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

Optimal. Leaf size=199 \[ -\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}-\frac{15 (a+b x) \sqrt{1-(a+b x)^2}}{64 b}+\frac{\sin ^{-1}(a+b x)^3}{8 b}+\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^2}{4 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{8 b}+\frac{\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)}{8 b}+\frac{9 \sin ^{-1}(a+b x)}{64 b} \]

[Out]

(-15*(a + b*x)*Sqrt[1 - (a + b*x)^2])/(64*b) - ((a + b*x)*(1 - (a + b*x)^2)^(3/2))/(32*b) + (9*ArcSin[a + b*x]
)/(64*b) - (3*(a + b*x)^2*ArcSin[a + b*x])/(8*b) + ((1 - (a + b*x)^2)^2*ArcSin[a + b*x])/(8*b) + (3*(a + b*x)*
Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(8*b) + ((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x]^2)/(4*b) +
ArcSin[a + b*x]^3/(8*b)

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Rubi [A]  time = 0.204458, antiderivative size = 199, normalized size of antiderivative = 1., number of steps used = 11, 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, 321, 216, 4677, 195} \[ -\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}-\frac{15 (a+b x) \sqrt{1-(a+b x)^2}}{64 b}+\frac{\sin ^{-1}(a+b x)^3}{8 b}+\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^2}{4 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{8 b}+\frac{\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)}{8 b}+\frac{9 \sin ^{-1}(a+b x)}{64 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-15*(a + b*x)*Sqrt[1 - (a + b*x)^2])/(64*b) - ((a + b*x)*(1 - (a + b*x)^2)^(3/2))/(32*b) + (9*ArcSin[a + b*x]
)/(64*b) - (3*(a + b*x)^2*ArcSin[a + b*x])/(8*b) + ((1 - (a + b*x)^2)^2*ArcSin[a + b*x])/(8*b) + (3*(a + b*x)*
Sqrt[1 - (a + b*x)^2]*ArcSin[a + b*x]^2)/(8*b) + ((a + b*x)*(1 - (a + b*x)^2)^(3/2)*ArcSin[a + b*x]^2)/(4*b) +
ArcSin[a + b*x]^3/(8*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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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)^2 \, dx &=\frac{\operatorname{Subst}\left (\int \left (1-x^2\right )^{3/2} \sin ^{-1}(x)^2 \, 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)^2}{4 b}-\frac{\operatorname{Subst}\left (\int x \left (1-x^2\right ) \sin ^{-1}(x) \, dx,x,a+b x\right )}{2 b}+\frac{3 \operatorname{Subst}\left (\int \sqrt{1-x^2} \sin ^{-1}(x)^2 \, dx,x,a+b x\right )}{4 b}\\ &=\frac{\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)}{8 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}+\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^2}{4 b}-\frac{\operatorname{Subst}\left (\int \left (1-x^2\right )^{3/2} \, dx,x,a+b x\right )}{8 b}+\frac{3 \operatorname{Subst}\left (\int \frac{\sin ^{-1}(x)^2}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{8 b}-\frac{3 \operatorname{Subst}\left (\int x \sin ^{-1}(x) \, dx,x,a+b x\right )}{4 b}\\ &=-\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{8 b}+\frac{\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)}{8 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}+\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^2}{4 b}+\frac{\sin ^{-1}(a+b x)^3}{8 b}-\frac{3 \operatorname{Subst}\left (\int \sqrt{1-x^2} \, dx,x,a+b x\right )}{32 b}+\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{8 b}\\ &=-\frac{15 (a+b x) \sqrt{1-(a+b x)^2}}{64 b}-\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{8 b}+\frac{\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)}{8 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}+\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^2}{4 b}+\frac{\sin ^{-1}(a+b x)^3}{8 b}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{64 b}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-x^2}} \, dx,x,a+b x\right )}{16 b}\\ &=-\frac{15 (a+b x) \sqrt{1-(a+b x)^2}}{64 b}-\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2}}{32 b}+\frac{9 \sin ^{-1}(a+b x)}{64 b}-\frac{3 (a+b x)^2 \sin ^{-1}(a+b x)}{8 b}+\frac{\left (1-(a+b x)^2\right )^2 \sin ^{-1}(a+b x)}{8 b}+\frac{3 (a+b x) \sqrt{1-(a+b x)^2} \sin ^{-1}(a+b x)^2}{8 b}+\frac{(a+b x) \left (1-(a+b x)^2\right )^{3/2} \sin ^{-1}(a+b x)^2}{4 b}+\frac{\sin ^{-1}(a+b x)^3}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.177784, size = 216, normalized size = 1.09 \[ \frac{\sqrt{-a^2-2 a b x-b^2 x^2+1} \left (6 a^2 b x+2 a^3+6 a b^2 x^2-17 a+2 b^3 x^3-17 b x\right )-8 \sqrt{-a^2-2 a b x-b^2 x^2+1} \left (6 a^2 b x+2 a^3+6 a b^2 x^2-5 a+2 b^3 x^3-5 b x\right ) \sin ^{-1}(a+b x)^2+8 b x \left (6 a^2 b x+4 a^3+4 a b^2 x^2-10 a+b^3 x^3-5 b x\right ) \sin ^{-1}(a+b x)+\left (8 a^4-40 a^2+17\right ) \sin ^{-1}(a+b x)+8 \sin ^{-1}(a+b x)^3}{64 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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) + (17 - 40*a
^2 + 8*a^4)*ArcSin[a + b*x] + 8*b*x*(-10*a + 4*a^3 - 5*b*x + 6*a^2*b*x + 4*a*b^2*x^2 + b^3*x^3)*ArcSin[a + b*x
] - 8*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]^2 + 8*ArcSin[a + b*x]^3)/(64*b)

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Maple [B]  time = 0.083, size = 515, normalized size = 2.6 \begin{align*}{\frac{1}{64\,b} \left ( -16\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}{x}^{3}{b}^{3}+8\,\arcsin \left ( bx+a \right ){x}^{4}{b}^{4}-48\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}{x}^{2}a{b}^{2}+32\,\arcsin \left ( bx+a \right ){x}^{3}a{b}^{3}-48\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}x{a}^{2}b+48\,\arcsin \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}+2\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}{x}^{3}{b}^{3}-16\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1} \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}{a}^{3}+32\,\arcsin \left ( bx+a \right ) x{a}^{3}b+6\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}{x}^{2}a{b}^{2}+40\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}xb-40\,\arcsin \left ( bx+a \right ){x}^{2}{b}^{2}+8\,\arcsin \left ( bx+a \right ){a}^{4}+6\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}x{a}^{2}b+40\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{2}\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}a-80\,\arcsin \left ( bx+a \right ) xab+2\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}{a}^{3}+8\, \left ( \arcsin \left ( bx+a \right ) \right ) ^{3}-40\,\arcsin \left ( bx+a \right ){a}^{2}-17\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}xb-17\,\sqrt{-{b}^{2}{x}^{2}-2\,xab-{a}^{2}+1}a+17\,\arcsin \left ( bx+a \right ) \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)^2,x)

[Out]

1/64*(-16*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x+a)^2*x^3*b^3+8*arcsin(b*x+a)*x^4*b^4-48*(-b^2*x^2-2*a*b*x-
a^2+1)^(1/2)*arcsin(b*x+a)^2*x^2*a*b^2+32*arcsin(b*x+a)*x^3*a*b^3-48*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*arcsin(b*x
+a)^2*x*a^2*b+48*arcsin(b*x+a)*x^2*a^2*b^2+2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x^3*b^3-16*(-b^2*x^2-2*a*b*x-a^2+1
)^(1/2)*arcsin(b*x+a)^2*a^3+32*arcsin(b*x+a)*x*a^3*b+6*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x^2*a*b^2+40*arcsin(b*x+
a)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x*b-40*arcsin(b*x+a)*x^2*b^2+8*arcsin(b*x+a)*a^4+6*(-b^2*x^2-2*a*b*x-a^2+1
)^(1/2)*x*a^2*b+40*arcsin(b*x+a)^2*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*a-80*arcsin(b*x+a)*x*a*b+2*(-b^2*x^2-2*a*b*x
-a^2+1)^(1/2)*a^3+8*arcsin(b*x+a)^3-40*arcsin(b*x+a)*a^2-17*(-b^2*x^2-2*a*b*x-a^2+1)^(1/2)*x*b-17*(-b^2*x^2-2*
a*b*x-a^2+1)^(1/2)*a+17*arcsin(b*x+a))/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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.44476, size = 433, normalized size = 2.18 \begin{align*} \frac{8 \, \arcsin \left (b x + a\right )^{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 ) +{\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} +{\left (6 \, a^{2} - 17\right )} b x - 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 )^{2} - 17 \, a\right )} \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}}{64 \, 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)^2,x, algorithm="fricas")

[Out]

1/64*(8*arcsin(b*x + a)^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*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 - 17)*b*x - 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)^2 - 17*a)*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1))/b

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Sympy [A]  time = 13.4809, size = 568, normalized size = 2.85 \begin{align*} \begin{cases} \frac{a^{4} \operatorname{asin}{\left (a + b x \right )}}{8 b} + \frac{a^{3} x \operatorname{asin}{\left (a + b x \right )}}{2} - \frac{a^{3} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{4 b} + \frac{a^{3} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32 b} + \frac{3 a^{2} b x^{2} \operatorname{asin}{\left (a + b x \right )}}{4} - \frac{3 a^{2} x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{4} + \frac{3 a^{2} x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32} - \frac{5 a^{2} \operatorname{asin}{\left (a + b x \right )}}{8 b} + \frac{a b^{2} x^{3} \operatorname{asin}{\left (a + b x \right )}}{2} - \frac{3 a b x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{4} + \frac{3 a b x^{2} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32} - \frac{5 a x \operatorname{asin}{\left (a + b x \right )}}{4} + \frac{5 a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{8 b} - \frac{17 a \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{64 b} + \frac{b^{3} x^{4} \operatorname{asin}{\left (a + b x \right )}}{8} - \frac{b^{2} x^{3} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{4} + \frac{b^{2} x^{3} \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{32} - \frac{5 b x^{2} \operatorname{asin}{\left (a + b x \right )}}{8} + \frac{5 x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1} \operatorname{asin}^{2}{\left (a + b x \right )}}{8} - \frac{17 x \sqrt{- a^{2} - 2 a b x - b^{2} x^{2} + 1}}{64} + \frac{\operatorname{asin}^{3}{\left (a + b x \right )}}{8 b} + \frac{17 \operatorname{asin}{\left (a + b x \right )}}{64 b} & \text{for}\: b \neq 0 \\x \left (1 - a^{2}\right )^{\frac{3}{2}} \operatorname{asin}^{2}{\left (a \right )} & \text{otherwise} \end{cases} \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)**2,x)

[Out]

Piecewise((a**4*asin(a + b*x)/(8*b) + a**3*x*asin(a + b*x)/2 - a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin
(a + b*x)**2/(4*b) + a**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(32*b) + 3*a**2*b*x**2*asin(a + b*x)/4 - 3*a**
2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/4 + 3*a**2*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/
32 - 5*a**2*asin(a + b*x)/(8*b) + a*b**2*x**3*asin(a + b*x)/2 - 3*a*b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 +
1)*asin(a + b*x)**2/4 + 3*a*b*x**2*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/32 - 5*a*x*asin(a + b*x)/4 + 5*a*sqrt
(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/(8*b) - 17*a*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/(64*b) +
b**3*x**4*asin(a + b*x)/8 - b**2*x**3*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*asin(a + b*x)**2/4 + b**2*x**3*sq
rt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/32 - 5*b*x**2*asin(a + b*x)/8 + 5*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)*
asin(a + b*x)**2/8 - 17*x*sqrt(-a**2 - 2*a*b*x - b**2*x**2 + 1)/64 + asin(a + b*x)**3/(8*b) + 17*asin(a + b*x)
/(64*b), Ne(b, 0)), (x*(1 - a**2)**(3/2)*asin(a)**2, True))

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Giac [A]  time = 1.26463, size = 306, normalized size = 1.54 \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 )^{2}}{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 )^{2}}{8 \, b} + \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )}^{2} \arcsin \left (b x + a\right )}{8 \, b} + \frac{\arcsin \left (b x + a\right )^{3}}{8 \, b} - \frac{{\left (-b^{2} x^{2} - 2 \, a b x - a^{2} + 1\right )}^{\frac{3}{2}}{\left (b x + a\right )}}{32 \, b} - \frac{3 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} \arcsin \left (b x + a\right )}{8 \, b} - \frac{15 \, \sqrt{-b^{2} x^{2} - 2 \, a b x - a^{2} + 1}{\left (b x + a\right )}}{64 \, b} - \frac{15 \, \arcsin \left (b x + a\right )}{64 \, 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)^2,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)^2/b + 3/8*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1
)*(b*x + a)*arcsin(b*x + a)^2/b + 1/8*(b^2*x^2 + 2*a*b*x + a^2 - 1)^2*arcsin(b*x + a)/b + 1/8*arcsin(b*x + a)^
3/b - 1/32*(-b^2*x^2 - 2*a*b*x - a^2 + 1)^(3/2)*(b*x + a)/b - 3/8*(b^2*x^2 + 2*a*b*x + a^2 - 1)*arcsin(b*x + a
)/b - 15/64*sqrt(-b^2*x^2 - 2*a*b*x - a^2 + 1)*(b*x + a)/b - 15/64*arcsin(b*x + a)/b