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

Optimal. Leaf size=120 \[ \frac{1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+6 e\right )}{45 c^5}+\frac{b \sqrt{1-c^2 x^2} \left (5 c^2 d+3 e\right )}{15 c^5}+\frac{b e \left (1-c^2 x^2\right )^{5/2}}{25 c^5} \]

[Out]

(b*(5*c^2*d + 3*e)*Sqrt[1 - c^2*x^2])/(15*c^5) - (b*(5*c^2*d + 6*e)*(1 - c^2*x^2)^(3/2))/(45*c^5) + (b*e*(1 -
c^2*x^2)^(5/2))/(25*c^5) + (d*x^3*(a + b*ArcSin[c*x]))/3 + (e*x^5*(a + b*ArcSin[c*x]))/5

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Rubi [A]  time = 0.127109, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {14, 4731, 12, 446, 77} \[ \frac{1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{b \left (1-c^2 x^2\right )^{3/2} \left (5 c^2 d+6 e\right )}{45 c^5}+\frac{b \sqrt{1-c^2 x^2} \left (5 c^2 d+3 e\right )}{15 c^5}+\frac{b e \left (1-c^2 x^2\right )^{5/2}}{25 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(5*c^2*d + 3*e)*Sqrt[1 - c^2*x^2])/(15*c^5) - (b*(5*c^2*d + 6*e)*(1 - c^2*x^2)^(3/2))/(45*c^5) + (b*e*(1 -
c^2*x^2)^(5/2))/(25*c^5) + (d*x^3*(a + b*ArcSin[c*x]))/3 + (e*x^5*(a + b*ArcSin[c*x]))/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin{align*} \int x^2 \left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{x^3 \left (5 d+3 e x^2\right )}{15 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{15} (b c) \int \frac{x^3 \left (5 d+3 e x^2\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{30} (b c) \operatorname{Subst}\left (\int \frac{x (5 d+3 e x)}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{30} (b c) \operatorname{Subst}\left (\int \left (\frac{5 c^2 d+3 e}{c^4 \sqrt{1-c^2 x}}+\frac{\left (-5 c^2 d-6 e\right ) \sqrt{1-c^2 x}}{c^4}+\frac{3 e \left (1-c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )\\ &=\frac{b \left (5 c^2 d+3 e\right ) \sqrt{1-c^2 x^2}}{15 c^5}-\frac{b \left (5 c^2 d+6 e\right ) \left (1-c^2 x^2\right )^{3/2}}{45 c^5}+\frac{b e \left (1-c^2 x^2\right )^{5/2}}{25 c^5}+\frac{1}{3} d x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} e x^5 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.0906024, size = 96, normalized size = 0.8 \[ \frac{1}{225} \left (15 a x^3 \left (5 d+3 e x^2\right )+\frac{b \sqrt{1-c^2 x^2} \left (c^4 \left (25 d x^2+9 e x^4\right )+2 c^2 \left (25 d+6 e x^2\right )+24 e\right )}{c^5}+15 b x^3 \sin ^{-1}(c x) \left (5 d+3 e x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*a*x^3*(5*d + 3*e*x^2) + (b*Sqrt[1 - c^2*x^2]*(24*e + 2*c^2*(25*d + 6*e*x^2) + c^4*(25*d*x^2 + 9*e*x^4)))/c
^5 + 15*b*x^3*(5*d + 3*e*x^2)*ArcSin[c*x])/225

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Maple [A]  time = 0.004, size = 161, normalized size = 1.3 \begin{align*}{\frac{1}{{c}^{3}} \left ({\frac{a}{{c}^{2}} \left ({\frac{e{c}^{5}{x}^{5}}{5}}+{\frac{{c}^{5}d{x}^{3}}{3}} \right ) }+{\frac{b}{{c}^{2}} \left ({\frac{\arcsin \left ( cx \right ) e{c}^{5}{x}^{5}}{5}}+{\frac{\arcsin \left ( cx \right ){c}^{5}d{x}^{3}}{3}}-{\frac{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{{c}^{2}d}{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)*(a+b*arcsin(c*x)),x)

[Out]

1/c^3*(a/c^2*(1/5*e*c^5*x^5+1/3*c^5*d*x^3)+b/c^2*(1/5*arcsin(c*x)*e*c^5*x^5+1/3*arcsin(c*x)*c^5*d*x^3-1/5*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*c^2*d*(-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 = 1.4469, size = 192, normalized size = 1.6 \begin{align*} \frac{1}{5} \, a e x^{5} + \frac{1}{3} \, a d 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 + \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 e \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*e*x^5 + 1/3*a*d*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 + 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*e

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Fricas [A]  time = 2.38698, size = 250, normalized size = 2.08 \begin{align*} \frac{45 \, a c^{5} e x^{5} + 75 \, a c^{5} d x^{3} + 15 \,{\left (3 \, b c^{5} e x^{5} + 5 \, b c^{5} d x^{3}\right )} \arcsin \left (c x\right ) +{\left (9 \, b c^{4} e x^{4} + 50 \, b c^{2} d +{\left (25 \, b c^{4} d + 12 \, b c^{2} e\right )} x^{2} + 24 \, b e\right )} \sqrt{-c^{2} x^{2} + 1}}{225 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/225*(45*a*c^5*e*x^5 + 75*a*c^5*d*x^3 + 15*(3*b*c^5*e*x^5 + 5*b*c^5*d*x^3)*arcsin(c*x) + (9*b*c^4*e*x^4 + 50*
b*c^2*d + (25*b*c^4*d + 12*b*c^2*e)*x^2 + 24*b*e)*sqrt(-c^2*x^2 + 1))/c^5

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Sympy [A]  time = 2.59258, size = 172, normalized size = 1.43 \begin{align*} \begin{cases} \frac{a d x^{3}}{3} + \frac{a e x^{5}}{5} + \frac{b d x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{b e x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{b d x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{b e x^{4} \sqrt{- c^{2} x^{2} + 1}}{25 c} + \frac{2 b d \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + \frac{4 b e x^{2} \sqrt{- c^{2} x^{2} + 1}}{75 c^{3}} + \frac{8 b e \sqrt{- c^{2} x^{2} + 1}}{75 c^{5}} & \text{for}\: c \neq 0 \\a \left (\frac{d x^{3}}{3} + \frac{e x^{5}}{5}\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)*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*d*x**3/3 + a*e*x**5/5 + b*d*x**3*asin(c*x)/3 + b*e*x**5*asin(c*x)/5 + b*d*x**2*sqrt(-c**2*x**2 +
1)/(9*c) + b*e*x**4*sqrt(-c**2*x**2 + 1)/(25*c) + 2*b*d*sqrt(-c**2*x**2 + 1)/(9*c**3) + 4*b*e*x**2*sqrt(-c**2*
x**2 + 1)/(75*c**3) + 8*b*e*sqrt(-c**2*x**2 + 1)/(75*c**5), Ne(c, 0)), (a*(d*x**3/3 + e*x**5/5), True))

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Giac [B]  time = 1.32404, size = 293, normalized size = 2.44 \begin{align*} \frac{1}{5} \, a x^{5} e + \frac{1}{3} \, a d x^{3} + \frac{{\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac{b d x \arcsin \left (c x\right )}{3 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b x \arcsin \left (c x\right ) e}{5 \, c^{4}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )} b x \arcsin \left (c x\right ) e}{5 \, c^{4}} - \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d}{9 \, c^{3}} + \frac{b x \arcsin \left (c x\right ) e}{5 \, c^{4}} + \frac{\sqrt{-c^{2} x^{2} + 1} b d}{3 \, c^{3}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b e}{25 \, c^{5}} - \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b e}{15 \, c^{5}} + \frac{\sqrt{-c^{2} x^{2} + 1} b e}{5 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*a*x^5*e + 1/3*a*d*x^3 + 1/3*(c^2*x^2 - 1)*b*d*x*arcsin(c*x)/c^2 + 1/3*b*d*x*arcsin(c*x)/c^2 + 1/5*(c^2*x^2
- 1)^2*b*x*arcsin(c*x)*e/c^4 + 2/5*(c^2*x^2 - 1)*b*x*arcsin(c*x)*e/c^4 - 1/9*(-c^2*x^2 + 1)^(3/2)*b*d/c^3 + 1
/5*b*x*arcsin(c*x)*e/c^4 + 1/3*sqrt(-c^2*x^2 + 1)*b*d/c^3 + 1/25*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*e/c^5 -
2/15*(-c^2*x^2 + 1)^(3/2)*b*e/c^5 + 1/5*sqrt(-c^2*x^2 + 1)*b*e/c^5