3.10 \(\int x^4 (d-c^2 d x^2)^2 (a+b \sin ^{-1}(c x)) \, dx\)

Optimal. Leaf size=186 \[ \frac{1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b d^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^5}-\frac{10 b d^2 \left (1-c^2 x^2\right )^{7/2}}{441 c^5}+\frac{b d^2 \left (1-c^2 x^2\right )^{5/2}}{525 c^5}+\frac{4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{945 c^5}+\frac{8 b d^2 \sqrt{1-c^2 x^2}}{315 c^5} \]

[Out]

(8*b*d^2*Sqrt[1 - c^2*x^2])/(315*c^5) + (4*b*d^2*(1 - c^2*x^2)^(3/2))/(945*c^5) + (b*d^2*(1 - c^2*x^2)^(5/2))/
(525*c^5) - (10*b*d^2*(1 - c^2*x^2)^(7/2))/(441*c^5) + (b*d^2*(1 - c^2*x^2)^(9/2))/(81*c^5) + (d^2*x^5*(a + b*
ArcSin[c*x]))/5 - (2*c^2*d^2*x^7*(a + b*ArcSin[c*x]))/7 + (c^4*d^2*x^9*(a + b*ArcSin[c*x]))/9

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Rubi [A]  time = 0.206582, antiderivative size = 186, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {270, 4687, 12, 1251, 897, 1153} \[ \frac{1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )+\frac{b d^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^5}-\frac{10 b d^2 \left (1-c^2 x^2\right )^{7/2}}{441 c^5}+\frac{b d^2 \left (1-c^2 x^2\right )^{5/2}}{525 c^5}+\frac{4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{945 c^5}+\frac{8 b d^2 \sqrt{1-c^2 x^2}}{315 c^5} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]),x]

[Out]

(8*b*d^2*Sqrt[1 - c^2*x^2])/(315*c^5) + (4*b*d^2*(1 - c^2*x^2)^(3/2))/(945*c^5) + (b*d^2*(1 - c^2*x^2)^(5/2))/
(525*c^5) - (10*b*d^2*(1 - c^2*x^2)^(7/2))/(441*c^5) + (b*d^2*(1 - c^2*x^2)^(9/2))/(81*c^5) + (d^2*x^5*(a + b*
ArcSin[c*x]))/5 - (2*c^2*d^2*x^7*(a + b*ArcSin[c*x]))/7 + (c^4*d^2*x^9*(a + b*ArcSin[c*x]))/9

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 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 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rule 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1153

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 -
b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]

Rubi steps

\begin{align*} \int x^4 \left (d-c^2 d x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=\frac{1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{d^2 x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{315 \sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{315} \left (b c d^2\right ) \int \frac{x^5 \left (63-90 c^2 x^2+35 c^4 x^4\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=\frac{1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{630} \left (b c d^2\right ) \operatorname{Subst}\left (\int \frac{x^2 \left (63-90 c^2 x+35 c^4 x^2\right )}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=\frac{1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{x^2}{c^2}\right )^2 \left (8+20 x^2+35 x^4\right ) \, dx,x,\sqrt{1-c^2 x^2}\right )}{315 c}\\ &=\frac{1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )+\frac{\left (b d^2\right ) \operatorname{Subst}\left (\int \left (\frac{8}{c^4}+\frac{4 x^2}{c^4}+\frac{3 x^4}{c^4}-\frac{50 x^6}{c^4}+\frac{35 x^8}{c^4}\right ) \, dx,x,\sqrt{1-c^2 x^2}\right )}{315 c}\\ &=\frac{8 b d^2 \sqrt{1-c^2 x^2}}{315 c^5}+\frac{4 b d^2 \left (1-c^2 x^2\right )^{3/2}}{945 c^5}+\frac{b d^2 \left (1-c^2 x^2\right )^{5/2}}{525 c^5}-\frac{10 b d^2 \left (1-c^2 x^2\right )^{7/2}}{441 c^5}+\frac{b d^2 \left (1-c^2 x^2\right )^{9/2}}{81 c^5}+\frac{1}{5} d^2 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{2}{7} c^2 d^2 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{1}{9} c^4 d^2 x^9 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.107498, size = 119, normalized size = 0.64 \[ \frac{d^2 \left (315 a c^5 x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right )+b \sqrt{1-c^2 x^2} \left (1225 c^8 x^8-2650 c^6 x^6+789 c^4 x^4+1052 c^2 x^2+2104\right )+315 b c^5 x^5 \left (35 c^4 x^4-90 c^2 x^2+63\right ) \sin ^{-1}(c x)\right )}{99225 c^5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d - c^2*d*x^2)^2*(a + b*ArcSin[c*x]),x]

[Out]

(d^2*(315*a*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4) + b*Sqrt[1 - c^2*x^2]*(2104 + 1052*c^2*x^2 + 789*c^4*x^4 -
2650*c^6*x^6 + 1225*c^8*x^8) + 315*b*c^5*x^5*(63 - 90*c^2*x^2 + 35*c^4*x^4)*ArcSin[c*x]))/(99225*c^5)

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Maple [A]  time = 0.01, size = 172, normalized size = 0.9 \begin{align*}{\frac{1}{{c}^{5}} \left ({d}^{2}a \left ({\frac{{c}^{9}{x}^{9}}{9}}-{\frac{2\,{c}^{7}{x}^{7}}{7}}+{\frac{{c}^{5}{x}^{5}}{5}} \right ) +{d}^{2}b \left ({\frac{\arcsin \left ( cx \right ){c}^{9}{x}^{9}}{9}}-{\frac{2\,\arcsin \left ( cx \right ){c}^{7}{x}^{7}}{7}}+{\frac{\arcsin \left ( cx \right ){c}^{5}{x}^{5}}{5}}+{\frac{{c}^{8}{x}^{8}}{81}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{106\,{c}^{6}{x}^{6}}{3969}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{263\,{c}^{4}{x}^{4}}{33075}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{1052\,{c}^{2}{x}^{2}}{99225}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{2104}{99225}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(-c^2*d*x^2+d)^2*(a+b*arcsin(c*x)),x)

[Out]

1/c^5*(d^2*a*(1/9*c^9*x^9-2/7*c^7*x^7+1/5*c^5*x^5)+d^2*b*(1/9*arcsin(c*x)*c^9*x^9-2/7*arcsin(c*x)*c^7*x^7+1/5*
arcsin(c*x)*c^5*x^5+1/81*c^8*x^8*(-c^2*x^2+1)^(1/2)-106/3969*c^6*x^6*(-c^2*x^2+1)^(1/2)+263/33075*c^4*x^4*(-c^
2*x^2+1)^(1/2)+1052/99225*c^2*x^2*(-c^2*x^2+1)^(1/2)+2104/99225*(-c^2*x^2+1)^(1/2)))

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Maxima [B]  time = 1.57597, size = 443, normalized size = 2.38 \begin{align*} \frac{1}{9} \, a c^{4} d^{2} x^{9} - \frac{2}{7} \, a c^{2} d^{2} x^{7} + \frac{1}{2835} \,{\left (315 \, x^{9} \arcsin \left (c x\right ) +{\left (\frac{35 \, \sqrt{-c^{2} x^{2} + 1} x^{8}}{c^{2}} + \frac{40 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac{48 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{6}} + \frac{64 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac{128 \, \sqrt{-c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{4} d^{2} + \frac{1}{5} \, a d^{2} x^{5} - \frac{2}{245} \,{\left (35 \, x^{7} \arcsin \left (c x\right ) +{\left (\frac{5 \, \sqrt{-c^{2} x^{2} + 1} x^{6}}{c^{2}} + \frac{6 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{6}} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d^{2} + \frac{1}{75} \,{\left (15 \, x^{5} \arcsin \left (c x\right ) +{\left (\frac{3 \, \sqrt{-c^{2} x^{2} + 1} x^{4}}{c^{2}} + \frac{4 \, \sqrt{-c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^2*(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

1/9*a*c^4*d^2*x^9 - 2/7*a*c^2*d^2*x^7 + 1/2835*(315*x^9*arcsin(c*x) + (35*sqrt(-c^2*x^2 + 1)*x^8/c^2 + 40*sqrt
(-c^2*x^2 + 1)*x^6/c^4 + 48*sqrt(-c^2*x^2 + 1)*x^4/c^6 + 64*sqrt(-c^2*x^2 + 1)*x^2/c^8 + 128*sqrt(-c^2*x^2 + 1
)/c^10)*c)*b*c^4*d^2 + 1/5*a*d^2*x^5 - 2/245*(35*x^7*arcsin(c*x) + (5*sqrt(-c^2*x^2 + 1)*x^6/c^2 + 6*sqrt(-c^2
*x^2 + 1)*x^4/c^4 + 8*sqrt(-c^2*x^2 + 1)*x^2/c^6 + 16*sqrt(-c^2*x^2 + 1)/c^8)*c)*b*c^2*d^2 + 1/75*(15*x^5*arcs
in(c*x) + (3*sqrt(-c^2*x^2 + 1)*x^4/c^2 + 4*sqrt(-c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(-c^2*x^2 + 1)/c^6)*c)*b*d^2

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Fricas [A]  time = 2.56715, size = 366, normalized size = 1.97 \begin{align*} \frac{11025 \, a c^{9} d^{2} x^{9} - 28350 \, a c^{7} d^{2} x^{7} + 19845 \, a c^{5} d^{2} x^{5} + 315 \,{\left (35 \, b c^{9} d^{2} x^{9} - 90 \, b c^{7} d^{2} x^{7} + 63 \, b c^{5} d^{2} x^{5}\right )} \arcsin \left (c x\right ) +{\left (1225 \, b c^{8} d^{2} x^{8} - 2650 \, b c^{6} d^{2} x^{6} + 789 \, b c^{4} d^{2} x^{4} + 1052 \, b c^{2} d^{2} x^{2} + 2104 \, b d^{2}\right )} \sqrt{-c^{2} x^{2} + 1}}{99225 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^2*(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

1/99225*(11025*a*c^9*d^2*x^9 - 28350*a*c^7*d^2*x^7 + 19845*a*c^5*d^2*x^5 + 315*(35*b*c^9*d^2*x^9 - 90*b*c^7*d^
2*x^7 + 63*b*c^5*d^2*x^5)*arcsin(c*x) + (1225*b*c^8*d^2*x^8 - 2650*b*c^6*d^2*x^6 + 789*b*c^4*d^2*x^4 + 1052*b*
c^2*d^2*x^2 + 2104*b*d^2)*sqrt(-c^2*x^2 + 1))/c^5

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Sympy [A]  time = 34.0927, size = 230, normalized size = 1.24 \begin{align*} \begin{cases} \frac{a c^{4} d^{2} x^{9}}{9} - \frac{2 a c^{2} d^{2} x^{7}}{7} + \frac{a d^{2} x^{5}}{5} + \frac{b c^{4} d^{2} x^{9} \operatorname{asin}{\left (c x \right )}}{9} + \frac{b c^{3} d^{2} x^{8} \sqrt{- c^{2} x^{2} + 1}}{81} - \frac{2 b c^{2} d^{2} x^{7} \operatorname{asin}{\left (c x \right )}}{7} - \frac{106 b c d^{2} x^{6} \sqrt{- c^{2} x^{2} + 1}}{3969} + \frac{b d^{2} x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{263 b d^{2} x^{4} \sqrt{- c^{2} x^{2} + 1}}{33075 c} + \frac{1052 b d^{2} x^{2} \sqrt{- c^{2} x^{2} + 1}}{99225 c^{3}} + \frac{2104 b d^{2} \sqrt{- c^{2} x^{2} + 1}}{99225 c^{5}} & \text{for}\: c \neq 0 \\\frac{a d^{2} x^{5}}{5} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(-c**2*d*x**2+d)**2*(a+b*asin(c*x)),x)

[Out]

Piecewise((a*c**4*d**2*x**9/9 - 2*a*c**2*d**2*x**7/7 + a*d**2*x**5/5 + b*c**4*d**2*x**9*asin(c*x)/9 + b*c**3*d
**2*x**8*sqrt(-c**2*x**2 + 1)/81 - 2*b*c**2*d**2*x**7*asin(c*x)/7 - 106*b*c*d**2*x**6*sqrt(-c**2*x**2 + 1)/396
9 + b*d**2*x**5*asin(c*x)/5 + 263*b*d**2*x**4*sqrt(-c**2*x**2 + 1)/(33075*c) + 1052*b*d**2*x**2*sqrt(-c**2*x**
2 + 1)/(99225*c**3) + 2104*b*d**2*sqrt(-c**2*x**2 + 1)/(99225*c**5), Ne(c, 0)), (a*d**2*x**5/5, True))

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Giac [A]  time = 1.30353, size = 383, normalized size = 2.06 \begin{align*} \frac{1}{9} \, a c^{4} d^{2} x^{9} - \frac{2}{7} \, a c^{2} d^{2} x^{7} + \frac{1}{5} \, a d^{2} x^{5} + \frac{{\left (c^{2} x^{2} - 1\right )}^{4} b d^{2} x \arcsin \left (c x\right )}{9 \, c^{4}} + \frac{10 \,{\left (c^{2} x^{2} - 1\right )}^{3} b d^{2} x \arcsin \left (c x\right )}{63 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} b d^{2} x \arcsin \left (c x\right )}{105 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{4} \sqrt{-c^{2} x^{2} + 1} b d^{2}}{81 \, c^{5}} - \frac{4 \,{\left (c^{2} x^{2} - 1\right )} b d^{2} x \arcsin \left (c x\right )}{315 \, c^{4}} + \frac{10 \,{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b d^{2}}{441 \, c^{5}} + \frac{8 \, b d^{2} x \arcsin \left (c x\right )}{315 \, c^{4}} + \frac{{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b d^{2}}{525 \, c^{5}} + \frac{4 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d^{2}}{945 \, c^{5}} + \frac{8 \, \sqrt{-c^{2} x^{2} + 1} b d^{2}}{315 \, c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(-c^2*d*x^2+d)^2*(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

1/9*a*c^4*d^2*x^9 - 2/7*a*c^2*d^2*x^7 + 1/5*a*d^2*x^5 + 1/9*(c^2*x^2 - 1)^4*b*d^2*x*arcsin(c*x)/c^4 + 10/63*(c
^2*x^2 - 1)^3*b*d^2*x*arcsin(c*x)/c^4 + 1/105*(c^2*x^2 - 1)^2*b*d^2*x*arcsin(c*x)/c^4 + 1/81*(c^2*x^2 - 1)^4*s
qrt(-c^2*x^2 + 1)*b*d^2/c^5 - 4/315*(c^2*x^2 - 1)*b*d^2*x*arcsin(c*x)/c^4 + 10/441*(c^2*x^2 - 1)^3*sqrt(-c^2*x
^2 + 1)*b*d^2/c^5 + 8/315*b*d^2*x*arcsin(c*x)/c^4 + 1/525*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^2/c^5 + 4/945
*(-c^2*x^2 + 1)^(3/2)*b*d^2/c^5 + 8/315*sqrt(-c^2*x^2 + 1)*b*d^2/c^5