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

Optimal. Leaf size=175 \[ -\frac{1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 x \left (a+b \sin ^{-1}(c x)\right )+\frac{b d^3 \left (1-c^2 x^2\right )^{7/2}}{49 c}+\frac{6 b d^3 \left (1-c^2 x^2\right )^{5/2}}{175 c}+\frac{8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{105 c}+\frac{16 b d^3 \sqrt{1-c^2 x^2}}{35 c} \]

[Out]

(16*b*d^3*Sqrt[1 - c^2*x^2])/(35*c) + (8*b*d^3*(1 - c^2*x^2)^(3/2))/(105*c) + (6*b*d^3*(1 - c^2*x^2)^(5/2))/(1
75*c) + (b*d^3*(1 - c^2*x^2)^(7/2))/(49*c) + d^3*x*(a + b*ArcSin[c*x]) - c^2*d^3*x^3*(a + b*ArcSin[c*x]) + (3*
c^4*d^3*x^5*(a + b*ArcSin[c*x]))/5 - (c^6*d^3*x^7*(a + b*ArcSin[c*x]))/7

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Rubi [A]  time = 0.171282, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {194, 4645, 12, 1799, 1850} \[ -\frac{1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+d^3 x \left (a+b \sin ^{-1}(c x)\right )+\frac{b d^3 \left (1-c^2 x^2\right )^{7/2}}{49 c}+\frac{6 b d^3 \left (1-c^2 x^2\right )^{5/2}}{175 c}+\frac{8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{105 c}+\frac{16 b d^3 \sqrt{1-c^2 x^2}}{35 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16*b*d^3*Sqrt[1 - c^2*x^2])/(35*c) + (8*b*d^3*(1 - c^2*x^2)^(3/2))/(105*c) + (6*b*d^3*(1 - c^2*x^2)^(5/2))/(1
75*c) + (b*d^3*(1 - c^2*x^2)^(7/2))/(49*c) + d^3*x*(a + b*ArcSin[c*x]) - c^2*d^3*x^3*(a + b*ArcSin[c*x]) + (3*
c^4*d^3*x^5*(a + b*ArcSin[c*x]))/5 - (c^6*d^3*x^7*(a + b*ArcSin[c*x]))/7

Rule 194

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 4645

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(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]] /; F
reeQ[{a, b, c, d, e}, 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 1799

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[x^((m - 1)/2)*SubstFor[x^2,
Pq, x]*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x^2] && IntegerQ[(m - 1)/2]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rubi steps

\begin{align*} \int \left (d-c^2 d x^2\right )^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac{d^3 x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{35 \sqrt{1-c^2 x^2}} \, dx\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{35} \left (b c d^3\right ) \int \frac{x \left (35-35 c^2 x^2+21 c^4 x^4-5 c^6 x^6\right )}{\sqrt{1-c^2 x^2}} \, dx\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{70} \left (b c d^3\right ) \operatorname{Subst}\left (\int \frac{35-35 c^2 x+21 c^4 x^2-5 c^6 x^3}{\sqrt{1-c^2 x}} \, dx,x,x^2\right )\\ &=d^3 x \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{70} \left (b c d^3\right ) \operatorname{Subst}\left (\int \left (\frac{16}{\sqrt{1-c^2 x}}+8 \sqrt{1-c^2 x}+6 \left (1-c^2 x\right )^{3/2}+5 \left (1-c^2 x\right )^{5/2}\right ) \, dx,x,x^2\right )\\ &=\frac{16 b d^3 \sqrt{1-c^2 x^2}}{35 c}+\frac{8 b d^3 \left (1-c^2 x^2\right )^{3/2}}{105 c}+\frac{6 b d^3 \left (1-c^2 x^2\right )^{5/2}}{175 c}+\frac{b d^3 \left (1-c^2 x^2\right )^{7/2}}{49 c}+d^3 x \left (a+b \sin ^{-1}(c x)\right )-c^2 d^3 x^3 \left (a+b \sin ^{-1}(c x)\right )+\frac{3}{5} c^4 d^3 x^5 \left (a+b \sin ^{-1}(c x)\right )-\frac{1}{7} c^6 d^3 x^7 \left (a+b \sin ^{-1}(c x)\right )\\ \end{align*}

Mathematica [A]  time = 0.205248, size = 119, normalized size = 0.68 \[ -\frac{d^3 \left (105 a c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right )+b \sqrt{1-c^2 x^2} \left (75 c^6 x^6-351 c^4 x^4+757 c^2 x^2-2161\right )+105 b c x \left (5 c^6 x^6-21 c^4 x^4+35 c^2 x^2-35\right ) \sin ^{-1}(c x)\right )}{3675 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^3*(105*a*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6) + b*Sqrt[1 - c^2*x^2]*(-2161 + 757*c^2*x^2 - 351*
c^4*x^4 + 75*c^6*x^6) + 105*b*c*x*(-35 + 35*c^2*x^2 - 21*c^4*x^4 + 5*c^6*x^6)*ArcSin[c*x]))/(3675*c)

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Maple [A]  time = 0.004, size = 164, normalized size = 0.9 \begin{align*}{\frac{1}{c} \left ( -{d}^{3}a \left ({\frac{{c}^{7}{x}^{7}}{7}}-{\frac{3\,{c}^{5}{x}^{5}}{5}}+{c}^{3}{x}^{3}-cx \right ) -{d}^{3}b \left ({\frac{\arcsin \left ( cx \right ){c}^{7}{x}^{7}}{7}}-{\frac{3\,\arcsin \left ( cx \right ){c}^{5}{x}^{5}}{5}}+{c}^{3}{x}^{3}\arcsin \left ( cx \right ) -cx\arcsin \left ( cx \right ) +{\frac{{c}^{6}{x}^{6}}{49}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{117\,{c}^{4}{x}^{4}}{1225}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{757\,{c}^{2}{x}^{2}}{3675}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{2161}{3675}\sqrt{-{c}^{2}{x}^{2}+1}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c*(-d^3*a*(1/7*c^7*x^7-3/5*c^5*x^5+c^3*x^3-c*x)-d^3*b*(1/7*arcsin(c*x)*c^7*x^7-3/5*arcsin(c*x)*c^5*x^5+c^3*x
^3*arcsin(c*x)-c*x*arcsin(c*x)+1/49*c^6*x^6*(-c^2*x^2+1)^(1/2)-117/1225*c^4*x^4*(-c^2*x^2+1)^(1/2)+757/3675*c^
2*x^2*(-c^2*x^2+1)^(1/2)-2161/3675*(-c^2*x^2+1)^(1/2)))

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Maxima [A]  time = 1.63586, size = 414, normalized size = 2.37 \begin{align*} -\frac{1}{7} \, a c^{6} d^{3} x^{7} + \frac{3}{5} \, a c^{4} d^{3} x^{5} - \frac{1}{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^{6} d^{3} + \frac{1}{25} \,{\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 c^{4} d^{3} - a c^{2} d^{3} x^{3} - \frac{1}{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 )} b c^{2} d^{3} + a d^{3} x + \frac{{\left (c x \arcsin \left (c x\right ) + \sqrt{-c^{2} x^{2} + 1}\right )} b d^{3}}{c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 - 1/245*(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^6*d^3 + 1/25*(15*x^5*ar
csin(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*c^4*
d^3 - a*c^2*d^3*x^3 - 1/3*(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))*b*c^
2*d^3 + a*d^3*x + (c*x*arcsin(c*x) + sqrt(-c^2*x^2 + 1))*b*d^3/c

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Fricas [A]  time = 2.19863, size = 367, normalized size = 2.1 \begin{align*} -\frac{525 \, a c^{7} d^{3} x^{7} - 2205 \, a c^{5} d^{3} x^{5} + 3675 \, a c^{3} d^{3} x^{3} - 3675 \, a c d^{3} x + 105 \,{\left (5 \, b c^{7} d^{3} x^{7} - 21 \, b c^{5} d^{3} x^{5} + 35 \, b c^{3} d^{3} x^{3} - 35 \, b c d^{3} x\right )} \arcsin \left (c x\right ) +{\left (75 \, b c^{6} d^{3} x^{6} - 351 \, b c^{4} d^{3} x^{4} + 757 \, b c^{2} d^{3} x^{2} - 2161 \, b d^{3}\right )} \sqrt{-c^{2} x^{2} + 1}}{3675 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3675*(525*a*c^7*d^3*x^7 - 2205*a*c^5*d^3*x^5 + 3675*a*c^3*d^3*x^3 - 3675*a*c*d^3*x + 105*(5*b*c^7*d^3*x^7 -
21*b*c^5*d^3*x^5 + 35*b*c^3*d^3*x^3 - 35*b*c*d^3*x)*arcsin(c*x) + (75*b*c^6*d^3*x^6 - 351*b*c^4*d^3*x^4 + 757
*b*c^2*d^3*x^2 - 2161*b*d^3)*sqrt(-c^2*x^2 + 1))/c

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Sympy [A]  time = 20.5782, size = 221, normalized size = 1.26 \begin{align*} \begin{cases} - \frac{a c^{6} d^{3} x^{7}}{7} + \frac{3 a c^{4} d^{3} x^{5}}{5} - a c^{2} d^{3} x^{3} + a d^{3} x - \frac{b c^{6} d^{3} x^{7} \operatorname{asin}{\left (c x \right )}}{7} - \frac{b c^{5} d^{3} x^{6} \sqrt{- c^{2} x^{2} + 1}}{49} + \frac{3 b c^{4} d^{3} x^{5} \operatorname{asin}{\left (c x \right )}}{5} + \frac{117 b c^{3} d^{3} x^{4} \sqrt{- c^{2} x^{2} + 1}}{1225} - b c^{2} d^{3} x^{3} \operatorname{asin}{\left (c x \right )} - \frac{757 b c d^{3} x^{2} \sqrt{- c^{2} x^{2} + 1}}{3675} + b d^{3} x \operatorname{asin}{\left (c x \right )} + \frac{2161 b d^{3} \sqrt{- c^{2} x^{2} + 1}}{3675 c} & \text{for}\: c \neq 0 \\a d^{3} x & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*c**6*d**3*x**7/7 + 3*a*c**4*d**3*x**5/5 - a*c**2*d**3*x**3 + a*d**3*x - b*c**6*d**3*x**7*asin(c*
x)/7 - b*c**5*d**3*x**6*sqrt(-c**2*x**2 + 1)/49 + 3*b*c**4*d**3*x**5*asin(c*x)/5 + 117*b*c**3*d**3*x**4*sqrt(-
c**2*x**2 + 1)/1225 - b*c**2*d**3*x**3*asin(c*x) - 757*b*c*d**3*x**2*sqrt(-c**2*x**2 + 1)/3675 + b*d**3*x*asin
(c*x) + 2161*b*d**3*sqrt(-c**2*x**2 + 1)/(3675*c), Ne(c, 0)), (a*d**3*x, True))

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Giac [A]  time = 1.235, size = 302, normalized size = 1.73 \begin{align*} -\frac{1}{7} \, a c^{6} d^{3} x^{7} + \frac{3}{5} \, a c^{4} d^{3} x^{5} - a c^{2} d^{3} x^{3} - \frac{1}{7} \,{\left (c^{2} x^{2} - 1\right )}^{3} b d^{3} x \arcsin \left (c x\right ) + \frac{6}{35} \,{\left (c^{2} x^{2} - 1\right )}^{2} b d^{3} x \arcsin \left (c x\right ) - \frac{8}{35} \,{\left (c^{2} x^{2} - 1\right )} b d^{3} x \arcsin \left (c x\right ) - \frac{{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b d^{3}}{49 \, c} + \frac{16}{35} \, b d^{3} x \arcsin \left (c x\right ) + \frac{6 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b d^{3}}{175 \, c} + a d^{3} x + \frac{8 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d^{3}}{105 \, c} + \frac{16 \, \sqrt{-c^{2} x^{2} + 1} b d^{3}}{35 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/7*a*c^6*d^3*x^7 + 3/5*a*c^4*d^3*x^5 - a*c^2*d^3*x^3 - 1/7*(c^2*x^2 - 1)^3*b*d^3*x*arcsin(c*x) + 6/35*(c^2*x
^2 - 1)^2*b*d^3*x*arcsin(c*x) - 8/35*(c^2*x^2 - 1)*b*d^3*x*arcsin(c*x) - 1/49*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 +
1)*b*d^3/c + 16/35*b*d^3*x*arcsin(c*x) + 6/175*(c^2*x^2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^3/c + a*d^3*x + 8/105*(-
c^2*x^2 + 1)^(3/2)*b*d^3/c + 16/35*sqrt(-c^2*x^2 + 1)*b*d^3/c