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

Optimal. Leaf size=242 \[ \frac{2 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{b d e x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}-\frac{d e \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2}+\frac{4 b e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{2 b e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}-\frac{d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e}+\frac{(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e}-\frac{4 b^2 e^2 x}{9 c^2}-2 b^2 d^2 x-\frac{1}{2} b^2 d e x^2-\frac{2}{27} b^2 e^2 x^3 \]

[Out]

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

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Rubi [A]  time = 0.479342, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.389, Rules used = {4743, 4763, 4641, 4677, 8, 4707, 30} \[ \frac{2 b d^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{b d e x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}-\frac{d e \left (a+b \sin ^{-1}(c x)\right )^2}{2 c^2}+\frac{4 b e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c^3}+\frac{2 b e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 c}-\frac{d^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e}+\frac{(d+e x)^3 \left (a+b \sin ^{-1}(c x)\right )^2}{3 e}-\frac{4 b^2 e^2 x}{9 c^2}-2 b^2 d^2 x-\frac{1}{2} b^2 d e x^2-\frac{2}{27} b^2 e^2 x^3 \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.386684, size = 249, normalized size = 1.03 \[ \frac{18 a^2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )+6 a b \sqrt{1-c^2 x^2} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )+6 b \sin ^{-1}(c x) \left (6 a c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-9 a c d e+b \sqrt{1-c^2 x^2} \left (c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )+4 e^2\right )\right )-b^2 c x \left (c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )+24 e^2\right )+9 b^2 c \sin ^{-1}(c x)^2 \left (6 c^2 d^2 x+3 d e \left (2 c^2 x^2-1\right )+2 c^2 e^2 x^3\right )}{54 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(18*a^2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) + 6*a*b*Sqrt[1 - c^2*x^2]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2
)) - b^2*c*x*(24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) + 6*b*(-9*a*c*d*e + 6*a*c^3*x*(3*d^2 + 3*d*e*x +
e^2*x^2) + b*Sqrt[1 - c^2*x^2]*(4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)))*ArcSin[c*x] + 9*b^2*c*(6*c^2*d^2*
x + 2*c^2*e^2*x^3 + 3*d*e*(-1 + 2*c^2*x^2))*ArcSin[c*x]^2)/(54*c^3)

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Maple [A]  time = 0.065, size = 420, normalized size = 1.7 \begin{align*}{\frac{1}{c} \left ({\frac{ \left ( ecx+dc \right ) ^{3}{a}^{2}}{3\,{c}^{2}e}}+{\frac{{b}^{2}}{{c}^{2}} \left ({c}^{2}{d}^{2} \left ( \left ( \arcsin \left ( cx \right ) \right ) ^{2}cx-2\,cx+2\,\arcsin \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1} \right ) +{\frac{dce}{2} \left ( 2\, \left ( \arcsin \left ( cx \right ) \right ) ^{2}{c}^{2}{x}^{2}+2\,\arcsin \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1}cx- \left ( \arcsin \left ( cx \right ) \right ) ^{2}-{c}^{2}{x}^{2} \right ) }+{\frac{{e}^{2}}{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 ) }+{e}^{2} \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}{{c}^{2}} \left ( 1/3\,\arcsin \left ( cx \right ){e}^{2}{c}^{3}{x}^{3}+e\arcsin \left ( cx \right ){c}^{3}{x}^{2}d+\arcsin \left ( cx \right ){c}^{3}x{d}^{2}+1/3\,{\frac{{c}^{3}{d}^{3}\arcsin \left ( cx \right ) }{e}}-1/3\,{\frac{{e}^{3} \left ( -1/3\,{c}^{2}{x}^{2}\sqrt{-{c}^{2}{x}^{2}+1}-2/3\,\sqrt{-{c}^{2}{x}^{2}+1} \right ) +3\,dc{e}^{2} \left ( -1/2\,cx\sqrt{-{c}^{2}{x}^{2}+1}+1/2\,\arcsin \left ( cx \right ) \right ) -3\,{d}^{2}{c}^{2}e\sqrt{-{c}^{2}{x}^{2}+1}+{c}^{3}{d}^{3}\arcsin \left ( cx \right ) }{e}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{3} \, a^{2} e^{2} x^{3} + b^{2} d^{2} x \arcsin \left (c x\right )^{2} + a^{2} d e x^{2} +{\left (2 \, x^{2} \arcsin \left (c x\right ) + c{\left (\frac{\sqrt{-c^{2} x^{2} + 1} x}{c^{2}} - \frac{\arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{2}}\right )}\right )} a b d e + \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^{2} - 2 \, b^{2} d^{2}{\left (x - \frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac{2 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a b d^{2}}{c} + \frac{1}{3} \,{\left (b^{2} e^{2} x^{3} + 3 \, b^{2} d e x^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + \int \frac{2 \,{\left (b^{2} c e^{2} x^{3} + 3 \, b^{2} c d e x^{2}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{3 \,{\left (c^{2} x^{2} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*e^2*x^3 + b^2*d^2*x*arcsin(c*x)^2 + a^2*d*e*x^2 + (2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^2 + 1)*x/c^2 - a
rcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*a*b*d*e + 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 - 2*b^2*d^2*(x - sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d^2*x + 2*(c*x*arcsi
n(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^2/c + 1/3*(b^2*e^2*x^3 + 3*b^2*d*e*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*
x + 1))^2 + integrate(2/3*(b^2*c*e^2*x^3 + 3*b^2*c*d*e*x^2)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x
+ 1)*sqrt(-c*x + 1))/(c^2*x^2 - 1), x)

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Fricas [A]  time = 2.42632, size = 633, normalized size = 2.62 \begin{align*} \frac{2 \,{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{3} e^{2} x^{3} + 27 \,{\left (2 \, a^{2} - b^{2}\right )} c^{3} d e x^{2} + 9 \,{\left (2 \, b^{2} c^{3} e^{2} x^{3} + 6 \, b^{2} c^{3} d e x^{2} + 6 \, b^{2} c^{3} d^{2} x - 3 \, b^{2} c d e\right )} \arcsin \left (c x\right )^{2} + 6 \,{\left (9 \,{\left (a^{2} - 2 \, b^{2}\right )} c^{3} d^{2} - 4 \, b^{2} c e^{2}\right )} x + 18 \,{\left (2 \, a b c^{3} e^{2} x^{3} + 6 \, a b c^{3} d e x^{2} + 6 \, a b c^{3} d^{2} x - 3 \, a b c d e\right )} \arcsin \left (c x\right ) + 6 \,{\left (2 \, a b c^{2} e^{2} x^{2} + 9 \, a b c^{2} d e x + 18 \, a b c^{2} d^{2} + 4 \, a b e^{2} +{\left (2 \, b^{2} c^{2} e^{2} x^{2} + 9 \, b^{2} c^{2} d e x + 18 \, b^{2} c^{2} d^{2} + 4 \, b^{2} e^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{54 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(2*(9*a^2 - 2*b^2)*c^3*e^2*x^3 + 27*(2*a^2 - b^2)*c^3*d*e*x^2 + 9*(2*b^2*c^3*e^2*x^3 + 6*b^2*c^3*d*e*x^2
+ 6*b^2*c^3*d^2*x - 3*b^2*c*d*e)*arcsin(c*x)^2 + 6*(9*(a^2 - 2*b^2)*c^3*d^2 - 4*b^2*c*e^2)*x + 18*(2*a*b*c^3*e
^2*x^3 + 6*a*b*c^3*d*e*x^2 + 6*a*b*c^3*d^2*x - 3*a*b*c*d*e)*arcsin(c*x) + 6*(2*a*b*c^2*e^2*x^2 + 9*a*b*c^2*d*e
*x + 18*a*b*c^2*d^2 + 4*a*b*e^2 + (2*b^2*c^2*e^2*x^2 + 9*b^2*c^2*d*e*x + 18*b^2*c^2*d^2 + 4*b^2*e^2)*arcsin(c*
x))*sqrt(-c^2*x^2 + 1))/c^3

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Sympy [A]  time = 2.16729, size = 454, normalized size = 1.88 \begin{align*} \begin{cases} a^{2} d^{2} x + a^{2} d e x^{2} + \frac{a^{2} e^{2} x^{3}}{3} + 2 a b d^{2} x \operatorname{asin}{\left (c x \right )} + 2 a b d e x^{2} \operatorname{asin}{\left (c x \right )} + \frac{2 a b e^{2} x^{3} \operatorname{asin}{\left (c x \right )}}{3} + \frac{2 a b d^{2} \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{a b d e x \sqrt{- c^{2} x^{2} + 1}}{c} + \frac{2 a b e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c} - \frac{a b d e \operatorname{asin}{\left (c x \right )}}{c^{2}} + \frac{4 a b e^{2} \sqrt{- c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d^{2} x \operatorname{asin}^{2}{\left (c x \right )} - 2 b^{2} d^{2} x + b^{2} d e x^{2} \operatorname{asin}^{2}{\left (c x \right )} - \frac{b^{2} d e x^{2}}{2} + \frac{b^{2} e^{2} x^{3} \operatorname{asin}^{2}{\left (c x \right )}}{3} - \frac{2 b^{2} e^{2} x^{3}}{27} + \frac{2 b^{2} d^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{c} + \frac{b^{2} d e x \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{c} + \frac{2 b^{2} e^{2} x^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{9 c} - \frac{b^{2} d e \operatorname{asin}^{2}{\left (c x \right )}}{2 c^{2}} - \frac{4 b^{2} e^{2} x}{9 c^{2}} + \frac{4 b^{2} e^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{9 c^{3}} & \text{for}\: c \neq 0 \\a^{2} \left (d^{2} x + d e x^{2} + \frac{e^{2} x^{3}}{3}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.28979, size = 655, normalized size = 2.71 \begin{align*} b^{2} d^{2} x \arcsin \left (c x\right )^{2} + 2 \, a b d^{2} x \arcsin \left (c x\right ) + \frac{1}{3} \, a^{2} x^{3} e^{2} + \frac{\sqrt{-c^{2} x^{2} + 1} b^{2} d x \arcsin \left (c x\right ) e}{c} + a^{2} d^{2} x - 2 \, b^{2} d^{2} x + \frac{{\left (c^{2} x^{2} - 1\right )} b^{2} x \arcsin \left (c x\right )^{2} e^{2}}{3 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )} b^{2} d \arcsin \left (c x\right )^{2} e}{c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b^{2} d^{2} \arcsin \left (c x\right )}{c} + \frac{\sqrt{-c^{2} x^{2} + 1} a b d x e}{c} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )} a b x \arcsin \left (c x\right ) e^{2}}{3 \, c^{2}} + \frac{b^{2} x \arcsin \left (c x\right )^{2} e^{2}}{3 \, c^{2}} + \frac{2 \,{\left (c^{2} x^{2} - 1\right )} a b d \arcsin \left (c x\right ) e}{c^{2}} + \frac{b^{2} d \arcsin \left (c x\right )^{2} e}{2 \, c^{2}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} a b d^{2}}{c} - \frac{2 \,{\left (c^{2} x^{2} - 1\right )} b^{2} x e^{2}}{27 \, c^{2}} + \frac{2 \, a b x \arcsin \left (c x\right ) e^{2}}{3 \, c^{2}} + \frac{{\left (c^{2} x^{2} - 1\right )} a^{2} d e}{c^{2}} - \frac{{\left (c^{2} x^{2} - 1\right )} b^{2} d e}{2 \, c^{2}} + \frac{a b d \arcsin \left (c x\right ) e}{c^{2}} - \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} \arcsin \left (c x\right ) e^{2}}{9 \, c^{3}} - \frac{14 \, b^{2} x e^{2}}{27 \, c^{2}} - \frac{b^{2} d e}{4 \, c^{2}} - \frac{2 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b e^{2}}{9 \, c^{3}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} b^{2} \arcsin \left (c x\right ) e^{2}}{3 \, c^{3}} + \frac{2 \, \sqrt{-c^{2} x^{2} + 1} a b e^{2}}{3 \, c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^2*d^2*x*arcsin(c*x)^2 + 2*a*b*d^2*x*arcsin(c*x) + 1/3*a^2*x^3*e^2 + sqrt(-c^2*x^2 + 1)*b^2*d*x*arcsin(c*x)*e
/c + a^2*d^2*x - 2*b^2*d^2*x + 1/3*(c^2*x^2 - 1)*b^2*x*arcsin(c*x)^2*e^2/c^2 + (c^2*x^2 - 1)*b^2*d*arcsin(c*x)
^2*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*b^2*d^2*arcsin(c*x)/c + sqrt(-c^2*x^2 + 1)*a*b*d*x*e/c + 2/3*(c^2*x^2 - 1)*a*b
*x*arcsin(c*x)*e^2/c^2 + 1/3*b^2*x*arcsin(c*x)^2*e^2/c^2 + 2*(c^2*x^2 - 1)*a*b*d*arcsin(c*x)*e/c^2 + 1/2*b^2*d
*arcsin(c*x)^2*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*a*b*d^2/c - 2/27*(c^2*x^2 - 1)*b^2*x*e^2/c^2 + 2/3*a*b*x*arcsin(c*
x)*e^2/c^2 + (c^2*x^2 - 1)*a^2*d*e/c^2 - 1/2*(c^2*x^2 - 1)*b^2*d*e/c^2 + a*b*d*arcsin(c*x)*e/c^2 - 2/9*(-c^2*x
^2 + 1)^(3/2)*b^2*arcsin(c*x)*e^2/c^3 - 14/27*b^2*x*e^2/c^2 - 1/4*b^2*d*e/c^2 - 2/9*(-c^2*x^2 + 1)^(3/2)*a*b*e
^2/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*b^2*arcsin(c*x)*e^2/c^3 + 2/3*sqrt(-c^2*x^2 + 1)*a*b*e^2/c^3