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

Optimal. Leaf size=150 \[ -\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac{35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac{35 b d^3 x \sqrt{1-c^2 x^2}}{1024 c}+\frac{35 b d^3 \sin ^{-1}(c x)}{1024 c^2} \]

[Out]

(35*b*d^3*x*Sqrt[1 - c^2*x^2])/(1024*c) + (35*b*d^3*x*(1 - c^2*x^2)^(3/2))/(1536*c) + (7*b*d^3*x*(1 - c^2*x^2)
^(5/2))/(384*c) + (b*d^3*x*(1 - c^2*x^2)^(7/2))/(64*c) + (35*b*d^3*ArcSin[c*x])/(1024*c^2) - (d^3*(1 - c^2*x^2
)^4*(a + b*ArcSin[c*x]))/(8*c^2)

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Rubi [A]  time = 0.0762765, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {4677, 195, 216} \[ -\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac{35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac{35 b d^3 x \sqrt{1-c^2 x^2}}{1024 c}+\frac{35 b d^3 \sin ^{-1}(c x)}{1024 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(35*b*d^3*x*Sqrt[1 - c^2*x^2])/(1024*c) + (35*b*d^3*x*(1 - c^2*x^2)^(3/2))/(1536*c) + (7*b*d^3*x*(1 - c^2*x^2)
^(5/2))/(384*c) + (b*d^3*x*(1 - c^2*x^2)^(7/2))/(64*c) + (35*b*d^3*ArcSin[c*x])/(1024*c^2) - (d^3*(1 - c^2*x^2
)^4*(a + b*ArcSin[c*x]))/(8*c^2)

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 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 \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^2}+\frac{\left (b d^3\right ) \int \left (1-c^2 x^2\right )^{7/2} \, dx}{8 c}\\ &=\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{\left (7 b d^3\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{64 c}\\ &=\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{\left (35 b d^3\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{384 c}\\ &=\frac{35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{\left (35 b d^3\right ) \int \sqrt{1-c^2 x^2} \, dx}{512 c}\\ &=\frac{35 b d^3 x \sqrt{1-c^2 x^2}}{1024 c}+\frac{35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}+\frac{\left (35 b d^3\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx}{1024 c}\\ &=\frac{35 b d^3 x \sqrt{1-c^2 x^2}}{1024 c}+\frac{35 b d^3 x \left (1-c^2 x^2\right )^{3/2}}{1536 c}+\frac{7 b d^3 x \left (1-c^2 x^2\right )^{5/2}}{384 c}+\frac{b d^3 x \left (1-c^2 x^2\right )^{7/2}}{64 c}+\frac{35 b d^3 \sin ^{-1}(c x)}{1024 c^2}-\frac{d^3 \left (1-c^2 x^2\right )^4 \left (a+b \sin ^{-1}(c x)\right )}{8 c^2}\\ \end{align*}

Mathematica [A]  time = 0.0807438, size = 110, normalized size = 0.73 \[ -\frac{d^3 \left (384 a \left (c^2 x^2-1\right )^4+b c x \sqrt{1-c^2 x^2} \left (48 c^6 x^6-200 c^4 x^4+326 c^2 x^2-279\right )+3 b \left (128 c^8 x^8-512 c^6 x^6+768 c^4 x^4-512 c^2 x^2+93\right ) \sin ^{-1}(c x)\right )}{3072 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^3*(384*a*(-1 + c^2*x^2)^4 + b*c*x*Sqrt[1 - c^2*x^2]*(-279 + 326*c^2*x^2 - 200*c^4*x^4 + 48*c^6*x^6) + 3*b*
(93 - 512*c^2*x^2 + 768*c^4*x^4 - 512*c^6*x^6 + 128*c^8*x^8)*ArcSin[c*x]))/(3072*c^2)

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Maple [A]  time = 0.004, size = 182, normalized size = 1.2 \begin{align*}{\frac{1}{{c}^{2}} \left ( -{d}^{3}a \left ({\frac{{c}^{8}{x}^{8}}{8}}-{\frac{{c}^{6}{x}^{6}}{2}}+{\frac{3\,{c}^{4}{x}^{4}}{4}}-{\frac{{c}^{2}{x}^{2}}{2}} \right ) -{d}^{3}b \left ({\frac{\arcsin \left ( cx \right ){c}^{8}{x}^{8}}{8}}-{\frac{\arcsin \left ( cx \right ){c}^{6}{x}^{6}}{2}}+{\frac{3\,{c}^{4}{x}^{4}\arcsin \left ( cx \right ) }{4}}-{\frac{{c}^{2}{x}^{2}\arcsin \left ( cx \right ) }{2}}+{\frac{{c}^{7}{x}^{7}}{64}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{25\,{c}^{5}{x}^{5}}{384}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{163\,{c}^{3}{x}^{3}}{1536}\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{93\,cx}{1024}\sqrt{-{c}^{2}{x}^{2}+1}}+{\frac{93\,\arcsin \left ( cx \right ) }{1024}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/c^2*(-d^3*a*(1/8*c^8*x^8-1/2*c^6*x^6+3/4*c^4*x^4-1/2*c^2*x^2)-d^3*b*(1/8*arcsin(c*x)*c^8*x^8-1/2*arcsin(c*x)
*c^6*x^6+3/4*c^4*x^4*arcsin(c*x)-1/2*c^2*x^2*arcsin(c*x)+1/64*c^7*x^7*(-c^2*x^2+1)^(1/2)-25/384*c^5*x^5*(-c^2*
x^2+1)^(1/2)+163/1536*c^3*x^3*(-c^2*x^2+1)^(1/2)-93/1024*c*x*(-c^2*x^2+1)^(1/2)+93/1024*arcsin(c*x)))

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Maxima [B]  time = 1.77141, size = 548, normalized size = 3.65 \begin{align*} -\frac{1}{8} \, a c^{6} d^{3} x^{8} + \frac{1}{2} \, a c^{4} d^{3} x^{6} - \frac{1}{3072} \,{\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^{6} d^{3} - \frac{3}{4} \, a c^{2} 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^{4} d^{3} - \frac{3}{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 c^{2} d^{3} + \frac{1}{2} \, a d^{3} x^{2} + \frac{1}{4} \,{\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 )} b d^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*a*c^6*d^3*x^8 + 1/2*a*c^4*d^3*x^6 - 1/3072*(384*x^8*arcsin(c*x) + (48*sqrt(-c^2*x^2 + 1)*x^7/c^2 + 56*sqr
t(-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^6*d^3 - 3/4*a*c^2*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^4*d^3 - 3/32*(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*ar
csin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^4))*c)*b*c^2*d^3 + 1/2*a*d^3*x^2 + 1/4*(2*x^2*arcsin(c*x) + c*(sqrt(-c^2*x^
2 + 1)*x/c^2 - arcsin(c^2*x/sqrt(c^2))/(sqrt(c^2)*c^2)))*b*d^3

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Fricas [A]  time = 2.12246, size = 404, normalized size = 2.69 \begin{align*} -\frac{384 \, a c^{8} d^{3} x^{8} - 1536 \, a c^{6} d^{3} x^{6} + 2304 \, a c^{4} d^{3} x^{4} - 1536 \, a c^{2} d^{3} x^{2} + 3 \,{\left (128 \, b c^{8} d^{3} x^{8} - 512 \, b c^{6} d^{3} x^{6} + 768 \, b c^{4} d^{3} x^{4} - 512 \, b c^{2} d^{3} x^{2} + 93 \, b d^{3}\right )} \arcsin \left (c x\right ) +{\left (48 \, b c^{7} d^{3} x^{7} - 200 \, b c^{5} d^{3} x^{5} + 326 \, b c^{3} d^{3} x^{3} - 279 \, b c d^{3} x\right )} \sqrt{-c^{2} x^{2} + 1}}{3072 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3072*(384*a*c^8*d^3*x^8 - 1536*a*c^6*d^3*x^6 + 2304*a*c^4*d^3*x^4 - 1536*a*c^2*d^3*x^2 + 3*(128*b*c^8*d^3*x
^8 - 512*b*c^6*d^3*x^6 + 768*b*c^4*d^3*x^4 - 512*b*c^2*d^3*x^2 + 93*b*d^3)*arcsin(c*x) + (48*b*c^7*d^3*x^7 - 2
00*b*c^5*d^3*x^5 + 326*b*c^3*d^3*x^3 - 279*b*c*d^3*x)*sqrt(-c^2*x^2 + 1))/c^2

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Sympy [A]  time = 17.4734, size = 253, normalized size = 1.69 \begin{align*} \begin{cases} - \frac{a c^{6} d^{3} x^{8}}{8} + \frac{a c^{4} d^{3} x^{6}}{2} - \frac{3 a c^{2} d^{3} x^{4}}{4} + \frac{a d^{3} x^{2}}{2} - \frac{b c^{6} d^{3} x^{8} \operatorname{asin}{\left (c x \right )}}{8} - \frac{b c^{5} d^{3} x^{7} \sqrt{- c^{2} x^{2} + 1}}{64} + \frac{b c^{4} d^{3} x^{6} \operatorname{asin}{\left (c x \right )}}{2} + \frac{25 b c^{3} d^{3} x^{5} \sqrt{- c^{2} x^{2} + 1}}{384} - \frac{3 b c^{2} d^{3} x^{4} \operatorname{asin}{\left (c x \right )}}{4} - \frac{163 b c d^{3} x^{3} \sqrt{- c^{2} x^{2} + 1}}{1536} + \frac{b d^{3} x^{2} \operatorname{asin}{\left (c x \right )}}{2} + \frac{93 b d^{3} x \sqrt{- c^{2} x^{2} + 1}}{1024 c} - \frac{93 b d^{3} \operatorname{asin}{\left (c x \right )}}{1024 c^{2}} & \text{for}\: c \neq 0 \\\frac{a d^{3} x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*c**6*d**3*x**8/8 + a*c**4*d**3*x**6/2 - 3*a*c**2*d**3*x**4/4 + a*d**3*x**2/2 - b*c**6*d**3*x**8*
asin(c*x)/8 - b*c**5*d**3*x**7*sqrt(-c**2*x**2 + 1)/64 + b*c**4*d**3*x**6*asin(c*x)/2 + 25*b*c**3*d**3*x**5*sq
rt(-c**2*x**2 + 1)/384 - 3*b*c**2*d**3*x**4*asin(c*x)/4 - 163*b*c*d**3*x**3*sqrt(-c**2*x**2 + 1)/1536 + b*d**3
*x**2*asin(c*x)/2 + 93*b*d**3*x*sqrt(-c**2*x**2 + 1)/(1024*c) - 93*b*d**3*asin(c*x)/(1024*c**2), Ne(c, 0)), (a
*d**3*x**2/2, True))

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Giac [A]  time = 1.24997, size = 227, normalized size = 1.51 \begin{align*} -\frac{{\left (c^{2} x^{2} - 1\right )}^{3} \sqrt{-c^{2} x^{2} + 1} b d^{3} x}{64 \, c} - \frac{{\left (c^{2} x^{2} - 1\right )}^{4} b d^{3} \arcsin \left (c x\right )}{8 \, c^{2}} + \frac{7 \,{\left (c^{2} x^{2} - 1\right )}^{2} \sqrt{-c^{2} x^{2} + 1} b d^{3} x}{384 \, c} - \frac{{\left (c^{2} x^{2} - 1\right )}^{4} a d^{3}}{8 \, c^{2}} + \frac{35 \,{\left (-c^{2} x^{2} + 1\right )}^{\frac{3}{2}} b d^{3} x}{1536 \, c} + \frac{35 \, \sqrt{-c^{2} x^{2} + 1} b d^{3} x}{1024 \, c} + \frac{35 \, b d^{3} \arcsin \left (c x\right )}{1024 \, c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/64*(c^2*x^2 - 1)^3*sqrt(-c^2*x^2 + 1)*b*d^3*x/c - 1/8*(c^2*x^2 - 1)^4*b*d^3*arcsin(c*x)/c^2 + 7/384*(c^2*x^
2 - 1)^2*sqrt(-c^2*x^2 + 1)*b*d^3*x/c - 1/8*(c^2*x^2 - 1)^4*a*d^3/c^2 + 35/1536*(-c^2*x^2 + 1)^(3/2)*b*d^3*x/c
+ 35/1024*sqrt(-c^2*x^2 + 1)*b*d^3*x/c + 35/1024*b*d^3*arcsin(c*x)/c^2