3.618 \(\int (d+e x^2)^3 (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=225 \[ d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 x \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{b e \left (1-c^2 x^2\right )^{3/2} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7}+\frac{b \sqrt{1-c^2 x^2} \left (35 c^4 d^2 e+35 c^6 d^3+21 c^2 d e^2+5 e^3\right )}{35 c^7}+\frac{3 b e^2 \left (1-c^2 x^2\right )^{5/2} \left (7 c^2 d+5 e\right )}{175 c^7}-\frac{b e^3 \left (1-c^2 x^2\right )^{7/2}}{49 c^7} \]

[Out]

(b*(35*c^6*d^3 + 35*c^4*d^2*e + 21*c^2*d*e^2 + 5*e^3)*Sqrt[1 - c^2*x^2])/(35*c^7) - (b*e*(35*c^4*d^2 + 42*c^2*
d*e + 15*e^2)*(1 - c^2*x^2)^(3/2))/(105*c^7) + (3*b*e^2*(7*c^2*d + 5*e)*(1 - c^2*x^2)^(5/2))/(175*c^7) - (b*e^
3*(1 - c^2*x^2)^(7/2))/(49*c^7) + d^3*x*(a + b*ArcSin[c*x]) + d^2*e*x^3*(a + b*ArcSin[c*x]) + (3*d*e^2*x^5*(a
+ b*ArcSin[c*x]))/5 + (e^3*x^7*(a + b*ArcSin[c*x]))/7

________________________________________________________________________________________

Rubi [A]  time = 0.250657, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {194, 4665, 12, 1799, 1850} \[ d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 x \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{b e \left (1-c^2 x^2\right )^{3/2} \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7}+\frac{b \sqrt{1-c^2 x^2} \left (35 c^4 d^2 e+35 c^6 d^3+21 c^2 d e^2+5 e^3\right )}{35 c^7}+\frac{3 b e^2 \left (1-c^2 x^2\right )^{5/2} \left (7 c^2 d+5 e\right )}{175 c^7}-\frac{b e^3 \left (1-c^2 x^2\right )^{7/2}}{49 c^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(35*c^6*d^3 + 35*c^4*d^2*e + 21*c^2*d*e^2 + 5*e^3)*Sqrt[1 - c^2*x^2])/(35*c^7) - (b*e*(35*c^4*d^2 + 42*c^2*
d*e + 15*e^2)*(1 - c^2*x^2)^(3/2))/(105*c^7) + (3*b*e^2*(7*c^2*d + 5*e)*(1 - c^2*x^2)^(5/2))/(175*c^7) - (b*e^
3*(1 - c^2*x^2)^(7/2))/(49*c^7) + d^3*x*(a + b*ArcSin[c*x]) + d^2*e*x^3*(a + b*ArcSin[c*x]) + (3*d*e^2*x^5*(a
+ b*ArcSin[c*x]))/5 + (e^3*x^7*(a + b*ArcSin[c*x]))/7

Rule 194

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

Rule 4665

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && NeQ[c^2*d + e, 0] && (IGtQ[p, 0] || ILtQ[p + 1/2, 0])

Rule 12

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

Rule 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1850

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])

Rubi steps

\begin{align*} \int \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{35 \sqrt{1-c^2 x^2}} \, dx\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{35} (b c) \int \frac{x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{70} (b c) \operatorname{Subst}\left (\int \frac{35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{70} (b c) \operatorname{Subst}\left (\int \left (\frac{35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3}{c^6 \sqrt{1-c^2 x}}-\frac{e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \sqrt{1-c^2 x}}{c^6}+\frac{3 e^2 \left (7 c^2 d+5 e\right ) \left (1-c^2 x\right )^{3/2}}{c^6}-\frac{5 e^3 \left (1-c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )\\ &=\frac{b \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right ) \sqrt{1-c^2 x^2}}{35 c^7}-\frac{b e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{105 c^7}+\frac{3 b e^2 \left (7 c^2 d+5 e\right ) \left (1-c^2 x^2\right )^{5/2}}{175 c^7}-\frac{b e^3 \left (1-c^2 x^2\right )^{7/2}}{49 c^7}+d^3 x \left (a+b \sin ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{7} e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.244473, size = 187, normalized size = 0.83 \[ \frac{105 a x \left (35 d^2 e x^2+35 d^3+21 d e^2 x^4+5 e^3 x^6\right )+\frac{b \sqrt{1-c^2 x^2} \left (c^6 \left (1225 d^2 e x^2+3675 d^3+441 d e^2 x^4+75 e^3 x^6\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+24 c^2 e^2 \left (49 d+5 e x^2\right )+240 e^3\right )}{c^7}+105 b x \sin ^{-1}(c x) \left (35 d^2 e x^2+35 d^3+21 d e^2 x^4+5 e^3 x^6\right )}{3675} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(105*a*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6) + (b*Sqrt[1 - c^2*x^2]*(240*e^3 + 24*c^2*e^2*(49*d
+ 5*e*x^2) + 2*c^4*e*(1225*d^2 + 294*d*e*x^2 + 45*e^2*x^4) + c^6*(3675*d^3 + 1225*d^2*e*x^2 + 441*d*e^2*x^4 +
75*e^3*x^6)))/c^7 + 105*b*x*(35*d^3 + 35*d^2*e*x^2 + 21*d*e^2*x^4 + 5*e^3*x^6)*ArcSin[c*x])/3675

________________________________________________________________________________________

Maple [A]  time = 0.005, size = 325, normalized size = 1.4 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{6}} \left ({\frac{{e}^{3}{c}^{7}{x}^{7}}{7}}+{\frac{3\,{c}^{7}d{e}^{2}{x}^{5}}{5}}+{c}^{7}{d}^{2}e{x}^{3}+{d}^{3}{c}^{7}x \right ) }+{\frac{b}{{c}^{6}} \left ({\frac{\arcsin \left ( cx \right ){e}^{3}{c}^{7}{x}^{7}}{7}}+{\frac{3\,\arcsin \left ( cx \right ){c}^{7}d{e}^{2}{x}^{5}}{5}}+\arcsin \left ( cx \right ){c}^{7}{d}^{2}e{x}^{3}+\arcsin \left ( cx \right ){d}^{3}{c}^{7}x-{\frac{{e}^{3}}{7} \left ( -{\frac{{c}^{6}{x}^{6}}{7}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{6\,{c}^{4}{x}^{4}}{35}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{8\,{c}^{2}{x}^{2}}{35}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{16}{35}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{3\,{c}^{2}d{e}^{2}}{5} \left ( -{\frac{{c}^{4}{x}^{4}}{5}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{4\,{c}^{2}{x}^{2}}{15}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{8}{15}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{c}^{4}{d}^{2}e \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) +{d}^{3}{c}^{6}\sqrt{-{c}^{2}{x}^{2}+1} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x^2+d)^3*(a+b*arcsin(c*x)),x)

[Out]

1/c*(a/c^6*(1/7*e^3*c^7*x^7+3/5*c^7*d*e^2*x^5+c^7*d^2*e*x^3+d^3*c^7*x)+b/c^6*(1/7*arcsin(c*x)*e^3*c^7*x^7+3/5*
arcsin(c*x)*c^7*d*e^2*x^5+arcsin(c*x)*c^7*d^2*e*x^3+arcsin(c*x)*d^3*c^7*x-1/7*e^3*(-1/7*c^6*x^6*(-c^2*x^2+1)^(
1/2)-6/35*c^4*x^4*(-c^2*x^2+1)^(1/2)-8/35*c^2*x^2*(-c^2*x^2+1)^(1/2)-16/35*(-c^2*x^2+1)^(1/2))-3/5*c^2*d*e^2*(
-1/5*c^4*x^4*(-c^2*x^2+1)^(1/2)-4/15*c^2*x^2*(-c^2*x^2+1)^(1/2)-8/15*(-c^2*x^2+1)^(1/2))-c^4*d^2*e*(-1/3*c^2*x
^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+d^3*c^6*(-c^2*x^2+1)^(1/2)))

________________________________________________________________________________________

Maxima [A]  time = 1.4679, size = 394, normalized size = 1.75 \begin{align*} \frac{1}{7} \, a e^{3} x^{7} + \frac{3}{5} \, a d e^{2} x^{5} + a d^{2} e x^{3} + \frac{1}{3} \,{\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^{2} e + \frac{1}{25} \,{\left (15 \, x^{5} \arcsin \left (c x\right ) +{\left (\frac{3 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d e^{2} + \frac{1}{245} \,{\left (35 \, x^{7} \arcsin \left (c x\right ) +{\left (\frac{5 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b e^{3} + a d^{3} x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d^{3}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

1/7*a*e^3*x^7 + 3/5*a*d*e^2*x^5 + a*d^2*e*x^3 + 1/3*(3*x^3*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqr
t(-c^2*x^2 + 1)/c^4))*b*d^2*e + 1/25*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1
)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*d*e^2 + 1/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 +
6*sqrt(-c^2*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*b*e^3 + a*d^3*x +
(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d^3/c

________________________________________________________________________________________

Fricas [A]  time = 2.05662, size = 537, normalized size = 2.39 \begin{align*} \frac{525 \, a c^{7} e^{3} x^{7} + 2205 \, a c^{7} d e^{2} x^{5} + 3675 \, a c^{7} d^{2} e x^{3} + 3675 \, a c^{7} d^{3} x + 105 \,{\left (5 \, b c^{7} e^{3} x^{7} + 21 \, b c^{7} d e^{2} x^{5} + 35 \, b c^{7} d^{2} e x^{3} + 35 \, b c^{7} d^{3} x\right )} \arcsin \left (c x\right ) +{\left (75 \, b c^{6} e^{3} x^{6} + 3675 \, b c^{6} d^{3} + 2450 \, b c^{4} d^{2} e + 1176 \, b c^{2} d e^{2} + 9 \,{\left (49 \, b c^{6} d e^{2} + 10 \, b c^{4} e^{3}\right )} x^{4} + 240 \, b e^{3} +{\left (1225 \, b c^{6} d^{2} e + 588 \, b c^{4} d e^{2} + 120 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt{-c^{2} x^{2} + 1}}{3675 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3675*(525*a*c^7*e^3*x^7 + 2205*a*c^7*d*e^2*x^5 + 3675*a*c^7*d^2*e*x^3 + 3675*a*c^7*d^3*x + 105*(5*b*c^7*e^3*
x^7 + 21*b*c^7*d*e^2*x^5 + 35*b*c^7*d^2*e*x^3 + 35*b*c^7*d^3*x)*arcsin(c*x) + (75*b*c^6*e^3*x^6 + 3675*b*c^6*d
^3 + 2450*b*c^4*d^2*e + 1176*b*c^2*d*e^2 + 9*(49*b*c^6*d*e^2 + 10*b*c^4*e^3)*x^4 + 240*b*e^3 + (1225*b*c^6*d^2
*e + 588*b*c^4*d*e^2 + 120*b*c^2*e^3)*x^2)*sqrt(-c^2*x^2 + 1))/c^7

________________________________________________________________________________________

Sympy [A]  time = 8.86437, size = 389, normalized size = 1.73 \begin{align*} \begin{cases} a d^{3} x + a d^{2} e x^{3} + \frac{3 a d e^{2} x^{5}}{5} + \frac{a e^{3} x^{7}}{7} + b d^{3} x \operatorname{asin}{\left (c x \right )} + b d^{2} e x^{3} \operatorname{asin}{\left (c x \right )} + \frac{3 b d e^{2} x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{b e^{3} x^{7} \operatorname{asin}{\left (c x \right )}}{7} + \frac{b d^{3} \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{b d^{2} e x^{2} \sqrt{- c^{2} x^{2} + 1}}{3 c} + \frac{3 b d e^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{25 c} + \frac{b e^{3} x^{6} \sqrt{- c^{2} x^{2} + 1}}{49 c} + \frac{2 b d^{2} e \sqrt{- c^{2} x^{2} + 1}}{3 c^{3}} + \frac{4 b d e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{25 c^{3}} + \frac{6 b e^{3} x^{4} \sqrt{- c^{2} x^{2} + 1}}{245 c^{3}} + \frac{8 b d e^{2} \sqrt{- c^{2} x^{2} + 1}}{25 c^{5}} + \frac{8 b e^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{245 c^{5}} + \frac{16 b e^{3} \sqrt{- c^{2} x^{2} + 1}}{245 c^{7}} & \text{for}\: c \neq 0 \\a \left (d^{3} x + d^{2} e x^{3} + \frac{3 d e^{2} x^{5}}{5} + \frac{e^{3} x^{7}}{7}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**3*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*d**3*x + a*d**2*e*x**3 + 3*a*d*e**2*x**5/5 + a*e**3*x**7/7 + b*d**3*x*asin(c*x) + b*d**2*e*x**3*a
sin(c*x) + 3*b*d*e**2*x**5*asin(c*x)/5 + b*e**3*x**7*asin(c*x)/7 + b*d**3*sqrt(-c**2*x**2 + 1)/c + b*d**2*e*x*
*2*sqrt(-c**2*x**2 + 1)/(3*c) + 3*b*d*e**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + b*e**3*x**6*sqrt(-c**2*x**2 + 1)
/(49*c) + 2*b*d**2*e*sqrt(-c**2*x**2 + 1)/(3*c**3) + 4*b*d*e**2*x**2*sqrt(-c**2*x**2 + 1)/(25*c**3) + 6*b*e**3
*x**4*sqrt(-c**2*x**2 + 1)/(245*c**3) + 8*b*d*e**2*sqrt(-c**2*x**2 + 1)/(25*c**5) + 8*b*e**3*x**2*sqrt(-c**2*x
**2 + 1)/(245*c**5) + 16*b*e**3*sqrt(-c**2*x**2 + 1)/(245*c**7), Ne(c, 0)), (a*(d**3*x + d**2*e*x**3 + 3*d*e**
2*x**5/5 + e**3*x**7/7), True))

________________________________________________________________________________________

Giac [B]  time = 1.2988, size = 633, normalized size = 2.81 \begin{align*} \frac{1}{7} \, a x^{7} e^{3} + \frac{3}{5} \, a d x^{5} e^{2} + a d^{2} x^{3} e + b d^{3} x \arcsin \left (c x\right ) + a d^{3} x + \frac{{\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right ) e}{c^{2}} + \frac{b d^{2} x \arcsin \left (c x\right ) e}{c^{2}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d^{3}}{c} + \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{2} b d x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d^{2} e}{3 \, c^{3}} + \frac{6 \,{\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d^{2} e}{c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b x \arcsin \left (c x\right ) e^{3}}{7 \, c^{6}} + \frac{3 \, b d x \arcsin \left (c x\right ) e^{2}}{5 \, c^{4}} + \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b d e^{2}}{25 \, c^{5}} + \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{2} b x \arcsin \left (c x\right ) e^{3}}{7 \, c^{6}} - \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d e^{2}}{5 \, c^{5}} + \frac{3 \,{\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right ) e^{3}}{7 \, c^{6}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b e^{3}}{49 \, c^{7}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} b d e^{2}}{5 \, c^{5}} + \frac{b x \arcsin \left (c x\right ) e^{3}}{7 \, c^{6}} + \frac{3 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b e^{3}}{35 \, c^{7}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b e^{3}}{7 \, c^{7}} + \frac{\sqrt{-c^{2} x^{2} + 1} b e^{3}}{7 \, c^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*a*x^7*e^3 + 3/5*a*d*x^5*e^2 + a*d^2*x^3*e + b*d^3*x*arcsin(c*x) + a*d^3*x + (c^2*x^2 - 1)*b*d^2*x*arcsin(c
*x)*e/c^2 + b*d^2*x*arcsin(c*x)*e/c^2 + sqrt(-c^2*x^2 + 1)*b*d^3/c + 3/5*(c^2*x^2 - 1)^2*b*d*x*arcsin(c*x)*e^2
/c^4 - 1/3*(-c^2*x^2 + 1)^(3/2)*b*d^2*e/c^3 + 6/5*(c^2*x^2 - 1)*b*d*x*arcsin(c*x)*e^2/c^4 + sqrt(-c^2*x^2 + 1)
*b*d^2*e/c^3 + 1/7*(c^2*x^2 - 1)^3*b*x*arcsin(c*x)*e^3/c^6 + 3/5*b*d*x*arcsin(c*x)*e^2/c^4 + 3/25*(c^2*x^2 - 1
)^2*sqrt(-c^2*x^2 + 1)*b*d*e^2/c^5 + 3/7*(c^2*x^2 - 1)^2*b*x*arcsin(c*x)*e^3/c^6 - 2/5*(-c^2*x^2 + 1)^(3/2)*b*
d*e^2/c^5 + 3/7*(c^2*x^2 - 1)*b*x*arcsin(c*x)*e^3/c^6 + 1/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*e^3/c^7 + 3/
5*sqrt(-c^2*x^2 + 1)*b*d*e^2/c^5 + 1/7*b*x*arcsin(c*x)*e^3/c^6 + 3/35*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^3
/c^7 - 1/7*(-c^2*x^2 + 1)^(3/2)*b*e^3/c^7 + 1/7*sqrt(-c^2*x^2 + 1)*b*e^3/c^7