3.9 \(\int (d+e x)^3 (a+b \sin ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=374 \[ \frac{3 b d^2 e x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac{3 d^2 e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac{2 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{b e^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{3 b e^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}-\frac{3 e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}-\frac{d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}+\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac{4 b^2 d e^2 x}{3 c^2}-\frac{3 b^2 e^3 x^2}{32 c^2}-\frac{3}{4} b^2 d^2 e x^2-2 b^2 d^3 x-\frac{2}{9} b^2 d e^2 x^3-\frac{1}{32} b^2 e^3 x^4 \]

[Out]

-2*b^2*d^3*x - (4*b^2*d*e^2*x)/(3*c^2) - (3*b^2*d^2*e*x^2)/4 - (3*b^2*e^3*x^2)/(32*c^2) - (2*b^2*d*e^2*x^3)/9
- (b^2*e^3*x^4)/32 + (2*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*d*e^2*Sqrt[1 - c^2*x^2]*(a + b*A
rcSin[c*x]))/(3*c^3) + (3*b*d^2*e*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c) + (3*b*e^3*x*Sqrt[1 - c^2*x^2
]*(a + b*ArcSin[c*x]))/(16*c^3) + (2*b*d*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c) + (b*e^3*x^3*Sqr
t[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) - (d^4*(a + b*ArcSin[c*x])^2)/(4*e) - (3*d^2*e*(a + b*ArcSin[c*x])^2
)/(4*c^2) - (3*e^3*(a + b*ArcSin[c*x])^2)/(32*c^4) + ((d + e*x)^4*(a + b*ArcSin[c*x])^2)/(4*e)

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Rubi [A]  time = 0.712387, antiderivative size = 374, normalized size of antiderivative = 1., number of steps used = 18, 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} \[ \frac{3 b d^2 e x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}-\frac{3 d^2 e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac{2 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{b e^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}+\frac{3 b e^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}-\frac{3 e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}-\frac{d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}+\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac{4 b^2 d e^2 x}{3 c^2}-\frac{3 b^2 e^3 x^2}{32 c^2}-\frac{3}{4} b^2 d^2 e x^2-2 b^2 d^3 x-\frac{2}{9} b^2 d e^2 x^3-\frac{1}{32} b^2 e^3 x^4 \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^3*(a + b*ArcSin[c*x])^2,x]

[Out]

-2*b^2*d^3*x - (4*b^2*d*e^2*x)/(3*c^2) - (3*b^2*d^2*e*x^2)/4 - (3*b^2*e^3*x^2)/(32*c^2) - (2*b^2*d*e^2*x^3)/9
- (b^2*e^3*x^4)/32 + (2*b*d^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/c + (4*b*d*e^2*Sqrt[1 - c^2*x^2]*(a + b*A
rcSin[c*x]))/(3*c^3) + (3*b*d^2*e*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*c) + (3*b*e^3*x*Sqrt[1 - c^2*x^2
]*(a + b*ArcSin[c*x]))/(16*c^3) + (2*b*d*e^2*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(3*c) + (b*e^3*x^3*Sqr
t[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(8*c) - (d^4*(a + b*ArcSin[c*x])^2)/(4*e) - (3*d^2*e*(a + b*ArcSin[c*x])^2
)/(4*c^2) - (3*e^3*(a + b*ArcSin[c*x])^2)/(32*c^4) + ((d + e*x)^4*(a + b*ArcSin[c*x])^2)/(4*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)^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac{(b c) \int \frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{2 e}\\ &=\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac{(b c) \int \left (\frac{d^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{4 d^3 e x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{6 d^2 e^2 x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{4 d e^3 x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}+\frac{e^4 x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}}\right ) \, dx}{2 e}\\ &=\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\left (2 b c d^3\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{\left (b c d^4\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 e}-\left (3 b c d^2 e\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\left (2 b c d e^2\right ) \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{2} \left (b c e^3\right ) \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{3 b d^2 e x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac{2 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{b e^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac{d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}+\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\left (2 b^2 d^3\right ) \int 1 \, dx-\frac{1}{2} \left (3 b^2 d^2 e\right ) \int x \, dx-\frac{\left (3 b d^2 e\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{2 c}-\frac{1}{3} \left (2 b^2 d e^2\right ) \int x^2 \, dx-\frac{\left (4 b d e^2\right ) \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{3 c}-\frac{1}{8} \left (b^2 e^3\right ) \int x^3 \, dx-\frac{\left (3 b e^3\right ) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{8 c}\\ &=-2 b^2 d^3 x-\frac{3}{4} b^2 d^2 e x^2-\frac{2}{9} b^2 d e^2 x^3-\frac{1}{32} b^2 e^3 x^4+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac{3 b d^2 e x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac{3 b e^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac{2 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{b e^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac{d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac{3 d^2 e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}+\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac{\left (4 b^2 d e^2\right ) \int 1 \, dx}{3 c^2}-\frac{\left (3 b e^3\right ) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{16 c^3}-\frac{\left (3 b^2 e^3\right ) \int x \, dx}{16 c^2}\\ &=-2 b^2 d^3 x-\frac{4 b^2 d e^2 x}{3 c^2}-\frac{3}{4} b^2 d^2 e x^2-\frac{3 b^2 e^3 x^2}{32 c^2}-\frac{2}{9} b^2 d e^2 x^3-\frac{1}{32} b^2 e^3 x^4+\frac{2 b d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c}+\frac{4 b d e^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^3}+\frac{3 b d^2 e x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 c}+\frac{3 b e^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{16 c^3}+\frac{2 b d e^2 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c}+\frac{b e^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{8 c}-\frac{d^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}-\frac{3 d^2 e \left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2}-\frac{3 e^3 \left (a+b \sin ^{-1}(c x)\right )^2}{32 c^4}+\frac{(d+e x)^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 e}\\ \end{align*}

Mathematica [A]  time = 0.571397, size = 355, normalized size = 0.95 \[ \frac{c \left (72 a^2 c^3 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )+6 a b \sqrt{1-c^2 x^2} \left (c^2 \left (72 d^2 e x+96 d^3+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )-b^2 c x \left (c^2 \left (216 d^2 e x+576 d^3+64 d e^2 x^2+9 e^3 x^3\right )+3 e^2 (128 d+9 e x)\right )\right )+6 b \sin ^{-1}(c x) \left (3 a \left (8 c^4 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )-24 c^2 d^2 e-3 e^3\right )+b c \sqrt{1-c^2 x^2} \left (c^2 \left (72 d^2 e x+96 d^3+32 d e^2 x^2+6 e^3 x^3\right )+e^2 (64 d+9 e x)\right )\right )+9 b^2 \sin ^{-1}(c x)^2 \left (8 c^4 x \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )-24 c^2 d^2 e-3 e^3\right )}{288 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^3*(a + b*ArcSin[c*x])^2,x]

[Out]

(c*(72*a^2*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 6*a*b*Sqrt[1 - c^2*x^2]*(e^2*(64*d + 9*e*x) + c
^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^3)) - b^2*c*x*(3*e^2*(128*d + 9*e*x) + c^2*(576*d^3 + 216*d^2
*e*x + 64*d*e^2*x^2 + 9*e^3*x^3))) + 6*b*(3*a*(-24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^
2 + e^3*x^3)) + b*c*Sqrt[1 - c^2*x^2]*(e^2*(64*d + 9*e*x) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3*x^
3)))*ArcSin[c*x] + 9*b^2*(-24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcSin[
c*x]^2)/(288*c^4)

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Maple [A]  time = 0.092, size = 660, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^3*(a+b*arcsin(c*x))^2,x)

[Out]

1/c*(1/4*(c*e*x+c*d)^4*a^2/c^3/e+b^2/c^3*(1/32*e^3*(8*arcsin(c*x)^2*c^4*x^4+4*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c
^3*x^3-16*arcsin(c*x)^2*c^2*x^2-c^4*x^4-10*arcsin(c*x)*(-c^2*x^2+1)^(1/2)*c*x+5*arcsin(c*x)^2+5*c^2*x^2-4)+3/4
*d^2*c^2*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)+1/9*d*c*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)+c^3*d^3*(arcsin(c*x)^2*c*x-2*c*x+2*arcsin(c*x)*(-c^2*x^2+1)^(1/2))+1/4*e^3*(2*arcsi
n(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)+3*d*c*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^3*(1/4*e^3*arcsin(c*x)*c^4*x^4+e^2*arcsin(c*x)*c^4*x^3*d+3/2*e*ar
csin(c*x)*c^4*x^2*d^2+arcsin(c*x)*c^4*x*d^3+1/4/e*arcsin(c*x)*c^4*d^4-1/4/e*(e^4*(-1/4*c^3*x^3*(-c^2*x^2+1)^(1
/2)-3/8*c*x*(-c^2*x^2+1)^(1/2)+3/8*arcsin(c*x))+4*d*c*e^3*(-1/3*c^2*x^2*(-c^2*x^2+1)^(1/2)-2/3*(-c^2*x^2+1)^(1
/2))+6*c^2*d^2*e^2*(-1/2*c*x*(-c^2*x^2+1)^(1/2)+1/2*arcsin(c*x))-4*c^3*d^3*e*(-c^2*x^2+1)^(1/2)+c^4*d^4*arcsin
(c*x))))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{4} \, a^{2} e^{3} x^{4} + a^{2} d e^{2} x^{3} + b^{2} d^{3} x \arcsin \left (c x\right )^{2} + \frac{3}{2} \, a^{2} d^{2} e x^{2} + \frac{3}{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^{2} e + \frac{2}{3} \,{\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 d e^{2} + \frac{1}{16} \,{\left (8 \, x^{4} \arcsin \left (c x\right ) +{\left (\frac{2 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{2}} + \frac{3 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{4}} - \frac{3 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{4}}\right )} c\right )} a b e^{3} - 2 \, b^{2} d^{3}{\left (x - \frac{\sqrt{-c^{2} x^{2} + 1} \arcsin \left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac{2 \,{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} a b d^{3}}{c} + \frac{1}{4} \,{\left (b^{2} e^{3} x^{4} + 4 \, b^{2} d e^{2} x^{3} + 6 \, b^{2} d^{2} e x^{2}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} + \int \frac{{\left (b^{2} c e^{3} x^{4} + 4 \, b^{2} c d e^{2} x^{3} + 6 \, b^{2} c d^{2} 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 )}{2 \,{\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)^3*(a+b*arcsin(c*x))^2,x, algorithm="maxima")

[Out]

1/4*a^2*e^3*x^4 + a^2*d*e^2*x^3 + b^2*d^3*x*arcsin(c*x)^2 + 3/2*a^2*d^2*e*x^2 + 3/2*(2*x^2*arcsin(c*x) + c*(sq
rt(-c^2*x^2 + 1)*x/c^2 - arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*a*b*d^2*e + 2/3*(3*x^3*arcsin(c*x) + c*(sqr
t(-c^2*x^2 + 1)*x^2/c^2 + 2*sqrt(-c^2*x^2 + 1)/c^4))*a*b*d*e^2 + 1/16*(8*x^4*arcsin(c*x) + (2*sqrt(-c^2*x^2 +
1)*x^3/c^2 + 3*sqrt(-c^2*x^2 + 1)*x/c^4 - 3*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*a*b*e^3 - 2*b^2*d^3*(x
- sqrt(-c^2*x^2 + 1)*arcsin(c*x)/c) + a^2*d^3*x + 2*(c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*a*b*d^3/c + 1/4*(b
^2*e^3*x^4 + 4*b^2*d*e^2*x^3 + 6*b^2*d^2*e*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + integrate(1/2*(
b^2*c*e^3*x^4 + 4*b^2*c*d*e^2*x^3 + 6*b^2*c*d^2*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 = 2.48604, size = 956, normalized size = 2.56 \begin{align*} \frac{9 \,{\left (8 \, a^{2} - b^{2}\right )} c^{4} e^{3} x^{4} + 32 \,{\left (9 \, a^{2} - 2 \, b^{2}\right )} c^{4} d e^{2} x^{3} + 27 \,{\left (8 \,{\left (2 \, a^{2} - b^{2}\right )} c^{4} d^{2} e - b^{2} c^{2} e^{3}\right )} x^{2} + 9 \,{\left (8 \, b^{2} c^{4} e^{3} x^{4} + 32 \, b^{2} c^{4} d e^{2} x^{3} + 48 \, b^{2} c^{4} d^{2} e x^{2} + 32 \, b^{2} c^{4} d^{3} x - 24 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} \arcsin \left (c x\right )^{2} + 96 \,{\left (3 \,{\left (a^{2} - 2 \, b^{2}\right )} c^{4} d^{3} - 4 \, b^{2} c^{2} d e^{2}\right )} x + 18 \,{\left (8 \, a b c^{4} e^{3} x^{4} + 32 \, a b c^{4} d e^{2} x^{3} + 48 \, a b c^{4} d^{2} e x^{2} + 32 \, a b c^{4} d^{3} x - 24 \, a b c^{2} d^{2} e - 3 \, a b e^{3}\right )} \arcsin \left (c x\right ) + 6 \,{\left (6 \, a b c^{3} e^{3} x^{3} + 32 \, a b c^{3} d e^{2} x^{2} + 96 \, a b c^{3} d^{3} + 64 \, a b c d e^{2} + 9 \,{\left (8 \, a b c^{3} d^{2} e + a b c e^{3}\right )} x +{\left (6 \, b^{2} c^{3} e^{3} x^{3} + 32 \, b^{2} c^{3} d e^{2} x^{2} + 96 \, b^{2} c^{3} d^{3} + 64 \, b^{2} c d e^{2} + 9 \,{\left (8 \, b^{2} c^{3} d^{2} e + b^{2} c e^{3}\right )} x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{288 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*arcsin(c*x))^2,x, algorithm="fricas")

[Out]

1/288*(9*(8*a^2 - b^2)*c^4*e^3*x^4 + 32*(9*a^2 - 2*b^2)*c^4*d*e^2*x^3 + 27*(8*(2*a^2 - b^2)*c^4*d^2*e - b^2*c^
2*e^3)*x^2 + 9*(8*b^2*c^4*e^3*x^4 + 32*b^2*c^4*d*e^2*x^3 + 48*b^2*c^4*d^2*e*x^2 + 32*b^2*c^4*d^3*x - 24*b^2*c^
2*d^2*e - 3*b^2*e^3)*arcsin(c*x)^2 + 96*(3*(a^2 - 2*b^2)*c^4*d^3 - 4*b^2*c^2*d*e^2)*x + 18*(8*a*b*c^4*e^3*x^4
+ 32*a*b*c^4*d*e^2*x^3 + 48*a*b*c^4*d^2*e*x^2 + 32*a*b*c^4*d^3*x - 24*a*b*c^2*d^2*e - 3*a*b*e^3)*arcsin(c*x) +
6*(6*a*b*c^3*e^3*x^3 + 32*a*b*c^3*d*e^2*x^2 + 96*a*b*c^3*d^3 + 64*a*b*c*d*e^2 + 9*(8*a*b*c^3*d^2*e + a*b*c*e^
3)*x + (6*b^2*c^3*e^3*x^3 + 32*b^2*c^3*d*e^2*x^2 + 96*b^2*c^3*d^3 + 64*b^2*c*d*e^2 + 9*(8*b^2*c^3*d^2*e + b^2*
c*e^3)*x)*arcsin(c*x))*sqrt(-c^2*x^2 + 1))/c^4

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Sympy [A]  time = 4.47399, size = 743, normalized size = 1.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(a+b*asin(c*x))**2,x)

[Out]

Piecewise((a**2*d**3*x + 3*a**2*d**2*e*x**2/2 + a**2*d*e**2*x**3 + a**2*e**3*x**4/4 + 2*a*b*d**3*x*asin(c*x) +
3*a*b*d**2*e*x**2*asin(c*x) + 2*a*b*d*e**2*x**3*asin(c*x) + a*b*e**3*x**4*asin(c*x)/2 + 2*a*b*d**3*sqrt(-c**2
*x**2 + 1)/c + 3*a*b*d**2*e*x*sqrt(-c**2*x**2 + 1)/(2*c) + 2*a*b*d*e**2*x**2*sqrt(-c**2*x**2 + 1)/(3*c) + a*b*
e**3*x**3*sqrt(-c**2*x**2 + 1)/(8*c) - 3*a*b*d**2*e*asin(c*x)/(2*c**2) + 4*a*b*d*e**2*sqrt(-c**2*x**2 + 1)/(3*
c**3) + 3*a*b*e**3*x*sqrt(-c**2*x**2 + 1)/(16*c**3) - 3*a*b*e**3*asin(c*x)/(16*c**4) + b**2*d**3*x*asin(c*x)**
2 - 2*b**2*d**3*x + 3*b**2*d**2*e*x**2*asin(c*x)**2/2 - 3*b**2*d**2*e*x**2/4 + b**2*d*e**2*x**3*asin(c*x)**2 -
2*b**2*d*e**2*x**3/9 + b**2*e**3*x**4*asin(c*x)**2/4 - b**2*e**3*x**4/32 + 2*b**2*d**3*sqrt(-c**2*x**2 + 1)*a
sin(c*x)/c + 3*b**2*d**2*e*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(2*c) + 2*b**2*d*e**2*x**2*sqrt(-c**2*x**2 + 1)*as
in(c*x)/(3*c) + b**2*e**3*x**3*sqrt(-c**2*x**2 + 1)*asin(c*x)/(8*c) - 3*b**2*d**2*e*asin(c*x)**2/(4*c**2) - 4*
b**2*d*e**2*x/(3*c**2) - 3*b**2*e**3*x**2/(32*c**2) + 4*b**2*d*e**2*sqrt(-c**2*x**2 + 1)*asin(c*x)/(3*c**3) +
3*b**2*e**3*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(16*c**3) - 3*b**2*e**3*asin(c*x)**2/(32*c**4), Ne(c, 0)), (a**2*
(d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4), True))

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Giac [B]  time = 1.3538, size = 1121, normalized size = 3. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(a+b*arcsin(c*x))^2,x, algorithm="giac")

[Out]

b^2*d^3*x*arcsin(c*x)^2 + 2*a*b*d^3*x*arcsin(c*x) + a^2*d*x^3*e^2 + 3/2*sqrt(-c^2*x^2 + 1)*b^2*d^2*x*arcsin(c*
x)*e/c + a^2*d^3*x - 2*b^2*d^3*x + (c^2*x^2 - 1)*b^2*d*x*arcsin(c*x)^2*e^2/c^2 + 3/2*(c^2*x^2 - 1)*b^2*d^2*arc
sin(c*x)^2*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*b^2*d^3*arcsin(c*x)/c + 3/2*sqrt(-c^2*x^2 + 1)*a*b*d^2*x*e/c + 2*(c^2*
x^2 - 1)*a*b*d*x*arcsin(c*x)*e^2/c^2 + b^2*d*x*arcsin(c*x)^2*e^2/c^2 + 3*(c^2*x^2 - 1)*a*b*d^2*arcsin(c*x)*e/c
^2 + 3/4*b^2*d^2*arcsin(c*x)^2*e/c^2 + 2*sqrt(-c^2*x^2 + 1)*a*b*d^3/c - 2/9*(c^2*x^2 - 1)*b^2*d*x*e^2/c^2 + 2*
a*b*d*x*arcsin(c*x)*e^2/c^2 + 3/2*(c^2*x^2 - 1)*a^2*d^2*e/c^2 - 3/4*(c^2*x^2 - 1)*b^2*d^2*e/c^2 + 3/2*a*b*d^2*
arcsin(c*x)*e/c^2 - 1/8*(-c^2*x^2 + 1)^(3/2)*b^2*x*arcsin(c*x)*e^3/c^3 - 2/3*(-c^2*x^2 + 1)^(3/2)*b^2*d*arcsin
(c*x)*e^2/c^3 + 1/4*(c^2*x^2 - 1)^2*b^2*arcsin(c*x)^2*e^3/c^4 - 14/9*b^2*d*x*e^2/c^2 - 3/8*b^2*d^2*e/c^2 - 1/8
*(-c^2*x^2 + 1)^(3/2)*a*b*x*e^3/c^3 + 5/16*sqrt(-c^2*x^2 + 1)*b^2*x*arcsin(c*x)*e^3/c^3 - 2/3*(-c^2*x^2 + 1)^(
3/2)*a*b*d*e^2/c^3 + 2*sqrt(-c^2*x^2 + 1)*b^2*d*arcsin(c*x)*e^2/c^3 + 1/2*(c^2*x^2 - 1)^2*a*b*arcsin(c*x)*e^3/
c^4 + 1/2*(c^2*x^2 - 1)*b^2*arcsin(c*x)^2*e^3/c^4 + 5/16*sqrt(-c^2*x^2 + 1)*a*b*x*e^3/c^3 + 2*sqrt(-c^2*x^2 +
1)*a*b*d*e^2/c^3 + 1/4*(c^2*x^2 - 1)^2*a^2*e^3/c^4 - 1/32*(c^2*x^2 - 1)^2*b^2*e^3/c^4 + (c^2*x^2 - 1)*a*b*arcs
in(c*x)*e^3/c^4 + 5/32*b^2*arcsin(c*x)^2*e^3/c^4 + 1/2*(c^2*x^2 - 1)*a^2*e^3/c^4 - 5/32*(c^2*x^2 - 1)*b^2*e^3/
c^4 + 5/16*a*b*arcsin(c*x)*e^3/c^4 - 17/256*b^2*e^3/c^4