3.238 \(\int \frac{x (a+b \sin ^{-1}(c x))^2}{\sqrt{d-c^2 d x^2}} \, dx\)

Optimal. Leaf size=146 \[ \frac{2 a b x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}+\frac{2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt{d-c^2 d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.1215, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {4677, 4619, 261} \[ \frac{2 a b x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{\sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 d}+\frac{2 b^2 \left (1-c^2 x^2\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 4619

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

Rule 261

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

Rubi steps

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

Mathematica [A]  time = 0.0856313, size = 86, normalized size = 0.59 \[ \frac{\left (c^2 x^2-1\right ) \left (a+b \sin ^{-1}(c x)\right )^2+2 b \sqrt{1-c^2 x^2} \left (a c x+b \sqrt{1-c^2 x^2}+b c x \sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.142, size = 316, normalized size = 2.2 \begin{align*} -{\frac{{a}^{2}}{{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}+{b}^{2} \left ( -{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2}-2+2\,i\arcsin \left ( cx \right ) }{2\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ({c}^{2}{x}^{2}-i\sqrt{-{c}^{2}{x}^{2}+1}xc-1 \right ) }-{\frac{ \left ( \arcsin \left ( cx \right ) \right ) ^{2}-2-2\,i\arcsin \left ( cx \right ) }{2\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( i\sqrt{-{c}^{2}{x}^{2}+1}xc+{c}^{2}{x}^{2}-1 \right ) } \right ) +2\,ab \left ( -1/2\,{\frac{\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ({c}^{2}{x}^{2}-i\sqrt{-{c}^{2}{x}^{2}+1}xc-1 \right ) \left ( \arcsin \left ( cx \right ) +i \right ) }{{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }}-1/2\,{\frac{\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( i\sqrt{-{c}^{2}{x}^{2}+1}xc+{c}^{2}{x}^{2}-1 \right ) \left ( \arcsin \left ( cx \right ) -i \right ) }{{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x)

[Out]

-a^2/c^2/d*(-c^2*d*x^2+d)^(1/2)+b^2*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^(1/2)*x*c-1)*(arcsin(
c*x)^2-2+2*I*arcsin(c*x))/c^2/d/(c^2*x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(a
rcsin(c*x)^2-2-2*I*arcsin(c*x))/c^2/d/(c^2*x^2-1))+2*a*b*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(c^2*x^2-I*(-c^2*x^2+1)^
(1/2)*x*c-1)*(arcsin(c*x)+I)/c^2/d/(c^2*x^2-1)-1/2*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)
*(arcsin(c*x)-I)/c^2/d/(c^2*x^2-1))

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Maxima [A]  time = 1.5907, size = 176, normalized size = 1.21 \begin{align*} 2 \, b^{2}{\left (\frac{x \arcsin \left (c x\right )}{c \sqrt{d}} + \frac{\sqrt{-c^{2} x^{2} + 1}}{c^{2} \sqrt{d}}\right )} + \frac{2 \, a b x}{c \sqrt{d}} - \frac{\sqrt{-c^{2} d x^{2} + d} b^{2} \arcsin \left (c x\right )^{2}}{c^{2} d} - \frac{2 \, \sqrt{-c^{2} d x^{2} + d} a b \arcsin \left (c x\right )}{c^{2} d} - \frac{\sqrt{-c^{2} d x^{2} + d} a^{2}}{c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.86209, size = 312, normalized size = 2.14 \begin{align*} -\frac{2 \, \sqrt{-c^{2} d x^{2} + d}{\left (b^{2} c x \arcsin \left (c x\right ) + a b c x\right )} \sqrt{-c^{2} x^{2} + 1} +{\left ({\left (a^{2} - 2 \, b^{2}\right )} c^{2} x^{2} +{\left (b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - a^{2} + 2 \, b^{2} + 2 \,{\left (a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{c^{4} d x^{2} - c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Exception raised: TypeError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)^2*x/sqrt(-c^2*d*x^2 + d), x)