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

Optimal. Leaf size=202 \[ -\frac{1}{18} b c d x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{b d x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{b d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 c^3}-\frac{d \left (a+b \sin ^{-1}(c x)\right )^2}{24 c^4}+\frac{1}{12} d x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{108} b^2 c^2 d x^6-\frac{b^2 d x^2}{24 c^2}-\frac{1}{72} b^2 d x^4 \]

[Out]

-(b^2*d*x^2)/(24*c^2) - (b^2*d*x^4)/72 + (b^2*c^2*d*x^6)/108 + (b*d*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(
12*c^3) + (b*d*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(18*c) - (b*c*d*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[
c*x]))/18 - (d*(a + b*ArcSin[c*x])^2)/(24*c^4) + (d*x^4*(a + b*ArcSin[c*x])^2)/12 + (d*x^4*(1 - c^2*x^2)*(a +
b*ArcSin[c*x])^2)/6

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Rubi [A]  time = 0.537396, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {4699, 4627, 4707, 4641, 30, 4697} \[ -\frac{1}{18} b c d x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{b d x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac{b d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 c^3}-\frac{d \left (a+b \sin ^{-1}(c x)\right )^2}{24 c^4}+\frac{1}{12} d x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{108} b^2 c^2 d x^6-\frac{b^2 d x^2}{24 c^2}-\frac{1}{72} b^2 d x^4 \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*d*x^2)/(24*c^2) - (b^2*d*x^4)/72 + (b^2*c^2*d*x^6)/108 + (b*d*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(
12*c^3) + (b*d*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(18*c) - (b*c*d*x^5*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[
c*x]))/18 - (d*(a + b*ArcSin[c*x])^2)/(24*c^4) + (d*x^4*(a + b*ArcSin[c*x])^2)/12 + (d*x^4*(1 - c^2*x^2)*(a +
b*ArcSin[c*x])^2)/6

Rule 4699

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[
((f*x)^(m + 1)*(d + e*x^2)^p*(a + b*ArcSin[c*x])^n)/(f*(m + 2*p + 1)), x] + (Dist[(2*d*p)/(m + 2*p + 1), Int[(
f*x)^m*(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])/(
f*(m + 2*p + 1)*(1 - c^2*x^2)^FracPart[p]), Int[(f*x)^(m + 1)*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n -
1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[m, -1]
&& (RationalQ[m] || EqQ[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 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 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 30

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

Rule 4697

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((
f*x)^(m + 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(f*(m + 2)), x] + (Dist[Sqrt[d + e*x^2]/((m + 2)*Sqrt[1 -
c^2*x^2]), Int[((f*x)^m*(a + b*ArcSin[c*x])^n)/Sqrt[1 - c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(f*(m
+ 2)*Sqrt[1 - c^2*x^2]), Int[(f*x)^(m + 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}
, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rubi steps

\begin{align*} \int x^3 \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac{1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{3} d \int x^3 \left (a+b \sin ^{-1}(c x)\right )^2 \, dx-\frac{1}{3} (b c d) \int x^4 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx\\ &=-\frac{1}{18} b c d x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{12} d x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{18} (b c d) \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx-\frac{1}{6} (b c d) \int \frac{x^4 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx+\frac{1}{18} \left (b^2 c^2 d\right ) \int x^5 \, dx\\ &=\frac{1}{108} b^2 c^2 d x^6+\frac{b d x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{1}{18} b c d x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{12} d x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{1}{72} \left (b^2 d\right ) \int x^3 \, dx-\frac{1}{24} \left (b^2 d\right ) \int x^3 \, dx-\frac{(b d) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{24 c}-\frac{(b d) \int \frac{x^2 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{1-c^2 x^2}} \, dx}{8 c}\\ &=-\frac{1}{72} b^2 d x^4+\frac{1}{108} b^2 c^2 d x^6+\frac{b d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 c^3}+\frac{b d x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{1}{18} b c d x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{12} d x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2-\frac{(b d) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{48 c^3}-\frac{(b d) \int \frac{a+b \sin ^{-1}(c x)}{\sqrt{1-c^2 x^2}} \, dx}{16 c^3}-\frac{\left (b^2 d\right ) \int x \, dx}{48 c^2}-\frac{\left (b^2 d\right ) \int x \, dx}{16 c^2}\\ &=-\frac{b^2 d x^2}{24 c^2}-\frac{1}{72} b^2 d x^4+\frac{1}{108} b^2 c^2 d x^6+\frac{b d x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{12 c^3}+\frac{b d x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}-\frac{1}{18} b c d x^5 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )-\frac{d \left (a+b \sin ^{-1}(c x)\right )^2}{24 c^4}+\frac{1}{12} d x^4 \left (a+b \sin ^{-1}(c x)\right )^2+\frac{1}{6} d x^4 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2\\ \end{align*}

Mathematica [A]  time = 0.161663, size = 192, normalized size = 0.95 \[ -\frac{d \left (9 a^2 \left (4 c^6 x^6-6 c^4 x^4+1\right )+6 a b c x \sqrt{1-c^2 x^2} \left (2 c^4 x^4-2 c^2 x^2-3\right )+6 b \sin ^{-1}(c x) \left (3 a \left (4 c^6 x^6-6 c^4 x^4+1\right )+b c x \sqrt{1-c^2 x^2} \left (2 c^4 x^4-2 c^2 x^2-3\right )\right )+b^2 c^2 x^2 \left (-2 c^4 x^4+3 c^2 x^2+9\right )+9 b^2 \left (4 c^6 x^6-6 c^4 x^4+1\right ) \sin ^{-1}(c x)^2\right )}{216 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(b^2*c^2*x^2*(9 + 3*c^2*x^2 - 2*c^4*x^4) + 6*a*b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^2*x^2 + 2*c^4*x^4) + 9*a^
2*(1 - 6*c^4*x^4 + 4*c^6*x^6) + 6*b*(b*c*x*Sqrt[1 - c^2*x^2]*(-3 - 2*c^2*x^2 + 2*c^4*x^4) + 3*a*(1 - 6*c^4*x^4
+ 4*c^6*x^6))*ArcSin[c*x] + 9*b^2*(1 - 6*c^4*x^4 + 4*c^6*x^6)*ArcSin[c*x]^2))/(216*c^4)

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Maple [A]  time = 0.044, size = 306, normalized size = 1.5 \begin{align*}{\frac{1}{{c}^{4}} \left ( -d{a}^{2} \left ({\frac{{c}^{6}{x}^{6}}{6}}-{\frac{{c}^{4}{x}^{4}}{4}} \right ) -d{b}^{2} \left ( -{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2}{c}^{4}{x}^{4}}{4}}+{\frac{\arcsin \left ( cx \right ) }{16} \left ( -2\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}-3\,cx\sqrt{-{c}^{2}{x}^{2}+1}+3\,\arcsin \left ( cx \right ) \right ) }-{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{24}}+{\frac{{c}^{4}{x}^{4}}{72}}+{\frac{{c}^{2}{x}^{2}}{24}}+{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2}{c}^{6}{x}^{6}}{6}}-{\frac{\arcsin \left ( cx \right ) }{144} \left ( -8\,{c}^{5}{x}^{5}\sqrt{-{c}^{2}{x}^{2}+1}-10\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}-15\,cx\sqrt{-{c}^{2}{x}^{2}+1}+15\,\arcsin \left ( cx \right ) \right ) }-{\frac{{c}^{6}{x}^{6}}{108}} \right ) -2\,dab \left ( 1/6\,\arcsin \left ( cx \right ){c}^{6}{x}^{6}-1/4\,{c}^{4}{x}^{4}\arcsin \left ( cx \right ) +1/36\,{c}^{5}{x}^{5}\sqrt{-{c}^{2}{x}^{2}+1}-1/36\,{c}^{3}{x}^{3}\sqrt{-{c}^{2}{x}^{2}+1}-1/24\,cx\sqrt{-{c}^{2}{x}^{2}+1}+1/24\,\arcsin \left ( cx \right ) \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(-d*a^2*(1/6*c^6*x^6-1/4*c^4*x^4)-d*b^2*(-1/4*arcsin(c*x)^2*c^4*x^4+1/16*arcsin(c*x)*(-2*c^3*x^3*(-c^2*x
^2+1)^(1/2)-3*c*x*(-c^2*x^2+1)^(1/2)+3*arcsin(c*x))-1/24*arcsin(c*x)^2+1/72*c^4*x^4+1/24*c^2*x^2+1/6*arcsin(c*
x)^2*c^6*x^6-1/144*arcsin(c*x)*(-8*c^5*x^5*(-c^2*x^2+1)^(1/2)-10*c^3*x^3*(-c^2*x^2+1)^(1/2)-15*c*x*(-c^2*x^2+1
)^(1/2)+15*arcsin(c*x))-1/108*c^6*x^6)-2*d*a*b*(1/6*arcsin(c*x)*c^6*x^6-1/4*c^4*x^4*arcsin(c*x)+1/36*c^5*x^5*(
-c^2*x^2+1)^(1/2)-1/36*c^3*x^3*(-c^2*x^2+1)^(1/2)-1/24*c*x*(-c^2*x^2+1)^(1/2)+1/24*arcsin(c*x)))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{6} \, a^{2} c^{2} d x^{6} + \frac{1}{4} \, a^{2} d x^{4} - \frac{1}{144} \,{\left (48 \, x^{6} \arcsin \left (c x\right ) +{\left (\frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{5}}{c^{2}} + \frac{10 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac{15 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{6}} - \frac{15 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{6}}\right )} c\right )} a b c^{2} d + \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 d - \frac{1}{12} \,{\left (2 \, b^{2} c^{2} d x^{6} - 3 \, b^{2} d x^{4}\right )} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )^{2} - \int \frac{{\left (2 \, b^{2} c^{3} d x^{6} - 3 \, b^{2} c d x^{4}\right )} \sqrt{c x + 1} \sqrt{-c x + 1} \arctan \left (c x, \sqrt{c x + 1} \sqrt{-c x + 1}\right )}{6 \,{\left (c^{2} x^{2} - 1\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6*a^2*c^2*d*x^6 + 1/4*a^2*d*x^4 - 1/144*(48*x^6*arcsin(c*x) + (8*sqrt(-c^2*x^2 + 1)*x^5/c^2 + 10*sqrt(-c^2*
x^2 + 1)*x^3/c^4 + 15*sqrt(-c^2*x^2 + 1)*x/c^6 - 15*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^6))*c)*a*b*c^2*d + 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*d - 1/12*(2*b^2*c^2*d*x^6 - 3*b^2*d*x^4)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2
- integrate(1/6*(2*b^2*c^3*d*x^6 - 3*b^2*c*d*x^4)*sqrt(c*x + 1)*sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqr
t(-c*x + 1))/(c^2*x^2 - 1), x)

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Fricas [A]  time = 1.90308, size = 470, normalized size = 2.33 \begin{align*} -\frac{2 \,{\left (18 \, a^{2} - b^{2}\right )} c^{6} d x^{6} - 3 \,{\left (18 \, a^{2} - b^{2}\right )} c^{4} d x^{4} + 9 \, b^{2} c^{2} d x^{2} + 9 \,{\left (4 \, b^{2} c^{6} d x^{6} - 6 \, b^{2} c^{4} d x^{4} + b^{2} d\right )} \arcsin \left (c x\right )^{2} + 18 \,{\left (4 \, a b c^{6} d x^{6} - 6 \, a b c^{4} d x^{4} + a b d\right )} \arcsin \left (c x\right ) + 6 \,{\left (2 \, a b c^{5} d x^{5} - 2 \, a b c^{3} d x^{3} - 3 \, a b c d x +{\left (2 \, b^{2} c^{5} d x^{5} - 2 \, b^{2} c^{3} d x^{3} - 3 \, b^{2} c d x\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} x^{2} + 1}}{216 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/216*(2*(18*a^2 - b^2)*c^6*d*x^6 - 3*(18*a^2 - b^2)*c^4*d*x^4 + 9*b^2*c^2*d*x^2 + 9*(4*b^2*c^6*d*x^6 - 6*b^2
*c^4*d*x^4 + b^2*d)*arcsin(c*x)^2 + 18*(4*a*b*c^6*d*x^6 - 6*a*b*c^4*d*x^4 + a*b*d)*arcsin(c*x) + 6*(2*a*b*c^5*
d*x^5 - 2*a*b*c^3*d*x^3 - 3*a*b*c*d*x + (2*b^2*c^5*d*x^5 - 2*b^2*c^3*d*x^3 - 3*b^2*c*d*x)*arcsin(c*x))*sqrt(-c
^2*x^2 + 1))/c^4

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Sympy [A]  time = 12.1833, size = 332, normalized size = 1.64 \begin{align*} \begin{cases} - \frac{a^{2} c^{2} d x^{6}}{6} + \frac{a^{2} d x^{4}}{4} - \frac{a b c^{2} d x^{6} \operatorname{asin}{\left (c x \right )}}{3} - \frac{a b c d x^{5} \sqrt{- c^{2} x^{2} + 1}}{18} + \frac{a b d x^{4} \operatorname{asin}{\left (c x \right )}}{2} + \frac{a b d x^{3} \sqrt{- c^{2} x^{2} + 1}}{18 c} + \frac{a b d x \sqrt{- c^{2} x^{2} + 1}}{12 c^{3}} - \frac{a b d \operatorname{asin}{\left (c x \right )}}{12 c^{4}} - \frac{b^{2} c^{2} d x^{6} \operatorname{asin}^{2}{\left (c x \right )}}{6} + \frac{b^{2} c^{2} d x^{6}}{108} - \frac{b^{2} c d x^{5} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{18} + \frac{b^{2} d x^{4} \operatorname{asin}^{2}{\left (c x \right )}}{4} - \frac{b^{2} d x^{4}}{72} + \frac{b^{2} d x^{3} \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{18 c} - \frac{b^{2} d x^{2}}{24 c^{2}} + \frac{b^{2} d x \sqrt{- c^{2} x^{2} + 1} \operatorname{asin}{\left (c x \right )}}{12 c^{3}} - \frac{b^{2} d \operatorname{asin}^{2}{\left (c x \right )}}{24 c^{4}} & \text{for}\: c \neq 0 \\\frac{a^{2} d x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a**2*c**2*d*x**6/6 + a**2*d*x**4/4 - a*b*c**2*d*x**6*asin(c*x)/3 - a*b*c*d*x**5*sqrt(-c**2*x**2 +
1)/18 + a*b*d*x**4*asin(c*x)/2 + a*b*d*x**3*sqrt(-c**2*x**2 + 1)/(18*c) + a*b*d*x*sqrt(-c**2*x**2 + 1)/(12*c**
3) - a*b*d*asin(c*x)/(12*c**4) - b**2*c**2*d*x**6*asin(c*x)**2/6 + b**2*c**2*d*x**6/108 - b**2*c*d*x**5*sqrt(-
c**2*x**2 + 1)*asin(c*x)/18 + b**2*d*x**4*asin(c*x)**2/4 - b**2*d*x**4/72 + b**2*d*x**3*sqrt(-c**2*x**2 + 1)*a
sin(c*x)/(18*c) - b**2*d*x**2/(24*c**2) + b**2*d*x*sqrt(-c**2*x**2 + 1)*asin(c*x)/(12*c**3) - b**2*d*asin(c*x)
**2/(24*c**4), Ne(c, 0)), (a**2*d*x**4/4, True))

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Giac [B]  time = 1.37161, size = 535, normalized size = 2.65 \begin{align*} -\frac{{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b^{2} d x \arcsin \left (c x\right )}{18 \, c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d \arcsin \left (c x\right )^{2}}{6 \, c^{4}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} a b d x}{18 \, c^{3}} + \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b^{2} d x \arcsin \left (c x\right )}{18 \, c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{3} a b d \arcsin \left (c x\right )}{3 \, c^{4}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d \arcsin \left (c x\right )^{2}}{4 \, c^{4}} + \frac{{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} a b d x}{18 \, c^{3}} + \frac{\sqrt{-c^{2} x^{2} + 1} b^{2} d x \arcsin \left (c x\right )}{12 \, c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{3} a^{2} d}{6 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{3} b^{2} d}{108 \, c^{4}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a b d \arcsin \left (c x\right )}{2 \, c^{4}} + \frac{\sqrt{-c^{2} x^{2} + 1} a b d x}{12 \, c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{2} a^{2} d}{4 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b^{2} d}{72 \, c^{4}} + \frac{b^{2} d \arcsin \left (c x\right )^{2}}{24 \, c^{4}} - \frac{{\left (c^{2} x^{2} - 1\right )} b^{2} d}{24 \, c^{4}} + \frac{a b d \arcsin \left (c x\right )}{12 \, c^{4}} - \frac{5 \, b^{2} d}{216 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b^2*d*x*arcsin(c*x)/c^3 - 1/6*(c^2*x^2 - 1)^3*b^2*d*arcsin(c*x)^2/c^4
- 1/18*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*a*b*d*x/c^3 + 1/18*(-c^2*x^2 + 1)^(3/2)*b^2*d*x*arcsin(c*x)/c^3 - 1
/3*(c^2*x^2 - 1)^3*a*b*d*arcsin(c*x)/c^4 - 1/4*(c^2*x^2 - 1)^2*b^2*d*arcsin(c*x)^2/c^4 + 1/18*(-c^2*x^2 + 1)^(
3/2)*a*b*d*x/c^3 + 1/12*sqrt(-c^2*x^2 + 1)*b^2*d*x*arcsin(c*x)/c^3 - 1/6*(c^2*x^2 - 1)^3*a^2*d/c^4 + 1/108*(c^
2*x^2 - 1)^3*b^2*d/c^4 - 1/2*(c^2*x^2 - 1)^2*a*b*d*arcsin(c*x)/c^4 + 1/12*sqrt(-c^2*x^2 + 1)*a*b*d*x/c^3 - 1/4
*(c^2*x^2 - 1)^2*a^2*d/c^4 + 1/72*(c^2*x^2 - 1)^2*b^2*d/c^4 + 1/24*b^2*d*arcsin(c*x)^2/c^4 - 1/24*(c^2*x^2 - 1
)*b^2*d/c^4 + 1/12*a*b*d*arcsin(c*x)/c^4 - 5/216*b^2*d/c^4