3.2.2 \(\int \frac {(f+g x+h x^2) (a+b \text {ArcSin}(c x))}{(d+e x)^3} \, dx\) [102]

Optimal. Leaf size=488 \[ \frac {b c \left (e^2 f-d e g+d^2 h\right ) \sqrt {1-c^2 x^2}}{2 e^2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {i b h \text {ArcSin}(c x)^2}{2 e^3}-\frac {\left (e^2 f-d e g+d^2 h\right ) (a+b \text {ArcSin}(c x))}{2 e^3 (d+e x)^2}-\frac {(e g-2 d h) (a+b \text {ArcSin}(c x))}{e^3 (d+e x)}-\frac {b c \left (2 e^2 (e g-2 d h)-c^2 d \left (e^2 f+d e g-3 d^2 h\right )\right ) \text {ArcTan}\left (\frac {e+c^2 d x}{\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}}\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b h \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b h \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {b h \text {ArcSin}(c x) \log (d+e x)}{e^3}+\frac {h (a+b \text {ArcSin}(c x)) \log (d+e x)}{e^3}-\frac {i b h \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {i b h \text {PolyLog}\left (2,\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.92, antiderivative size = 488, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 14, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {712, 4837, 12, 6874, 821, 739, 210, 222, 2451, 4825, 4615, 2221, 2317, 2438} \begin {gather*} -\frac {(a+b \text {ArcSin}(c x)) \left (d^2 h-d e g+e^2 f\right )}{2 e^3 (d+e x)^2}-\frac {(e g-2 d h) (a+b \text {ArcSin}(c x))}{e^3 (d+e x)}+\frac {h \log (d+e x) (a+b \text {ArcSin}(c x))}{e^3}-\frac {i b h \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d-\sqrt {c^2 d^2-e^2}}\right )}{e^3}-\frac {i b h \text {Li}_2\left (\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )}{e^3}+\frac {b h \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^3}+\frac {b h \text {ArcSin}(c x) \log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{\sqrt {c^2 d^2-e^2}+c d}\right )}{e^3}-\frac {b h \text {ArcSin}(c x) \log (d+e x)}{e^3}-\frac {i b h \text {ArcSin}(c x)^2}{2 e^3}-\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 (2 e^2 (e g-2 d h)-c^2 d \left (-3 d^2 h+d e g+e^2 f\right )\right )}{2 e^3 \left (c^2 d^2-e^2\right )^{3/2}}+\frac {b c \sqrt {1-c^2 x^2} \left (d^2 h-d e g+e^2 f\right )}{2 e^2 \left (c^2 d^2-e^2\right ) (d+e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 712

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

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 821

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(e*f - d*g
))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Dist[(c*d*f + a*e*g)/(c*d^2 + a*e
^2), Int[(d + e*x)^(m + 1)*(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
] && EqQ[Simplify[m + 2*p + 3], 0]

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

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Mathematica [C] Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
time = 4.39, size = 939, normalized size = 1.92 \begin {gather*} -\frac {a \left (e^2 f-d e g+d^2 h\right )}{2 e^3 (d+e x)^2}+\frac {a (-e g+2 d h)}{e^3 (d+e x)}+\frac {b 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}} (d+e x) F_1\left (2;\frac {1}{2},\frac {1}{2};3;\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}}-2 e \text {ArcSin}(c x)\right )}{4 e^2 (d+e x)^2}+\frac {a h \log (d+e x)}{e^3}+b g \left (\frac {-\frac {\text {ArcSin}(c x)}{d+e x}+\frac {c \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}}}{e^2}-\frac {d \left (\frac {c \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {ArcSin}(c x)}{e (d+e x)^2}-\frac {i c^3 d \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) e (c d+e) \sqrt {c^2 d^2-e^2}}\right )}{2 e}\right )-\frac {b h \left (-\frac {c d^2 e \sqrt {1-c^2 x^2}}{\left (c^2 d^2-e^2\right ) (d+e x)}+\frac {d^2 \text {ArcSin}(c x)}{(d+e x)^2}-\frac {4 d \text {ArcSin}(c x)}{d+e x}+\frac {4 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}}+\frac {i c^3 d^3 \left (\log (4)+\log \left (\frac {e^2 \sqrt {c^2 d^2-e^2} \left (i e+i c^2 d x+\sqrt {c^2 d^2-e^2} \sqrt {1-c^2 x^2}\right )}{c^3 d (d+e x)}\right )\right )}{(c d-e) (c d+e) \sqrt {c^2 d^2-e^2}}+i \left (\text {ArcSin}(c x) \left (\text {ArcSin}(c x)+2 i \left (\log \left (1+\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+\log \left (1-\frac {i e e^{i \text {ArcSin}(c x)}}{c d+\sqrt {c^2 d^2-e^2}}\right )\right )\right )+2 \text {PolyLog}\left (2,-\frac {i e e^{i \text {ArcSin}(c x)}}{-c d+\sqrt {c^2 d^2-e^2}}\right )+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 )\right )}{2 e^3} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*(a*(e^2*f - d*e*g + d^2*h))/(e^3*(d + e*x)^2) + (a*(-(e*g) + 2*d*h))/(e^3*(d + e*x)) + (b*f*(-((c*Sqrt[(e
*(-Sqrt[c^(-2)] + x))/(d + e*x)]*Sqrt[(e*(Sqrt[c^(-2)] + x))/(d + e*x)]*(d + e*x)*AppellF1[2, 1/2, 1/2, 3, (d
- Sqrt[c^(-2)]*e)/(d + e*x), (d + Sqrt[c^(-2)]*e)/(d + e*x)])/Sqrt[1 - c^2*x^2]) - 2*e*ArcSin[c*x]))/(4*e^2*(d
+ e*x)^2) + (a*h*Log[d + e*x])/e^3 + b*g*((-(ArcSin[c*x]/(d + e*x)) + (c*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^2 - (d*((c*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x)) -
ArcSin[c*x]/(e*(d + e*x)^2) - (I*c^3*d*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2
- e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*(d + e*x))]))/((c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2])))/(2*e)) - (b*h*(-
((c*d^2*e*Sqrt[1 - c^2*x^2])/((c^2*d^2 - e^2)*(d + e*x))) + (d^2*ArcSin[c*x])/(d + e*x)^2 - (4*d*ArcSin[c*x])/
(d + e*x) + (4*c*d*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] + (I*c^3
*d^3*(Log[4] + Log[(e^2*Sqrt[c^2*d^2 - e^2]*(I*e + I*c^2*d*x + Sqrt[c^2*d^2 - e^2]*Sqrt[1 - c^2*x^2]))/(c^3*d*
(d + e*x))]))/((c*d - e)*(c*d + e)*Sqrt[c^2*d^2 - e^2]) + I*(ArcSin[c*x]*(ArcSin[c*x] + (2*I)*(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])
])) + 2*PolyLog[2, ((-I)*e*E^(I*ArcSin[c*x]))/(-(c*d) + Sqrt[c^2*d^2 - e^2])] + 2*PolyLog[2, (I*e*E^(I*ArcSin[
c*x]))/(c*d + Sqrt[c^2*d^2 - e^2])])))/(2*e^3)

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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. 2731 vs. \(2 (489 ) = 978\).
time = 2.84, size = 2732, normalized size = 5.60

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(2732\)
default	\(\text {Expression too large to display}\)	\(2732\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(-2*b*c^3/(c^2*d^2-e^2)/(c*e*x+c*d)^2*arcsin(c*x)*d*h*x+1/2*b*c^4/(c^2*d^2-e^2)/(c*e*x+c*d)^2*e*(-c^2*x^2+
1)^(1/2)*f*x+b*c^3/(c^2*d^2-e^2)/(c*e*x+c*d)^2*e*arcsin(c*x)*g*x+2*b*c^5/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^2*arcsi
n(c*x)*d^3*h*x-b*c^5/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*arcsin(c*x)*d^2*g*x+I*b*c^5/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*d
^2*g*x+1/2*b*c^4/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*(-c^2*x^2+1)^(1/2)*d^2*h*x-I*b*c^5/(c^2*d^2-e^2)/(c*e*x+c*d)^2/
e^2*d^3*h*x-1/2*I*b*c^5/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*d^2*h*x^2-1/2*a*c^3/e^3/(c*e*x+c*d)^2*d^2*h-4*I*b*c^2/e/
(c^2*d^2-e^2)^(3/2)*d*h*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))-I*b*c^4/e^2/
(c^2*d^2-e^2)^(3/2)*d^2*g*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))-1/2*I*b*c^
5/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^3*d^4*h+1/2*I*b*c^5/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^2*d^3*g-1/2*b*c^5/(c^2*d^2-e
^2)/(c*e*x+c*d)^2/e^2*d^3*g*arcsin(c*x)-1/2*b*c^5/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*d^2*f*arcsin(c*x)+3*I*b*c^4/e^
3/(c^2*d^2-e^2)^(3/2)*h*d^3*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*c^
3/e/(c^2*d^2-e^2)^2*d^2*h*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^3/(c^2*d^2-e^2)*d^2*h*arcsin(c*x)^2+2*I*b*c^3/e/(c^2*d^2-e^2)^2*d^2*h*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)))-3/2*b*c^3/(c^2*d^2-e^2)/(c*e*x+c*d)^
2/e*d^2*h*arcsin(c*x)-2*b*c^3/e/(c^2*d^2-e^2)^2*d^2*h*arcsin(c*x)*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*c^3/e/(c^2*d^2-e^2)^2*d^2*h*arcsin(c*x)*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^5/e^3/(c^2*d^2-e^2)^2*h*d^4*arcsin
(c*x)*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^5/e^3/(c^
2*d^2-e^2)^2*h*d^4*arcsin(c*x)*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)))+3/2*b*c^5/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^3*d^4*h*arcsin(c*x)-I*b*c^4/e/(c^2*d^2-e^2)^(3/2)*d*f*arct
anh(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))+1/2*b*c^4/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e^
2*(-c^2*x^2+1)^(1/2)*d^3*h-1/2*b*c^4/(c^2*d^2-e^2)/(c*e*x+c*d)^2/e*(-c^2*x^2+1)^(1/2)*d^2*g-1/2*I*b*c^5/(c^2*d
^2-e^2)/(c*e*x+c*d)^2/e*d^2*f-I*b*c^5/e^3/(c^2*d^2-e^2)^2*h*d^4*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^5/e^3/(c^2*d^2-e^2)^2*h*d^4*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)))+a*c*h/e^3*ln(c*e*x+c*d)-1/2*a*c^3/e/(c*e*x+c
*d)^2*f-a*c^2/e^2/(c*e*x+c*d)*g+1/2*a*c^3/e^2/(c*e*x+c*d)^2*d*g+2*a*c^2/e^3/(c*e*x+c*d)*d*h+1/2*I*b*c*h*arcsin
(c*x)^2/e^3+2*I*b*c^2/(c^2*d^2-e^2)^(3/2)*g*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))-1/2*I*b*c^5/(c^2*d^2-e^2)/(c*e*x+c*d)^2*e*f*x^2-I*b*c^5/(c^2*d^2-e^2)/(c*e*x+c*d)^2*d*f*x+1/2*I*b*c^5/
(c^2*d^2-e^2)/(c*e*x+c*d)^2*d*g*x^2-1/2*b*c^4/(c^2*d^2-e^2)/(c*e*x+c*d)^2*(-c^2*x^2+1)^(1/2)*d*g*x+1/2*b*c^4/(
c^2*d^2-e^2)/(c*e*x+c*d)^2*(-c^2*x^2+1)^(1/2)*d*f+1/2*b*c^3/(c^2*d^2-e^2)/(c*e*x+c*d)^2*g*arcsin(c*x)*d+I*b*c/
e/(c^2*d^2-e^2)*h*arcsin(c*x)^2+b*c*e/(c^2*d^2-e^2)^2*h*arcsin(c*x)*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*e/(c^2*d^2-e^2)^2*h*arcsin(c*x)*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)))+1/2*b*c^3/(c^2*d^2-e^2)/(c*e*x+c*d)^2*e*f*ar
csin(c*x)-I*b*c*e/(c^2*d^2-e^2)^2*h*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*e/(c^2*d^2-e^2)^2*h*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))))

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

[Out]

integral((a*h*x^2 + a*g*x + a*f + (b*h*x^2 + b*g*x + b*f)*arcsin(c*x))/(x^3*e^3 + 3*d*x^2*e^2 + 3*d^2*x*e + d^
3), 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}\right )}{\left (d + e x\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate((h*x^2 + g*x + f)*(b*arcsin(c*x) + a)/(e*x + d)^3, 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 (h\,x^2+g\,x+f\right )}{{\left (d+e\,x\right )}^3} \,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))/(d + e*x)^3,x)

[Out]

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

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