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

Optimal. Leaf size=45 \[ \frac{a x^2}{2}+\frac{b \sqrt{1-c^2 x^4}}{2 c}+\frac{1}{2} b x^2 \sin ^{-1}\left (c x^2\right ) \]

[Out]

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

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Rubi [A]  time = 0.0388268, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6715, 4619, 261} \[ \frac{a x^2}{2}+\frac{b \sqrt{1-c^2 x^4}}{2 c}+\frac{1}{2} b x^2 \sin ^{-1}\left (c x^2\right ) \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSin[c*x^2]),x]

[Out]

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

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

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 x \left (a+b \sin ^{-1}\left (c x^2\right )\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \left (a+b \sin ^{-1}(c x)\right ) \, dx,x,x^2\right )\\ &=\frac{a x^2}{2}+\frac{1}{2} b \operatorname{Subst}\left (\int \sin ^{-1}(c x) \, dx,x,x^2\right )\\ &=\frac{a x^2}{2}+\frac{1}{2} b x^2 \sin ^{-1}\left (c x^2\right )-\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{x}{\sqrt{1-c^2 x^2}} \, dx,x,x^2\right )\\ &=\frac{a x^2}{2}+\frac{b \sqrt{1-c^2 x^4}}{2 c}+\frac{1}{2} b x^2 \sin ^{-1}\left (c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0084366, size = 43, normalized size = 0.96 \[ \frac{a x^2}{2}+\frac{1}{2} b \left (\frac{\sqrt{1-c^2 x^4}}{c}+x^2 \sin ^{-1}\left (c x^2\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*ArcSin[c*x^2]),x]

[Out]

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

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Maple [A]  time = 0.002, size = 39, normalized size = 0.9 \begin{align*}{\frac{1}{2\,c} \left ( a{x}^{2}c+b \left ({x}^{2}c\arcsin \left ( c{x}^{2} \right ) +\sqrt{-{c}^{2}{x}^{4}+1} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.47575, size = 50, normalized size = 1.11 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{{\left (c x^{2} \arcsin \left (c x^{2}\right ) + \sqrt{-c^{2} x^{4} + 1}\right )} b}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.51839, size = 86, normalized size = 1.91 \begin{align*} \frac{b c x^{2} \arcsin \left (c x^{2}\right ) + a c x^{2} + \sqrt{-c^{2} x^{4} + 1} b}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.248994, size = 42, normalized size = 0.93 \begin{align*} \begin{cases} \frac{a x^{2}}{2} + \frac{b x^{2} \operatorname{asin}{\left (c x^{2} \right )}}{2} + \frac{b \sqrt{- c^{2} x^{4} + 1}}{2 c} & \text{for}\: c \neq 0 \\\frac{a x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15716, size = 51, normalized size = 1.13 \begin{align*} \frac{a c x^{2} +{\left (c x^{2} \arcsin \left (c x^{2}\right ) + \sqrt{-c^{2} x^{4} + 1}\right )} b}{2 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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