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

Optimal. Leaf size=245 \[ -\frac{3 (a+b x)^4}{128 b}+\frac{51 (a+b x)^2}{128 b}-\frac{9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{\left (1-(a+b x)^2\right )^{3/2} (a+b x) \sin ^{-1}(a+b x)^3}{4 b}+\frac{3 \sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)^3}{8 b}-\frac{3 \left (1-(a+b x)^2\right )^{3/2} (a+b x) \sin ^{-1}(a+b x)}{32 b}-\frac{45 \sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{64 b}+\frac{3 \sin ^{-1}(a+b x)^4}{32 b}+\frac{3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{27 \sin ^{-1}(a+b x)^2}{128 b} \]

[Out]

(51*(a + b*x)^2)/(128*b) - (3*(a + b*x)^4)/(128*b) - (45*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(64*
b) - (3*(a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x])/(32*b) + (27*ArcSin[a + b*x]^2)/(128*b) - (9*(a + b
*x)^2*ArcSin[a + b*x]^2)/(16*b) + (3*(1 - (a + b*x)^2)^2*ArcSin[a + b*x]^2)/(16*b) + (3*(a + b*x)*Sqrt[1 - (a
+ b*x)^2]*ArcSin[a + b*x]^3)/(8*b) + ((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x]^3)/(4*b) + (3*ArcSin[a
+ b*x]^4)/(32*b)

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Rubi [A]  time = 0.324746, antiderivative size = 245, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 9, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {4807, 4649, 4647, 4641, 4627, 4707, 30, 4677, 14} \[ -\frac{3 (a+b x)^4}{128 b}+\frac{51 (a+b x)^2}{128 b}-\frac{9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{\left (1-(a+b x)^2\right )^{3/2} (a+b x) \sin ^{-1}(a+b x)^3}{4 b}+\frac{3 \sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)^3}{8 b}-\frac{3 \left (1-(a+b x)^2\right )^{3/2} (a+b x) \sin ^{-1}(a+b x)}{32 b}-\frac{45 \sqrt{1-(a+b x)^2} (a+b x) \sin ^{-1}(a+b x)}{64 b}+\frac{3 \sin ^{-1}(a+b x)^4}{32 b}+\frac{3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{27 \sin ^{-1}(a+b x)^2}{128 b} \]

Antiderivative was successfully verified.

[In]

Int[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^3,x]

[Out]

(51*(a + b*x)^2)/(128*b) - (3*(a + b*x)^4)/(128*b) - (45*(a + b*x)*Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x])/(64*
b) - (3*(a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x])/(32*b) + (27*ArcSin[a + b*x]^2)/(128*b) - (9*(a + b
*x)^2*ArcSin[a + b*x]^2)/(16*b) + (3*(1 - (a + b*x)^2)^2*ArcSin[a + b*x]^2)/(16*b) + (3*(a + b*x)*Sqrt[1 - (a
+ b*x)^2]*ArcSin[a + b*x]^3)/(8*b) + ((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x]^3)/(4*b) + (3*ArcSin[a
+ b*x]^4)/(32*b)

Rule 4807

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(p_.), x_Symbol] :> Di
st[1/d, Subst[Int[(-(C/d^2) + (C*x^2)/d^2)^p*(a + b*ArcSin[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A,
B, C, n, p}, x] && EqQ[B*(1 - c^2) + 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]

Rule 4649

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

Rule 4647

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

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 4627

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

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]

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 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]]

Rubi steps

\begin{align*} \int \left (1-a^2-2 a b x-b^2 x^2\right )^{3/2} \sin ^{-1}(a+b x)^3 \, dx &=\frac{\operatorname{Subst}\left (\int \left (1-x^2\right )^{3/2} \sin ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}-\frac{3 \operatorname{Subst}\left (\int x \left (1-x^2\right ) \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{4 b}+\frac{3 \operatorname{Subst}\left (\int \sqrt{1-x^2} \sin ^{-1}(x)^3 \, dx,x,a+b x\right )}{4 b}\\ &=\frac{3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}-\frac{3 \operatorname{Subst}\left (\int \left (1-x^2\right )^{3/2} \sin ^{-1}(x) \, dx,x,a+b x\right )}{8 b}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)^3}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{8 b}-\frac{9 \operatorname{Subst}\left (\int x \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{8 b}\\ &=-\frac{3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{32 b}-\frac{9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}+\frac{3 \sin ^{-1}(a+b x)^4}{32 b}+\frac{3 \operatorname{Subst}\left (\int x \left (1-x^2\right ) \, dx,x,a+b x\right )}{32 b}-\frac{9 \operatorname{Subst}\left (\int \sqrt{1-x^2} \sin ^{-1}(x) \, dx,x,a+b x\right )}{32 b}+\frac{9 \operatorname{Subst}\left (\int \frac{x^2 \sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{8 b}\\ &=-\frac{45 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{64 b}-\frac{3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{32 b}-\frac{9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}+\frac{3 \sin ^{-1}(a+b x)^4}{32 b}+\frac{3 \operatorname{Subst}\left (\int \left (x-x^3\right ) \, dx,x,a+b x\right )}{32 b}+\frac{9 \operatorname{Subst}(\int x \, dx,x,a+b x)}{64 b}-\frac{9 \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{64 b}+\frac{9 \operatorname{Subst}(\int x \, dx,x,a+b x)}{16 b}+\frac{9 \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{16 b}\\ &=\frac{51 (a+b x)^2}{128 b}-\frac{3 (a+b x)^4}{128 b}-\frac{45 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)}{64 b}-\frac{3 (a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)}{32 b}+\frac{27 \sin ^{-1}(a+b x)^2}{128 b}-\frac{9 (a+b x)^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{3 \left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)^2}{16 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^3}{8 b}+\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^3}{4 b}+\frac{3 \sin ^{-1}(a+b x)^4}{32 b}\\ \end{align*}

Mathematica [A]  time = 0.216258, size = 272, normalized size = 1.11 \[ \frac{3 \left (17-6 a^2\right ) b^2 x^2-16 \sqrt{-a^2-2 a b x-b^2 x^2+1} \left (6 a^2 b x+2 a^3+6 a b^2 x^2-5 a+2 b^3 x^3-5 b x\right ) \sin ^{-1}(a+b x)^3+3 \left (8 a^2 \left (6 b^2 x^2-5\right )+32 a^3 b x+8 a^4+16 a b x \left (2 b^2 x^2-5\right )+8 b^4 x^4-40 b^2 x^2+17\right ) \sin ^{-1}(a+b x)^2+6 \sqrt{-a^2-2 a b x-b^2 x^2+1} \left (6 a^2 b x+2 a^3+6 a b^2 x^2-17 a+2 b^3 x^3-17 b x\right ) \sin ^{-1}(a+b x)+6 a \left (17-2 a^2\right ) b x-12 a b^3 x^3+12 \sin ^{-1}(a+b x)^4-3 b^4 x^4}{128 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - a^2 - 2*a*b*x - b^2*x^2)^(3/2)*ArcSin[a + b*x]^3,x]

[Out]

(6*a*(17 - 2*a^2)*b*x + 3*(17 - 6*a^2)*b^2*x^2 - 12*a*b^3*x^3 - 3*b^4*x^4 + 6*Sqrt[1 - a^2 - 2*a*b*x - b^2*x^2
]*(-17*a + 2*a^3 - 17*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x] + 3*(17 + 8*a^4 + 32*a^3*b*x
- 40*b^2*x^2 + 8*b^4*x^4 + 16*a*b*x*(-5 + 2*b^2*x^2) + 8*a^2*(-5 + 6*b^2*x^2))*ArcSin[a + b*x]^2 - 16*Sqrt[1 -
a^2 - 2*a*b*x - b^2*x^2]*(-5*a + 2*a^3 - 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSin[a + b*x]^3 + 12*
ArcSin[a + b*x]^4)/(128*b)

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Maple [B]  time = 0.094, size = 628, normalized size = 2.6 \begin{align*}{\frac{1}{128\,b} \left ( -12+51\,{a}^{2}-240\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}xab-102\,\arcsin \left ( bx+a \right ) \sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}xb+80\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{3}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}xb-96\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \left ( \arcsin \left ( bx+a \right ) \right ) ^{3}x{a}^{2}b+12\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{4}+96\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}{x}^{3}a{b}^{3}+144\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}{x}^{2}{a}^{2}{b}^{2}+96\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}x{a}^{3}b-32\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \left ( \arcsin \left ( bx+a \right ) \right ) ^{3}{x}^{3}{b}^{3}+12\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}\arcsin \left ( bx+a \right ){x}^{3}{b}^{3}-120\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}{x}^{2}{b}^{2}-18\,{x}^{2}{a}^{2}{b}^{2}+36\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}\arcsin \left ( bx+a \right ){x}^{2}a{b}^{2}+36\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}\arcsin \left ( bx+a \right ) x{a}^{2}b-96\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \left ( \arcsin \left ( bx+a \right ) \right ) ^{3}{x}^{2}a{b}^{2}-12\,{x}^{3}a{b}^{3}-12\,x{a}^{3}b+24\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}{x}^{4}{b}^{4}-32\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \left ( \arcsin \left ( bx+a \right ) \right ) ^{3}{a}^{3}+12\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}\arcsin \left ( bx+a \right ){a}^{3}+80\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{3}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}a-102\,\arcsin \left ( bx+a \right ) \sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}a-3\,{a}^{4}+102\,xab-3\,{x}^{4}{b}^{4}+24\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}{a}^{4}-120\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}{a}^{2}+51\,{b}^{2}{x}^{2}+51\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x)

[Out]

1/128*(-12+51*a^2-240*arcsin(b*x+a)^2*x*a*b-102*arcsin(b*x+a)*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x*b+80*arcsin(b*x
+a)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x*b-96*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*x*a^2*b+12*arcsin(b
*x+a)^4+96*arcsin(b*x+a)^2*x^3*a*b^3+144*arcsin(b*x+a)^2*x^2*a^2*b^2+96*arcsin(b*x+a)^2*x*a^3*b-32*(-b^2*x^2-2
*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*x^3*b^3+12*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)*x^3*b^3-120*arcsin
(b*x+a)^2*x^2*b^2-18*x^2*a^2*b^2+36*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)*x^2*a*b^2+36*(-b^2*x^2-2*a*b*
x-a^2+1)^(1/2)*arcsin(b*x+a)*x*a^2*b-96*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*x^2*a*b^2-12*x^3*a*b^3-
12*x*a^3*b+24*arcsin(b*x+a)^2*x^4*b^4-32*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^3*a^3+12*(-b^2*x^2-2*a*b
*x-a^2+1)^(1/2)*arcsin(b*x+a)*a^3+80*arcsin(b*x+a)^3*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a-102*arcsin(b*x+a)*(-b^2*
x^2-2*a*b*x-a^2+1)^(1/2)*a-3*a^4+102*x*a*b-3*x^4*b^4+24*arcsin(b*x+a)^2*a^4-120*arcsin(b*x+a)^2*a^2+51*b^2*x^2
+51*arcsin(b*x+a)^2)/b

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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((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.50564, size = 574, normalized size = 2.34 \begin{align*} -\frac{3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} + 3 \,{\left (6 \, a^{2} - 17\right )} b^{2} x^{2} - 12 \, \arcsin \left (b x + a\right )^{4} + 6 \,{\left (2 \, a^{3} - 17 \, a\right )} b x - 3 \,{\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \,{\left (6 \, a^{2} - 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \,{\left (2 \, a^{3} - 5 \, a\right )} b x - 40 \, a^{2} + 17\right )} \arcsin \left (b x + a\right )^{2} + 2 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (8 \,{\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} +{\left (6 \, a^{2} - 5\right )} b x - 5 \, a\right )} \arcsin \left (b x + a\right )^{3} - 3 \,{\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} +{\left (6 \, a^{2} - 17\right )} b x - 17 \, a\right )} \arcsin \left (b x + a\right )\right )}}{128 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/128*(3*b^4*x^4 + 12*a*b^3*x^3 + 3*(6*a^2 - 17)*b^2*x^2 - 12*arcsin(b*x + a)^4 + 6*(2*a^3 - 17*a)*b*x - 3*(8
*b^4*x^4 + 32*a*b^3*x^3 + 8*(6*a^2 - 5)*b^2*x^2 + 8*a^4 + 16*(2*a^3 - 5*a)*b*x - 40*a^2 + 17)*arcsin(b*x + a)^
2 + 2*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(8*(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 - 5)*b*x - 5*a)*arcsin(b
*x + a)^3 - 3*(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 - 17)*b*x - 17*a)*arcsin(b*x + a)))/b

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Sympy [A]  time = 26.9102, size = 694, normalized size = 2.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b**2*x**2-2*a*b*x-a**2+1)**(3/2)*asin(b*x+a)**3,x)

[Out]

Piecewise((3*a**4*asin(a + b*x)**2/(16*b) + 3*a**3*x*asin(a + b*x)**2/4 - 3*a**3*x/32 - a**3*sqrt(-a**2 - 2*a*
b*x - b**2*x**2 + 1)*asin(a + b*x)**3/(4*b) + 3*a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/(32*b
) + 9*a**2*b*x**2*asin(a + b*x)**2/8 - 9*a**2*b*x**2/64 - 3*a**2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(
a + b*x)**3/4 + 9*a**2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/32 - 15*a**2*asin(a + b*x)**2/(16
*b) + 3*a*b**2*x**3*asin(a + b*x)**2/4 - 3*a*b**2*x**3/32 - 3*a*b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*a
sin(a + b*x)**3/4 + 9*a*b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/32 - 15*a*x*asin(a + b*x)**
2/8 + 51*a*x/64 + 5*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3/(8*b) - 51*a*sqrt(-a**2 - 2*a*b*x
- b**2*x**2 + 1)*asin(a + b*x)/(64*b) + 3*b**3*x**4*asin(a + b*x)**2/16 - 3*b**3*x**4/128 - b**2*x**3*sqrt(-a
**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**3/4 + 3*b**2*x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a +
b*x)/32 - 15*b*x**2*asin(a + b*x)**2/16 + 51*b*x**2/128 + 5*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a +
b*x)**3/8 - 51*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)/64 + 3*asin(a + b*x)**4/(32*b) + 51*asin(
a + b*x)**2/(128*b), Ne(b, 0)), (x*(1 - a**2)**(3/2)*asin(a)**3, True))

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Giac [A]  time = 1.31244, size = 400, normalized size = 1.63 \begin{align*} \frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}}{\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{4 \, b} + \frac{3 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )} \arcsin \left (b x + a\right )^{3}}{8 \, b} + \frac{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2} \arcsin \left (b x + a\right )^{2}}{16 \, b} + \frac{3 \, \arcsin \left (b x + a\right )^{4}}{32 \, b} - \frac{3 \,{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}}{\left (b x + a\right )} \arcsin \left (b x + a\right )}{32 \, b} - \frac{9 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )^{2}}{16 \, b} - \frac{45 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )} \arcsin \left (b x + a\right )}{64 \, b} - \frac{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2}}{128 \, b} - \frac{45 \, \arcsin \left (b x + a\right )^{2}}{128 \, b} + \frac{45 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}}{128 \, b} + \frac{189}{1024 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b^2*x^2-2*a*b*x-a^2+1)^(3/2)*arcsin(b*x+a)^3,x, algorithm="giac")

[Out]

1/4*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)*arcsin(b*x + a)^3/b + 3/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1
)*(b*x + a)*arcsin(b*x + a)^3/b + 3/16*(b^2*x^2 + 2*a*b*x + a^2 - 1)^2*arcsin(b*x + a)^2/b + 3/32*arcsin(b*x +
a)^4/b - 3/32*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)*arcsin(b*x + a)/b - 9/16*(b^2*x^2 + 2*a*b*x + a^
2 - 1)*arcsin(b*x + a)^2/b - 45/64*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)*arcsin(b*x + a)/b - 3/128*(b^2
*x^2 + 2*a*b*x + a^2 - 1)^2/b - 45/128*arcsin(b*x + a)^2/b + 45/128*(b^2*x^2 + 2*a*b*x + a^2 - 1)/b + 189/1024
/b