x+c*d)+2*I*c^4*b^2*arcsin(c*x)/(c^2*d^2-e^2)/(c*e*x+
c*d)^2/e*g*d^2*x+c^2*b^2*arcsin(c*x)^2/(c^2*d^2-e^2)/(c*e*x+c*d)^2*e*x*g-1/2*c^4*b^2*arcsin(c*x)^2/(c^2*d^2-e^
2)/(c*e*x+c*d)^2/e^2*d^3*g-1/2*c^4*b^2*arcsin(c*x)^2/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*d^2*f+c^3*b^2*arcsin(c*x)/(
c^2*d^2-e^2)/(c*e*x+c*d)^2*(-c^2*x^2+1)^(1/2)*d*f+c^2*a*b*arcsin(c*x)/e^2/(c*e*x+c*d)^2*d*g+c^2*a*b/e/(c^2*d^2
-e^2)/(c*x+d*c/e)*(-(c*x+d*c/e)^2+2*d*c/e*(c*x+d*c/e)-(c^2*d^2-e^2)/e^2)^(1/2)*f-1/2*c^2*a^2/e/(c*e*x+c*d)^2*f
-c^3*b^2*arcsin(c*x)/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*(-c^2*x^2+1)^(1/2)*d^2*g

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*arcsin(c*x))^2/(e*x+d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e-c*d>0)', see `assume?` for m
ore details)Is e-c*d positive, negative or zero?

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (f+g\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d+e\,x\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((f + g*x)*(a + b*asin(c*x))^2)/(d + e*x)^3,x)

[Out]

int(((f + g*x)*(a + b*asin(c*x))^2)/(d + e*x)^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (f + g x\right )}{\left (d + e x\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x+f)*(a+b*asin(c*x))**2/(e*x+d)**3,x)

[Out]

Integral((a + b*asin(c*x))**2*(f + g*x)/(d + e*x)**3, x)

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