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

Optimal. Leaf size=140 \[ \frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{2 b e^2 \sqrt{1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}+\frac{4 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}-\frac{2 b^2 e^2 (c+d x)^3}{27 d}-\frac{4}{9} b^2 e^2 x \]

[Out]

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

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Rubi [A]  time = 0.206665, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {4805, 12, 4627, 4707, 4677, 8, 30} \[ \frac{e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2}{3 d}+\frac{2 b e^2 \sqrt{1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}+\frac{4 b e^2 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )}{9 d}-\frac{2 b^2 e^2 (c+d x)^3}{27 d}-\frac{4}{9} b^2 e^2 x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 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 30

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

Rubi steps

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

Mathematica [A]  time = 0.193641, size = 112, normalized size = 0.8 \[ \frac{e^2 \left ((c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^2-\frac{2}{9} b \left (-3 \sqrt{1-(c+d x)^2} (c+d x)^2 \left (a+b \sin ^{-1}(c+d x)\right )-6 \sqrt{1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )+b (c+d x)^3+6 b d x\right )\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e^2*((c + d*x)^3*(a + b*ArcSin[c + d*x])^2 - (2*b*(6*b*d*x + b*(c + d*x)^3 - 6*Sqrt[1 - (c + d*x)^2]*(a + b*A
rcSin[c + d*x]) - 3*(c + d*x)^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])))/9))/(3*d)

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Maple [A]  time = 0.031, size = 152, normalized size = 1.1 \begin{align*}{\frac{1}{d} \left ({\frac{{e}^{2} \left ( dx+c \right ) ^{3}{a}^{2}}{3}}+{e}^{2}{b}^{2} \left ({\frac{ \left ( \arcsin \left ( dx+c \right ) \right ) ^{2} \left ( dx+c \right ) ^{3}}{3}}+{\frac{2\,\arcsin \left ( dx+c \right ) \left ( \left ( dx+c \right ) ^{2}+2 \right ) }{9}\sqrt{1- \left ( dx+c \right ) ^{2}}}-{\frac{2\, \left ( dx+c \right ) ^{3}}{27}}-{\frac{4\,dx}{9}}-{\frac{4\,c}{9}} \right ) +2\,{e}^{2}ab \left ( 1/3\, \left ( dx+c \right ) ^{3}\arcsin \left ( dx+c \right ) +1/9\, \left ( dx+c \right ) ^{2}\sqrt{1- \left ( dx+c \right ) ^{2}}+2/9\,\sqrt{1- \left ( dx+c \right ) ^{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.48297, size = 651, normalized size = 4.65 \begin{align*} \frac{{\left (9 \, a^{2} - 2 \, b^{2}\right )} d^{3} e^{2} x^{3} + 3 \,{\left (9 \, a^{2} - 2 \, b^{2}\right )} c d^{2} e^{2} x^{2} + 3 \,{\left ({\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{2} - 4 \, b^{2}\right )} d e^{2} x + 9 \,{\left (b^{2} d^{3} e^{2} x^{3} + 3 \, b^{2} c d^{2} e^{2} x^{2} + 3 \, b^{2} c^{2} d e^{2} x + b^{2} c^{3} e^{2}\right )} \arcsin \left (d x + c\right )^{2} + 18 \,{\left (a b d^{3} e^{2} x^{3} + 3 \, a b c d^{2} e^{2} x^{2} + 3 \, a b c^{2} d e^{2} x + a b c^{3} e^{2}\right )} \arcsin \left (d x + c\right ) + 6 \,{\left (a b d^{2} e^{2} x^{2} + 2 \, a b c d e^{2} x +{\left (a b c^{2} + 2 \, a b\right )} e^{2} +{\left (b^{2} d^{2} e^{2} x^{2} + 2 \, b^{2} c d e^{2} x +{\left (b^{2} c^{2} + 2 \, b^{2}\right )} e^{2}\right )} \arcsin \left (d x + c\right )\right )} \sqrt{-d^{2} x^{2} - 2 \, c d x - c^{2} + 1}}{27 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.85857, size = 610, normalized size = 4.36 \begin{align*} \begin{cases} a^{2} c^{2} e^{2} x + a^{2} c d e^{2} x^{2} + \frac{a^{2} d^{2} e^{2} x^{3}}{3} + \frac{2 a b c^{3} e^{2} \operatorname{asin}{\left (c + d x \right )}}{3 d} + 2 a b c^{2} e^{2} x \operatorname{asin}{\left (c + d x \right )} + \frac{2 a b c^{2} e^{2} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{9 d} + 2 a b c d e^{2} x^{2} \operatorname{asin}{\left (c + d x \right )} + \frac{4 a b c e^{2} x \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{9} + \frac{2 a b d^{2} e^{2} x^{3} \operatorname{asin}{\left (c + d x \right )}}{3} + \frac{2 a b d e^{2} x^{2} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{9} + \frac{4 a b e^{2} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1}}{9 d} + \frac{b^{2} c^{3} e^{2} \operatorname{asin}^{2}{\left (c + d x \right )}}{3 d} + b^{2} c^{2} e^{2} x \operatorname{asin}^{2}{\left (c + d x \right )} - \frac{2 b^{2} c^{2} e^{2} x}{9} + \frac{2 b^{2} c^{2} e^{2} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname{asin}{\left (c + d x \right )}}{9 d} + b^{2} c d e^{2} x^{2} \operatorname{asin}^{2}{\left (c + d x \right )} - \frac{2 b^{2} c d e^{2} x^{2}}{9} + \frac{4 b^{2} c e^{2} x \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname{asin}{\left (c + d x \right )}}{9} + \frac{b^{2} d^{2} e^{2} x^{3} \operatorname{asin}^{2}{\left (c + d x \right )}}{3} - \frac{2 b^{2} d^{2} e^{2} x^{3}}{27} + \frac{2 b^{2} d e^{2} x^{2} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname{asin}{\left (c + d x \right )}}{9} - \frac{4 b^{2} e^{2} x}{9} + \frac{4 b^{2} e^{2} \sqrt{- c^{2} - 2 c d x - d^{2} x^{2} + 1} \operatorname{asin}{\left (c + d x \right )}}{9 d} & \text{for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname{asin}{\left (c \right )}\right )^{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.28405, size = 355, normalized size = 2.54 \begin{align*} \frac{{\left ({\left (d x + c\right )}^{2} - 1\right )}{\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right )^{2} e^{2}}{3 \, d} + \frac{{\left (d x + c\right )}^{3} a^{2} e^{2}}{3 \, d} + \frac{2 \,{\left ({\left (d x + c\right )}^{2} - 1\right )}{\left (d x + c\right )} a b \arcsin \left (d x + c\right ) e^{2}}{3 \, d} + \frac{{\left (d x + c\right )} b^{2} \arcsin \left (d x + c\right )^{2} e^{2}}{3 \, d} - \frac{2 \,{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac{3}{2}} b^{2} \arcsin \left (d x + c\right ) e^{2}}{9 \, d} - \frac{2 \,{\left ({\left (d x + c\right )}^{2} - 1\right )}{\left (d x + c\right )} b^{2} e^{2}}{27 \, d} + \frac{2 \,{\left (d x + c\right )} a b \arcsin \left (d x + c\right ) e^{2}}{3 \, d} - \frac{2 \,{\left (-{\left (d x + c\right )}^{2} + 1\right )}^{\frac{3}{2}} a b e^{2}}{9 \, d} + \frac{2 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} b^{2} \arcsin \left (d x + c\right ) e^{2}}{3 \, d} - \frac{14 \,{\left (d x + c\right )} b^{2} e^{2}}{27 \, d} + \frac{2 \, \sqrt{-{\left (d x + c\right )}^{2} + 1} a b e^{2}}{3 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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