3.131 \(\int x^3 \sin ^{-1}(a+b x)^2 \, dx\)

Optimal. Leaf size=343 \[ \frac{2 a^3 x}{b^3}-\frac{3 a^2 (a+b x)^2}{4 b^4}-\frac{a^4 \sin ^{-1}(a+b x)^2}{4 b^4}-\frac{2 a^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^4}-\frac{3 a^2 \sin ^{-1}(a+b x)^2}{4 b^4}+\frac{3 a^2 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{2 b^4}+\frac{2 a (a+b x)^3}{9 b^4}+\frac{4 a x}{3 b^3}-\frac{(a+b x)^4}{32 b^4}-\frac{3 (a+b x)^2}{32 b^4}-\frac{2 a (a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}-\frac{4 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}-\frac{3 \sin ^{-1}(a+b x)^2}{32 b^4}+\frac{(a+b x)^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{8 b^4}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{16 b^4}+\frac{1}{4} x^4 \sin ^{-1}(a+b x)^2 \]

[Out]

(4*a*x)/(3*b^3) + (2*a^3*x)/b^3 - (3*(a + b*x)^2)/(32*b^4) - (3*a^2*(a + b*x)^2)/(4*b^4) + (2*a*(a + b*x)^3)/(
9*b^4) - (a + b*x)^4/(32*b^4) - (4*a*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(3*b^4) - (2*a^3*Sqrt[1 - (a + b*x
)^2]*ArcSin[a + b*x])/b^4 + (3*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(16*b^4) + (3*a^2*(a + b*x)*Sq
rt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(2*b^4) - (2*a*(a + b*x)^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(3*b^4)
+ ((a + b*x)^3*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(8*b^4) - (3*ArcSin[a + b*x]^2)/(32*b^4) - (3*a^2*ArcSi
n[a + b*x]^2)/(4*b^4) - (a^4*ArcSin[a + b*x]^2)/(4*b^4) + (x^4*ArcSin[a + b*x]^2)/4

________________________________________________________________________________________

Rubi [A]  time = 0.595695, antiderivative size = 343, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {4805, 4743, 4763, 4641, 4677, 8, 4707, 30} \[ \frac{2 a^3 x}{b^3}-\frac{3 a^2 (a+b x)^2}{4 b^4}-\frac{a^4 \sin ^{-1}(a+b x)^2}{4 b^4}-\frac{2 a^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^4}-\frac{3 a^2 \sin ^{-1}(a+b x)^2}{4 b^4}+\frac{3 a^2 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{2 b^4}+\frac{2 a (a+b x)^3}{9 b^4}+\frac{4 a x}{3 b^3}-\frac{(a+b x)^4}{32 b^4}-\frac{3 (a+b x)^2}{32 b^4}-\frac{2 a (a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}-\frac{4 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}-\frac{3 \sin ^{-1}(a+b x)^2}{32 b^4}+\frac{(a+b x)^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{8 b^4}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{16 b^4}+\frac{1}{4} x^4 \sin ^{-1}(a+b x)^2 \]

Antiderivative was successfully verified.

[In]

Int[x^3*ArcSin[a + b*x]^2,x]

[Out]

(4*a*x)/(3*b^3) + (2*a^3*x)/b^3 - (3*(a + b*x)^2)/(32*b^4) - (3*a^2*(a + b*x)^2)/(4*b^4) + (2*a*(a + b*x)^3)/(
9*b^4) - (a + b*x)^4/(32*b^4) - (4*a*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(3*b^4) - (2*a^3*Sqrt[1 - (a + b*x
)^2]*ArcSin[a + b*x])/b^4 + (3*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(16*b^4) + (3*a^2*(a + b*x)*Sq
rt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(2*b^4) - (2*a*(a + b*x)^2*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(3*b^4)
+ ((a + b*x)^3*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(8*b^4) - (3*ArcSin[a + b*x]^2)/(32*b^4) - (3*a^2*ArcSi
n[a + b*x]^2)/(4*b^4) - (a^4*ArcSin[a + b*x]^2)/(4*b^4) + (x^4*ArcSin[a + b*x]^2)/4

Rule 4805

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Dist[1/d, Subst[I
nt[((d*e - c*f)/d + (f*x)/d)^m*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]

Rule 4743

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
a + b*ArcSin[c*x])^n)/(e*(m + 1)), x] - Dist[(b*c*n)/(e*(m + 1)), Int[((d + e*x)^(m + 1)*(a + b*ArcSin[c*x])^(
n - 1))/Sqrt[1 - c^2*x^2], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 4763

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g},
x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 ||
(n == 1 && p > -1) || (m == 2 && p < -2))

Rule 4641

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(a + b*ArcSin[c*x])^
(n + 1)/(b*c*Sqrt[d]*(n + 1)), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c^2*d + e, 0] && GtQ[d, 0] && NeQ[n,
-1]

Rule 4677

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[((d + e*x^2)^
(p + 1)*(a + b*ArcSin[c*x])^n)/(2*e*(p + 1)), x] + Dist[(b*n*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p + 1
)*(1 - c^2*x^2)^FracPart[p]), Int[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, b,
c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x^3 \sin ^{-1}(a+b x)^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (-\frac{a}{b}+\frac{x}{b}\right )^3 \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{b}\\ &=\frac{1}{4} x^4 \sin ^{-1}(a+b x)^2-\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (-\frac{a}{b}+\frac{x}{b}\right )^4 \sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )\\ &=\frac{1}{4} x^4 \sin ^{-1}(a+b x)^2-\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^4 \sin ^{-1}(x)}{b^4 \sqrt{1-x^2}}-\frac{4 a^3 x \sin ^{-1}(x)}{b^4 \sqrt{1-x^2}}+\frac{6 a^2 x^2 \sin ^{-1}(x)}{b^4 \sqrt{1-x^2}}-\frac{4 a x^3 \sin ^{-1}(x)}{b^4 \sqrt{1-x^2}}+\frac{x^4 \sin ^{-1}(x)}{b^4 \sqrt{1-x^2}}\right ) \, dx,x,a+b x\right )\\ &=\frac{1}{4} x^4 \sin ^{-1}(a+b x)^2-\frac{\operatorname{Subst}\left (\int \frac{x^4 \sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{2 b^4}+\frac{(2 a) \operatorname{Subst}\left (\int \frac{x^3 \sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{b^4}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{x^2 \sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{b^4}+\frac{\left (2 a^3\right ) \operatorname{Subst}\left (\int \frac{x \sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{b^4}-\frac{a^4 \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{2 b^4}\\ &=-\frac{2 a^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^4}+\frac{3 a^2 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{2 b^4}-\frac{2 a (a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}+\frac{(a+b x)^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{8 b^4}-\frac{a^4 \sin ^{-1}(a+b x)^2}{4 b^4}+\frac{1}{4} x^4 \sin ^{-1}(a+b x)^2-\frac{\operatorname{Subst}\left (\int x^3 \, dx,x,a+b x\right )}{8 b^4}-\frac{3 \operatorname{Subst}\left (\int \frac{x^2 \sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{8 b^4}+\frac{(2 a) \operatorname{Subst}\left (\int x^2 \, dx,x,a+b x\right )}{3 b^4}+\frac{(4 a) \operatorname{Subst}\left (\int \frac{x \sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{3 b^4}-\frac{\left (3 a^2\right ) \operatorname{Subst}(\int x \, dx,x,a+b x)}{2 b^4}-\frac{\left (3 a^2\right ) \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{2 b^4}+\frac{\left (2 a^3\right ) \operatorname{Subst}(\int 1 \, dx,x,a+b x)}{b^4}\\ &=\frac{2 a^3 x}{b^3}-\frac{3 a^2 (a+b x)^2}{4 b^4}+\frac{2 a (a+b x)^3}{9 b^4}-\frac{(a+b x)^4}{32 b^4}-\frac{4 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}-\frac{2 a^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^4}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{16 b^4}+\frac{3 a^2 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{2 b^4}-\frac{2 a (a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}+\frac{(a+b x)^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{8 b^4}-\frac{3 a^2 \sin ^{-1}(a+b x)^2}{4 b^4}-\frac{a^4 \sin ^{-1}(a+b x)^2}{4 b^4}+\frac{1}{4} x^4 \sin ^{-1}(a+b x)^2-\frac{3 \operatorname{Subst}(\int x \, dx,x,a+b x)}{16 b^4}-\frac{3 \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{16 b^4}+\frac{(4 a) \operatorname{Subst}(\int 1 \, dx,x,a+b x)}{3 b^4}\\ &=\frac{4 a x}{3 b^3}+\frac{2 a^3 x}{b^3}-\frac{3 (a+b x)^2}{32 b^4}-\frac{3 a^2 (a+b x)^2}{4 b^4}+\frac{2 a (a+b x)^3}{9 b^4}-\frac{(a+b x)^4}{32 b^4}-\frac{4 a \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}-\frac{2 a^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{b^4}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{16 b^4}+\frac{3 a^2 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{2 b^4}-\frac{2 a (a+b x)^2 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{3 b^4}+\frac{(a+b x)^3 \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{8 b^4}-\frac{3 \sin ^{-1}(a+b x)^2}{32 b^4}-\frac{3 a^2 \sin ^{-1}(a+b x)^2}{4 b^4}-\frac{a^4 \sin ^{-1}(a+b x)^2}{4 b^4}+\frac{1}{4} x^4 \sin ^{-1}(a+b x)^2\\ \end{align*}

Mathematica [A]  time = 0.2067, size = 148, normalized size = 0.43 \[ \frac{b x \left (-78 a^2 b x+300 a^3+a \left (28 b^2 x^2+330\right )-9 b x \left (b^2 x^2+3\right )\right )-9 \left (8 a^4+24 a^2-8 b^4 x^4+3\right ) \sin ^{-1}(a+b x)^2-6 \sqrt{-a^2-2 a b x-b^2 x^2+1} \left (-26 a^2 b x+50 a^3+14 a b^2 x^2+55 a-6 b^3 x^3-9 b x\right ) \sin ^{-1}(a+b x)}{288 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*ArcSin[a + b*x]^2,x]

[Out]

(b*x*(300*a^3 - 78*a^2*b*x - 9*b*x*(3 + b^2*x^2) + a*(330 + 28*b^2*x^2)) - 6*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2]
*(55*a + 50*a^3 - 9*b*x - 26*a^2*b*x + 14*a*b^2*x^2 - 6*b^3*x^3)*ArcSin[a + b*x] - 9*(3 + 24*a^2 + 8*a^4 - 8*b
^4*x^4)*ArcSin[a + b*x]^2)/(288*b^4)

________________________________________________________________________________________

Maple [A]  time = 0.084, size = 435, normalized size = 1.3 \begin{align*}{\frac{1}{{b}^{4}} \left ({\frac{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2} \left ( -1+ \left ( bx+a \right ) ^{2} \right ) ^{2}}{4}}-{\frac{\arcsin \left ( bx+a \right ) }{16} \left ( -2\, \left ( bx+a \right ) ^{3}\sqrt{1- \left ( bx+a \right ) ^{2}}+5\, \left ( bx+a \right ) \sqrt{1- \left ( bx+a \right ) ^{2}}+3\,\arcsin \left ( bx+a \right ) \right ) }-{\frac{5\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}}{32}}-{\frac{ \left ( -1+ \left ( bx+a \right ) ^{2} \right ) ^{2}}{32}}-{\frac{5\, \left ( bx+a \right ) ^{2}}{32}}-{\frac{3}{32}}+{\frac{3\,{a}^{2}}{4} \left ( 2\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2} \left ( bx+a \right ) ^{2}+2\,\arcsin \left ( bx+a \right ) \sqrt{1- \left ( bx+a \right ) ^{2}} \left ( bx+a \right ) - \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}- \left ( bx+a \right ) ^{2} \right ) }-{\frac{a}{9} \left ( 9\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2} \left ( bx+a \right ) ^{3}+6\,\arcsin \left ( bx+a \right ) \sqrt{1- \left ( bx+a \right ) ^{2}} \left ( bx+a \right ) ^{2}-27\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2} \left ( bx+a \right ) -2\, \left ( bx+a \right ) ^{3}-42\,\sqrt{1- \left ( bx+a \right ) ^{2}}\arcsin \left ( bx+a \right ) +42\,bx+42\,a \right ) }-{a}^{3} \left ( \left ( \arcsin \left ( bx+a \right ) \right ) ^{2} \left ( bx+a \right ) -2\,bx-2\,a+2\,\sqrt{1- \left ( bx+a \right ) ^{2}}\arcsin \left ( bx+a \right ) \right ) +{\frac{ \left ( \arcsin \left ( bx+a \right ) \right ) ^{2} \left ( -1+ \left ( bx+a \right ) ^{2} \right ) }{2}}+{\frac{\arcsin \left ( bx+a \right ) }{2} \left ( \left ( bx+a \right ) \sqrt{1- \left ( bx+a \right ) ^{2}}+\arcsin \left ( bx+a \right ) \right ) }-3\,a \left ( \left ( \arcsin \left ( bx+a \right ) \right ) ^{2} \left ( bx+a \right ) -2\,bx-2\,a+2\,\sqrt{1- \left ( bx+a \right ) ^{2}}\arcsin \left ( bx+a \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*arcsin(b*x+a)^2,x)

[Out]

1/b^4*(1/4*arcsin(b*x+a)^2*(-1+(b*x+a)^2)^2-1/16*arcsin(b*x+a)*(-2*(b*x+a)^3*(1-(b*x+a)^2)^(1/2)+5*(b*x+a)*(1-
(b*x+a)^2)^(1/2)+3*arcsin(b*x+a))-5/32*arcsin(b*x+a)^2-1/32*(-1+(b*x+a)^2)^2-5/32*(b*x+a)^2-3/32+3/4*a^2*(2*ar
csin(b*x+a)^2*(b*x+a)^2+2*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)*(b*x+a)-arcsin(b*x+a)^2-(b*x+a)^2)-1/9*a*(9*arcsin
(b*x+a)^2*(b*x+a)^3+6*arcsin(b*x+a)*(1-(b*x+a)^2)^(1/2)*(b*x+a)^2-27*arcsin(b*x+a)^2*(b*x+a)-2*(b*x+a)^3-42*(1
-(b*x+a)^2)^(1/2)*arcsin(b*x+a)+42*b*x+42*a)-a^3*(arcsin(b*x+a)^2*(b*x+a)-2*b*x-2*a+2*(1-(b*x+a)^2)^(1/2)*arcs
in(b*x+a))+1/2*arcsin(b*x+a)^2*(-1+(b*x+a)^2)+1/2*arcsin(b*x+a)*((b*x+a)*(1-(b*x+a)^2)^(1/2)+arcsin(b*x+a))-3*
a*(arcsin(b*x+a)^2*(b*x+a)-2*b*x-2*a+2*(1-(b*x+a)^2)^(1/2)*arcsin(b*x+a)))

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 2.65182, size = 352, normalized size = 1.03 \begin{align*} -\frac{9 \, b^{4} x^{4} - 28 \, a b^{3} x^{3} + 3 \,{\left (26 \, a^{2} + 9\right )} b^{2} x^{2} - 30 \,{\left (10 \, a^{3} + 11 \, a\right )} b x - 9 \,{\left (8 \, b^{4} x^{4} - 8 \, a^{4} - 24 \, a^{2} - 3\right )} \arcsin \left (b x + a\right )^{2} - 6 \,{\left (6 \, b^{3} x^{3} - 14 \, a b^{2} x^{2} - 50 \, a^{3} +{\left (26 \, a^{2} + 9\right )} b x - 55 \, a\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1} \arcsin \left (b x + a\right )}{288 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/288*(9*b^4*x^4 - 28*a*b^3*x^3 + 3*(26*a^2 + 9)*b^2*x^2 - 30*(10*a^3 + 11*a)*b*x - 9*(8*b^4*x^4 - 8*a^4 - 24
*a^2 - 3)*arcsin(b*x + a)^2 - 6*(6*b^3*x^3 - 14*a*b^2*x^2 - 50*a^3 + (26*a^2 + 9)*b*x - 55*a)*sqrt(-b^2*x^2 -
2*a*b*x - a^2 + 1)*arcsin(b*x + a))/b^4

________________________________________________________________________________________

Sympy [A]  time = 4.15582, size = 366, normalized size = 1.07 \begin{align*} \begin{cases} - \frac{a^{4} \operatorname{asin}^{2}{\left (a + b x \right )}}{4 b^{4}} + \frac{25 a^{3} x}{24 b^{3}} - \frac{25 a^{3} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}{\left (a + b x \right )}}{24 b^{4}} - \frac{13 a^{2} x^{2}}{48 b^{2}} + \frac{13 a^{2} x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}{\left (a + b x \right )}}{24 b^{3}} - \frac{3 a^{2} \operatorname{asin}^{2}{\left (a + b x \right )}}{4 b^{4}} + \frac{7 a x^{3}}{72 b} - \frac{7 a x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}{\left (a + b x \right )}}{24 b^{2}} + \frac{55 a x}{48 b^{3}} - \frac{55 a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}{\left (a + b x \right )}}{48 b^{4}} + \frac{x^{4} \operatorname{asin}^{2}{\left (a + b x \right )}}{4} - \frac{x^{4}}{32} + \frac{x^{3} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}{\left (a + b x \right )}}{8 b} - \frac{3 x^{2}}{32 b^{2}} + \frac{3 x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}{\left (a + b x \right )}}{16 b^{3}} - \frac{3 \operatorname{asin}^{2}{\left (a + b x \right )}}{32 b^{4}} & \text{for}\: b \neq 0 \\\frac{x^{4} \operatorname{asin}^{2}{\left (a \right )}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*asin(b*x+a)**2,x)

[Out]

Piecewise((-a**4*asin(a + b*x)**2/(4*b**4) + 25*a**3*x/(24*b**3) - 25*a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 +
1)*asin(a + b*x)/(24*b**4) - 13*a**2*x**2/(48*b**2) + 13*a**2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a +
b*x)/(24*b**3) - 3*a**2*asin(a + b*x)**2/(4*b**4) + 7*a*x**3/(72*b) - 7*a*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x*
*2 + 1)*asin(a + b*x)/(24*b**2) + 55*a*x/(48*b**3) - 55*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/
(48*b**4) + x**4*asin(a + b*x)**2/4 - x**4/32 + x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(8*b)
- 3*x**2/(32*b**2) + 3*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(16*b**3) - 3*asin(a + b*x)**2/(
32*b**4), Ne(b, 0)), (x**4*asin(a)**2/4, True))

________________________________________________________________________________________

Giac [A]  time = 1.25373, size = 594, normalized size = 1.73 \begin{align*} -\frac{{\left (b x + a\right )} a^{3} \arcsin \left (b x + a\right )^{2}}{b^{4}} - \frac{{\left ({\left (b x + a\right )}^{2} - 1\right )}{\left (b x + a\right )} a \arcsin \left (b x + a\right )^{2}}{b^{4}} + \frac{3 \,{\left ({\left (b x + a\right )}^{2} - 1\right )} a^{2} \arcsin \left (b x + a\right )^{2}}{2 \, b^{4}} + \frac{3 \, \sqrt{-{\left (b x + a\right )}^{2} + 1}{\left (b x + a\right )} a^{2} \arcsin \left (b x + a\right )}{2 \, b^{4}} - \frac{2 \, \sqrt{-{\left (b x + a\right )}^{2} + 1} a^{3} \arcsin \left (b x + a\right )}{b^{4}} + \frac{2 \,{\left (b x + a\right )} a^{3}}{b^{4}} + \frac{{\left ({\left (b x + a\right )}^{2} - 1\right )}^{2} \arcsin \left (b x + a\right )^{2}}{4 \, b^{4}} - \frac{{\left (b x + a\right )} a \arcsin \left (b x + a\right )^{2}}{b^{4}} + \frac{3 \, a^{2} \arcsin \left (b x + a\right )^{2}}{4 \, b^{4}} - \frac{{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac{3}{2}}{\left (b x + a\right )} \arcsin \left (b x + a\right )}{8 \, b^{4}} + \frac{2 \,{\left (-{\left (b x + a\right )}^{2} + 1\right )}^{\frac{3}{2}} a \arcsin \left (b x + a\right )}{3 \, b^{4}} + \frac{2 \,{\left ({\left (b x + a\right )}^{2} - 1\right )}{\left (b x + a\right )} a}{9 \, b^{4}} - \frac{3 \,{\left ({\left (b x + a\right )}^{2} - 1\right )} a^{2}}{4 \, b^{4}} + \frac{{\left ({\left (b x + a\right )}^{2} - 1\right )} \arcsin \left (b x + a\right )^{2}}{2 \, b^{4}} + \frac{5 \, \sqrt{-{\left (b x + a\right )}^{2} + 1}{\left (b x + a\right )} \arcsin \left (b x + a\right )}{16 \, b^{4}} - \frac{2 \, \sqrt{-{\left (b x + a\right )}^{2} + 1} a \arcsin \left (b x + a\right )}{b^{4}} - \frac{{\left ({\left (b x + a\right )}^{2} - 1\right )}^{2}}{32 \, b^{4}} + \frac{14 \,{\left (b x + a\right )} a}{9 \, b^{4}} - \frac{3 \, a^{2}}{8 \, b^{4}} + \frac{5 \, \arcsin \left (b x + a\right )^{2}}{32 \, b^{4}} - \frac{5 \,{\left ({\left (b x + a\right )}^{2} - 1\right )}}{32 \, b^{4}} - \frac{17}{256 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(b*x + a)*a^3*arcsin(b*x + a)^2/b^4 - ((b*x + a)^2 - 1)*(b*x + a)*a*arcsin(b*x + a)^2/b^4 + 3/2*((b*x + a)^2
- 1)*a^2*arcsin(b*x + a)^2/b^4 + 3/2*sqrt(-(b*x + a)^2 + 1)*(b*x + a)*a^2*arcsin(b*x + a)/b^4 - 2*sqrt(-(b*x +
a)^2 + 1)*a^3*arcsin(b*x + a)/b^4 + 2*(b*x + a)*a^3/b^4 + 1/4*((b*x + a)^2 - 1)^2*arcsin(b*x + a)^2/b^4 - (b*
x + a)*a*arcsin(b*x + a)^2/b^4 + 3/4*a^2*arcsin(b*x + a)^2/b^4 - 1/8*(-(b*x + a)^2 + 1)^(3/2)*(b*x + a)*arcsin
(b*x + a)/b^4 + 2/3*(-(b*x + a)^2 + 1)^(3/2)*a*arcsin(b*x + a)/b^4 + 2/9*((b*x + a)^2 - 1)*(b*x + a)*a/b^4 - 3
/4*((b*x + a)^2 - 1)*a^2/b^4 + 1/2*((b*x + a)^2 - 1)*arcsin(b*x + a)^2/b^4 + 5/16*sqrt(-(b*x + a)^2 + 1)*(b*x
+ a)*arcsin(b*x + a)/b^4 - 2*sqrt(-(b*x + a)^2 + 1)*a*arcsin(b*x + a)/b^4 - 1/32*((b*x + a)^2 - 1)^2/b^4 + 14/
9*(b*x + a)*a/b^4 - 3/8*a^2/b^4 + 5/32*arcsin(b*x + a)^2/b^4 - 5/32*((b*x + a)^2 - 1)/b^4 - 17/256/b^4