3.1.99 \(\int (d+e x) (f+g x+h x^2) (a+b \text {ArcSin}(c x)) \, dx\) [99]

Optimal. Leaf size=223 \[ \frac {b (e g+d h) x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b e h x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {b \left (32 \left (9 c^2 d f+2 e g+2 d h\right )+9 \left (8 c^2 (e f+d g)+3 e h\right ) x\right ) \sqrt {1-c^2 x^2}}{288 c^3}-\frac {b \left (8 c^2 (e f+d g)+3 e h\right ) \text {ArcSin}(c x)}{32 c^4}+d f x (a+b \text {ArcSin}(c x))+\frac {1}{2} (e f+d g) x^2 (a+b \text {ArcSin}(c x))+\frac {1}{3} (e g+d h) x^3 (a+b \text {ArcSin}(c x))+\frac {1}{4} e h x^4 (a+b \text {ArcSin}(c x)) \]

[Out]

-1/32*b*(8*c^2*(d*g+e*f)+3*e*h)*arcsin(c*x)/c^4+d*f*x*(a+b*arcsin(c*x))+1/2*(d*g+e*f)*x^2*(a+b*arcsin(c*x))+1/
3*(d*h+e*g)*x^3*(a+b*arcsin(c*x))+1/4*e*h*x^4*(a+b*arcsin(c*x))+1/9*b*(d*h+e*g)*x^2*(-c^2*x^2+1)^(1/2)/c+1/16*
b*e*h*x^3*(-c^2*x^2+1)^(1/2)/c+1/288*b*(288*c^2*d*f+64*d*h+64*e*g+9*(8*c^2*(d*g+e*f)+3*e*h)*x)*(-c^2*x^2+1)^(1
/2)/c^3

________________________________________________________________________________________

Rubi [A]
time = 0.31, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {4833, 12, 1823, 794, 222} \begin {gather*} \frac {1}{2} x^2 (d g+e f) (a+b \text {ArcSin}(c x))+\frac {1}{3} x^3 (d h+e g) (a+b \text {ArcSin}(c x))+d f x (a+b \text {ArcSin}(c x))+\frac {1}{4} e h x^4 (a+b \text {ArcSin}(c x))-\frac {b \text {ArcSin}(c x) \left (8 c^2 (d g+e f)+3 e h\right )}{32 c^4}+\frac {b x^2 \sqrt {1-c^2 x^2} (d h+e g)}{9 c}+\frac {b e h x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {b \sqrt {1-c^2 x^2} \left (9 x \left (8 c^2 (d g+e f)+3 e h\right )+32 \left (9 c^2 d f+2 d h+2 e g\right )\right )}{288 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(e*g + d*h)*x^2*Sqrt[1 - c^2*x^2])/(9*c) + (b*e*h*x^3*Sqrt[1 - c^2*x^2])/(16*c) + (b*(32*(9*c^2*d*f + 2*e*g
+ 2*d*h) + 9*(8*c^2*(e*f + d*g) + 3*e*h)*x)*Sqrt[1 - c^2*x^2])/(288*c^3) - (b*(8*c^2*(e*f + d*g) + 3*e*h)*Arc
Sin[c*x])/(32*c^4) + d*f*x*(a + b*ArcSin[c*x]) + ((e*f + d*g)*x^2*(a + b*ArcSin[c*x]))/2 + ((e*g + d*h)*x^3*(a
+ b*ArcSin[c*x]))/3 + (e*h*x^4*(a + b*ArcSin[c*x]))/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 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) \left (f+g x+h x^2\right ) \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+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x \left (12 d f+6 (e f+d g) x+4 (e g+d h) x^2+3 e h x^3\right )}{12 \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+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{12} (b c) \int \frac {x \left (12 d f+6 (e f+d g) x+4 (e g+d h) x^2+3 e h x^3\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {b e h x^3 \sqrt {1-c^2 x^2}}{16 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+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \int \frac {x \left (-48 c^2 d f-3 \left (8 c^2 (e f+d g)+3 e h\right ) x-16 c^2 (e g+d h) x^2\right )}{\sqrt {1-c^2 x^2}} \, dx}{48 c}\\ &=\frac {b (e g+d h) x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b e h x^3 \sqrt {1-c^2 x^2}}{16 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+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {b \int \frac {x \left (16 c^2 \left (9 c^2 d f+2 e g+2 d h\right )+9 c^2 \left (8 c^2 (e f+d g)+3 e h\right ) x\right )}{\sqrt {1-c^2 x^2}} \, dx}{144 c^3}\\ &=\frac {b (e g+d h) x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b e h x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {b \left (32 \left (9 c^2 d f+2 e g+2 d h\right )+9 \left (8 c^2 (e f+d g)+3 e h\right ) x\right ) \sqrt {1-c^2 x^2}}{288 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+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )-\frac {\left (b \left (8 c^2 (e f+d g)+3 e h\right )\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c^3}\\ &=\frac {b (e g+d h) x^2 \sqrt {1-c^2 x^2}}{9 c}+\frac {b e h x^3 \sqrt {1-c^2 x^2}}{16 c}+\frac {b \left (32 \left (9 c^2 d f+2 e g+2 d h\right )+9 \left (8 c^2 (e f+d g)+3 e h\right ) x\right ) \sqrt {1-c^2 x^2}}{288 c^3}-\frac {b \left (8 c^2 (e f+d g)+3 e h\right ) \sin ^{-1}(c x)}{32 c^4}+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+d h) x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{4} e h x^4 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.18, size = 186, normalized size = 0.83 \begin {gather*} \frac {24 a c^4 x (2 d (6 f+x (3 g+2 h x))+e x (6 f+x (4 g+3 h x)))+b c \sqrt {1-c^2 x^2} \left (64 e g+64 d h+27 e h x+2 c^2 \left (4 d \left (36 f+9 g x+4 h x^2\right )+e x \left (36 f+16 g x+9 h x^2\right )\right )\right )+3 b \left (-24 c^2 (e f+d g)-9 e h+8 c^4 x \left (2 d \left (6 f+3 g x+2 h x^2\right )+e x \left (6 f+4 g x+3 h x^2\right )\right )\right ) \text {ArcSin}(c x)}{288 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(24*a*c^4*x*(2*d*(6*f + x*(3*g + 2*h*x)) + e*x*(6*f + x*(4*g + 3*h*x))) + b*c*Sqrt[1 - c^2*x^2]*(64*e*g + 64*d
*h + 27*e*h*x + 2*c^2*(4*d*(36*f + 9*g*x + 4*h*x^2) + e*x*(36*f + 16*g*x + 9*h*x^2))) + 3*b*(-24*c^2*(e*f + d*
g) - 9*e*h + 8*c^4*x*(2*d*(6*f + 3*g*x + 2*h*x^2) + e*x*(6*f + 4*g*x + 3*h*x^2)))*ArcSin[c*x])/(288*c^4)

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 307, normalized size = 1.38

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {e h \,c^{4} x^{4}}{4}+\frac {\left (c d h +c e g \right ) c^{3} x^{3}}{3}+\frac {\left (c^{2} d g +c^{2} e f \right ) c^{2} x^{2}}{2}+d \,c^{4} f x \right )}{c^{3}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e h \,c^{4} x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{4} d h \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} e g \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} d g \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{4} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d \,c^{4} f x -\frac {e h \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {\left (4 c d h +4 c e g \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{12}-\frac {\left (6 c^{2} d g +6 c^{2} e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{12}+d \,c^{3} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{3}}}{c}\)	\(307\)
default	\(\frac {\frac {a \left (\frac {e h \,c^{4} x^{4}}{4}+\frac {\left (c d h +c e g \right ) c^{3} x^{3}}{3}+\frac {\left (c^{2} d g +c^{2} e f \right ) c^{2} x^{2}}{2}+d \,c^{4} f x \right )}{c^{3}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e h \,c^{4} x^{4}}{4}+\frac {\arcsin \left (c x \right ) c^{4} d h \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} e g \,x^{3}}{3}+\frac {\arcsin \left (c x \right ) c^{4} d g \,x^{2}}{2}+\frac {\arcsin \left (c x \right ) c^{4} e f \,x^{2}}{2}+\arcsin \left (c x \right ) d \,c^{4} f x -\frac {e h \left (-\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {-c^{2} x^{2}+1}}{8}+\frac {3 \arcsin \left (c x \right )}{8}\right )}{4}-\frac {\left (4 c d h +4 c e g \right ) \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{12}-\frac {\left (6 c^{2} d g +6 c^{2} e f \right ) \left (-\frac {c x \sqrt {-c^{2} x^{2}+1}}{2}+\frac {\arcsin \left (c x \right )}{2}\right )}{12}+d \,c^{3} f \sqrt {-c^{2} x^{2}+1}\right )}{c^{3}}}{c}\)	\(307\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a/c^3*(1/4*e*h*c^4*x^4+1/3*(c*d*h+c*e*g)*c^3*x^3+1/2*(c^2*d*g+c^2*e*f)*c^2*x^2+d*c^4*f*x)+b/c^3*(1/4*arcs
in(c*x)*e*h*c^4*x^4+1/3*arcsin(c*x)*c^4*d*h*x^3+1/3*arcsin(c*x)*c^4*e*g*x^3+1/2*arcsin(c*x)*c^4*d*g*x^2+1/2*ar
csin(c*x)*c^4*e*f*x^2+arcsin(c*x)*d*c^4*f*x-1/4*e*h*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1/2)-3/8*c*x*(-c^2*x^2+1)^(1/2
)+3/8*arcsin(c*x))-1/12*(4*c*d*h+4*c*e*g)*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))-1/12*(6*c^2
*d*g+6*c^2*e*f)*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))+d*c^3*f*(-c^2*x^2+1)^(1/2)))

________________________________________________________________________________________

Maxima [A]
time = 0.48, size = 340, normalized size = 1.52 \begin {gather*} \frac {1}{4} \, a h x^{4} e + \frac {1}{3} \, a d h x^{3} + \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 + \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 d h + 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 {1}{32} \, {\left (8 \, x^{4} \arcsin \left (c x\right ) + {\left (\frac {2 \, \sqrt {-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} x}{c^{4}} - \frac {3 \, \arcsin \left (c x\right )}{c^{5}}\right )} c\right )} b h 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)*(h*x^2+g*x+f)*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

1/4*a*h*x^4*e + 1/3*a*d*h*x^3 + 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*(sq
rt(-c^2*x^2 + 1)*x/c^2 - arcsin(c*x)/c^3))*b*d*g + 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*d*h + 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 + 1/32
*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 + 1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c*x)/c^5)*c)*b*h*e
+ (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d*f/c

________________________________________________________________________________________

Fricas [A]
time = 2.71, size = 256, normalized size = 1.15 \begin {gather*} \frac {96 \, a c^{4} d h x^{3} + 144 \, a c^{4} d g x^{2} + 288 \, a c^{4} d f x + 3 \, {\left (32 \, b c^{4} d h x^{3} + 48 \, b c^{4} d g x^{2} + 96 \, b c^{4} d f x - 24 \, b c^{2} d g + {\left (24 \, b c^{4} h x^{4} + 32 \, b c^{4} g x^{3} + 48 \, b c^{4} f x^{2} - 24 \, b c^{2} f - 9 \, b h\right )} e\right )} \arcsin \left (c x\right ) + 24 \, {\left (3 \, a c^{4} h x^{4} + 4 \, a c^{4} g x^{3} + 6 \, a c^{4} f x^{2}\right )} e + {\left (32 \, b c^{3} d h x^{2} + 72 \, b c^{3} d g x + 288 \, b c^{3} d f + 64 \, b c d h + {\left (18 \, b c^{3} h x^{3} + 32 \, b c^{3} g x^{2} + 64 \, b c g + 9 \, {\left (8 \, b c^{3} f + 3 \, b c h\right )} x\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{288 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(96*a*c^4*d*h*x^3 + 144*a*c^4*d*g*x^2 + 288*a*c^4*d*f*x + 3*(32*b*c^4*d*h*x^3 + 48*b*c^4*d*g*x^2 + 96*b*
c^4*d*f*x - 24*b*c^2*d*g + (24*b*c^4*h*x^4 + 32*b*c^4*g*x^3 + 48*b*c^4*f*x^2 - 24*b*c^2*f - 9*b*h)*e)*arcsin(c
*x) + 24*(3*a*c^4*h*x^4 + 4*a*c^4*g*x^3 + 6*a*c^4*f*x^2)*e + (32*b*c^3*d*h*x^2 + 72*b*c^3*d*g*x + 288*b*c^3*d*
f + 64*b*c*d*h + (18*b*c^3*h*x^3 + 32*b*c^3*g*x^2 + 64*b*c*g + 9*(8*b*c^3*f + 3*b*c*h)*x)*e)*sqrt(-c^2*x^2 + 1
))/c^4

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 449 vs. \(2 (209) = 418\).
time = 0.38, size = 449, normalized size = 2.01 \begin {gather*} \begin {cases} a d f x + \frac {a d g x^{2}}{2} + \frac {a d h x^{3}}{3} + \frac {a e f x^{2}}{2} + \frac {a e g x^{3}}{3} + \frac {a e h x^{4}}{4} + 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 d h x^{3} \operatorname {asin}{\left (c x \right )}}{3} + \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 e h x^{4} \operatorname {asin}{\left (c x \right )}}{4} + \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 d h x^{2} \sqrt {- c^{2} x^{2} + 1}}{9 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 e h x^{3} \sqrt {- c^{2} x^{2} + 1}}{16 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 d h \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {2 b e g \sqrt {- c^{2} x^{2} + 1}}{9 c^{3}} + \frac {3 b e h x \sqrt {- c^{2} x^{2} + 1}}{32 c^{3}} - \frac {3 b e h \operatorname {asin}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a \left (d f x + \frac {d g x^{2}}{2} + \frac {d h x^{3}}{3} + \frac {e f x^{2}}{2} + \frac {e g x^{3}}{3} + \frac {e h x^{4}}{4}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d*f*x + a*d*g*x**2/2 + a*d*h*x**3/3 + a*e*f*x**2/2 + a*e*g*x**3/3 + a*e*h*x**4/4 + b*d*f*x*asin(c
*x) + b*d*g*x**2*asin(c*x)/2 + b*d*h*x**3*asin(c*x)/3 + b*e*f*x**2*asin(c*x)/2 + b*e*g*x**3*asin(c*x)/3 + b*e*
h*x**4*asin(c*x)/4 + b*d*f*sqrt(-c**2*x**2 + 1)/c + b*d*g*x*sqrt(-c**2*x**2 + 1)/(4*c) + b*d*h*x**2*sqrt(-c**2
*x**2 + 1)/(9*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*e*h*x**3*sqr
t(-c**2*x**2 + 1)/(16*c) - b*d*g*asin(c*x)/(4*c**2) - b*e*f*asin(c*x)/(4*c**2) + 2*b*d*h*sqrt(-c**2*x**2 + 1)/
(9*c**3) + 2*b*e*g*sqrt(-c**2*x**2 + 1)/(9*c**3) + 3*b*e*h*x*sqrt(-c**2*x**2 + 1)/(32*c**3) - 3*b*e*h*asin(c*x
)/(32*c**4), Ne(c, 0)), (a*(d*f*x + d*g*x**2/2 + d*h*x**3/3 + e*f*x**2/2 + e*g*x**3/3 + e*h*x**4/4), True))

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (200) = 400\).
time = 0.43, size = 448, normalized size = 2.01 \begin {gather*} \frac {1}{4} \, a e h x^{4} + \frac {1}{3} \, a e g x^{3} + \frac {1}{3} \, a d h 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 {{\left (c^{2} x^{2} - 1\right )} b d h 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 {b d h 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 )}^{\frac {3}{2}} b e h x}{16 \, c^{3}} + \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 )}^{2} b e h \arcsin \left (c x\right )}{4 \, c^{4}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e g}{9 \, c^{3}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d h}{9 \, c^{3}} + \frac {5 \, \sqrt {-c^{2} x^{2} + 1} b e h x}{32 \, c^{3}} + \frac {{\left (c^{2} x^{2} - 1\right )} b e h \arcsin \left (c x\right )}{2 \, c^{4}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e g}{3 \, c^{3}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d h}{3 \, c^{3}} + \frac {5 \, b e h \arcsin \left (c x\right )}{32 \, c^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*a*e*h*x^4 + 1/3*a*e*g*x^3 + 1/3*a*d*h*x^3 + b*d*f*x*arcsin(c*x) + a*d*f*x + 1/3*(c^2*x^2 - 1)*b*e*g*x*arcs
in(c*x)/c^2 + 1/3*(c^2*x^2 - 1)*b*d*h*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 + 1/3*b*d*h*x*arcsin(c*x)/c^2 + sqrt(-c^2*x^2 + 1)*b*d*f/c - 1/16*(-c^2*x^2 + 1)^(3/2)*b*e
*h*x/c^3 + 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*a
rcsin(c*x)/c^2 + 1/4*(c^2*x^2 - 1)^2*b*e*h*arcsin(c*x)/c^4 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*e*g/c^3 - 1/9*(-c^2*x^
2 + 1)^(3/2)*b*d*h/c^3 + 5/32*sqrt(-c^2*x^2 + 1)*b*e*h*x/c^3 + 1/2*(c^2*x^2 - 1)*b*e*h*arcsin(c*x)/c^4 + 1/3*s
qrt(-c^2*x^2 + 1)*b*e*g/c^3 + 1/3*sqrt(-c^2*x^2 + 1)*b*d*h/c^3 + 5/32*b*e*h*arcsin(c*x)/c^4

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________