3.623 \(\int (d+e x^2)^4 (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=317 \[ \frac{6}{5} d^2 e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sin ^{-1}(c x)\right )+d^4 x \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{2 b e^2 \left (1-c^2 x^2\right )^{5/2} \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{525 c^9}-\frac{4 b e \left (1-c^2 x^2\right )^{3/2} \left (189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2+35 e^3\right )}{945 c^9}+\frac{b \sqrt{1-c^2 x^2} \left (378 c^4 d^2 e^2+420 c^6 d^3 e+315 c^8 d^4+180 c^2 d e^3+35 e^4\right )}{315 c^9}-\frac{4 b e^3 \left (1-c^2 x^2\right )^{7/2} \left (9 c^2 d+7 e\right )}{441 c^9}+\frac{b e^4 \left (1-c^2 x^2\right )^{9/2}}{81 c^9} \]

[Out]

(b*(315*c^8*d^4 + 420*c^6*d^3*e + 378*c^4*d^2*e^2 + 180*c^2*d*e^3 + 35*e^4)*Sqrt[1 - c^2*x^2])/(315*c^9) - (4*
b*e*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*(1 - c^2*x^2)^(3/2))/(945*c^9) + (2*b*e^2*(63*c^4*d
^2 + 90*c^2*d*e + 35*e^2)*(1 - c^2*x^2)^(5/2))/(525*c^9) - (4*b*e^3*(9*c^2*d + 7*e)*(1 - c^2*x^2)^(7/2))/(441*
c^9) + (b*e^4*(1 - c^2*x^2)^(9/2))/(81*c^9) + d^4*x*(a + b*ArcSin[c*x]) + (4*d^3*e*x^3*(a + b*ArcSin[c*x]))/3
+ (6*d^2*e^2*x^5*(a + b*ArcSin[c*x]))/5 + (4*d*e^3*x^7*(a + b*ArcSin[c*x]))/7 + (e^4*x^9*(a + b*ArcSin[c*x]))/
9

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Rubi [A]  time = 0.340063, antiderivative size = 317, 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} \[ \frac{6}{5} d^2 e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sin ^{-1}(c x)\right )+d^4 x \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{2 b e^2 \left (1-c^2 x^2\right )^{5/2} \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{525 c^9}-\frac{4 b e \left (1-c^2 x^2\right )^{3/2} \left (189 c^4 d^2 e+105 c^6 d^3+135 c^2 d e^2+35 e^3\right )}{945 c^9}+\frac{b \sqrt{1-c^2 x^2} \left (378 c^4 d^2 e^2+420 c^6 d^3 e+315 c^8 d^4+180 c^2 d e^3+35 e^4\right )}{315 c^9}-\frac{4 b e^3 \left (1-c^2 x^2\right )^{7/2} \left (9 c^2 d+7 e\right )}{441 c^9}+\frac{b e^4 \left (1-c^2 x^2\right )^{9/2}}{81 c^9} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2)^4*(a + b*ArcSin[c*x]),x]

[Out]

(b*(315*c^8*d^4 + 420*c^6*d^3*e + 378*c^4*d^2*e^2 + 180*c^2*d*e^3 + 35*e^4)*Sqrt[1 - c^2*x^2])/(315*c^9) - (4*
b*e*(105*c^6*d^3 + 189*c^4*d^2*e + 135*c^2*d*e^2 + 35*e^3)*(1 - c^2*x^2)^(3/2))/(945*c^9) + (2*b*e^2*(63*c^4*d
^2 + 90*c^2*d*e + 35*e^2)*(1 - c^2*x^2)^(5/2))/(525*c^9) - (4*b*e^3*(9*c^2*d + 7*e)*(1 - c^2*x^2)^(7/2))/(441*
c^9) + (b*e^4*(1 - c^2*x^2)^(9/2))/(81*c^9) + d^4*x*(a + b*ArcSin[c*x]) + (4*d^3*e*x^3*(a + b*ArcSin[c*x]))/3
+ (6*d^2*e^2*x^5*(a + b*ArcSin[c*x]))/5 + (4*d*e^3*x^7*(a + b*ArcSin[c*x]))/7 + (e^4*x^9*(a + b*ArcSin[c*x]))/
9

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 )^4 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^4 x \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{315 \sqrt{1-c^2 x^2}} \, dx\\ &=d^4 x \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{315} (b c) \int \frac{x \left (315 d^4+420 d^3 e x^2+378 d^2 e^2 x^4+180 d e^3 x^6+35 e^4 x^8\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=d^4 x \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{630} (b c) \operatorname{Subst}\left (\int \frac{315 d^4+420 d^3 e x+378 d^2 e^2 x^2+180 d e^3 x^3+35 e^4 x^4}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=d^4 x \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{630} (b c) \operatorname{Subst}\left (\int \left (\frac{315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4}{c^8 \sqrt{1-c^2 x}}-\frac{4 e \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}}{c^8}+\frac{6 e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x\right )^{3/2}}{c^8}-\frac{20 e^3 \left (9 c^2 d+7 e\right ) \left (1-c^2 x\right )^{5/2}}{c^8}+\frac{35 e^4 \left (1-c^2 x\right )^{7/2}}{c^8}\right ) \, dx,x,x^2\right )\\ &=\frac{b \left (315 c^8 d^4+420 c^6 d^3 e+378 c^4 d^2 e^2+180 c^2 d e^3+35 e^4\right ) \sqrt{1-c^2 x^2}}{315 c^9}-\frac{4 b e \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right ) \left (1-c^2 x^2\right )^{3/2}}{945 c^9}+\frac{2 b e^2 \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \left (1-c^2 x^2\right )^{5/2}}{525 c^9}-\frac{4 b e^3 \left (9 c^2 d+7 e\right ) \left (1-c^2 x^2\right )^{7/2}}{441 c^9}+\frac{b e^4 \left (1-c^2 x^2\right )^{9/2}}{81 c^9}+d^4 x \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{3} d^3 e x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{6}{5} d^2 e^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{4}{7} d e^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} e^4 x^9 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.306752, size = 260, normalized size = 0.82 \[ \frac{315 a x \left (378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4+180 d e^3 x^6+35 e^4 x^8\right )+\frac{b \sqrt{1-c^2 x^2} \left (c^8 \left (23814 d^2 e^2 x^4+44100 d^3 e x^2+99225 d^4+8100 d e^3 x^6+1225 e^4 x^8\right )+8 c^6 e \left (3969 d^2 e x^2+11025 d^3+1215 d e^2 x^4+175 e^3 x^6\right )+48 c^4 e^2 \left (1323 d^2+270 d e x^2+35 e^2 x^4\right )+320 c^2 e^3 \left (81 d+7 e x^2\right )+4480 e^4\right )}{c^9}+315 b x \sin ^{-1}(c x) \left (378 d^2 e^2 x^4+420 d^3 e x^2+315 d^4+180 d e^3 x^6+35 e^4 x^8\right )}{99225} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2)^4*(a + b*ArcSin[c*x]),x]

[Out]

(315*a*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x^8) + (b*Sqrt[1 - c^2*x^2]*(4480
*e^4 + 320*c^2*e^3*(81*d + 7*e*x^2) + 48*c^4*e^2*(1323*d^2 + 270*d*e*x^2 + 35*e^2*x^4) + 8*c^6*e*(11025*d^3 +
3969*d^2*e*x^2 + 1215*d*e^2*x^4 + 175*e^3*x^6) + c^8*(99225*d^4 + 44100*d^3*e*x^2 + 23814*d^2*e^2*x^4 + 8100*d
*e^3*x^6 + 1225*e^4*x^8)))/c^9 + 315*b*x*(315*d^4 + 420*d^3*e*x^2 + 378*d^2*e^2*x^4 + 180*d*e^3*x^6 + 35*e^4*x
^8)*ArcSin[c*x])/99225

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Maple [A]  time = 0.004, size = 465, normalized size = 1.5 \begin{align*}{\frac{1}{c} \left ({\frac{a}{{c}^{8}} \left ({\frac{{e}^{4}{c}^{9}{x}^{9}}{9}}+{\frac{4\,{c}^{9}d{e}^{3}{x}^{7}}{7}}+{\frac{6\,{c}^{9}{d}^{2}{e}^{2}{x}^{5}}{5}}+{\frac{4\,{c}^{9}{d}^{3}e{x}^{3}}{3}}+{c}^{9}{d}^{4}x \right ) }+{\frac{b}{{c}^{8}} \left ({\frac{\arcsin \left ( cx \right ){e}^{4}{c}^{9}{x}^{9}}{9}}+{\frac{4\,\arcsin \left ( cx \right ){c}^{9}d{e}^{3}{x}^{7}}{7}}+{\frac{6\,\arcsin \left ( cx \right ){c}^{9}{d}^{2}{e}^{2}{x}^{5}}{5}}+{\frac{4\,\arcsin \left ( cx \right ){c}^{9}{d}^{3}e{x}^{3}}{3}}+\arcsin \left ( cx \right ){c}^{9}{d}^{4}x-{\frac{{e}^{4}}{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{4\,{c}^{2}d{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{6\,{c}^{4}{d}^{2}{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 ) }-{\frac{4\,{c}^{6}{d}^{3}e}{3} \left ( -{\frac{{c}^{2}{x}^{2}}{3}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2}{3}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) }+{c}^{8}{d}^{4}\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)^4*(a+b*arcsin(c*x)),x)

[Out]

1/c*(a/c^8*(1/9*e^4*c^9*x^9+4/7*c^9*d*e^3*x^7+6/5*c^9*d^2*e^2*x^5+4/3*c^9*d^3*e*x^3+c^9*d^4*x)+b/c^8*(1/9*arcs
in(c*x)*e^4*c^9*x^9+4/7*arcsin(c*x)*c^9*d*e^3*x^7+6/5*arcsin(c*x)*c^9*d^2*e^2*x^5+4/3*arcsin(c*x)*c^9*d^3*e*x^
3+arcsin(c*x)*c^9*d^4*x-1/9*e^4*(-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))-4/7*c^2*d*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))-
6/5*c^4*d^2*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))-4/3*
c^6*d^3*e*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+c^8*d^4*(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.49512, size = 572, normalized size = 1.8 \begin{align*} \frac{1}{9} \, a e^{4} x^{9} + \frac{4}{7} \, a d e^{3} x^{7} + \frac{6}{5} \, a d^{2} e^{2} x^{5} + \frac{4}{3} \, a d^{3} e x^{3} + \frac{4}{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} e + \frac{2}{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^{2} + \frac{4}{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^{3} + \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^{4} + a d^{4} x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d^{4}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^4*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

1/9*a*e^4*x^9 + 4/7*a*d*e^3*x^7 + 6/5*a*d^2*e^2*x^5 + 4/3*a*d^3*e*x^3 + 4/9*(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*e + 2/25*(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^2 + 4/245*(35*x^7*arcsin(c*x) + (5*sq
rt(-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^3 + 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*sqrt(-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^4 + a*
d^4*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d^4/c

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Fricas [A]  time = 2.48234, size = 782, normalized size = 2.47 \begin{align*} \frac{11025 \, a c^{9} e^{4} x^{9} + 56700 \, a c^{9} d e^{3} x^{7} + 119070 \, a c^{9} d^{2} e^{2} x^{5} + 132300 \, a c^{9} d^{3} e x^{3} + 99225 \, a c^{9} d^{4} x + 315 \,{\left (35 \, b c^{9} e^{4} x^{9} + 180 \, b c^{9} d e^{3} x^{7} + 378 \, b c^{9} d^{2} e^{2} x^{5} + 420 \, b c^{9} d^{3} e x^{3} + 315 \, b c^{9} d^{4} x\right )} \arcsin \left (c x\right ) +{\left (1225 \, b c^{8} e^{4} x^{8} + 99225 \, b c^{8} d^{4} + 88200 \, b c^{6} d^{3} e + 63504 \, b c^{4} d^{2} e^{2} + 25920 \, b c^{2} d e^{3} + 100 \,{\left (81 \, b c^{8} d e^{3} + 14 \, b c^{6} e^{4}\right )} x^{6} + 4480 \, b e^{4} + 6 \,{\left (3969 \, b c^{8} d^{2} e^{2} + 1620 \, b c^{6} d e^{3} + 280 \, b c^{4} e^{4}\right )} x^{4} + 4 \,{\left (11025 \, b c^{8} d^{3} e + 7938 \, b c^{6} d^{2} e^{2} + 3240 \, b c^{4} d e^{3} + 560 \, b c^{2} e^{4}\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((e*x^2+d)^4*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

1/99225*(11025*a*c^9*e^4*x^9 + 56700*a*c^9*d*e^3*x^7 + 119070*a*c^9*d^2*e^2*x^5 + 132300*a*c^9*d^3*e*x^3 + 992
25*a*c^9*d^4*x + 315*(35*b*c^9*e^4*x^9 + 180*b*c^9*d*e^3*x^7 + 378*b*c^9*d^2*e^2*x^5 + 420*b*c^9*d^3*e*x^3 + 3
15*b*c^9*d^4*x)*arcsin(c*x) + (1225*b*c^8*e^4*x^8 + 99225*b*c^8*d^4 + 88200*b*c^6*d^3*e + 63504*b*c^4*d^2*e^2
+ 25920*b*c^2*d*e^3 + 100*(81*b*c^8*d*e^3 + 14*b*c^6*e^4)*x^6 + 4480*b*e^4 + 6*(3969*b*c^8*d^2*e^2 + 1620*b*c^
6*d*e^3 + 280*b*c^4*e^4)*x^4 + 4*(11025*b*c^8*d^3*e + 7938*b*c^6*d^2*e^2 + 3240*b*c^4*d*e^3 + 560*b*c^2*e^4)*x
^2)*sqrt(-c^2*x^2 + 1))/c^9

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Sympy [A]  time = 25.2348, size = 593, normalized size = 1.87 \begin{align*} \begin{cases} a d^{4} x + \frac{4 a d^{3} e x^{3}}{3} + \frac{6 a d^{2} e^{2} x^{5}}{5} + \frac{4 a d e^{3} x^{7}}{7} + \frac{a e^{4} x^{9}}{9} + b d^{4} x \operatorname{asin}{\left (c x \right )} + \frac{4 b d^{3} e x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{6 b d^{2} e^{2} x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{4 b d e^{3} x^{7} \operatorname{asin}{\left (c x \right )}}{7} + \frac{b e^{4} x^{9} \operatorname{asin}{\left (c x \right )}}{9} + \frac{b d^{4} \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{4 b d^{3} e x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{6 b d^{2} e^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{25 c} + \frac{4 b d e^{3} x^{6} \sqrt{- c^{2} x^{2} + 1}}{49 c} + \frac{b e^{4} x^{8} \sqrt{- c^{2} x^{2} + 1}}{81 c} + \frac{8 b d^{3} e \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{8 b d^{2} e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{25 c^{3}} + \frac{24 b d e^{3} x^{4} \sqrt{- c^{2} x^{2} + 1}}{245 c^{3}} + \frac{8 b e^{4} x^{6} \sqrt{- c^{2} x^{2} + 1}}{567 c^{3}} + \frac{16 b d^{2} e^{2} \sqrt{- c^{2} x^{2} + 1}}{25 c^{5}} + \frac{32 b d e^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{245 c^{5}} + \frac{16 b e^{4} x^{4} \sqrt{- c^{2} x^{2} + 1}}{945 c^{5}} + \frac{64 b d e^{3} \sqrt{- c^{2} x^{2} + 1}}{245 c^{7}} + \frac{64 b e^{4} x^{2} \sqrt{- c^{2} x^{2} + 1}}{2835 c^{7}} + \frac{128 b e^{4} \sqrt{- c^{2} x^{2} + 1}}{2835 c^{9}} & \text{for}\: c \neq 0 \\a \left (d^{4} x + \frac{4 d^{3} e x^{3}}{3} + \frac{6 d^{2} e^{2} x^{5}}{5} + \frac{4 d e^{3} x^{7}}{7} + \frac{e^{4} x^{9}}{9}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x**2+d)**4*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*d**4*x + 4*a*d**3*e*x**3/3 + 6*a*d**2*e**2*x**5/5 + 4*a*d*e**3*x**7/7 + a*e**4*x**9/9 + b*d**4*x*
asin(c*x) + 4*b*d**3*e*x**3*asin(c*x)/3 + 6*b*d**2*e**2*x**5*asin(c*x)/5 + 4*b*d*e**3*x**7*asin(c*x)/7 + b*e**
4*x**9*asin(c*x)/9 + b*d**4*sqrt(-c**2*x**2 + 1)/c + 4*b*d**3*e*x**2*sqrt(-c**2*x**2 + 1)/(9*c) + 6*b*d**2*e**
2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + 4*b*d*e**3*x**6*sqrt(-c**2*x**2 + 1)/(49*c) + b*e**4*x**8*sqrt(-c**2*x**2
+ 1)/(81*c) + 8*b*d**3*e*sqrt(-c**2*x**2 + 1)/(9*c**3) + 8*b*d**2*e**2*x**2*sqrt(-c**2*x**2 + 1)/(25*c**3) +
24*b*d*e**3*x**4*sqrt(-c**2*x**2 + 1)/(245*c**3) + 8*b*e**4*x**6*sqrt(-c**2*x**2 + 1)/(567*c**3) + 16*b*d**2*e
**2*sqrt(-c**2*x**2 + 1)/(25*c**5) + 32*b*d*e**3*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) + 16*b*e**4*x**4*sqrt(-c
**2*x**2 + 1)/(945*c**5) + 64*b*d*e**3*sqrt(-c**2*x**2 + 1)/(245*c**7) + 64*b*e**4*x**2*sqrt(-c**2*x**2 + 1)/(
2835*c**7) + 128*b*e**4*sqrt(-c**2*x**2 + 1)/(2835*c**9), Ne(c, 0)), (a*(d**4*x + 4*d**3*e*x**3/3 + 6*d**2*e**
2*x**5/5 + 4*d*e**3*x**7/7 + e**4*x**9/9), True))

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Giac [B]  time = 1.39186, size = 1004, normalized size = 3.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x^2+d)^4*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

1/9*a*x^9*e^4 + 4/7*a*d*x^7*e^3 + 6/5*a*d^2*x^5*e^2 + 4/3*a*d^3*x^3*e + b*d^4*x*arcsin(c*x) + a*d^4*x + 4/3*(c
^2*x^2 - 1)*b*d^3*x*arcsin(c*x)*e/c^2 + 4/3*b*d^3*x*arcsin(c*x)*e/c^2 + sqrt(-c^2*x^2 + 1)*b*d^4/c + 6/5*(c^2*
x^2 - 1)^2*b*d^2*x*arcsin(c*x)*e^2/c^4 - 4/9*(-c^2*x^2 + 1)^(3/2)*b*d^3*e/c^3 + 12/5*(c^2*x^2 - 1)*b*d^2*x*arc
sin(c*x)*e^2/c^4 + 4/3*sqrt(-c^2*x^2 + 1)*b*d^3*e/c^3 + 4/7*(c^2*x^2 - 1)^3*b*d*x*arcsin(c*x)*e^3/c^6 + 6/5*b*
d^2*x*arcsin(c*x)*e^2/c^4 + 6/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^2*e^2/c^5 + 12/7*(c^2*x^2 - 1)^2*b*d*x
*arcsin(c*x)*e^3/c^6 - 4/5*(-c^2*x^2 + 1)^(3/2)*b*d^2*e^2/c^5 + 1/9*(c^2*x^2 - 1)^4*b*x*arcsin(c*x)*e^4/c^8 +
12/7*(c^2*x^2 - 1)*b*d*x*arcsin(c*x)*e^3/c^6 + 4/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*d*e^3/c^7 + 6/5*sqrt(
-c^2*x^2 + 1)*b*d^2*e^2/c^5 + 4/9*(c^2*x^2 - 1)^3*b*x*arcsin(c*x)*e^4/c^8 + 4/7*b*d*x*arcsin(c*x)*e^3/c^6 + 12
/35*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*e^3/c^7 + 2/3*(c^2*x^2 - 1)^2*b*x*arcsin(c*x)*e^4/c^8 + 1/81*(c^2*x
^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b*e^4/c^9 - 4/7*(-c^2*x^2 + 1)^(3/2)*b*d*e^3/c^7 + 4/9*(c^2*x^2 - 1)*b*x*arcsin(c
*x)*e^4/c^8 + 4/63*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*e^4/c^9 + 4/7*sqrt(-c^2*x^2 + 1)*b*d*e^3/c^7 + 1/9*b*x
*arcsin(c*x)*e^4/c^8 + 2/15*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^4/c^9 - 4/27*(-c^2*x^2 + 1)^(3/2)*b*e^4/c^9
+ 1/9*sqrt(-c^2*x^2 + 1)*b*e^4/c^9