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

Optimal. Leaf size=206 \[ \frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 c^4}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^4}-\frac{b d^3 x \left (1-c^2 x^2\right )^{9/2}}{100 c^3}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{7/2}}{1600 c^3}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{9600 c^3}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{7680 c^3}+\frac{49 b d^3 x \sqrt{1-c^2 x^2}}{5120 c^3}+\frac{49 b d^3 \sin ^{-1}(c x)}{5120 c^4} \]

[Out]

(49*b*d^3*x*Sqrt[1 - c^2*x^2])/(5120*c^3) + (49*b*d^3*x*(1 - c^2*x^2)^(3/2))/(7680*c^3) + (49*b*d^3*x*(1 - c^2
*x^2)^(5/2))/(9600*c^3) + (7*b*d^3*x*(1 - c^2*x^2)^(7/2))/(1600*c^3) - (b*d^3*x*(1 - c^2*x^2)^(9/2))/(100*c^3)
+ (49*b*d^3*ArcSin[c*x])/(5120*c^4) - (d^3*(1 - c^2*x^2)^4*(a + b*ArcSin[c*x]))/(8*c^4) + (d^3*(1 - c^2*x^2)^
5*(a + b*ArcSin[c*x]))/(10*c^4)

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Rubi [A]  time = 0.178894, antiderivative size = 206, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.28, Rules used = {266, 43, 4687, 12, 388, 195, 216} \[ \frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 c^4}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^4}-\frac{b d^3 x \left (1-c^2 x^2\right )^{9/2}}{100 c^3}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{7/2}}{1600 c^3}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{9600 c^3}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{7680 c^3}+\frac{49 b d^3 x \sqrt{1-c^2 x^2}}{5120 c^3}+\frac{49 b d^3 \sin ^{-1}(c x)}{5120 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(49*b*d^3*x*Sqrt[1 - c^2*x^2])/(5120*c^3) + (49*b*d^3*x*(1 - c^2*x^2)^(3/2))/(7680*c^3) + (49*b*d^3*x*(1 - c^2
*x^2)^(5/2))/(9600*c^3) + (7*b*d^3*x*(1 - c^2*x^2)^(7/2))/(1600*c^3) - (b*d^3*x*(1 - c^2*x^2)^(9/2))/(100*c^3)
+ (49*b*d^3*ArcSin[c*x])/(5120*c^4) - (d^3*(1 - c^2*x^2)^4*(a + b*ArcSin[c*x]))/(8*c^4) + (d^3*(1 - c^2*x^2)^
5*(a + b*ArcSin[c*x]))/(10*c^4)

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 4687

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = I
ntHide[(f*x)^m*(d + e*x^2)^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 -
c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 12

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

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 216

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

Rubi steps

\begin{align*} \int x^3 \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 c^4}-(b c) \int \frac{d^3 \left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^{7/2}}{40 c^4} \, dx\\ &=-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 c^4}-\frac{\left (b d^3\right ) \int \left (-1-4 c^2 x^2\right ) \left (1-c^2 x^2\right )^{7/2} \, dx}{40 c^3}\\ &=-\frac{b d^3 x \left (1-c^2 x^2\right )^{9/2}}{100 c^3}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 c^4}+\frac{\left (7 b d^3\right ) \int \left (1-c^2 x^2\right )^{7/2} \, dx}{200 c^3}\\ &=\frac{7 b d^3 x \left (1-c^2 x^2\right )^{7/2}}{1600 c^3}-\frac{b d^3 x \left (1-c^2 x^2\right )^{9/2}}{100 c^3}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{1600 c^3}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{9600 c^3}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{7/2}}{1600 c^3}-\frac{b d^3 x \left (1-c^2 x^2\right )^{9/2}}{100 c^3}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{1920 c^3}\\ &=\frac{49 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{7680 c^3}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{9600 c^3}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{7/2}}{1600 c^3}-\frac{b d^3 x \left (1-c^2 x^2\right )^{9/2}}{100 c^3}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3\right ) \int \sqrt{1-c^2 x^2} \, dx}{2560 c^3}\\ &=\frac{49 b d^3 x \sqrt{1-c^2 x^2}}{5120 c^3}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{7680 c^3}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{9600 c^3}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{7/2}}{1600 c^3}-\frac{b d^3 x \left (1-c^2 x^2\right )^{9/2}}{100 c^3}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 c^4}+\frac{\left (49 b d^3\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{5120 c^3}\\ &=\frac{49 b d^3 x \sqrt{1-c^2 x^2}}{5120 c^3}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{7680 c^3}+\frac{49 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{9600 c^3}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{7/2}}{1600 c^3}-\frac{b d^3 x \left (1-c^2 x^2\right )^{9/2}}{100 c^3}+\frac{49 b d^3 \sin ^{-1}(c x)}{5120 c^4}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^4}+\frac{d^3 \left (1-c^2 x^2\right )^5 \left (a+b \sin ^{-1}(c x)\right )}{10 c^4}\\ \end{align*}

Mathematica [A]  time = 0.188653, size = 139, normalized size = 0.67 \[ \frac{d^3 \left (-1920 a c^4 x^4 \left (4 c^6 x^6-15 c^4 x^4+20 c^2 x^2-10\right )+b c x \sqrt{1-c^2 x^2} \left (-768 c^8 x^8+2736 c^6 x^6-3208 c^4 x^4+790 c^2 x^2+1185\right )-15 b \left (512 c^{10} x^{10}-1920 c^8 x^8+2560 c^6 x^6-1280 c^4 x^4+79\right ) \sin ^{-1}(c x)\right )}{76800 c^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(d^3*(-1920*a*c^4*x^4*(-10 + 20*c^2*x^2 - 15*c^4*x^4 + 4*c^6*x^6) + b*c*x*Sqrt[1 - c^2*x^2]*(1185 + 790*c^2*x^
2 - 3208*c^4*x^4 + 2736*c^6*x^6 - 768*c^8*x^8) - 15*b*(79 - 1280*c^4*x^4 + 2560*c^6*x^6 - 1920*c^8*x^8 + 512*c
^10*x^10)*ArcSin[c*x]))/(76800*c^4)

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Maple [A]  time = 0.013, size = 202, normalized size = 1. \begin{align*}{\frac{1}{{c}^{4}} \left ( -{d}^{3}a \left ({\frac{{c}^{10}{x}^{10}}{10}}-{\frac{3\,{c}^{8}{x}^{8}}{8}}+{\frac{{c}^{6}{x}^{6}}{2}}-{\frac{{c}^{4}{x}^{4}}{4}} \right ) -{d}^{3}b \left ({\frac{\arcsin \left ( cx \right ){c}^{10}{x}^{10}}{10}}-{\frac{3\,\arcsin \left ( cx \right ){c}^{8}{x}^{8}}{8}}+{\frac{\arcsin \left ( cx \right ){c}^{6}{x}^{6}}{2}}-{\frac{{c}^{4}{x}^{4}\arcsin \left ( cx \right ) }{4}}+{\frac{{c}^{9}{x}^{9}}{100}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{57\,{c}^{7}{x}^{7}}{1600}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{401\,{c}^{5}{x}^{5}}{9600}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{79\,{c}^{3}{x}^{3}}{7680}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{79\,cx}{5120}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{79\,\arcsin \left ( cx \right ) }{5120}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^4*(-d^3*a*(1/10*c^10*x^10-3/8*c^8*x^8+1/2*c^6*x^6-1/4*c^4*x^4)-d^3*b*(1/10*arcsin(c*x)*c^10*x^10-3/8*arcsi
n(c*x)*c^8*x^8+1/2*arcsin(c*x)*c^6*x^6-1/4*c^4*x^4*arcsin(c*x)+1/100*c^9*x^9*(-c^2*x^2+1)^(1/2)-57/1600*c^7*x^
7*(-c^2*x^2+1)^(1/2)+401/9600*c^5*x^5*(-c^2*x^2+1)^(1/2)-79/7680*c^3*x^3*(-c^2*x^2+1)^(1/2)-79/5120*c*x*(-c^2*
x^2+1)^(1/2)+79/5120*arcsin(c*x)))

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Maxima [B]  time = 1.64879, size = 657, normalized size = 3.19 \begin{align*} -\frac{1}{10} \, a c^{6} d^{3} x^{10} + \frac{3}{8} \, a c^{4} d^{3} x^{8} - \frac{1}{2} \, a c^{2} d^{3} x^{6} - \frac{1}{12800} \,{\left (1280 \, x^{10} \arcsin \left (c x\right ) +{\left (\frac{128 \, \sqrt{-c^{2} x^{2} + 1} x^{9}}{c^{2}} + \frac{144 \, \sqrt{-c^{2} x^{2} + 1} x^{7}}{c^{4}} + \frac{168 \, \sqrt{-c^{2} x^{2} + 1} x^{5}}{c^{6}} + \frac{210 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{8}} + \frac{315 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{10}} - \frac{315 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{10}}\right )} c\right )} b c^{6} d^{3} + \frac{1}{1024} \,{\left (384 \, x^{8} \arcsin \left (c x\right ) +{\left (\frac{48 \, \sqrt{-c^{2} x^{2} + 1} x^{7}}{c^{2}} + \frac{56 \, \sqrt{-c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac{70 \, \sqrt{-c^{2} x^{2} + 1} x^{3}}{c^{6}} + \frac{105 \, \sqrt{-c^{2} x^{2} + 1} x}{c^{8}} - \frac{105 \, \arcsin \left (\frac{c^{2} x}{\sqrt{c^{2}}}\right )}{\sqrt{c^{2}} c^{8}}\right )} c\right )} b c^{4} d^{3} + \frac{1}{4} \, a d^{3} x^{4} - \frac{1}{96} \,{\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 )} b c^{2} d^{3} + \frac{1}{32} \,{\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 )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*a*c^6*d^3*x^10 + 3/8*a*c^4*d^3*x^8 - 1/2*a*c^2*d^3*x^6 - 1/12800*(1280*x^10*arcsin(c*x) + (128*sqrt(-c^2
*x^2 + 1)*x^9/c^2 + 144*sqrt(-c^2*x^2 + 1)*x^7/c^4 + 168*sqrt(-c^2*x^2 + 1)*x^5/c^6 + 210*sqrt(-c^2*x^2 + 1)*x
^3/c^8 + 315*sqrt(-c^2*x^2 + 1)*x/c^10 - 315*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^10))*c)*b*c^6*d^3 + 1/1024*(
384*x^8*arcsin(c*x) + (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56*sqrt(-c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(-c^2*x^2 + 1)*x
^3/c^6 + 105*sqrt(-c^2*x^2 + 1)*x/c^8 - 105*arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^8))*c)*b*c^4*d^3 + 1/4*a*d^3*
x^4 - 1/96*(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)*b*c^2*d^3 + 1/32*(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)*b*d^3

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Fricas [A]  time = 2.08545, size = 454, normalized size = 2.2 \begin{align*} -\frac{7680 \, a c^{10} d^{3} x^{10} - 28800 \, a c^{8} d^{3} x^{8} + 38400 \, a c^{6} d^{3} x^{6} - 19200 \, a c^{4} d^{3} x^{4} + 15 \,{\left (512 \, b c^{10} d^{3} x^{10} - 1920 \, b c^{8} d^{3} x^{8} + 2560 \, b c^{6} d^{3} x^{6} - 1280 \, b c^{4} d^{3} x^{4} + 79 \, b d^{3}\right )} \arcsin \left (c x\right ) +{\left (768 \, b c^{9} d^{3} x^{9} - 2736 \, b c^{7} d^{3} x^{7} + 3208 \, b c^{5} d^{3} x^{5} - 790 \, b c^{3} d^{3} x^{3} - 1185 \, b c d^{3} x\right )} \sqrt{-c^{2} x^{2} + 1}}{76800 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/76800*(7680*a*c^10*d^3*x^10 - 28800*a*c^8*d^3*x^8 + 38400*a*c^6*d^3*x^6 - 19200*a*c^4*d^3*x^4 + 15*(512*b*c
^10*d^3*x^10 - 1920*b*c^8*d^3*x^8 + 2560*b*c^6*d^3*x^6 - 1280*b*c^4*d^3*x^4 + 79*b*d^3)*arcsin(c*x) + (768*b*c
^9*d^3*x^9 - 2736*b*c^7*d^3*x^7 + 3208*b*c^5*d^3*x^5 - 790*b*c^3*d^3*x^3 - 1185*b*c*d^3*x)*sqrt(-c^2*x^2 + 1))
/c^4

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Sympy [A]  time = 69.1025, size = 280, normalized size = 1.36 \begin{align*} \begin{cases} - \frac{a c^{6} d^{3} x^{10}}{10} + \frac{3 a c^{4} d^{3} x^{8}}{8} - \frac{a c^{2} d^{3} x^{6}}{2} + \frac{a d^{3} x^{4}}{4} - \frac{b c^{6} d^{3} x^{10} \operatorname{asin}{\left (c x \right )}}{10} - \frac{b c^{5} d^{3} x^{9} \sqrt{- c^{2} x^{2} + 1}}{100} + \frac{3 b c^{4} d^{3} x^{8} \operatorname{asin}{\left (c x \right )}}{8} + \frac{57 b c^{3} d^{3} x^{7} \sqrt{- c^{2} x^{2} + 1}}{1600} - \frac{b c^{2} d^{3} x^{6} \operatorname{asin}{\left (c x \right )}}{2} - \frac{401 b c d^{3} x^{5} \sqrt{- c^{2} x^{2} + 1}}{9600} + \frac{b d^{3} x^{4} \operatorname{asin}{\left (c x \right )}}{4} + \frac{79 b d^{3} x^{3} \sqrt{- c^{2} x^{2} + 1}}{7680 c} + \frac{79 b d^{3} x \sqrt{- c^{2} x^{2} + 1}}{5120 c^{3}} - \frac{79 b d^{3} \operatorname{asin}{\left (c x \right )}}{5120 c^{4}} & \text{for}\: c \neq 0 \\\frac{a d^{3} 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)**3*(a+b*asin(c*x)),x)

[Out]

Piecewise((-a*c**6*d**3*x**10/10 + 3*a*c**4*d**3*x**8/8 - a*c**2*d**3*x**6/2 + a*d**3*x**4/4 - b*c**6*d**3*x**
10*asin(c*x)/10 - b*c**5*d**3*x**9*sqrt(-c**2*x**2 + 1)/100 + 3*b*c**4*d**3*x**8*asin(c*x)/8 + 57*b*c**3*d**3*
x**7*sqrt(-c**2*x**2 + 1)/1600 - b*c**2*d**3*x**6*asin(c*x)/2 - 401*b*c*d**3*x**5*sqrt(-c**2*x**2 + 1)/9600 +
b*d**3*x**4*asin(c*x)/4 + 79*b*d**3*x**3*sqrt(-c**2*x**2 + 1)/(7680*c) + 79*b*d**3*x*sqrt(-c**2*x**2 + 1)/(512
0*c**3) - 79*b*d**3*asin(c*x)/(5120*c**4), Ne(c, 0)), (a*d**3*x**4/4, True))

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Giac [A]  time = 1.26654, size = 331, normalized size = 1.61 \begin{align*} -\frac{{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt{-c^{2} x^{2} + 1} b d^{3} x}{100 \, c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{5} b d^{3} \arcsin \left (c x\right )}{10 \, c^{4}} - \frac{7 \,{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b d^{3} x}{1600 \, c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{5} a d^{3}}{10 \, c^{4}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{4} b d^{3} \arcsin \left (c x\right )}{8 \, c^{4}} + \frac{49 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b d^{3} x}{9600 \, c^{3}} - \frac{{\left (c^{2} x^{2} - 1\right )}^{4} a d^{3}}{8 \, c^{4}} + \frac{49 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d^{3} x}{7680 \, c^{3}} + \frac{49 \, \sqrt{-c^{2} x^{2} + 1} b d^{3} x}{5120 \, c^{3}} + \frac{49 \, b d^{3} \arcsin \left (c x\right )}{5120 \, c^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/100*(c^2*x^2 - 1)^4*sqrt(-c^2*x^2 + 1)*b*d^3*x/c^3 - 1/10*(c^2*x^2 - 1)^5*b*d^3*arcsin(c*x)/c^4 - 7/1600*(c
^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*d^3*x/c^3 - 1/10*(c^2*x^2 - 1)^5*a*d^3/c^4 - 1/8*(c^2*x^2 - 1)^4*b*d^3*arcs
in(c*x)/c^4 + 49/9600*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^3*x/c^3 - 1/8*(c^2*x^2 - 1)^4*a*d^3/c^4 + 49/7680
*(-c^2*x^2 + 1)^(3/2)*b*d^3*x/c^3 + 49/5120*sqrt(-c^2*x^2 + 1)*b*d^3*x/c^3 + 49/5120*b*d^3*arcsin(c*x)/c^4