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

Optimal. Leaf size=287 \[ \frac{3}{5} d^2 e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{b e \left (1-c^2 x^2\right )^{5/2} \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{525 c^9}-\frac{b \left (1-c^2 x^2\right )^{3/2} \left (378 c^4 d^2 e+105 c^6 d^3+405 c^2 d e^2+140 e^3\right )}{945 c^9}+\frac{b \sqrt{1-c^2 x^2} \left (189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2+35 e^3\right )}{315 c^9}-\frac{b e^2 \left (1-c^2 x^2\right )^{7/2} \left (27 c^2 d+28 e\right )}{441 c^9}+\frac{b e^3 \left (1-c^2 x^2\right )^{9/2}}{81 c^9} \]

[Out]

(b*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*Sqrt[1 - c^2*x^2])/(315*c^9) - (b*(105*c^6*d^3 + 378
*c^4*d^2*e + 405*c^2*d*e^2 + 140*e^3)*(1 - c^2*x^2)^(3/2))/(945*c^9) + (b*e*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2
)*(1 - c^2*x^2)^(5/2))/(525*c^9) - (b*e^2*(27*c^2*d + 28*e)*(1 - c^2*x^2)^(7/2))/(441*c^9) + (b*e^3*(1 - c^2*x
^2)^(9/2))/(81*c^9) + (d^3*x^3*(a + b*ArcSin[c*x]))/3 + (3*d^2*e*x^5*(a + b*ArcSin[c*x]))/5 + (3*d*e^2*x^7*(a
+ b*ArcSin[c*x]))/7 + (e^3*x^9*(a + b*ArcSin[c*x]))/9

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Rubi [A]  time = 0.373168, antiderivative size = 287, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {270, 4731, 12, 1799, 1620} \[ \frac{3}{5} d^2 e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{b e \left (1-c^2 x^2\right )^{5/2} \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{525 c^9}-\frac{b \left (1-c^2 x^2\right )^{3/2} \left (378 c^4 d^2 e+105 c^6 d^3+405 c^2 d e^2+140 e^3\right )}{945 c^9}+\frac{b \sqrt{1-c^2 x^2} \left (189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2+35 e^3\right )}{315 c^9}-\frac{b e^2 \left (1-c^2 x^2\right )^{7/2} \left (27 c^2 d+28 e\right )}{441 c^9}+\frac{b e^3 \left (1-c^2 x^2\right )^{9/2}}{81 c^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*Sqrt[1 - c^2*x^2])/(315*c^9) - (b*(105*c^6*d^3 + 378
*c^4*d^2*e + 405*c^2*d*e^2 + 140*e^3)*(1 - c^2*x^2)^(3/2))/(945*c^9) + (b*e*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2
)*(1 - c^2*x^2)^(5/2))/(525*c^9) - (b*e^2*(27*c^2*d + 28*e)*(1 - c^2*x^2)^(7/2))/(441*c^9) + (b*e^3*(1 - c^2*x
^2)^(9/2))/(81*c^9) + (d^3*x^3*(a + b*ArcSin[c*x]))/3 + (3*d^2*e*x^5*(a + b*ArcSin[c*x]))/5 + (3*d*e^2*x^7*(a
+ b*ArcSin[c*x]))/7 + (e^3*x^9*(a + b*ArcSin[c*x]))/9

Rule 270

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

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u =
IntHide[(f*x)^m*(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]] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p] && (GtQ[p, 0] ||
(IGtQ[(m - 1)/2, 0] && LeQ[m + p, 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 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin{align*} \int x^2 \left (d+e x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )}{315 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{315} (b c) \int \frac{x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{630} (b c) \operatorname{Subst}\left (\int \frac{x \left (105 d^3+189 d^2 e x+135 d e^2 x^2+35 e^3 x^3\right )}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{630} (b c) \operatorname{Subst}\left (\int \left (\frac{105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3}{c^8 \sqrt{1-c^2 x}}+\frac{\left (-105 c^6 d^3-378 c^4 d^2 e-405 c^2 d e^2-140 e^3\right ) \sqrt{1-c^2 x}}{c^8}+\frac{3 e \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x\right )^{3/2}}{c^8}-\frac{5 e^2 \left (27 c^2 d+28 e\right ) \left (1-c^2 x\right )^{5/2}}{c^8}+\frac{35 e^3 \left (1-c^2 x\right )^{7/2}}{c^8}\right ) \, dx,x,x^2\right )\\ &=\frac{b \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \sqrt{1-c^2 x^2}}{315 c^9}-\frac{b \left (105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right ) \left (1-c^2 x^2\right )^{3/2}}{945 c^9}+\frac{b e \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^{5/2}}{525 c^9}-\frac{b e^2 \left (27 c^2 d+28 e\right ) \left (1-c^2 x^2\right )^{7/2}}{441 c^9}+\frac{b e^3 \left (1-c^2 x^2\right )^{9/2}}{81 c^9}+\frac{1}{3} d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} d^2 e x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{7} d e^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^3 x^9 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.226405, size = 231, normalized size = 0.8 \[ \frac{315 a x^3 \left (189 d^2 e x^2+105 d^3+135 d e^2 x^4+35 e^3 x^6\right )+\frac{b \sqrt{1-c^2 x^2} \left (c^8 \left (11907 d^2 e x^4+11025 d^3 x^2+6075 d e^2 x^6+1225 e^3 x^8\right )+2 c^6 \left (7938 d^2 e x^2+11025 d^3+3645 d e^2 x^4+700 e^3 x^6\right )+24 c^4 e \left (1323 d^2+405 d e x^2+70 e^2 x^4\right )+80 c^2 e^2 \left (243 d+28 e x^2\right )+4480 e^3\right )}{c^9}+315 b x^3 \sin ^{-1}(c x) \left (189 d^2 e x^2+105 d^3+135 d e^2 x^4+35 e^3 x^6\right )}{99225} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(315*a*x^3*(105*d^3 + 189*d^2*e*x^2 + 135*d*e^2*x^4 + 35*e^3*x^6) + (b*Sqrt[1 - c^2*x^2]*(4480*e^3 + 80*c^2*e^
2*(243*d + 28*e*x^2) + 24*c^4*e*(1323*d^2 + 405*d*e*x^2 + 70*e^2*x^4) + 2*c^6*(11025*d^3 + 7938*d^2*e*x^2 + 36
45*d*e^2*x^4 + 700*e^3*x^6) + c^8*(11025*d^3*x^2 + 11907*d^2*e*x^4 + 6075*d*e^2*x^6 + 1225*e^3*x^8)))/c^9 + 31
5*b*x^3*(105*d^3 + 189*d^2*e*x^2 + 135*d*e^2*x^4 + 35*e^3*x^6)*ArcSin[c*x])/99225

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Maple [A]  time = 0.006, size = 417, normalized size = 1.5 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{a}{{c}^{6}} \left ({\frac{{e}^{3}{c}^{9}{x}^{9}}{9}}+{\frac{3\,{c}^{9}d{e}^{2}{x}^{7}}{7}}+{\frac{3\,{c}^{9}{d}^{2}e{x}^{5}}{5}}+{\frac{{d}^{3}{c}^{9}{x}^{3}}{3}} \right ) }+{\frac{b}{{c}^{6}} \left ({\frac{\arcsin \left ( cx \right ){e}^{3}{c}^{9}{x}^{9}}{9}}+{\frac{3\,\arcsin \left ( cx \right ){c}^{9}d{e}^{2}{x}^{7}}{7}}+{\frac{3\,\arcsin \left ( cx \right ){c}^{9}{d}^{2}e{x}^{5}}{5}}+{\frac{\arcsin \left ( cx \right ){d}^{3}{c}^{9}{x}^{3}}{3}}-{\frac{{e}^{3}}{9} \left ( -{\frac{{c}^{8}{x}^{8}}{9}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{8\,{c}^{6}{x}^{6}}{63}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{16\,{c}^{4}{x}^{4}}{105}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{64\,{c}^{2}{x}^{2}}{315}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{128}{315}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }-{\frac{3\,{c}^{2}d{e}^{2}}{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}^{4}{d}^{2}e}{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 ) }-{\frac{{d}^{3}{c}^{6}}{3} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^3*(a/c^6*(1/9*e^3*c^9*x^9+3/7*c^9*d*e^2*x^7+3/5*c^9*d^2*e*x^5+1/3*d^3*c^9*x^3)+b/c^6*(1/9*arcsin(c*x)*e^3*
c^9*x^9+3/7*arcsin(c*x)*c^9*d*e^2*x^7+3/5*arcsin(c*x)*c^9*d^2*e*x^5+1/3*arcsin(c*x)*d^3*c^9*x^3-1/9*e^3*(-1/9*
c^8*x^8*(-c^2*x^2+1)^(1/2)-8/63*c^6*x^6*(-c^2*x^2+1)^(1/2)-16/105*c^4*x^4*(-c^2*x^2+1)^(1/2)-64/315*c^2*x^2*(-
c^2*x^2+1)^(1/2)-128/315*(-c^2*x^2+1)^(1/2))-3/7*c^2*d*e^2*(-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^4*d^2*e*(-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))-1/3*d^3*c^6*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)
-2/3*(-c^2*x^2+1)^(1/2))))

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Maxima [A]  time = 1.47401, size = 518, normalized size = 1.8 \begin{align*} \frac{1}{9} \, a e^{3} x^{9} + \frac{3}{7} \, a d e^{2} x^{7} + \frac{3}{5} \, a d^{2} e x^{5} + \frac{1}{3} \, a d^{3} x^{3} + \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^{3} + \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^{2} e + \frac{3}{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 d e^{2} + \frac{1}{2835} \,{\left (315 \, x^{9} \arcsin \left (c x\right ) +{\left (\frac{35 \, \sqrt{-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac{40 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac{48 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac{64 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac{128 \, \sqrt{-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b e^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*e^3*x^9 + 3/7*a*d*e^2*x^7 + 3/5*a*d^2*e*x^5 + 1/3*a*d^3*x^3 + 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^3 + 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^2*e + 3/245*(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*d*e^2 + 1/2835*(315*x^9*arcsin(c*x) + (35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqrt(-c^2*x^2 + 1)*x^6/c^4 + 48*s
qrt(-c^2*x^2 + 1)*x^4/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 + 1)/c^10)*c)*b*e^3

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Fricas [A]  time = 2.06325, size = 679, normalized size = 2.37 \begin{align*} \frac{11025 \, a c^{9} e^{3} x^{9} + 42525 \, a c^{9} d e^{2} x^{7} + 59535 \, a c^{9} d^{2} e x^{5} + 33075 \, a c^{9} d^{3} x^{3} + 315 \,{\left (35 \, b c^{9} e^{3} x^{9} + 135 \, b c^{9} d e^{2} x^{7} + 189 \, b c^{9} d^{2} e x^{5} + 105 \, b c^{9} d^{3} x^{3}\right )} \arcsin \left (c x\right ) +{\left (1225 \, b c^{8} e^{3} x^{8} + 22050 \, b c^{6} d^{3} + 31752 \, b c^{4} d^{2} e + 25 \,{\left (243 \, b c^{8} d e^{2} + 56 \, b c^{6} e^{3}\right )} x^{6} + 19440 \, b c^{2} d e^{2} + 3 \,{\left (3969 \, b c^{8} d^{2} e + 2430 \, b c^{6} d e^{2} + 560 \, b c^{4} e^{3}\right )} x^{4} + 4480 \, b e^{3} +{\left (11025 \, b c^{8} d^{3} + 15876 \, b c^{6} d^{2} e + 9720 \, b c^{4} d e^{2} + 2240 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt{-c^{2} x^{2} + 1}}{99225 \, c^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/99225*(11025*a*c^9*e^3*x^9 + 42525*a*c^9*d*e^2*x^7 + 59535*a*c^9*d^2*e*x^5 + 33075*a*c^9*d^3*x^3 + 315*(35*b
*c^9*e^3*x^9 + 135*b*c^9*d*e^2*x^7 + 189*b*c^9*d^2*e*x^5 + 105*b*c^9*d^3*x^3)*arcsin(c*x) + (1225*b*c^8*e^3*x^
8 + 22050*b*c^6*d^3 + 31752*b*c^4*d^2*e + 25*(243*b*c^8*d*e^2 + 56*b*c^6*e^3)*x^6 + 19440*b*c^2*d*e^2 + 3*(396
9*b*c^8*d^2*e + 2430*b*c^6*d*e^2 + 560*b*c^4*e^3)*x^4 + 4480*b*e^3 + (11025*b*c^8*d^3 + 15876*b*c^6*d^2*e + 97
20*b*c^4*d*e^2 + 2240*b*c^2*e^3)*x^2)*sqrt(-c^2*x^2 + 1))/c^9

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Sympy [A]  time = 24.8269, size = 525, normalized size = 1.83 \begin{align*} \begin{cases} \frac{a d^{3} x^{3}}{3} + \frac{3 a d^{2} e x^{5}}{5} + \frac{3 a d e^{2} x^{7}}{7} + \frac{a e^{3} x^{9}}{9} + \frac{b d^{3} x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{3 b d^{2} e x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{3 b d e^{2} x^{7} \operatorname{asin}{\left (c x \right )}}{7} + \frac{b e^{3} x^{9} \operatorname{asin}{\left (c x \right )}}{9} + \frac{b d^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{3 b d^{2} e x^{4} \sqrt{- c^{2} x^{2} + 1}}{25 c} + \frac{3 b d e^{2} x^{6} \sqrt{- c^{2} x^{2} + 1}}{49 c} + \frac{b e^{3} x^{8} \sqrt{- c^{2} x^{2} + 1}}{81 c} + \frac{2 b d^{3} \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{4 b d^{2} e x^{2} \sqrt{- c^{2} x^{2} + 1}}{25 c^{3}} + \frac{18 b d e^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{245 c^{3}} + \frac{8 b e^{3} x^{6} \sqrt{- c^{2} x^{2} + 1}}{567 c^{3}} + \frac{8 b d^{2} e \sqrt{- c^{2} x^{2} + 1}}{25 c^{5}} + \frac{24 b d e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{245 c^{5}} + \frac{16 b e^{3} x^{4} \sqrt{- c^{2} x^{2} + 1}}{945 c^{5}} + \frac{48 b d e^{2} \sqrt{- c^{2} x^{2} + 1}}{245 c^{7}} + \frac{64 b e^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{2835 c^{7}} + \frac{128 b e^{3} \sqrt{- c^{2} x^{2} + 1}}{2835 c^{9}} & \text{for}\: c \neq 0 \\a \left (\frac{d^{3} x^{3}}{3} + \frac{3 d^{2} e x^{5}}{5} + \frac{3 d e^{2} x^{7}}{7} + \frac{e^{3} x^{9}}{9}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**3*x**3/3 + 3*a*d**2*e*x**5/5 + 3*a*d*e**2*x**7/7 + a*e**3*x**9/9 + b*d**3*x**3*asin(c*x)/3 + 3
*b*d**2*e*x**5*asin(c*x)/5 + 3*b*d*e**2*x**7*asin(c*x)/7 + b*e**3*x**9*asin(c*x)/9 + b*d**3*x**2*sqrt(-c**2*x*
*2 + 1)/(9*c) + 3*b*d**2*e*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + 3*b*d*e**2*x**6*sqrt(-c**2*x**2 + 1)/(49*c) + b*
e**3*x**8*sqrt(-c**2*x**2 + 1)/(81*c) + 2*b*d**3*sqrt(-c**2*x**2 + 1)/(9*c**3) + 4*b*d**2*e*x**2*sqrt(-c**2*x*
*2 + 1)/(25*c**3) + 18*b*d*e**2*x**4*sqrt(-c**2*x**2 + 1)/(245*c**3) + 8*b*e**3*x**6*sqrt(-c**2*x**2 + 1)/(567
*c**3) + 8*b*d**2*e*sqrt(-c**2*x**2 + 1)/(25*c**5) + 24*b*d*e**2*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) + 16*b*e
**3*x**4*sqrt(-c**2*x**2 + 1)/(945*c**5) + 48*b*d*e**2*sqrt(-c**2*x**2 + 1)/(245*c**7) + 64*b*e**3*x**2*sqrt(-
c**2*x**2 + 1)/(2835*c**7) + 128*b*e**3*sqrt(-c**2*x**2 + 1)/(2835*c**9), Ne(c, 0)), (a*(d**3*x**3/3 + 3*d**2*
e*x**5/5 + 3*d*e**2*x**7/7 + e**3*x**9/9), True))

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Giac [B]  time = 1.28987, size = 942, normalized size = 3.28 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*x^9*e^3 + 3/7*a*d*x^7*e^2 + 3/5*a*d^2*x^5*e + 1/3*a*d^3*x^3 + 1/3*(c^2*x^2 - 1)*b*d^3*x*arcsin(c*x)/c^2
+ 1/3*b*d^3*x*arcsin(c*x)/c^2 + 3/5*(c^2*x^2 - 1)^2*b*d^2*x*arcsin(c*x)*e/c^4 + 6/5*(c^2*x^2 - 1)*b*d^2*x*arcs
in(c*x)*e/c^4 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*d^3/c^3 + 3/7*(c^2*x^2 - 1)^3*b*d*x*arcsin(c*x)*e^2/c^6 + 3/5*b*d^2
*x*arcsin(c*x)*e/c^4 + 1/3*sqrt(-c^2*x^2 + 1)*b*d^3/c^3 + 3/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^2*e/c^5
+ 9/7*(c^2*x^2 - 1)^2*b*d*x*arcsin(c*x)*e^2/c^6 - 2/5*(-c^2*x^2 + 1)^(3/2)*b*d^2*e/c^5 + 1/9*(c^2*x^2 - 1)^4*b
*x*arcsin(c*x)*e^3/c^8 + 9/7*(c^2*x^2 - 1)*b*d*x*arcsin(c*x)*e^2/c^6 + 3/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)
*b*d*e^2/c^7 + 3/5*sqrt(-c^2*x^2 + 1)*b*d^2*e/c^5 + 4/9*(c^2*x^2 - 1)^3*b*x*arcsin(c*x)*e^3/c^8 + 3/7*b*d*x*ar
csin(c*x)*e^2/c^6 + 9/35*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*e^2/c^7 + 2/3*(c^2*x^2 - 1)^2*b*x*arcsin(c*x)*
e^3/c^8 + 1/81*(c^2*x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b*e^3/c^9 - 3/7*(-c^2*x^2 + 1)^(3/2)*b*d*e^2/c^7 + 4/9*(c^2*
x^2 - 1)*b*x*arcsin(c*x)*e^3/c^8 + 4/63*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*e^3/c^9 + 3/7*sqrt(-c^2*x^2 + 1)*
b*d*e^2/c^7 + 1/9*b*x*arcsin(c*x)*e^3/c^8 + 2/15*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^3/c^9 - 4/27*(-c^2*x^2
+ 1)^(3/2)*b*e^3/c^9 + 1/9*sqrt(-c^2*x^2 + 1)*b*e^3/c^9