3.1.90 \(\int (d+e x) (f+g x) (a+b \text {ArcSin}(c x)) \, dx\) [90]

Optimal. Leaf size=148 \[ \frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (9 c^2 d f+2 e g\right )+9 c^2 (e f+d g) x\right ) \sqrt {1-c^2 x^2}}{36 c^3}-\frac {b (e f+d g) \text {ArcSin}(c x)}{4 c^2}+d f x (a+b \text {ArcSin}(c x))+\frac {1}{2} (e f+d g) x^2 (a+b \text {ArcSin}(c x))+\frac {1}{3} e g x^3 (a+b \text {ArcSin}(c x)) \]

[Out]

-1/4*b*(d*g+e*f)*arcsin(c*x)/c^2+d*f*x*(a+b*arcsin(c*x))+1/2*(d*g+e*f)*x^2*(a+b*arcsin(c*x))+1/3*e*g*x^3*(a+b*
arcsin(c*x))+1/9*b*e*g*x^2*(-c^2*x^2+1)^(1/2)/c+1/36*b*(36*c^2*d*f+8*e*g+9*c^2*(d*g+e*f)*x)*(-c^2*x^2+1)^(1/2)
/c^3

________________________________________________________________________________________

Rubi [A]
time = 0.14, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {4833, 12, 1823, 794, 222} \begin {gather*} \frac {1}{2} x^2 (d g+e f) (a+b \text {ArcSin}(c x))+d f x (a+b \text {ArcSin}(c x))+\frac {1}{3} e g x^3 (a+b \text {ArcSin}(c x))-\frac {b \text {ArcSin}(c x) (d g+e f)}{4 c^2}+\frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \sqrt {1-c^2 x^2} \left (9 c^2 x (d g+e f)+4 \left (9 c^2 d f+2 e g\right )\right )}{36 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*e*g*x^2*Sqrt[1 - c^2*x^2])/(9*c) + (b*(4*(9*c^2*d*f + 2*e*g) + 9*c^2*(e*f + d*g)*x)*Sqrt[1 - c^2*x^2])/(36*
c^3) - (b*(e*f + d*g)*ArcSin[c*x])/(4*c^2) + d*f*x*(a + b*ArcSin[c*x]) + ((e*f + d*g)*x^2*(a + b*ArcSin[c*x]))
/2 + (e*g*x^3*(a + b*ArcSin[c*x]))/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 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 794

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((e*f + d*g)*(2*p
+ 3) + 2*e*g*(p + 1)*x)*((a + c*x^2)^(p + 1)/(2*c*(p + 1)*(2*p + 3))), x] - Dist[(a*e*g - c*d*f*(2*p + 3))/(c*
(2*p + 3)), Int[(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] &&  !LeQ[p, -1]

Rule 1823

Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x,
Expon[Pq, x]]}, Simp[f*(c*x)^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*c^(q - 1)*(m + q + 2*p + 1))), x] + Dist[1/(
b*(m + q + 2*p + 1)), Int[(c*x)^m*(a + b*x^2)^p*ExpandToSum[b*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*x^q
- a*f*(m + q - 1)*x^(q - 2), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, m, p}, x]
&& PolyQ[Pq, x] && ( !IGtQ[m, 0] || IGtQ[p + 1/2, -1])

Rule 4833

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*(Px_), x_Symbol] :> With[{u = IntHide[ExpandExpression[Px, x], x]}, Dis
t[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}, x] && PolynomialQ[Px, x]

Rubi steps

\begin {align*} \int (d+e x) (f+g x) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e g x^3 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (6 d f+3 (e f+d g) x+2 e g x^2\right )}{6 \sqrt {1-c^2 x^2}} \, dx\\ &=d f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e g x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{6} (b c) \int \frac {x \left (6 d f+3 (e f+d g) x+2 e g x^2\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e g x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-2 \left (9 c^2 d f+2 e g\right )-9 c^2 (e f+d g) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{18 c}\\ &=\frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (9 c^2 d f+2 e g\right )+9 c^2 (e f+d g) x\right ) \sqrt {1-c^2 x^2}}{36 c^3}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e g x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {(b (e f+d g)) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 c}\\ &=\frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b \left (4 \left (9 c^2 d f+2 e g\right )+9 c^2 (e f+d g) x\right ) \sqrt {1-c^2 x^2}}{36 c^3}-\frac {b (e f+d g) \sin ^{-1}(c x)}{4 c^2}+d f x \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{2} (e f+d g) x^2 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} e g x^3 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.13, size = 138, normalized size = 0.93 \begin {gather*} \frac {6 a c^3 x (3 d (2 f+g x)+e x (3 f+2 g x))+b \sqrt {1-c^2 x^2} \left (8 e g+c^2 (9 d (4 f+g x)+e x (9 f+4 g x))\right )+3 b c \left (12 c^2 d f x+4 c^2 e g x^3+3 d g \left (-1+2 c^2 x^2\right )+e f \left (-3+6 c^2 x^2\right )\right ) \text {ArcSin}(c x)}{36 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(6*a*c^3*x*(3*d*(2*f + g*x) + e*x*(3*f + 2*g*x)) + b*Sqrt[1 - c^2*x^2]*(8*e*g + c^2*(9*d*(4*f + g*x) + e*x*(9*
f + 4*g*x))) + 3*b*c*(12*c^2*d*f*x + 4*c^2*e*g*x^3 + 3*d*g*(-1 + 2*c^2*x^2) + e*f*(-3 + 6*c^2*x^2))*ArcSin[c*x
])/(36*c^3)

________________________________________________________________________________________

Maple [A]
time = 0.01, size = 198, normalized size = 1.34

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {e g \,c^{3} x^{3}}{3}+\frac {\left (c d g +c e f \right ) c^{2} x^{2}}{2}+d \,c^{3} f x \right )}{c^{2}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e g \,c^{3} x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{3} d g \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{3} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d \,c^{3} f x -\frac {e g \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {\left (3 c d g +3 c e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{6}+d \,c^{2} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}}{c}\)	\(198\)
default	\(\frac {\frac {a \left (\frac {e g \,c^{3} x^{3}}{3}+\frac {\left (c d g +c e f \right ) c^{2} x^{2}}{2}+d \,c^{3} f x \right )}{c^{2}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e g \,c^{3} x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{3} d g \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{3} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d \,c^{3} f x -\frac {e g \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {\left (3 c d g +3 c e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{6}+d \,c^{2} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}}{c}\)	\(198\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(g*x+f)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c*(a/c^2*(1/3*e*g*c^3*x^3+1/2*(c*d*g+c*e*f)*c^2*x^2+d*c^3*f*x)+b/c^2*(1/3*arcsin(c*x)*e*g*c^3*x^3+1/2*arcsin
(c*x)*c^3*d*g*x^2+1/2*arcsin(c*x)*c^3*e*f*x^2+arcsin(c*x)*d*c^3*f*x-1/3*e*g*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2
/3*(-c^2*x^2+1)^(1/2))-1/6*(3*c*d*g+3*c*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+d*c^2*f*(-c^2*x^2+1
)^(1/2)))

________________________________________________________________________________________

Maxima [A]
time = 0.47, size = 203, normalized size = 1.37 \begin {gather*} \frac {1}{3} \, a g x^{3} e + \frac {1}{2} \, a d g x^{2} + \frac {1}{2} \, a f x^{2} e + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d g + a d f x + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b f e + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b g e + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d f}{c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*g*x^3*e + 1/2*a*d*g*x^2 + 1/2*a*f*x^2*e + 1/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(
c*x)/c^3))*b*d*g + a*d*f*x + 1/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*b*f*e +
1/9*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*b*g*e + (c*x*arcsin(c*x) +
sqrt(-c^2*x^2 + 1))*b*d*f/c

________________________________________________________________________________________

Fricas [A]
time = 1.94, size = 170, normalized size = 1.15 \begin {gather*} \frac {18 \, a c^{3} d g x^{2} + 36 \, a c^{3} d f x + 3 \, {\left (6 \, b c^{3} d g x^{2} + 12 \, b c^{3} d f x - 3 \, b c d g + {\left (4 \, b c^{3} g x^{3} + 6 \, b c^{3} f x^{2} - 3 \, b c f\right )} e\right )} \arcsin \left (c x\right ) + 6 \, {\left (2 \, a c^{3} g x^{3} + 3 \, a c^{3} f x^{2}\right )} e + {\left (9 \, b c^{2} d g x + 36 \, b c^{2} d f + {\left (4 \, b c^{2} g x^{2} + 9 \, b c^{2} f x + 8 \, b g\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{36 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/36*(18*a*c^3*d*g*x^2 + 36*a*c^3*d*f*x + 3*(6*b*c^3*d*g*x^2 + 12*b*c^3*d*f*x - 3*b*c*d*g + (4*b*c^3*g*x^3 + 6
*b*c^3*f*x^2 - 3*b*c*f)*e)*arcsin(c*x) + 6*(2*a*c^3*g*x^3 + 3*a*c^3*f*x^2)*e + (9*b*c^2*d*g*x + 36*b*c^2*d*f +
(4*b*c^2*g*x^2 + 9*b*c^2*f*x + 8*b*g)*e)*sqrt(-c^2*x^2 + 1))/c^3

________________________________________________________________________________________

Sympy [A]
time = 0.21, size = 267, normalized size = 1.80 \begin {gather*} \begin {cases} a d f x + \frac {a d g x^{2}}{2} + \frac {a e f x^{2}}{2} + \frac {a e g x^{3}}{3} + b d f x \operatorname {asin}{\left (c x \right )} + \frac {b d g x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e f x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e g x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d f \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d g x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b e f x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b e g x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {b d g \operatorname {asin}{\left (c x \right )}}{4 c^{2}} - \frac {b e f \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {2 b e g \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d f x + \frac {d g x^{2}}{2} + \frac {e f x^{2}}{2} + \frac {e g x^{3}}{3}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d*f*x + a*d*g*x**2/2 + a*e*f*x**2/2 + a*e*g*x**3/3 + b*d*f*x*asin(c*x) + b*d*g*x**2*asin(c*x)/2 +
b*e*f*x**2*asin(c*x)/2 + b*e*g*x**3*asin(c*x)/3 + b*d*f*sqrt(-c**2*x**2 + 1)/c + b*d*g*x*sqrt(-c**2*x**2 + 1)
/(4*c) + b*e*f*x*sqrt(-c**2*x**2 + 1)/(4*c) + b*e*g*x**2*sqrt(-c**2*x**2 + 1)/(9*c) - b*d*g*asin(c*x)/(4*c**2)
- b*e*f*asin(c*x)/(4*c**2) + 2*b*e*g*sqrt(-c**2*x**2 + 1)/(9*c**3), Ne(c, 0)), (a*(d*f*x + d*g*x**2/2 + e*f*x
**2/2 + e*g*x**3/3), True))

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 259, normalized size = 1.75 \begin {gather*} \frac {1}{3} \, a e g x^{3} + b d f x \arcsin \left (c x\right ) + a d f x + \frac {{\left (c^{2} x^{2} - 1\right )} b e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e f x}{4 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} b d g x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b e f \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {b e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d f}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a e f}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d g}{2 \, c^{2}} + \frac {b e f \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {b d g \arcsin \left (c x\right )}{4 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e g}{9 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e g}{3 \, c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*e*g*x^3 + b*d*f*x*arcsin(c*x) + a*d*f*x + 1/3*(c^2*x^2 - 1)*b*e*g*x*arcsin(c*x)/c^2 + 1/4*sqrt(-c^2*x^2
+ 1)*b*e*f*x/c + 1/4*sqrt(-c^2*x^2 + 1)*b*d*g*x/c + 1/2*(c^2*x^2 - 1)*b*e*f*arcsin(c*x)/c^2 + 1/2*(c^2*x^2 - 1
)*b*d*g*arcsin(c*x)/c^2 + 1/3*b*e*g*x*arcsin(c*x)/c^2 + sqrt(-c^2*x^2 + 1)*b*d*f/c + 1/2*(c^2*x^2 - 1)*a*e*f/c
^2 + 1/2*(c^2*x^2 - 1)*a*d*g/c^2 + 1/4*b*e*f*arcsin(c*x)/c^2 + 1/4*b*d*g*arcsin(c*x)/c^2 - 1/9*(-c^2*x^2 + 1)^
(3/2)*b*e*g/c^3 + 1/3*sqrt(-c^2*x^2 + 1)*b*e*g/c^3

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \left \{\begin {array}{cl} \frac {a\,x^2\,\left (d\,g+e\,f\right )}{2}+a\,d\,f\,x+b\,e\,g\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}+\frac {x^3\,\mathrm {asin}\left (c\,x\right )}{3}\right )+\frac {a\,e\,g\,x^3}{3}+\frac {b\,d\,f\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c}+\frac {b\,d\,g\,\left (\frac {\mathrm {asin}\left (c\,x\right )\,\left (2\,c^2\,x^2-1\right )}{4}+\frac {c\,x\,\sqrt {1-c^2\,x^2}}{4}\right )}{c^2}+\frac {b\,e\,f\,\left (\frac {\mathrm {asin}\left (c\,x\right )\,\left (2\,c^2\,x^2-1\right )}{4}+\frac {c\,x\,\sqrt {1-c^2\,x^2}}{4}\right )}{c^2} & \text {\ if\ \ }0<c\\ \int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d+e\,x\right ) \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f + g*x)*(a + b*asin(c*x))*(d + e*x),x)

[Out]

piecewise(0 < c, (a*x^2*(d*g + e*f))/2 + a*d*f*x + b*e*g*(((1/c^2 - x^2)^(1/2)*(2/c^2 + x^2))/9 + (x^3*asin(c*
x))/3) + (a*e*g*x^3)/3 + (b*d*f*((- c^2*x^2 + 1)^(1/2) + c*x*asin(c*x)))/c + (b*d*g*((asin(c*x)*(2*c^2*x^2 - 1
))/4 + (c*x*(- c^2*x^2 + 1)^(1/2))/4))/c^2 + (b*e*f*((asin(c*x)*(2*c^2*x^2 - 1))/4 + (c*x*(- c^2*x^2 + 1)^(1/2
))/4))/c^2, ~0 < c, int((f + g*x)*(a + b*asin(c*x))*(d + e*x), x))

________________________________________________________________________________________