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

Optimal. Leaf size=241 \[ \frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {2 b e \left (1-c^2 x^2\right )^{7/2} \left (9 c^2 d+14 e\right )}{441 c^9}+\frac {b e^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^9}+\frac {b \left (1-c^2 x^2\right )^{5/2} \left (21 c^4 d^2+90 c^2 d e+70 e^2\right )}{525 c^9}-\frac {2 b \left (1-c^2 x^2\right )^{3/2} \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{945 c^9}+\frac {b \sqrt {1-c^2 x^2} \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{315 c^9} \]

[Out]

-2/945*b*(63*c^4*d^2+135*c^2*d*e+70*e^2)*(-c^2*x^2+1)^(3/2)/c^9+1/525*b*(21*c^4*d^2+90*c^2*d*e+70*e^2)*(-c^2*x
^2+1)^(5/2)/c^9-2/441*b*e*(9*c^2*d+14*e)*(-c^2*x^2+1)^(7/2)/c^9+1/81*b*e^2*(-c^2*x^2+1)^(9/2)/c^9+1/5*d^2*x^5*
(a+b*arcsin(c*x))+2/7*d*e*x^7*(a+b*arcsin(c*x))+1/9*e^2*x^9*(a+b*arcsin(c*x))+1/315*b*(63*c^4*d^2+90*c^2*d*e+3
5*e^2)*(-c^2*x^2+1)^(1/2)/c^9

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Rubi [A]  time = 0.32, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {270, 4731, 12, 1251, 897, 1153} \[ \frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \left (1-c^2 x^2\right )^{5/2} \left (21 c^4 d^2+90 c^2 d e+70 e^2\right )}{525 c^9}-\frac {2 b \left (1-c^2 x^2\right )^{3/2} \left (63 c^4 d^2+135 c^2 d e+70 e^2\right )}{945 c^9}+\frac {b \sqrt {1-c^2 x^2} \left (63 c^4 d^2+90 c^2 d e+35 e^2\right )}{315 c^9}-\frac {2 b e \left (1-c^2 x^2\right )^{7/2} \left (9 c^2 d+14 e\right )}{441 c^9}+\frac {b e^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^9} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(63*c^4*d^2 + 90*c^2*d*e + 35*e^2)*Sqrt[1 - c^2*x^2])/(315*c^9) - (2*b*(63*c^4*d^2 + 135*c^2*d*e + 70*e^2)*
(1 - c^2*x^2)^(3/2))/(945*c^9) + (b*(21*c^4*d^2 + 90*c^2*d*e + 70*e^2)*(1 - c^2*x^2)^(5/2))/(525*c^9) - (2*b*e
*(9*c^2*d + 14*e)*(1 - c^2*x^2)^(7/2))/(441*c^9) + (b*e^2*(1 - c^2*x^2)^(9/2))/(81*c^9) + (d^2*x^5*(a + b*ArcS
in[c*x]))/5 + (2*d*e*x^7*(a + b*ArcSin[c*x]))/7 + (e^2*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 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 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

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]))

Rubi steps

\begin {align*} \int x^4 \left (d+e x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{315 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{315} (b c) \int \frac {x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{630} (b c) \operatorname {Subst}\left (\int \frac {x^2 \left (63 d^2+90 d e x+35 e^2 x^2\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {x^2}{c^2}\right )^2 \left (\frac {63 c^4 d^2+90 c^2 d e+35 e^2}{c^4}-\frac {\left (90 c^2 d e+70 e^2\right ) x^2}{c^4}+\frac {35 e^2 x^4}{c^4}\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{315 c}\\ &=\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {b \operatorname {Subst}\left (\int \left (\frac {63 c^4 d^2+90 c^2 d e+35 e^2}{c^8}-\frac {2 \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) x^2}{c^8}+\frac {3 \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) x^4}{c^8}-\frac {10 e \left (9 c^2 d+14 e\right ) x^6}{c^8}+\frac {35 e^2 x^8}{c^8}\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{315 c}\\ &=\frac {b \left (63 c^4 d^2+90 c^2 d e+35 e^2\right ) \sqrt {1-c^2 x^2}}{315 c^9}-\frac {2 b \left (63 c^4 d^2+135 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^{3/2}}{945 c^9}+\frac {b \left (21 c^4 d^2+90 c^2 d e+70 e^2\right ) \left (1-c^2 x^2\right )^{5/2}}{525 c^9}-\frac {2 b e \left (9 c^2 d+14 e\right ) \left (1-c^2 x^2\right )^{7/2}}{441 c^9}+\frac {b e^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^9}+\frac {1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {2}{7} d e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{9} e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 187, normalized size = 0.78 \[ \frac {315 a x^5 \left (63 d^2+90 d e x^2+35 e^2 x^4\right )+\frac {b \sqrt {1-c^2 x^2} \left (c^8 \left (3969 d^2 x^4+4050 d e x^6+1225 e^2 x^8\right )+4 c^6 \left (1323 d^2 x^2+1215 d e x^4+350 e^2 x^6\right )+24 c^4 \left (441 d^2+270 d e x^2+70 e^2 x^4\right )+160 c^2 e \left (81 d+14 e x^2\right )+4480 e^2\right )}{c^9}+315 b x^5 \sin ^{-1}(c x) \left (63 d^2+90 d e x^2+35 e^2 x^4\right )}{99225} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(315*a*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4) + (b*Sqrt[1 - c^2*x^2]*(4480*e^2 + 160*c^2*e*(81*d + 14*e*x^2) +
24*c^4*(441*d^2 + 270*d*e*x^2 + 70*e^2*x^4) + 4*c^6*(1323*d^2*x^2 + 1215*d*e*x^4 + 350*e^2*x^6) + c^8*(3969*d
^2*x^4 + 4050*d*e*x^6 + 1225*e^2*x^8)))/c^9 + 315*b*x^5*(63*d^2 + 90*d*e*x^2 + 35*e^2*x^4)*ArcSin[c*x])/99225

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fricas [A]  time = 0.60, size = 219, normalized size = 0.91 \[ \frac {11025 \, a c^{9} e^{2} x^{9} + 28350 \, a c^{9} d e x^{7} + 19845 \, a c^{9} d^{2} x^{5} + 315 \, {\left (35 \, b c^{9} e^{2} x^{9} + 90 \, b c^{9} d e x^{7} + 63 \, b c^{9} d^{2} x^{5}\right )} \arcsin \left (c x\right ) + {\left (1225 \, b c^{8} e^{2} x^{8} + 10584 \, b c^{4} d^{2} + 50 \, {\left (81 \, b c^{8} d e + 28 \, b c^{6} e^{2}\right )} x^{6} + 12960 \, b c^{2} d e + 3 \, {\left (1323 \, b c^{8} d^{2} + 1620 \, b c^{6} d e + 560 \, b c^{4} e^{2}\right )} x^{4} + 4480 \, b e^{2} + 4 \, {\left (1323 \, b c^{6} d^{2} + 1620 \, b c^{4} d e + 560 \, b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{99225 \, c^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/99225*(11025*a*c^9*e^2*x^9 + 28350*a*c^9*d*e*x^7 + 19845*a*c^9*d^2*x^5 + 315*(35*b*c^9*e^2*x^9 + 90*b*c^9*d*
e*x^7 + 63*b*c^9*d^2*x^5)*arcsin(c*x) + (1225*b*c^8*e^2*x^8 + 10584*b*c^4*d^2 + 50*(81*b*c^8*d*e + 28*b*c^6*e^
2)*x^6 + 12960*b*c^2*d*e + 3*(1323*b*c^8*d^2 + 1620*b*c^6*d*e + 560*b*c^4*e^2)*x^4 + 4480*b*e^2 + 4*(1323*b*c^
6*d^2 + 1620*b*c^4*d*e + 560*b*c^2*e^2)*x^2)*sqrt(-c^2*x^2 + 1))/c^9

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giac [B]  time = 0.44, size = 596, normalized size = 2.47 \[ \frac {1}{9} \, a x^{9} e^{2} + \frac {2}{7} \, a d x^{7} e + \frac {1}{5} \, a d^{2} x^{5} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} b d x \arcsin \left (c x\right ) e}{7 \, c^{6}} + \frac {b d^{2} x \arcsin \left (c x\right )}{5 \, c^{4}} + \frac {6 \, {\left (c^{2} x^{2} - 1\right )}^{2} b d x \arcsin \left (c x\right ) e}{7 \, c^{6}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d^{2}}{25 \, c^{5}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} b x \arcsin \left (c x\right ) e^{2}}{9 \, c^{8}} + \frac {6 \, {\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right ) e}{7 \, c^{6}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d^{2}}{15 \, c^{5}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b d e}{49 \, c^{7}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{3} b x \arcsin \left (c x\right ) e^{2}}{9 \, c^{8}} + \frac {2 \, b d x \arcsin \left (c x\right ) e}{7 \, c^{6}} + \frac {\sqrt {-c^{2} x^{2} + 1} b d^{2}}{5 \, c^{5}} + \frac {6 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b d e}{35 \, c^{7}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} b x \arcsin \left (c x\right ) e^{2}}{3 \, c^{8}} + \frac {{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt {-c^{2} x^{2} + 1} b e^{2}}{81 \, c^{9}} - \frac {2 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d e}{7 \, c^{7}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right ) e^{2}}{9 \, c^{8}} + \frac {4 \, {\left (c^{2} x^{2} - 1\right )}^{3} \sqrt {-c^{2} x^{2} + 1} b e^{2}}{63 \, c^{9}} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d e}{7 \, c^{7}} + \frac {b x \arcsin \left (c x\right ) e^{2}}{9 \, c^{8}} + \frac {2 \, {\left (c^{2} x^{2} - 1\right )}^{2} \sqrt {-c^{2} x^{2} + 1} b e^{2}}{15 \, c^{9}} - \frac {4 \, {\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b e^{2}}{27 \, c^{9}} + \frac {\sqrt {-c^{2} x^{2} + 1} b e^{2}}{9 \, c^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 339, normalized size = 1.41 \[ \frac {\frac {a \left (\frac {1}{9} e^{2} c^{9} x^{9}+\frac {2}{7} c^{9} e d \,x^{7}+\frac {1}{5} d^{2} c^{9} x^{5}\right )}{c^{4}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{2} c^{9} x^{9}}{9}+\frac {2 \arcsin \left (c x \right ) c^{9} e d \,x^{7}}{7}+\frac {\arcsin \left (c x \right ) d^{2} c^{9} x^{5}}{5}-\frac {e^{2} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{9}-\frac {2 c^{2} e d \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-\frac {d^{2} c^{4} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{4}}}{c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x^2+d)^2*(a+b*arcsin(c*x)),x)

[Out]

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

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maxima [A]  time = 1.85, size = 314, normalized size = 1.30 \[ \frac {1}{9} \, a e^{2} x^{9} + \frac {2}{7} \, a d e x^{7} + \frac {1}{5} \, a d^{2} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} + \frac {2}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b d e + \frac {1}{2835} \, {\left (315 \, x^{9} \arcsin \left (c x\right ) + {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b e^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*a*e^2*x^9 + 2/7*a*d*e*x^7 + 1/5*a*d^2*x^5 + 1/75*(15*x^5*arcsin(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*s
qrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*d^2 + 2/245*(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 + 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^2

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 15.97, size = 415, normalized size = 1.72 \[ \begin {cases} \frac {a d^{2} x^{5}}{5} + \frac {2 a d e x^{7}}{7} + \frac {a e^{2} x^{9}}{9} + \frac {b d^{2} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {2 b d e x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {b e^{2} x^{9} \operatorname {asin}{\left (c x \right )}}{9} + \frac {b d^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {2 b d e x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} + \frac {b e^{2} x^{8} \sqrt {- c^{2} x^{2} + 1}}{81 c} + \frac {4 b d^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {12 b d e x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} + \frac {8 b e^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{567 c^{3}} + \frac {8 b d^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + \frac {16 b d e x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} + \frac {16 b e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{945 c^{5}} + \frac {32 b d e \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} + \frac {64 b e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{2835 c^{7}} + \frac {128 b e^{2} \sqrt {- c^{2} x^{2} + 1}}{2835 c^{9}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{2} x^{5}}{5} + \frac {2 d e x^{7}}{7} + \frac {e^{2} x^{9}}{9}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a*d**2*x**5/5 + 2*a*d*e*x**7/7 + a*e**2*x**9/9 + b*d**2*x**5*asin(c*x)/5 + 2*b*d*e*x**7*asin(c*x)/7
+ b*e**2*x**9*asin(c*x)/9 + b*d**2*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + 2*b*d*e*x**6*sqrt(-c**2*x**2 + 1)/(49*c
) + b*e**2*x**8*sqrt(-c**2*x**2 + 1)/(81*c) + 4*b*d**2*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) + 12*b*d*e*x**4*sqr
t(-c**2*x**2 + 1)/(245*c**3) + 8*b*e**2*x**6*sqrt(-c**2*x**2 + 1)/(567*c**3) + 8*b*d**2*sqrt(-c**2*x**2 + 1)/(
75*c**5) + 16*b*d*e*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5) + 16*b*e**2*x**4*sqrt(-c**2*x**2 + 1)/(945*c**5) + 32
*b*d*e*sqrt(-c**2*x**2 + 1)/(245*c**7) + 64*b*e**2*x**2*sqrt(-c**2*x**2 + 1)/(2835*c**7) + 128*b*e**2*sqrt(-c*
*2*x**2 + 1)/(2835*c**9), Ne(c, 0)), (a*(d**2*x**5/5 + 2*d*e*x**7/7 + e**2*x**9/9), True))

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