3.614 \(\int x^4 (d+e x^2)^3 (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=341 \[ \frac {1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )+\frac {b e^2 \left (1-c^2 x^2\right )^{9/2} \left (11 c^2 d+15 e\right )}{297 c^{11}}-\frac {b e^3 \left (1-c^2 x^2\right )^{11/2}}{121 c^{11}}-\frac {b e \left (1-c^2 x^2\right )^{7/2} \left (99 c^4 d^2+308 c^2 d e+210 e^2\right )}{1617 c^{11}}+\frac {b \left (1-c^2 x^2\right )^{5/2} \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right )}{1925 c^{11}}-\frac {b \left (1-c^2 x^2\right )^{3/2} \left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right )}{3465 c^{11}}+\frac {b \sqrt {1-c^2 x^2} \left (231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right )}{1155 c^{11}} \]

[Out]

-1/3465*b*(462*c^6*d^3+1485*c^4*d^2*e+1540*c^2*d*e^2+525*e^3)*(-c^2*x^2+1)^(3/2)/c^11+1/1925*b*(77*c^6*d^3+495
*c^4*d^2*e+770*c^2*d*e^2+350*e^3)*(-c^2*x^2+1)^(5/2)/c^11-1/1617*b*e*(99*c^4*d^2+308*c^2*d*e+210*e^2)*(-c^2*x^
2+1)^(7/2)/c^11+1/297*b*e^2*(11*c^2*d+15*e)*(-c^2*x^2+1)^(9/2)/c^11-1/121*b*e^3*(-c^2*x^2+1)^(11/2)/c^11+1/5*d
^3*x^5*(a+b*arcsin(c*x))+3/7*d^2*e*x^7*(a+b*arcsin(c*x))+1/3*d*e^2*x^9*(a+b*arcsin(c*x))+1/11*e^3*x^11*(a+b*ar
csin(c*x))+1/1155*b*(231*c^6*d^3+495*c^4*d^2*e+385*c^2*d*e^2+105*e^3)*(-c^2*x^2+1)^(1/2)/c^11

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Rubi [A]  time = 0.43, antiderivative size = 341, 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 = {270, 4731, 12, 1799, 1620} \[ \frac {3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-\frac {b e \left (1-c^2 x^2\right )^{7/2} \left (99 c^4 d^2+308 c^2 d e+210 e^2\right )}{1617 c^{11}}+\frac {b \left (1-c^2 x^2\right )^{5/2} \left (495 c^4 d^2 e+77 c^6 d^3+770 c^2 d e^2+350 e^3\right )}{1925 c^{11}}-\frac {b \left (1-c^2 x^2\right )^{3/2} \left (1485 c^4 d^2 e+462 c^6 d^3+1540 c^2 d e^2+525 e^3\right )}{3465 c^{11}}+\frac {b \sqrt {1-c^2 x^2} \left (495 c^4 d^2 e+231 c^6 d^3+385 c^2 d e^2+105 e^3\right )}{1155 c^{11}}+\frac {b e^2 \left (1-c^2 x^2\right )^{9/2} \left (11 c^2 d+15 e\right )}{297 c^{11}}-\frac {b e^3 \left (1-c^2 x^2\right )^{11/2}}{121 c^{11}} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]

[Out]

(b*(231*c^6*d^3 + 495*c^4*d^2*e + 385*c^2*d*e^2 + 105*e^3)*Sqrt[1 - c^2*x^2])/(1155*c^11) - (b*(462*c^6*d^3 +
1485*c^4*d^2*e + 1540*c^2*d*e^2 + 525*e^3)*(1 - c^2*x^2)^(3/2))/(3465*c^11) + (b*(77*c^6*d^3 + 495*c^4*d^2*e +
770*c^2*d*e^2 + 350*e^3)*(1 - c^2*x^2)^(5/2))/(1925*c^11) - (b*e*(99*c^4*d^2 + 308*c^2*d*e + 210*e^2)*(1 - c^
2*x^2)^(7/2))/(1617*c^11) + (b*e^2*(11*c^2*d + 15*e)*(1 - c^2*x^2)^(9/2))/(297*c^11) - (b*e^3*(1 - c^2*x^2)^(1
1/2))/(121*c^11) + (d^3*x^5*(a + b*ArcSin[c*x]))/5 + (3*d^2*e*x^7*(a + b*ArcSin[c*x]))/7 + (d*e^2*x^9*(a + b*A
rcSin[c*x]))/3 + (e^3*x^11*(a + b*ArcSin[c*x]))/11

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 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]

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 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 )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{1155 \sqrt {1-c^2 x^2}} \, dx\\ &=\frac {1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-\frac {(b c) \int \frac {x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{\sqrt {1-c^2 x^2}} \, dx}{1155}\\ &=\frac {1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-\frac {(b c) \operatorname {Subst}\left (\int \frac {x^2 \left (231 d^3+495 d^2 e x+385 d e^2 x^2+105 e^3 x^3\right )}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{2310}\\ &=\frac {1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )-\frac {(b c) \operatorname {Subst}\left (\int \left (\frac {231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3}{c^{10} \sqrt {1-c^2 x}}+\frac {\left (-462 c^6 d^3-1485 c^4 d^2 e-1540 c^2 d e^2-525 e^3\right ) \sqrt {1-c^2 x}}{c^{10}}+\frac {3 \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right ) \left (1-c^2 x\right )^{3/2}}{c^{10}}-\frac {5 e \left (99 c^4 d^2+308 c^2 d e+210 e^2\right ) \left (1-c^2 x\right )^{5/2}}{c^{10}}+\frac {35 e^2 \left (11 c^2 d+15 e\right ) \left (1-c^2 x\right )^{7/2}}{c^{10}}-\frac {105 e^3 \left (1-c^2 x\right )^{9/2}}{c^{10}}\right ) \, dx,x,x^2\right )}{2310}\\ &=\frac {b \left (231 c^6 d^3+495 c^4 d^2 e+385 c^2 d e^2+105 e^3\right ) \sqrt {1-c^2 x^2}}{1155 c^{11}}-\frac {b \left (462 c^6 d^3+1485 c^4 d^2 e+1540 c^2 d e^2+525 e^3\right ) \left (1-c^2 x^2\right )^{3/2}}{3465 c^{11}}+\frac {b \left (77 c^6 d^3+495 c^4 d^2 e+770 c^2 d e^2+350 e^3\right ) \left (1-c^2 x^2\right )^{5/2}}{1925 c^{11}}-\frac {b e \left (99 c^4 d^2+308 c^2 d e+210 e^2\right ) \left (1-c^2 x^2\right )^{7/2}}{1617 c^{11}}+\frac {b e^2 \left (11 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^{9/2}}{297 c^{11}}-\frac {b e^3 \left (1-c^2 x^2\right )^{11/2}}{121 c^{11}}+\frac {1}{5} d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac {3}{7} d^2 e x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{3} d e^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac {1}{11} e^3 x^{11} \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 271, normalized size = 0.79 \[ \frac {3465 a x^5 \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )+\frac {b \sqrt {1-c^2 x^2} \left (c^{10} x^4 \left (160083 d^3+245025 d^2 e x^2+148225 d e^2 x^4+33075 e^3 x^6\right )+2 c^8 \left (106722 d^3 x^2+147015 d^2 e x^4+84700 d e^2 x^6+18375 e^3 x^8\right )+24 c^6 \left (17787 d^3+16335 d^2 e x^2+8470 d e^2 x^4+1750 e^3 x^6\right )+80 c^4 e \left (9801 d^2+3388 d e x^2+630 e^2 x^4\right )+4480 c^2 e^2 \left (121 d+15 e x^2\right )+134400 e^3\right )}{c^{11}}+3465 b x^5 \sin ^{-1}(c x) \left (231 d^3+495 d^2 e x^2+385 d e^2 x^4+105 e^3 x^6\right )}{4002075} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x^2)^3*(a + b*ArcSin[c*x]),x]

[Out]

(3465*a*x^5*(231*d^3 + 495*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6) + (b*Sqrt[1 - c^2*x^2]*(134400*e^3 + 4480*
c^2*e^2*(121*d + 15*e*x^2) + 80*c^4*e*(9801*d^2 + 3388*d*e*x^2 + 630*e^2*x^4) + 24*c^6*(17787*d^3 + 16335*d^2*
e*x^2 + 8470*d*e^2*x^4 + 1750*e^3*x^6) + c^10*x^4*(160083*d^3 + 245025*d^2*e*x^2 + 148225*d*e^2*x^4 + 33075*e^
3*x^6) + 2*c^8*(106722*d^3*x^2 + 147015*d^2*e*x^4 + 84700*d*e^2*x^6 + 18375*e^3*x^8)))/c^11 + 3465*b*x^5*(231*
d^3 + 495*d^2*e*x^2 + 385*d*e^2*x^4 + 105*e^3*x^6)*ArcSin[c*x])/4002075

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fricas [A]  time = 0.69, size = 322, normalized size = 0.94 \[ \frac {363825 \, a c^{11} e^{3} x^{11} + 1334025 \, a c^{11} d e^{2} x^{9} + 1715175 \, a c^{11} d^{2} e x^{7} + 800415 \, a c^{11} d^{3} x^{5} + 3465 \, {\left (105 \, b c^{11} e^{3} x^{11} + 385 \, b c^{11} d e^{2} x^{9} + 495 \, b c^{11} d^{2} e x^{7} + 231 \, b c^{11} d^{3} x^{5}\right )} \arcsin \left (c x\right ) + {\left (33075 \, b c^{10} e^{3} x^{10} + 426888 \, b c^{6} d^{3} + 1225 \, {\left (121 \, b c^{10} d e^{2} + 30 \, b c^{8} e^{3}\right )} x^{8} + 784080 \, b c^{4} d^{2} e + 25 \, {\left (9801 \, b c^{10} d^{2} e + 6776 \, b c^{8} d e^{2} + 1680 \, b c^{6} e^{3}\right )} x^{6} + 542080 \, b c^{2} d e^{2} + 3 \, {\left (53361 \, b c^{10} d^{3} + 98010 \, b c^{8} d^{2} e + 67760 \, b c^{6} d e^{2} + 16800 \, b c^{4} e^{3}\right )} x^{4} + 134400 \, b e^{3} + 4 \, {\left (53361 \, b c^{8} d^{3} + 98010 \, b c^{6} d^{2} e + 67760 \, b c^{4} d e^{2} + 16800 \, b c^{2} e^{3}\right )} x^{2}\right )} \sqrt {-c^{2} x^{2} + 1}}{4002075 \, c^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

1/4002075*(363825*a*c^11*e^3*x^11 + 1334025*a*c^11*d*e^2*x^9 + 1715175*a*c^11*d^2*e*x^7 + 800415*a*c^11*d^3*x^
5 + 3465*(105*b*c^11*e^3*x^11 + 385*b*c^11*d*e^2*x^9 + 495*b*c^11*d^2*e*x^7 + 231*b*c^11*d^3*x^5)*arcsin(c*x)
+ (33075*b*c^10*e^3*x^10 + 426888*b*c^6*d^3 + 1225*(121*b*c^10*d*e^2 + 30*b*c^8*e^3)*x^8 + 784080*b*c^4*d^2*e
+ 25*(9801*b*c^10*d^2*e + 6776*b*c^8*d*e^2 + 1680*b*c^6*e^3)*x^6 + 542080*b*c^2*d*e^2 + 3*(53361*b*c^10*d^3 +
98010*b*c^8*d^2*e + 67760*b*c^6*d*e^2 + 16800*b*c^4*e^3)*x^4 + 134400*b*e^3 + 4*(53361*b*c^8*d^3 + 98010*b*c^6
*d^2*e + 67760*b*c^4*d*e^2 + 16800*b*c^2*e^3)*x^2)*sqrt(-c^2*x^2 + 1))/c^11

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giac [B]  time = 0.43, size = 928, normalized size = 2.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

1/11*a*x^11*e^3 + 1/3*a*d*x^9*e^2 + 3/7*a*d^2*x^7*e + 1/5*a*d^3*x^5 + 1/5*(c^2*x^2 - 1)^2*b*d^3*x*arcsin(c*x)/
c^4 + 2/5*(c^2*x^2 - 1)*b*d^3*x*arcsin(c*x)/c^4 + 3/7*(c^2*x^2 - 1)^3*b*d^2*x*arcsin(c*x)*e/c^6 + 1/5*b*d^3*x*
arcsin(c*x)/c^4 + 9/7*(c^2*x^2 - 1)^2*b*d^2*x*arcsin(c*x)*e/c^6 + 1/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^
3/c^5 + 1/3*(c^2*x^2 - 1)^4*b*d*x*arcsin(c*x)*e^2/c^8 + 9/7*(c^2*x^2 - 1)*b*d^2*x*arcsin(c*x)*e/c^6 - 2/15*(-c
^2*x^2 + 1)^(3/2)*b*d^3/c^5 + 3/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*d^2*e/c^7 + 4/3*(c^2*x^2 - 1)^3*b*d*x*
arcsin(c*x)*e^2/c^8 + 3/7*b*d^2*x*arcsin(c*x)*e/c^6 + 1/5*sqrt(-c^2*x^2 + 1)*b*d^3/c^5 + 9/35*(c^2*x^2 - 1)^2*
sqrt(-c^2*x^2 + 1)*b*d^2*e/c^7 + 1/11*(c^2*x^2 - 1)^5*b*x*arcsin(c*x)*e^3/c^10 + 2*(c^2*x^2 - 1)^2*b*d*x*arcsi
n(c*x)*e^2/c^8 + 1/27*(c^2*x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b*d*e^2/c^9 - 3/7*(-c^2*x^2 + 1)^(3/2)*b*d^2*e/c^7 +
5/11*(c^2*x^2 - 1)^4*b*x*arcsin(c*x)*e^3/c^10 + 4/3*(c^2*x^2 - 1)*b*d*x*arcsin(c*x)*e^2/c^8 + 4/21*(c^2*x^2 -
1)^3*sqrt(-c^2*x^2 + 1)*b*d*e^2/c^9 + 3/7*sqrt(-c^2*x^2 + 1)*b*d^2*e/c^7 + 10/11*(c^2*x^2 - 1)^3*b*x*arcsin(c*
x)*e^3/c^10 + 1/3*b*d*x*arcsin(c*x)*e^2/c^8 + 1/121*(c^2*x^2 - 1)^5*sqrt(-c^2*x^2 + 1)*b*e^3/c^11 + 2/5*(c^2*x
^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d*e^2/c^9 + 10/11*(c^2*x^2 - 1)^2*b*x*arcsin(c*x)*e^3/c^10 + 5/99*(c^2*x^2 - 1)
^4*sqrt(-c^2*x^2 + 1)*b*e^3/c^11 - 4/9*(-c^2*x^2 + 1)^(3/2)*b*d*e^2/c^9 + 5/11*(c^2*x^2 - 1)*b*x*arcsin(c*x)*e
^3/c^10 + 10/77*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*e^3/c^11 + 1/3*sqrt(-c^2*x^2 + 1)*b*d*e^2/c^9 + 1/11*b*x*
arcsin(c*x)*e^3/c^10 + 2/11*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e^3/c^11 - 5/33*(-c^2*x^2 + 1)^(3/2)*b*e^3/c^
11 + 1/11*sqrt(-c^2*x^2 + 1)*b*e^3/c^11

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maple [A]  time = 0.00, size = 497, normalized size = 1.46 \[ \frac {\frac {a \left (\frac {1}{11} e^{3} c^{11} x^{11}+\frac {1}{3} c^{11} d \,e^{2} x^{9}+\frac {3}{7} c^{11} d^{2} e \,x^{7}+\frac {1}{5} c^{11} x^{5} d^{3}\right )}{c^{6}}+\frac {b \left (\frac {\arcsin \left (c x \right ) e^{3} c^{11} x^{11}}{11}+\frac {\arcsin \left (c x \right ) c^{11} d \,e^{2} x^{9}}{3}+\frac {3 \arcsin \left (c x \right ) c^{11} d^{2} e \,x^{7}}{7}+\frac {\arcsin \left (c x \right ) c^{11} x^{5} d^{3}}{5}-\frac {e^{3} \left (-\frac {c^{10} x^{10} \sqrt {-c^{2} x^{2}+1}}{11}-\frac {10 c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{99}-\frac {80 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{693}-\frac {32 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{231}-\frac {128 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{693}-\frac {256 \sqrt {-c^{2} x^{2}+1}}{693}\right )}{11}-\frac {c^{2} d \,e^{2} \left (-\frac {c^{8} x^{8} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {8 c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{63}-\frac {16 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{105}-\frac {64 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{315}-\frac {128 \sqrt {-c^{2} x^{2}+1}}{315}\right )}{3}-\frac {3 c^{4} d^{2} e \left (-\frac {c^{6} x^{6} \sqrt {-c^{2} x^{2}+1}}{7}-\frac {6 c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {8 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{35}-\frac {16 \sqrt {-c^{2} x^{2}+1}}{35}\right )}{7}-\frac {d^{3} c^{6} \left (-\frac {c^{4} x^{4} \sqrt {-c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{15}-\frac {8 \sqrt {-c^{2} x^{2}+1}}{15}\right )}{5}\right )}{c^{6}}}{c^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x^2+d)^3*(a+b*arcsin(c*x)),x)

[Out]

1/c^5*(a/c^6*(1/11*e^3*c^11*x^11+1/3*c^11*d*e^2*x^9+3/7*c^11*d^2*e*x^7+1/5*c^11*x^5*d^3)+b/c^6*(1/11*arcsin(c*
x)*e^3*c^11*x^11+1/3*arcsin(c*x)*c^11*d*e^2*x^9+3/7*arcsin(c*x)*c^11*d^2*e*x^7+1/5*arcsin(c*x)*c^11*x^5*d^3-1/
11*e^3*(-1/11*c^10*x^10*(-c^2*x^2+1)^(1/2)-10/99*c^8*x^8*(-c^2*x^2+1)^(1/2)-80/693*c^6*x^6*(-c^2*x^2+1)^(1/2)-
32/231*c^4*x^4*(-c^2*x^2+1)^(1/2)-128/693*c^2*x^2*(-c^2*x^2+1)^(1/2)-256/693*(-c^2*x^2+1)^(1/2))-1/3*c^2*d*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))-3/7*c^4*d^2*e*(-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^3*c^6*(-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 = 0.75, size = 465, normalized size = 1.36 \[ \frac {1}{11} \, a e^{3} x^{11} + \frac {1}{3} \, a d e^{2} x^{9} + \frac {3}{7} \, a d^{2} e x^{7} + \frac {1}{5} \, a d^{3} x^{5} + \frac {1}{75} \, {\left (15 \, x^{5} \arcsin \left (c x\right ) + {\left (\frac {3 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{3} + \frac {3}{245} \, {\left (35 \, x^{7} \arcsin \left (c x\right ) + {\left (\frac {5 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b d^{2} e + \frac {1}{945} \, {\left (315 \, x^{9} \arcsin \left (c x\right ) + {\left (\frac {35 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac {40 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac {64 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b d e^{2} + \frac {1}{7623} \, {\left (693 \, x^{11} \arcsin \left (c x\right ) + {\left (\frac {63 \, \sqrt {-c^{2} x^{2} + 1} x^{10}}{c^{2}} + \frac {70 \, \sqrt {-c^{2} x^{2} + 1} x^{8}}{c^{4}} + \frac {80 \, \sqrt {-c^{2} x^{2} + 1} x^{6}}{c^{6}} + \frac {96 \, \sqrt {-c^{2} x^{2} + 1} x^{4}}{c^{8}} + \frac {128 \, \sqrt {-c^{2} x^{2} + 1} x^{2}}{c^{10}} + \frac {256 \, \sqrt {-c^{2} x^{2} + 1}}{c^{12}}\right )} c\right )} b e^{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x^2+d)^3*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

1/11*a*e^3*x^11 + 1/3*a*d*e^2*x^9 + 3/7*a*d^2*e*x^7 + 1/5*a*d^3*x^5 + 1/75*(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^3 + 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^2*e + 1/945*(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
*d*e^2 + 1/7623*(693*x^11*arcsin(c*x) + (63*sqrt(-c^2*x^2 + 1)*x^10/c^2 + 70*sqrt(-c^2*x^2 + 1)*x^8/c^4 + 80*s
qrt(-c^2*x^2 + 1)*x^6/c^6 + 96*sqrt(-c^2*x^2 + 1)*x^4/c^8 + 128*sqrt(-c^2*x^2 + 1)*x^2/c^10 + 256*sqrt(-c^2*x^
2 + 1)/c^12)*c)*b*e^3

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*asin(c*x))*(d + e*x^2)^3,x)

[Out]

int(x^4*(a + b*asin(c*x))*(d + e*x^2)^3, x)

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sympy [A]  time = 38.57, size = 631, normalized size = 1.85 \[ \begin {cases} \frac {a d^{3} x^{5}}{5} + \frac {3 a d^{2} e x^{7}}{7} + \frac {a d e^{2} x^{9}}{3} + \frac {a e^{3} x^{11}}{11} + \frac {b d^{3} x^{5} \operatorname {asin}{\left (c x \right )}}{5} + \frac {3 b d^{2} e x^{7} \operatorname {asin}{\left (c x \right )}}{7} + \frac {b d e^{2} x^{9} \operatorname {asin}{\left (c x \right )}}{3} + \frac {b e^{3} x^{11} \operatorname {asin}{\left (c x \right )}}{11} + \frac {b d^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{25 c} + \frac {3 b d^{2} e x^{6} \sqrt {- c^{2} x^{2} + 1}}{49 c} + \frac {b d e^{2} x^{8} \sqrt {- c^{2} x^{2} + 1}}{27 c} + \frac {b e^{3} x^{10} \sqrt {- c^{2} x^{2} + 1}}{121 c} + \frac {4 b d^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{75 c^{3}} + \frac {18 b d^{2} e x^{4} \sqrt {- c^{2} x^{2} + 1}}{245 c^{3}} + \frac {8 b d e^{2} x^{6} \sqrt {- c^{2} x^{2} + 1}}{189 c^{3}} + \frac {10 b e^{3} x^{8} \sqrt {- c^{2} x^{2} + 1}}{1089 c^{3}} + \frac {8 b d^{3} \sqrt {- c^{2} x^{2} + 1}}{75 c^{5}} + \frac {24 b d^{2} e x^{2} \sqrt {- c^{2} x^{2} + 1}}{245 c^{5}} + \frac {16 b d e^{2} x^{4} \sqrt {- c^{2} x^{2} + 1}}{315 c^{5}} + \frac {80 b e^{3} x^{6} \sqrt {- c^{2} x^{2} + 1}}{7623 c^{5}} + \frac {48 b d^{2} e \sqrt {- c^{2} x^{2} + 1}}{245 c^{7}} + \frac {64 b d e^{2} x^{2} \sqrt {- c^{2} x^{2} + 1}}{945 c^{7}} + \frac {32 b e^{3} x^{4} \sqrt {- c^{2} x^{2} + 1}}{2541 c^{7}} + \frac {128 b d e^{2} \sqrt {- c^{2} x^{2} + 1}}{945 c^{9}} + \frac {128 b e^{3} x^{2} \sqrt {- c^{2} x^{2} + 1}}{7623 c^{9}} + \frac {256 b e^{3} \sqrt {- c^{2} x^{2} + 1}}{7623 c^{11}} & \text {for}\: c \neq 0 \\a \left (\frac {d^{3} x^{5}}{5} + \frac {3 d^{2} e x^{7}}{7} + \frac {d e^{2} x^{9}}{3} + \frac {e^{3} x^{11}}{11}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x**2+d)**3*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*d**3*x**5/5 + 3*a*d**2*e*x**7/7 + a*d*e**2*x**9/3 + a*e**3*x**11/11 + b*d**3*x**5*asin(c*x)/5 + 3
*b*d**2*e*x**7*asin(c*x)/7 + b*d*e**2*x**9*asin(c*x)/3 + b*e**3*x**11*asin(c*x)/11 + b*d**3*x**4*sqrt(-c**2*x*
*2 + 1)/(25*c) + 3*b*d**2*e*x**6*sqrt(-c**2*x**2 + 1)/(49*c) + b*d*e**2*x**8*sqrt(-c**2*x**2 + 1)/(27*c) + b*e
**3*x**10*sqrt(-c**2*x**2 + 1)/(121*c) + 4*b*d**3*x**2*sqrt(-c**2*x**2 + 1)/(75*c**3) + 18*b*d**2*e*x**4*sqrt(
-c**2*x**2 + 1)/(245*c**3) + 8*b*d*e**2*x**6*sqrt(-c**2*x**2 + 1)/(189*c**3) + 10*b*e**3*x**8*sqrt(-c**2*x**2
+ 1)/(1089*c**3) + 8*b*d**3*sqrt(-c**2*x**2 + 1)/(75*c**5) + 24*b*d**2*e*x**2*sqrt(-c**2*x**2 + 1)/(245*c**5)
+ 16*b*d*e**2*x**4*sqrt(-c**2*x**2 + 1)/(315*c**5) + 80*b*e**3*x**6*sqrt(-c**2*x**2 + 1)/(7623*c**5) + 48*b*d*
*2*e*sqrt(-c**2*x**2 + 1)/(245*c**7) + 64*b*d*e**2*x**2*sqrt(-c**2*x**2 + 1)/(945*c**7) + 32*b*e**3*x**4*sqrt(
-c**2*x**2 + 1)/(2541*c**7) + 128*b*d*e**2*sqrt(-c**2*x**2 + 1)/(945*c**9) + 128*b*e**3*x**2*sqrt(-c**2*x**2 +
1)/(7623*c**9) + 256*b*e**3*sqrt(-c**2*x**2 + 1)/(7623*c**11), Ne(c, 0)), (a*(d**3*x**5/5 + 3*d**2*e*x**7/7 +
d*e**2*x**9/3 + e**3*x**11/11), True))

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