3.1.4 \(\int x (d-c^2 d x^2) (a+b \text {ArcSin}(c x)) \, dx\) [4]

Optimal. Leaf size=90 \[ \frac {3 b d x \sqrt {1-c^2 x^2}}{32 c}+\frac {b d x \left (1-c^2 x^2\right )^{3/2}}{16 c}+\frac {3 b d \text {ArcSin}(c x)}{32 c^2}-\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{4 c^2} \]

[Out]

1/16*b*d*x*(-c^2*x^2+1)^(3/2)/c+3/32*b*d*arcsin(c*x)/c^2-1/4*d*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))/c^2+3/32*b*d*x
*(-c^2*x^2+1)^(1/2)/c

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Rubi [A]
time = 0.03, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {4767, 201, 222} \begin {gather*} -\frac {d \left (1-c^2 x^2\right )^2 (a+b \text {ArcSin}(c x))}{4 c^2}+\frac {3 b d \text {ArcSin}(c x)}{32 c^2}+\frac {b d x \left (1-c^2 x^2\right )^{3/2}}{16 c}+\frac {3 b d x \sqrt {1-c^2 x^2}}{32 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*b*d*x*Sqrt[1 - c^2*x^2])/(32*c) + (b*d*x*(1 - c^2*x^2)^(3/2))/(16*c) + (3*b*d*ArcSin[c*x])/(32*c^2) - (d*(1
- c^2*x^2)^2*(a + b*ArcSin[c*x]))/(4*c^2)

Rule 201

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 222

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]

Rule 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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]

Rubi steps

\begin {align*} \int x \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac {d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{4 c^2}+\frac {(b d) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{4 c}\\ &=\frac {b d x \left (1-c^2 x^2\right )^{3/2}}{16 c}-\frac {d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{4 c^2}+\frac {(3 b d) \int \sqrt {1-c^2 x^2} \, dx}{16 c}\\ &=\frac {3 b d x \sqrt {1-c^2 x^2}}{32 c}+\frac {b d x \left (1-c^2 x^2\right )^{3/2}}{16 c}-\frac {d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{4 c^2}+\frac {(3 b d) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 c}\\ &=\frac {3 b d x \sqrt {1-c^2 x^2}}{32 c}+\frac {b d x \left (1-c^2 x^2\right )^{3/2}}{16 c}+\frac {3 b d \sin ^{-1}(c x)}{32 c^2}-\frac {d \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right )}{4 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.06, size = 77, normalized size = 0.86 \begin {gather*} -\frac {d \left (c x \left (8 a c x \left (-2+c^2 x^2\right )+b \sqrt {1-c^2 x^2} \left (-5+2 c^2 x^2\right )\right )+b \left (5-16 c^2 x^2+8 c^4 x^4\right ) \text {ArcSin}(c x)\right )}{32 c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/32*(d*(c*x*(8*a*c*x*(-2 + c^2*x^2) + b*Sqrt[1 - c^2*x^2]*(-5 + 2*c^2*x^2)) + b*(5 - 16*c^2*x^2 + 8*c^4*x^4)
*ArcSin[c*x]))/c^2

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Maple [A]
time = 0.06, size = 92, normalized size = 1.02

method	result	size
derivativedivides	\(\frac {-\frac {d \left (c^{2} x^{2}-1\right )^{2} a}{4}-d b \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {5 \arcsin \left (c x \right )}{32}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{32}\right )}{c^{2}}\)	\(92\)
default	\(\frac {-\frac {d \left (c^{2} x^{2}-1\right )^{2} a}{4}-d b \left (\frac {c^{4} x^{4} \arcsin \left (c x \right )}{4}-\frac {c^{2} x^{2} \arcsin \left (c x \right )}{2}+\frac {5 \arcsin \left (c x \right )}{32}+\frac {c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}}{16}-\frac {5 c x \sqrt {-c^{2} x^{2}+1}}{32}\right )}{c^{2}}\)	\(92\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(-c^2*d*x^2+d)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)

[Out]

1/c^2*(-1/4*d*(c^2*x^2-1)^2*a-d*b*(1/4*c^4*x^4*arcsin(c*x)-1/2*c^2*x^2*arcsin(c*x)+5/32*arcsin(c*x)+1/16*c^3*x
^3*(-c^2*x^2+1)^(1/2)-5/32*c*x*(-c^2*x^2+1)^(1/2)))

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Maxima [A]
time = 0.48, size = 128, normalized size = 1.42 \begin {gather*} -\frac {1}{4} \, a c^{2} d x^{4} - \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 (c x\right )}{c^{5}}\right )} c\right )} b c^{2} d + \frac {1}{2} \, a d 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 (c x\right )}{c^{3}}\right )}\right )} b d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 2.46, size = 86, normalized size = 0.96 \begin {gather*} -\frac {8 \, a c^{4} d x^{4} - 16 \, a c^{2} d x^{2} + {\left (8 \, b c^{4} d x^{4} - 16 \, b c^{2} d x^{2} + 5 \, b d\right )} \arcsin \left (c x\right ) + {\left (2 \, b c^{3} d x^{3} - 5 \, b c d x\right )} \sqrt {-c^{2} x^{2} + 1}}{32 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(8*a*c^4*d*x^4 - 16*a*c^2*d*x^2 + (8*b*c^4*d*x^4 - 16*b*c^2*d*x^2 + 5*b*d)*arcsin(c*x) + (2*b*c^3*d*x^3
- 5*b*c*d*x)*sqrt(-c^2*x^2 + 1))/c^2

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Sympy [A]
time = 0.26, size = 117, normalized size = 1.30 \begin {gather*} \begin {cases} - \frac {a c^{2} d x^{4}}{4} + \frac {a d x^{2}}{2} - \frac {b c^{2} d x^{4} \operatorname {asin}{\left (c x \right )}}{4} - \frac {b c d x^{3} \sqrt {- c^{2} x^{2} + 1}}{16} + \frac {b d x^{2} \operatorname {asin}{\left (c x \right )}}{2} + \frac {5 b d x \sqrt {- c^{2} x^{2} + 1}}{32 c} - \frac {5 b d \operatorname {asin}{\left (c x \right )}}{32 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d x^{2}}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-a*c**2*d*x**4/4 + a*d*x**2/2 - b*c**2*d*x**4*asin(c*x)/4 - b*c*d*x**3*sqrt(-c**2*x**2 + 1)/16 + b*
d*x**2*asin(c*x)/2 + 5*b*d*x*sqrt(-c**2*x**2 + 1)/(32*c) - 5*b*d*asin(c*x)/(32*c**2), Ne(c, 0)), (a*d*x**2/2,
True))

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Giac [A]
time = 0.43, size = 100, normalized size = 1.11 \begin {gather*} -\frac {1}{4} \, a c^{2} d x^{4} + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d x}{16 \, c} - \frac {{\left (c^{2} x^{2} - 1\right )}^{2} b d \arcsin \left (c x\right )}{4 \, c^{2}} + \frac {3 \, \sqrt {-c^{2} x^{2} + 1} b d x}{32 \, c} + \frac {{\left (c^{2} x^{2} - 1\right )} a d}{2 \, c^{2}} + \frac {3 \, b d \arcsin \left (c x\right )}{32 \, c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*a*c^2*d*x^4 + 1/16*(-c^2*x^2 + 1)^(3/2)*b*d*x/c - 1/4*(c^2*x^2 - 1)^2*b*d*arcsin(c*x)/c^2 + 3/32*sqrt(-c^
2*x^2 + 1)*b*d*x/c + 1/2*(c^2*x^2 - 1)*a*d/c^2 + 3/32*b*d*arcsin(c*x)/c^2

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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