3.2.10 \(\int \frac {(f+g x+h x^2+i x^3) (a+b \text {ArcSin}(c x))}{(d+e x)^2} \, dx\) [110]

Optimal. Leaf size=617 \[ \frac {b (e h-2 d i) \sqrt {1-c^2 x^2}}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2}-\frac {b i \text {ArcSin}(c x)}{4 c^2 e^2}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \text {ArcSin}(c x)^2}{2 e^4}+\frac {(e h-2 d i) x (a+b \text {ArcSin}(c x))}{e^3}+\frac {i x^2 (a+b \text {ArcSin}(c x))}{2 e^2}-\frac {\left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) (a+b \text {ArcSin}(c x))}{e^4 (d+e x)}+\frac {b c \left (e^3 f-d e^2 g+d^2 e h-d^3 i\right ) \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {b \left (e^2 g-2 d e h+3 d^2 i\right ) \text {ArcSin}(c x) \log (d+e x)}{e^4}+\frac {\left (e^2 g-2 d e h+3 d^2 i\right ) (a+b \text {ArcSin}(c x)) \log (d+e x)}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (e^2 g-2 d e h+3 d^2 i\right ) \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4} \]

[Out]

-1/4*b*i*arcsin(c*x)/c^2/e^2-1/2*I*b*(3*d^2*i-2*d*e*h+e^2*g)*arcsin(c*x)^2/e^4+(-2*d*i+e*h)*x*(a+b*arcsin(c*x)
)/e^3+1/2*i*x^2*(a+b*arcsin(c*x))/e^2-(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*(a+b*arcsin(c*x))/e^4/(e*x+d)-b*(3*d^2*i-
2*d*e*h+e^2*g)*arcsin(c*x)*ln(e*x+d)/e^4+(3*d^2*i-2*d*e*h+e^2*g)*(a+b*arcsin(c*x))*ln(e*x+d)/e^4+b*(3*d^2*i-2*
d*e*h+e^2*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^4+b*(3*d^2*i-2*d*e*h
+e^2*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^4-I*b*(3*d^2*i-2*d*e*h+e^
2*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^4-I*b*(3*d^2*i-2*d*e*h+e^2*g)*polyl
og(2,I*e*(I*c*x+(-c^2*x^2+1)^(1/2))/(c*d+(c^2*d^2-e^2)^(1/2)))/e^4+b*c*(-d^3*i+d^2*e*h-d*e^2*g+e^3*f)*arctan((
c^2*d*x+e)/(c^2*d^2-e^2)^(1/2)/(-c^2*x^2+1)^(1/2))/e^4/(c^2*d^2-e^2)^(1/2)+b*(-2*d*i+e*h)*(-c^2*x^2+1)^(1/2)/c
/e^3+1/4*b*i*x*(-c^2*x^2+1)^(1/2)/c/e^2

________________________________________________________________________________________

Rubi [A]
time = 1.12, antiderivative size = 617, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 15, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {1864, 4837, 12, 6874, 267, 327, 222, 739, 210, 2451, 4825, 4615, 2221, 2317, 2438} \begin {gather*} \frac {\log (d+e x) (a+b \text {ArcSin}(c x)) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}-\frac {(a+b \text {ArcSin}(c x)) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4 (d+e x)}+\frac {x (e h-2 d i) (a+b \text {ArcSin}(c x))}{e^3}+\frac {i x^2 (a+b \text {ArcSin}(c x))}{2 e^2}-\frac {i b \left (3 d^2 i-2 d e h+e^2 g\right ) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (3 d^2 i-2 d e h+e^2 g\right ) \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \text {ArcSin}(c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}+\frac {b \text {ArcSin}(c x) \left (3 d^2 i-2 d e h+e^2 g\right ) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^4}-\frac {b i \text {ArcSin}(c x)}{4 c^2 e^2}-\frac {i b \text {ArcSin}(c x)^2 \left (3 d^2 i-2 d e h+e^2 g\right )}{2 e^4}-\frac {b \text {ArcSin}(c x) \log (d+e x) \left (3 d^2 i-2 d e h+e^2 g\right )}{e^4}+\frac {b c \text {ArcTan}\left (\frac {c^2 d x+e}{\sqrt {1-c^2 x^2} \sqrt {c^2 d^2-e^2}}\right ) \left (d^3 (-i)+d^2 e h-d e^2 g+e^3 f\right )}{e^4 \sqrt {c^2 d^2-e^2}}+\frac {b \sqrt {1-c^2 x^2} (e h-2 d i)}{c e^3}+\frac {b i x \sqrt {1-c^2 x^2}}{4 c e^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 210

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

Rule 222

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 267

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

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 739

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 1864

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 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 2451

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/Sqrt[(f_) + (g_.)*(x_)^2], x_Symbol] :> With[{u = Int
Hide[1/Sqrt[f + g*x^2], x]}, Simp[u*(a + b*Log[c*(d + e*x)^n]), x] - Dist[b*e*n, Int[SimplifyIntegrand[u/(d +
e*x), x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && GtQ[f, 0]

Rule 4615

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

Rule 4825

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

Rule 4837

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[Px*(d
+ e*x)^m, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]
] /; FreeQ[{a, b, c, d, e, m}, x] && PolynomialQ[Px, x]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {\left (f+g x+h x^2+110 x^3\right ) \left (a+b \sin ^{-1}(c x)\right )}{(d+e x)^2} \, dx &=-\frac {(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-(b c) \int \frac {110 d^3-d^2 e (h+220 x)+d e^2 (g+(h-165 x) x)+e^3 \left (-f+x^2 (h+55 x)\right )+\left (330 d^2+e^2 g-2 d e h\right ) (d+e x) \log (d+e x)}{e^4 (d+e x) \sqrt {1-c^2 x^2}} \, dx\\ &=-\frac {(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {110 d^3-d^2 e (h+220 x)+d e^2 (g+(h-165 x) x)+e^3 \left (-f+x^2 (h+55 x)\right )+\left (330 d^2+e^2 g-2 d e h\right ) (d+e x) \log (d+e x)}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^4}\\ &=-\frac {(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \left (\frac {110 d^3-e^3 f+d e^2 g-d^2 e h-d e (220 d-e h) x-e^2 (165 d-e h) x^2+55 e^3 x^3}{(d+e x) \sqrt {1-c^2 x^2}}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \log (d+e x)}{\sqrt {1-c^2 x^2}}\right ) \, dx}{e^4}\\ &=-\frac {(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(b c) \int \frac {110 d^3-e^3 f+d e^2 g-d^2 e h-d e (220 d-e h) x-e^2 (165 d-e h) x^2+55 e^3 x^3}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^4}-\frac {\left (b c \left (330 d^2+e^2 g-2 d e h\right )\right ) \int \frac {\log (d+e x)}{\sqrt {1-c^2 x^2}} \, dx}{e^4}\\ &=\frac {55 b (d+e x) \sqrt {1-c^2 x^2}}{2 c e^3}-\frac {(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {b \int \frac {-e^3 \left (55 d e^2+2 c^2 \left (110 d^3-e^3 f+d e^2 g-d^2 e h\right )\right )-e^4 \left (55 e^2-c^2 d (495 d-2 e h)\right ) x+c^2 e^5 (495 d-2 e h) x^2}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 c e^7}+\frac {\left (b c \left (330 d^2+e^2 g-2 d e h\right )\right ) \int \frac {\sin ^{-1}(c x)}{c d+c e x} \, dx}{e^3}\\ &=-\frac {b (495 d-2 e h) \sqrt {1-c^2 x^2}}{2 c e^3}+\frac {55 b (d+e x) \sqrt {1-c^2 x^2}}{2 c e^3}-\frac {(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {b \int \frac {c^2 e^5 \left (55 d e^2+2 c^2 \left (110 d^3-e^3 f+d e^2 g-d^2 e h\right )\right )+55 c^2 e^8 x}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{2 c^3 e^9}+\frac {\left (b c \left (330 d^2+e^2 g-2 d e h\right )\right ) \text {Subst}\left (\int \frac {x \cos (x)}{c^2 d+c e \sin (x)} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}\\ &=-\frac {b (495 d-2 e h) \sqrt {1-c^2 x^2}}{2 c e^3}+\frac {55 b (d+e x) \sqrt {1-c^2 x^2}}{2 c e^3}-\frac {i b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x)^2}{2 e^4}-\frac {(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {(55 b) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{2 c e^2}+\frac {\left (b c \left (330 d^2+e^2 g-2 d e h\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d-c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}+\frac {\left (b c \left (330 d^2+e^2 g-2 d e h\right )\right ) \text {Subst}\left (\int \frac {e^{i x} x}{c^2 d+c \sqrt {c^2 d^2-e^2}-i c e e^{i x}} \, dx,x,\sin ^{-1}(c x)\right )}{e^3}-\frac {\left (b c \left (110 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \int \frac {1}{(d+e x) \sqrt {1-c^2 x^2}} \, dx}{e^4}\\ &=-\frac {b (495 d-2 e h) \sqrt {1-c^2 x^2}}{2 c e^3}+\frac {55 b (d+e x) \sqrt {1-c^2 x^2}}{2 c e^3}-\frac {55 b \sin ^{-1}(c x)}{2 c^2 e^2}-\frac {i b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x)^2}{2 e^4}-\frac {(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}+\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \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^4}+\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \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^4}-\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {\left (b \left (330 d^2+e^2 g-2 d e h\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^4}-\frac {\left (b \left (330 d^2+e^2 g-2 d e h\right )\right ) \text {Subst}\left (\int \log \left (1-\frac {i c e e^{i x}}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{e^4}+\frac {\left (b c \left (110 d^3-e^3 f+d e^2 g-d^2 e h\right )\right ) \text {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^4}\\ &=-\frac {b (495 d-2 e h) \sqrt {1-c^2 x^2}}{2 c e^3}+\frac {55 b (d+e x) \sqrt {1-c^2 x^2}}{2 c e^3}-\frac {55 b \sin ^{-1}(c x)}{2 c^2 e^2}-\frac {i b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x)^2}{2 e^4}-\frac {(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {b c \left (110 d^3-e^3 f+d e^2 g-d^2 e h\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^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \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^4}+\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \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^4}-\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}+\frac {\left (i b \left (330 d^2+e^2 g-2 d e h\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d-c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^4}+\frac {\left (i b \left (330 d^2+e^2 g-2 d e h\right )\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {i c e x}{c^2 d+c \sqrt {c^2 d^2-e^2}}\right )}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{e^4}\\ &=-\frac {b (495 d-2 e h) \sqrt {1-c^2 x^2}}{2 c e^3}+\frac {55 b (d+e x) \sqrt {1-c^2 x^2}}{2 c e^3}-\frac {55 b \sin ^{-1}(c x)}{2 c^2 e^2}-\frac {i b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x)^2}{2 e^4}-\frac {(220 d-e h) x \left (a+b \sin ^{-1}(c x)\right )}{e^3}+\frac {55 x^2 \left (a+b \sin ^{-1}(c x)\right )}{e^2}+\frac {\left (110 d^3-e^3 f+d e^2 g-d^2 e h\right ) \left (a+b \sin ^{-1}(c x)\right )}{e^4 (d+e x)}-\frac {b c \left (110 d^3-e^3 f+d e^2 g-d^2 e h\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^4 \sqrt {c^2 d^2-e^2}}+\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \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^4}+\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \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^4}-\frac {b \left (330 d^2+e^2 g-2 d e h\right ) \sin ^{-1}(c x) \log (d+e x)}{e^4}+\frac {\left (330 d^2+e^2 g-2 d e h\right ) \left (a+b \sin ^{-1}(c x)\right ) \log (d+e x)}{e^4}-\frac {i b \left (330 d^2+e^2 g-2 d e h\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^4}-\frac {i b \left (330 d^2+e^2 g-2 d e h\right ) \text {Li}_2\left (\frac {i e e^{i \sin ^{-1}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^4}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 3.40, size = 1168, normalized size = 1.89 \begin {gather*} \frac {2 a e (e h-2 d i) x+a e^2 i x^2+\frac {2 a \left (-e^3 f+d e^2 g-d^2 e h+d^3 i\right )}{d+e x}-2 b e^2 f \left (\frac {c \sqrt {\frac {e \left (-\sqrt {\frac {1}{c^2}}+x\right )}{d+e x}} \sqrt {\frac {e \left (\sqrt {\frac {1}{c^2}}+x\right )}{d+e x}} F_1\left (1;\frac {1}{2},\frac {1}{2};2;\frac {d-\sqrt {\frac {1}{c^2}} e}{d+e x},\frac {d+\sqrt {\frac {1}{c^2}} e}{d+e x}\right )}{\sqrt {1-c^2 x^2}}+\frac {e \text {ArcSin}(c x)}{d+e x}\right )+2 a \left (e^2 g-2 d e h+3 d^2 i\right ) \log (d+e x)-6 i b d^2 i \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+b e^2 g \left (\frac {2 d \text {ArcSin}(c x)}{d+e x}-i \text {ArcSin}(c x)^2-\frac {2 c d \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}+2 \text {ArcSin}(c x) \log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )-2 i \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-2 i \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )+2 b e h \left (\frac {e \sqrt {1-c^2 x^2}}{c}+e x \text {ArcSin}(c x)-\frac {d^2 \text {ArcSin}(c x)}{d+e x}+i d \text {ArcSin}(c x)^2+\frac {c d^2 \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}-2 d \text {ArcSin}(c x) \log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )-2 d \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )+2 i d \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+2 i d \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )+\frac {1}{2} b i \left (-6 i d^2 \text {ArcSin}(c x)^2-\frac {4 c d^3 \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{\sqrt {c^2 d^2-e^2}}+\frac {e \left (c (-8 d+e x) \sqrt {1-c^2 x^2}-2 e \text {ArcTan}\left (\frac {c x}{-1+\sqrt {1-c^2 x^2}}\right )\right )}{c^2}+2 \text {ArcSin}(c x) \left (-4 d e x+e^2 x^2+\frac {2 d^3}{d+e x}+6 d^2 \log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+6 d^2 \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )-12 i d^2 \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )}{2 e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a*e*(e*h - 2*d*i)*x + a*e^2*i*x^2 + (2*a*(-(e^3*f) + d*e^2*g - d^2*e*h + d^3*i))/(d + e*x) - 2*b*e^2*f*((c*
Sqrt[(e*(-Sqrt[c^(-2)] + x))/(d + e*x)]*Sqrt[(e*(Sqrt[c^(-2)] + x))/(d + e*x)]*AppellF1[1, 1/2, 1/2, 2, (d - S
qrt[c^(-2)]*e)/(d + e*x), (d + Sqrt[c^(-2)]*e)/(d + e*x)])/Sqrt[1 - c^2*x^2] + (e*ArcSin[c*x])/(d + e*x)) + 2*
a*(e^2*g - 2*d*e*h + 3*d^2*i)*Log[d + e*x] - (6*I)*b*d^2*i*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqr
t[c^2*d^2 - e^2])] + b*e^2*g*((2*d*ArcSin[c*x])/(d + e*x) - I*ArcSin[c*x]^2 - (2*c*d*ArcTan[(e + c^2*d*x)/(Sqr
t[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*d^2 - e^2] + 2*ArcSin[c*x]*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c
*d) + Sqrt[c^2*d^2 - e^2])] + 2*ArcSin[c*x]*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] - (2*
I)*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - (2*I)*PolyLog[2, (I*e*E^(I*ArcSin[c
*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) + 2*b*e*h*((e*Sqrt[1 - c^2*x^2])/c + e*x*ArcSin[c*x] - (d^2*ArcSin[c*x])/(
d + e*x) + I*d*ArcSin[c*x]^2 + (c*d^2*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c^2*
d^2 - e^2] - 2*d*ArcSin[c*x]*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] - 2*d*ArcSin[c*x]
*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])] + (2*I)*d*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/
(-(c*d) + Sqrt[c^2*d^2 - e^2])] + (2*I)*d*PolyLog[2, (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) + (
b*i*((-6*I)*d^2*ArcSin[c*x]^2 - (4*c*d^3*ArcTan[(e + c^2*d*x)/(Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2])])/Sqrt[c
^2*d^2 - e^2] + (e*(c*(-8*d + e*x)*Sqrt[1 - c^2*x^2] - 2*e*ArcTan[(c*x)/(-1 + Sqrt[1 - c^2*x^2])]))/c^2 + 2*Ar
cSin[c*x]*(-4*d*e*x + e^2*x^2 + (2*d^3)/(d + e*x) + 6*d^2*Log[1 + (I*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d
^2 - e^2])] + 6*d^2*Log[1 - (I*e*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]) - (12*I)*d^2*PolyLog[2, (I*e
*E^(I*ArcSin[c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])]))/2)/(2*e^4)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2985 vs. \(2 (616 ) = 1232\).
time = 3.27, size = 2986, normalized size = 4.84

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2986\)
default	\(\text {Expression too large to display}\)	\(2986\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((i*x^3+h*x^2+g*x+f)*(a+b*arcsin(c*x))/(e*x+d)^2,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(c*d-%e>0)', see `assume?` for
more details

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/2*(2*a*h*x^2 + 2*I*a*x^3 + 2*a*g*x + 2*a*f + (-I*b*h*x^2 + b*x^3 - I*b*g*x - I*b*f)*log(-2*c^2*x^2
- 2*sqrt(c^2*x^2 - 1)*c*x + 1))/(x^2*e^2 + 2*d*x*e + d^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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