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

Optimal. Leaf size=156 \[ \frac{2 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{2 b e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{4 b e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{4 b^2 e x}{9 c^2}-2 b^2 d x-\frac{2}{27} b^2 e x^3 \]

[Out]

-2*b^2*d*x - (4*b^2*e*x)/(9*c^2) - (2*b^2*e*x^3)/27 + (2*b*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*e
*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (2*b*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + d*
x*(a + b*ArcSin[c*x])^2 + (e*x^3*(a + b*ArcSin[c*x])^2)/3

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Rubi [A]  time = 0.261411, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {4667, 4619, 4677, 8, 4627, 4707, 30} \[ \frac{2 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{2 b e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+\frac{4 b e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{4 b^2 e x}{9 c^2}-2 b^2 d x-\frac{2}{27} b^2 e x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-2*b^2*d*x - (4*b^2*e*x)/(9*c^2) - (2*b^2*e*x^3)/27 + (2*b*d*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*e
*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c^3) + (2*b*e*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*c) + d*
x*(a + b*ArcSin[c*x])^2 + (e*x^3*(a + b*ArcSin[c*x])^2)/3

Rule 4667

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a
+ b*ArcSin[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[c^2*d + e, 0] && IntegerQ[p]
&& (GtQ[p, 0] || IGtQ[n, 0])

Rule 4619

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

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 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]

Rubi steps

\begin{align*} \int \left (d+e x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\int \left (d \left (a+b \sin ^{-1}(c x)\right )^2+e x^2 \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx\\ &=d \int \left (a+b \sin ^{-1}(c x)\right )^2 \, dx+e \int x^2 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx\\ &=d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^2-(2 b c d) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{3} (2 b c e) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{2 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{2 b e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^2-\left (2 b^2 d\right ) \int 1 \, dx-\frac{1}{9} \left (2 b^2 e\right ) \int x^2 \, dx-\frac{(4 b e) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{9 c}\\ &=-2 b^2 d x-\frac{2}{27} b^2 e x^3+\frac{2 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{2 b e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^2-\frac{\left (4 b^2 e\right ) \int 1 \, dx}{9 c^2}\\ &=-2 b^2 d x-\frac{4 b^2 e x}{9 c^2}-\frac{2}{27} b^2 e x^3+\frac{2 b d \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{2 b e x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}+d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.24365, size = 148, normalized size = 0.95 \[ -2 b d \left (b x-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}\right )-\frac{2}{27} b e \left (-\frac{3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{6 \left (\frac{b x}{c}-\frac{\sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c^2}\right )}{c}+b x^3\right )+d x \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} e x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \]

Antiderivative was successfully verified.

[In]

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

[Out]

d*x*(a + b*ArcSin[c*x])^2 + (e*x^3*(a + b*ArcSin[c*x])^2)/3 - 2*b*d*(b*x - (Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*
x]))/c) - (2*b*e*(b*x^3 - (3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (6*((b*x)/c - (Sqrt[1 - c^2*x^2]*(
a + b*ArcSin[c*x]))/c^2))/c))/27

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Maple [A]  time = 0.049, size = 276, normalized size = 1.8 \begin{align*}{\frac{1}{c} \left ({\frac{{a}^{2}}{{c}^{2}} \left ({\frac{{c}^{3}{x}^{3}e}{3}}+d{c}^{3}x \right ) }+{\frac{{b}^{2}}{{c}^{2}} \left ({\frac{e}{27} \left ( 9\,{c}^{3}{x}^{3} \left ( \arcsin \left ( cx \right ) \right ) ^{2}+6\,\arcsin \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1}{c}^{2}{x}^{2}-27\, \left ( \arcsin \left ( cx \right ) \right ) ^{2}cx-2\,{c}^{3}{x}^{3}-42\,\arcsin \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1}+42\,cx \right ) }+{c}^{2}d \left ( \left ( \arcsin \left ( cx \right ) \right ) ^{2}cx-2\,cx+2\,\arcsin \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1} \right ) +e \left ( \left ( \arcsin \left ( cx \right ) \right ) ^{2}cx-2\,cx+2\,\arcsin \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1} \right ) \right ) }+2\,{\frac{ab \left ( 1/3\,\arcsin \left ( cx \right ){c}^{3}{x}^{3}e+\arcsin \left ( cx \right ) d{c}^{3}x-1/3\,e \left ( -1/3\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}-2/3\,\sqrt{-{c}^{2}{x}^{2}+1} \right ) +{c}^{2}d\sqrt{-{c}^{2}{x}^{2}+1} \right ) }{{c}^{2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(a^2/c^2*(1/3*c^3*x^3*e+d*c^3*x)+b^2/c^2*(1/27*e*(9*c^3*x^3*arcsin(c*x)^2+6*arcsin(c*x)*(-c^2*x^2+1)^(1/2)
*c^2*x^2-27*arcsin(c*x)^2*c*x-2*c^3*x^3-42*arcsin(c*x)*(-c^2*x^2+1)^(1/2)+42*c*x)+c^2*d*(arcsin(c*x)^2*c*x-2*c
*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+e*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2)))+2*a*b/c^2*(
1/3*arcsin(c*x)*c^3*x^3*e+arcsin(c*x)*d*c^3*x-1/3*e*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1/2))+c
^2*d*(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.44661, size = 298, normalized size = 1.91 \begin{align*} \frac{1}{3} \, b^{2} e x^{3} \arcsin \left (c x\right )^{2} + \frac{1}{3} \, a^{2} e x^{3} + b^{2} d x \arcsin \left (c x\right )^{2} + \frac{2}{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 )} a b e + \frac{2}{27} \,{\left (3 \, c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1}}{c^{4}}\right )} \arcsin \left (c x\right ) - \frac{c^{2} x^{3} + 6 \, x}{c^{2}}\right )} b^{2} e - 2 \, b^{2} d{\left (x - \frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d x + \frac{2 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a b d}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*b^2*e*x^3*arcsin(c*x)^2 + 1/3*a^2*e*x^3 + b^2*d*x*arcsin(c*x)^2 + 2/9*(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))*a*b*e + 2/27*(3*c*(sqrt(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 +
1)/c^4)*arcsin(c*x) - (c^2*x^3 + 6*x)/c^2)*b^2*e - 2*b^2*d*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d*x +
2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d/c

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Fricas [A]  time = 2.03491, size = 400, normalized size = 2.56 \begin{align*} \frac{{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} e x^{3} + 9 \,{\left (b^{2} c^{3} e x^{3} + 3 \, b^{2} c^{3} d x\right )} \arcsin \left (c x\right )^{2} + 3 \,{\left (9 \,{\left (a^{2} - 2 \, b^{2}\right )} c^{3} d - 4 \, b^{2} c e\right )} x + 18 \,{\left (a b c^{3} e x^{3} + 3 \, a b c^{3} d x\right )} \arcsin \left (c x\right ) + 6 \,{\left (a b c^{2} e x^{2} + 9 \, a b c^{2} d + 2 \, a b e +{\left (b^{2} c^{2} e x^{2} + 9 \, b^{2} c^{2} d + 2 \, b^{2} e\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{27 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/27*((9*a^2 - 2*b^2)*c^3*e*x^3 + 9*(b^2*c^3*e*x^3 + 3*b^2*c^3*d*x)*arcsin(c*x)^2 + 3*(9*(a^2 - 2*b^2)*c^3*d -
4*b^2*c*e)*x + 18*(a*b*c^3*e*x^3 + 3*a*b*c^3*d*x)*arcsin(c*x) + 6*(a*b*c^2*e*x^2 + 9*a*b*c^2*d + 2*a*b*e + (b
^2*c^2*e*x^2 + 9*b^2*c^2*d + 2*b^2*e)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^3

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Sympy [A]  time = 1.59377, size = 279, normalized size = 1.79 \begin{align*} \begin{cases} a^{2} d x + \frac{a^{2} e x^{3}}{3} + 2 a b d x \operatorname{asin}{\left (c x \right )} + \frac{2 a b e x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{2 a b d \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{2 a b e x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} + \frac{4 a b e \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d x \operatorname{asin}^{2}{\left (c x \right )} - 2 b^{2} d x + \frac{b^{2} e x^{3} \operatorname{asin}^{2}{\left (c x \right )}}{3} - \frac{2 b^{2} e x^{3}}{27} + \frac{2 b^{2} d \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{c} + \frac{2 b^{2} e x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{9 c} - \frac{4 b^{2} e x}{9 c^{2}} + \frac{4 b^{2} e \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{9 c^{3}} & \text{for}\: c \neq 0 \\a^{2} \left (d x + \frac{e x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**2*d*x + a**2*e*x**3/3 + 2*a*b*d*x*asin(c*x) + 2*a*b*e*x**3*asin(c*x)/3 + 2*a*b*d*sqrt(-c**2*x**2
+ 1)/c + 2*a*b*e*x**2*sqrt(-c**2*x**2 + 1)/(9*c) + 4*a*b*e*sqrt(-c**2*x**2 + 1)/(9*c**3) + b**2*d*x*asin(c*x)
**2 - 2*b**2*d*x + b**2*e*x**3*asin(c*x)**2/3 - 2*b**2*e*x**3/27 + 2*b**2*d*sqrt(-c**2*x**2 + 1)*asin(c*x)/c +
2*b**2*e*x**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(9*c) - 4*b**2*e*x/(9*c**2) + 4*b**2*e*sqrt(-c**2*x**2 + 1)*asin
(c*x)/(9*c**3), Ne(c, 0)), (a**2*(d*x + e*x**3/3), True))

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Giac [B]  time = 1.32157, size = 400, normalized size = 2.56 \begin{align*} b^{2} d x \arcsin \left (c x\right )^{2} + \frac{1}{3} \, a^{2} x^{3} e + 2 \, a b d x \arcsin \left (c x\right ) + \frac{{\left (c^{2} x^{2} - 1\right )} b^{2} x \arcsin \left (c x\right )^{2} e}{3 \, c^{2}} + a^{2} d x - 2 \, b^{2} d x + \frac{2 \,{\left (c^{2} x^{2} - 1\right )} a b x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac{b^{2} x \arcsin \left (c x\right )^{2} e}{3 \, c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d \arcsin \left (c x\right )}{c} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )} b^{2} x e}{27 \, c^{2}} + \frac{2 \, a b x \arcsin \left (c x\right ) e}{3 \, c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} a b d}{c} - \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} \arcsin \left (c x\right ) e}{9 \, c^{3}} - \frac{14 \, b^{2} x e}{27 \, c^{2}} - \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b e}{9 \, c^{3}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b^{2} \arcsin \left (c x\right ) e}{3 \, c^{3}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} a b e}{3 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*d*x*arcsin(c*x)^2 + 1/3*a^2*x^3*e + 2*a*b*d*x*arcsin(c*x) + 1/3*(c^2*x^2 - 1)*b^2*x*arcsin(c*x)^2*e/c^2 +
a^2*d*x - 2*b^2*d*x + 2/3*(c^2*x^2 - 1)*a*b*x*arcsin(c*x)*e/c^2 + 1/3*b^2*x*arcsin(c*x)^2*e/c^2 + 2*sqrt(-c^2*
x^2 + 1)*b^2*d*arcsin(c*x)/c - 2/27*(c^2*x^2 - 1)*b^2*x*e/c^2 + 2/3*a*b*x*arcsin(c*x)*e/c^2 + 2*sqrt(-c^2*x^2
+ 1)*a*b*d/c - 2/9*(-c^2*x^2 + 1)^(3/2)*b^2*arcsin(c*x)*e/c^3 - 14/27*b^2*x*e/c^2 - 2/9*(-c^2*x^2 + 1)^(3/2)*a
*b*e/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*b^2*arcsin(c*x)*e/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*a*b*e/c^3