\(\int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx\) [33]

Optimal result

Integrand size = 19, antiderivative size = 162 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {b \left (9 c^2 d f+2 e g\right ) \sqrt {1-c^2 x^2}}{9 c^3}+\frac {b (e f+d g) x \sqrt {1-c^2 x^2}}{4 c}+\frac {b e g x^2 \sqrt {1-c^2 x^2}}{9 c}-\frac {b (e f+d g) \arcsin (c x)}{4 c^2}+d f x (a+b \arcsin (c x))+\frac {1}{2} (e f+d g) x^2 (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x)) \] Output:

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

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.85 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {6 a c^3 x (3 d (2 f+g x)+e x (3 f+2 g x))+b \sqrt {1-c^2 x^2} \left (8 e g+c^2 (9 d (4 f+g x)+e x (9 f+4 g x))\right )+3 b c \left (12 c^2 d f x+4 c^2 e g x^3+3 d g \left (-1+2 c^2 x^2\right )+e f \left (-3+6 c^2 x^2\right )\right ) \arcsin (c x)}{36 c^3} \] Input:

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

Output:

(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)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.04, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {5248, 27, 2340, 25, 533, 27, 455, 223}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx\)

\(\Big \downarrow \) 5248

\(\displaystyle -b c \int \frac {x \left (2 e g x^2+3 (e f+d g) x+6 d f\right )}{6 \sqrt {1-c^2 x^2}}dx+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{6} b c \int \frac {x \left (2 e g x^2+3 (e f+d g) x+6 d f\right )}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 2340

\(\displaystyle -\frac {1}{6} b c \left (-\frac {\int -\frac {x \left (9 (e f+d g) x c^2+2 \left (9 d f c^2+2 e g\right )\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2 e g x^2 \sqrt {1-c^2 x^2}}{3 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {1}{6} b c \left (\frac {\int \frac {x \left (9 (e f+d g) x c^2+2 \left (9 d f c^2+2 e g\right )\right )}{\sqrt {1-c^2 x^2}}dx}{3 c^2}-\frac {2 e g x^2 \sqrt {1-c^2 x^2}}{3 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 533

\(\displaystyle -\frac {1}{6} b c \left (\frac {\frac {\int \frac {c^2 \left (9 (e f+d g)+4 \left (9 d f c^2+2 e g\right ) x\right )}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {9}{2} x \sqrt {1-c^2 x^2} (d g+e f)}{3 c^2}-\frac {2 e g x^2 \sqrt {1-c^2 x^2}}{3 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{6} b c \left (\frac {\frac {1}{2} \int \frac {9 (e f+d g)+4 \left (9 d f c^2+2 e g\right ) x}{\sqrt {1-c^2 x^2}}dx-\frac {9}{2} x \sqrt {1-c^2 x^2} (d g+e f)}{3 c^2}-\frac {2 e g x^2 \sqrt {1-c^2 x^2}}{3 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 455

\(\displaystyle -\frac {1}{6} b c \left (\frac {\frac {1}{2} \left (9 (d g+e f) \int \frac {1}{\sqrt {1-c^2 x^2}}dx-4 \sqrt {1-c^2 x^2} \left (\frac {2 e g}{c^2}+9 d f\right )\right )-\frac {9}{2} x \sqrt {1-c^2 x^2} (d g+e f)}{3 c^2}-\frac {2 e g x^2 \sqrt {1-c^2 x^2}}{3 c^2}\right )+\frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {1}{2} x^2 (d g+e f) (a+b \arcsin (c x))+d f x (a+b \arcsin (c x))+\frac {1}{3} e g x^3 (a+b \arcsin (c x))-\frac {1}{6} b c \left (\frac {\frac {1}{2} \left (\frac {9 \arcsin (c x) (d g+e f)}{c}-4 \sqrt {1-c^2 x^2} \left (\frac {2 e g}{c^2}+9 d f\right )\right )-\frac {9}{2} x \sqrt {1-c^2 x^2} (d g+e f)}{3 c^2}-\frac {2 e g x^2 \sqrt {1-c^2 x^2}}{3 c^2}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 223

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 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 533

Int[(x_)^(m_.)*((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 Simp[d*x^m*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 2))), x] - Simp[1/(b*(m + 2* 
p + 2))   Int[x^(m - 1)*(a + b*x^2)^p*Simp[a*d*m - b*c*(m + 2*p + 2)*x, x], 
 x], x] /; FreeQ[{a, b, c, d, p}, x] && IGtQ[m, 0] && GtQ[p, -1] && Integer 
Q[2*p]
 

rule 2340

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] + Simp[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 5248

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

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.09

method	result	size
parts	\(a \left (\frac {e g \,x^{3}}{3}+\frac {\left (d g +e f \right ) x^{2}}{2}+d f x \right )+\frac {b \left (\frac {c \arcsin \left (c x \right ) e g \,x^{3}}{3}+\frac {c \arcsin \left (c x \right ) x^{2} d g}{2}+\frac {c \arcsin \left (c x \right ) x^{2} e f}{2}+\arcsin \left (c x \right ) d f c x -\frac {3 c \left (d g +e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )+2 e g \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )-6 d \,c^{2} f \sqrt {-c^{2} x^{2}+1}}{6 c^{2}}\right )}{c}\)	\(176\)
derivativedivides	\(\frac {\frac {a \left (\frac {e g \,c^{3} x^{3}}{3}+\frac {\left (c d g +c e f \right ) c^{2} x^{2}}{2}+d \,c^{3} f x \right )}{c^{2}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e g \,c^{3} x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{3} d g \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{3} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d \,c^{3} f x -\frac {c \left (d g +e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}-\frac {e g \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+d \,c^{2} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}}{c}\)	\(195\)
default	\(\frac {\frac {a \left (\frac {e g \,c^{3} x^{3}}{3}+\frac {\left (c d g +c e f \right ) c^{2} x^{2}}{2}+d \,c^{3} f x \right )}{c^{2}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e g \,c^{3} x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{3} d g \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{3} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d \,c^{3} f x -\frac {c \left (d g +e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{2}-\frac {e g \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}+d \,c^{2} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{2}}}{c}\)	\(195\)
orering	\(\frac {\left (20 c^{4} e^{2} g^{2} x^{5}+52 c^{4} d e \,g^{2} x^{4}+52 c^{4} e^{2} f g \,x^{4}+27 c^{4} d^{2} g^{2} x^{3}+174 c^{4} d e f g \,x^{3}+27 c^{4} e^{2} f^{2} x^{3}+90 c^{4} d^{2} f g \,x^{2}+90 c^{4} d e \,f^{2} x^{2}+36 c^{4} d^{2} f^{2} x +8 c^{2} e^{2} g^{2} x^{3}-23 c^{2} d e \,g^{2} x^{2}-23 c^{2} e^{2} f g \,x^{2}-18 c^{2} d^{2} g^{2} x -108 c^{2} d e f g x -18 c^{2} e^{2} f^{2} x -45 c^{2} d^{2} f g -45 c^{2} d e \,f^{2}-16 e^{2} g^{2} x -8 d e \,g^{2}-8 e^{2} f g \right ) \left (a +b \arcsin \left (c x \right )\right )}{36 c^{4} \left (g x +f \right ) \left (e x +d \right )}-\frac {\left (4 c^{2} e g \,x^{2}+9 c^{2} d g x +9 c^{2} e f x +36 c^{2} d f +8 e g \right ) \left (c x -1\right ) \left (c x +1\right ) \left (e \left (g x +f \right ) \left (a +b \arcsin \left (c x \right )\right )+\left (e x +d \right ) g \left (a +b \arcsin \left (c x \right )\right )+\frac {\left (e x +d \right ) \left (g x +f \right ) b c}{\sqrt {-c^{2} x^{2}+1}}\right )}{36 c^{4} \left (g x +f \right ) \left (e x +d \right )}\)	\(388\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.99 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {12 \, a c^{3} e g x^{3} + 36 \, a c^{3} d f x + 18 \, {\left (a c^{3} e f + a c^{3} d g\right )} x^{2} + 3 \, {\left (4 \, b c^{3} e g x^{3} + 12 \, b c^{3} d f x - 3 \, b c e f - 3 \, b c d g + 6 \, {\left (b c^{3} e f + b c^{3} d g\right )} x^{2}\right )} \arcsin \left (c x\right ) + {\left (4 \, b c^{2} e g x^{2} + 36 \, b c^{2} d f + 8 \, b e g + 9 \, {\left (b c^{2} e f + b c^{2} d g\right )} x\right )} \sqrt {-c^{2} x^{2} + 1}}{36 \, c^{3}} \] Input:

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

Output:

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

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.65 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\begin {cases} a d f x + \frac {a d g x^{2}}{2} + \frac {a e f x^{2}}{2} + \frac {a e g x^{3}}{3} + b d f x \operatorname {asin}{\left (c x \right )} + \frac {b d g x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e f x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {b e g x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b d f \sqrt {- c^{2} x^{2} + 1}}{c} + \frac {b d g x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b e f x \sqrt {- c^{2} x^{2} + 1}}{4 c} + \frac {b e g x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 c} - \frac {b d g \operatorname {asin}{\left (c x \right )}}{4 c^{2}} - \frac {b e f \operatorname {asin}{\left (c x \right )}}{4 c^{2}} + \frac {2 b e g \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} & \text {for}\: c \neq 0 \\a \left (d f x + \frac {d g x^{2}}{2} + \frac {e f x^{2}}{2} + \frac {e g x^{3}}{3}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

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*a 
sin(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*sq 
rt(-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))
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.23 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {1}{3} \, a e g x^{3} + \frac {1}{2} \, a e f x^{2} + \frac {1}{2} \, a d g x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b e f + \frac {1}{4} \, {\left (2 \, x^{2} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x}{c^{2}} - \frac {\arcsin \left (c x\right )}{c^{3}}\right )}\right )} b d g + \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \left (c x\right ) + c {\left (\frac {\sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} b e g + a d f x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d f}{c} \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.60 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {1}{3} \, a e g x^{3} + b d f x \arcsin \left (c x\right ) + a d f x + \frac {{\left (c^{2} x^{2} - 1\right )} b e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e f x}{4 \, c} + \frac {\sqrt {-c^{2} x^{2} + 1} b d g x}{4 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} b e f \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} b d g \arcsin \left (c x\right )}{2 \, c^{2}} + \frac {b e g x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d f}{c} + \frac {{\left (c^{2} x^{2} - 1\right )} a e f}{2 \, c^{2}} + \frac {{\left (c^{2} x^{2} - 1\right )} a d g}{2 \, c^{2}} + \frac {b e f \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {b d g \arcsin \left (c x\right )}{4 \, c^{2}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e g}{9 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e g}{3 \, c^{3}} \] Input:

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

Output:

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(-1)]

Timed out. \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\left \{\begin {array}{cl} \frac {a\,x^2\,\left (d\,g+e\,f\right )}{2}+a\,d\,f\,x+b\,e\,g\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}+\frac {x^3\,\mathrm {asin}\left (c\,x\right )}{3}\right )+\frac {a\,e\,g\,x^3}{3}+\frac {b\,d\,f\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c}+\frac {b\,d\,g\,\left (\frac {\mathrm {asin}\left (c\,x\right )\,\left (2\,c^2\,x^2-1\right )}{4}+\frac {c\,x\,\sqrt {1-c^2\,x^2}}{4}\right )}{c^2}+\frac {b\,e\,f\,\left (\frac {\mathrm {asin}\left (c\,x\right )\,\left (2\,c^2\,x^2-1\right )}{4}+\frac {c\,x\,\sqrt {1-c^2\,x^2}}{4}\right )}{c^2} & \text {\ if\ \ }0<c\\ \int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d+e\,x\right ) \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \] Input:

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

Output:

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))
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.38 \[ \int (d+e x) (f+g x) (a+b \arcsin (c x)) \, dx=\frac {36 \mathit {asin} \left (c x \right ) b \,c^{3} d f x +18 \mathit {asin} \left (c x \right ) b \,c^{3} d g \,x^{2}+18 \mathit {asin} \left (c x \right ) b \,c^{3} e f \,x^{2}+12 \mathit {asin} \left (c x \right ) b \,c^{3} e g \,x^{3}-9 \mathit {asin} \left (c x \right ) b c d g -9 \mathit {asin} \left (c x \right ) b c e f +36 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d f +9 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} d g x +9 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} e f x +4 \sqrt {-c^{2} x^{2}+1}\, b \,c^{2} e g \,x^{2}+8 \sqrt {-c^{2} x^{2}+1}\, b e g +36 a \,c^{3} d f x +18 a \,c^{3} d g \,x^{2}+18 a \,c^{3} e f \,x^{2}+12 a \,c^{3} e g \,x^{3}}{36 c^{3}} \] Input:

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

Output:

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