3.7.16 \(\int x^2 (d+e x^2)^3 (a+b \text {ArcSin}(c x)) \, dx\) [616]

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

[Out]

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

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Rubi [A]
time = 0.26, antiderivative size = 287, normalized size of antiderivative = 1.00, 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 = {276, 4815, 12, 1813, 1634} \begin {gather*} \frac {1}{3} d^3 x^3 (a+b \text {ArcSin}(c x))+\frac {3}{5} d^2 e x^5 (a+b \text {ArcSin}(c x))+\frac {3}{7} d e^2 x^7 (a+b \text {ArcSin}(c x))+\frac {1}{9} e^3 x^9 (a+b \text {ArcSin}(c x))-\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}+\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 (105 c^6 d^3+378 c^4 d^2 e+405 c^2 d e^2+140 e^3\right )}{945 c^9}+\frac {b \sqrt {1-c^2 x^2} \left (105 c^6 d^3+189 c^4 d^2 e+135 c^2 d e^2+35 e^3\right )}{315 c^9} \end {gather*}

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 12

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

Rule 276

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 1634

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]

Rule 1813

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 4815

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]))

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) \text {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) \text {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*}

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Mathematica [A]
time = 0.15, size = 231, normalized size = 0.80 \begin {gather*} \frac {315 a x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right )+\frac {b \sqrt {1-c^2 x^2} \left (4480 e^3+80 c^2 e^2 \left (243 d+28 e x^2\right )+24 c^4 e \left (1323 d^2+405 d e x^2+70 e^2 x^4\right )+2 c^6 \left (11025 d^3+7938 d^2 e x^2+3645 d e^2 x^4+700 e^3 x^6\right )+c^8 \left (11025 d^3 x^2+11907 d^2 e x^4+6075 d e^2 x^6+1225 e^3 x^8\right )\right )}{c^9}+315 b x^3 \left (105 d^3+189 d^2 e x^2+135 d e^2 x^4+35 e^3 x^6\right ) \text {ArcSin}(c x)}{99225} \end {gather*}

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 = 0.14, size = 417, normalized size = 1.45

method	result	size
derivativedivides	\(\frac {\frac {a \left (\frac {1}{3} d^{3} c^{9} x^{3}+\frac {3}{5} d^{2} c^{9} e \,x^{5}+\frac {3}{7} d \,c^{9} e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{3} c^{9} x^{3}}{3}+\frac {3 \arcsin \left (c x \right ) d^{2} c^{9} e \,x^{5}}{5}+\frac {3 \arcsin \left (c x \right ) d \,c^{9} e^{2} x^{7}}{7}+\frac {\arcsin \left (c x \right ) e^{3} c^{9} x^{9}}{9}-\frac {d^{3} c^{6} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {3 d^{2} c^{4} e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {3 d \,c^{2} e^{2} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-\frac {e^{3} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}\right )}{c^{6}}}{c^{3}}\)	\(417\)
default	\(\frac {\frac {a \left (\frac {1}{3} d^{3} c^{9} x^{3}+\frac {3}{5} d^{2} c^{9} e \,x^{5}+\frac {3}{7} d \,c^{9} e^{2} x^{7}+\frac {1}{9} e^{3} c^{9} x^{9}\right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) d^{3} c^{9} x^{3}}{3}+\frac {3 \arcsin \left (c x \right ) d^{2} c^{9} e \,x^{5}}{5}+\frac {3 \arcsin \left (c x \right ) d \,c^{9} e^{2} x^{7}}{7}+\frac {\arcsin \left (c x \right ) e^{3} c^{9} x^{9}}{9}-\frac {d^{3} c^{6} \left (-\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {-c^{2} x^{2}+1}}{3}\right )}{3}-\frac {3 d^{2} c^{4} e \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}-\frac {3 d \,c^{2} e^{2} \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-\frac {e^{3} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}\right )}{c^{6}}}{c^{3}}\)	\(417\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 382, normalized size = 1.33 \begin {gather*} \frac {1}{9} \, a x^{9} e^{3} + \frac {3}{7} \, a d x^{7} e^{2} + \frac {3}{5} \, a d^{2} x^{5} e + \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 {gather*}

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*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 + 1/9*(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 = 2.40, size = 270, normalized size = 0.94 \begin {gather*} \frac {11025 \, a c^{9} x^{9} e^{3} + 42525 \, a c^{9} d x^{7} e^{2} + 59535 \, a c^{9} d^{2} x^{5} e + 33075 \, a c^{9} d^{3} x^{3} + 315 \, {\left (35 \, b c^{9} x^{9} e^{3} + 135 \, b c^{9} d x^{7} e^{2} + 189 \, b c^{9} d^{2} x^{5} e + 105 \, b c^{9} d^{3} x^{3}\right )} \arcsin \left (c x\right ) + {\left (11025 \, b c^{8} d^{3} x^{2} + 22050 \, b c^{6} d^{3} + 35 \, {\left (35 \, b c^{8} x^{8} + 40 \, b c^{6} x^{6} + 48 \, b c^{4} x^{4} + 64 \, b c^{2} x^{2} + 128 \, b\right )} e^{3} + 1215 \, {\left (5 \, b c^{8} d x^{6} + 6 \, b c^{6} d x^{4} + 8 \, b c^{4} d x^{2} + 16 \, b c^{2} d\right )} e^{2} + 3969 \, {\left (3 \, b c^{8} d^{2} x^{4} + 4 \, b c^{6} d^{2} x^{2} + 8 \, b c^{4} d^{2}\right )} e\right )} \sqrt {-c^{2} x^{2} + 1}}{99225 \, c^{9}} \end {gather*}

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*x^9*e^3 + 42525*a*c^9*d*x^7*e^2 + 59535*a*c^9*d^2*x^5*e + 33075*a*c^9*d^3*x^3 + 315*(35*b
*c^9*x^9*e^3 + 135*b*c^9*d*x^7*e^2 + 189*b*c^9*d^2*x^5*e + 105*b*c^9*d^3*x^3)*arcsin(c*x) + (11025*b*c^8*d^3*x
^2 + 22050*b*c^6*d^3 + 35*(35*b*c^8*x^8 + 40*b*c^6*x^6 + 48*b*c^4*x^4 + 64*b*c^2*x^2 + 128*b)*e^3 + 1215*(5*b*
c^8*d*x^6 + 6*b*c^6*d*x^4 + 8*b*c^4*d*x^2 + 16*b*c^2*d)*e^2 + 3969*(3*b*c^8*d^2*x^4 + 4*b*c^6*d^2*x^2 + 8*b*c^
4*d^2)*e)*sqrt(-c^2*x^2 + 1))/c^9

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Sympy [A]
time = 1.69, size = 525, normalized size = 1.83 \begin {gather*} \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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 711 vs. \(2 (259) = 518\).
time = 0.42, size = 711, normalized size = 2.48 \begin {gather*} \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 {{\left (c^{2} x^{2} - 1\right )} b d^{3} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {b d^{3} x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} e x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {6 \, {\left (c^{2} x^{2} - 1\right )} b d^{2} e x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{3} b d e^{2} x \arcsin \left (c x\right )}{7 \, c^{6}} - \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{3}}{9 \, c^{3}} + \frac {3 \, b d^{2} e x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {9 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d e^{2} x \arcsin \left (c x\right )}{7 \, c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b e^{3} x \arcsin \left (c x\right )}{9 \, c^{8}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{3}}{3 \, c^{3}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2} e}{25 \, c^{5}} + \frac {9 \, {\left (c^{2} x^{2} - 1\right )} b d e^{2} x \arcsin \left (c x\right )}{7 \, c^{6}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{3} b e^{3} x \arcsin \left (c x\right )}{9 \, c^{8}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2} e}{5 \, c^{5}} + \frac {3 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d e^{2}}{49 \, c^{7}} + \frac {3 \, b d e^{2} x \arcsin \left (c x\right )}{7 \, c^{6}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b e^{3} x \arcsin \left (c x\right )}{3 \, c^{8}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d^{2} e}{5 \, c^{5}} + \frac {9 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d e^{2}}{35 \, c^{7}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b e^{3}}{81 \, c^{9}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b e^{3} x \arcsin \left (c x\right )}{9 \, c^{8}} - \frac {3 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e^{2}}{7 \, c^{7}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b e^{3}}{63 \, c^{9}} + \frac {b e^{3} x \arcsin \left (c x\right )}{9 \, c^{8}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d e^{2}}{7 \, c^{7}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{3}}{15 \, c^{9}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{3}}{27 \, c^{9}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{3}}{9 \, c^{9}} \end {gather*}

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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