3.4.88 \(\int x (a+b \text {ArcSin}(c+d x^2)) \, dx\) [388]

Optimal. Leaf size=57 \[ \frac {a x^2}{2}+\frac {b \sqrt {1-\left (c+d x^2\right )^2}}{2 d}+\frac {b \left (c+d x^2\right ) \text {ArcSin}\left (c+d x^2\right )}{2 d} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {6847, 4887, 4715, 267} \begin {gather*} \frac {a x^2}{2}+\frac {b \left (c+d x^2\right ) \text {ArcSin}\left (c+d x^2\right )}{2 d}+\frac {b \sqrt {1-\left (c+d x^2\right )^2}}{2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a + b*ArcSin[c + d*x^2]),x]

[Out]

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

Rule 267

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]

Rule 4715

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 4887

Int[((a_.) + ArcSin[(c_) + (d_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Dist[1/d, Subst[Int[(a + b*ArcSin[x])^n, x],
x, c + d*x], x] /; FreeQ[{a, b, c, d, n}, x]

Rule 6847

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]

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(183\) vs. \(2(57)=114\).
time = 0.11, size = 183, normalized size = 3.21 \begin {gather*} \frac {a x^2}{2}+\frac {1}{2} b x^2 \text {ArcSin}\left (c+d x^2\right )+\frac {b \left (2 d \sqrt {1-c^2-2 c d x^2-d^2 x^4}+2 c d \text {ArcTan}\left (\frac {\sqrt {-d^2} x^2-\sqrt {1-c^2-2 c d x^2-d^2 x^4}}{c}\right )+c \sqrt {-d^2} \log \left (-1+2 c d x^2+2 d^2 x^4+2 \sqrt {-d^2} x^2 \sqrt {1-c^2-2 c d x^2-d^2 x^4}\right )\right )}{4 d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.07, size = 50, normalized size = 0.88

method	result	size
derivativedivides	\(\frac {\left (d \,x^{2}+c \right ) a +b \left (\left (d \,x^{2}+c \right ) \arcsin \left (d \,x^{2}+c \right )+\sqrt {1-\left (d \,x^{2}+c \right )^{2}}\right )}{2 d}\)	\(50\)
default	\(\frac {\left (d \,x^{2}+c \right ) a +b \left (\left (d \,x^{2}+c \right ) \arcsin \left (d \,x^{2}+c \right )+\sqrt {1-\left (d \,x^{2}+c \right )^{2}}\right )}{2 d}\)	\(50\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.47, size = 45, normalized size = 0.79 \begin {gather*} \frac {1}{2} \, a x^{2} + \frac {{\left ({\left (d x^{2} + c\right )} \arcsin \left (d x^{2} + c\right ) + \sqrt {-{\left (d x^{2} + c\right )}^{2} + 1}\right )} b}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 3.24, size = 57, normalized size = 1.00 \begin {gather*} \frac {a d x^{2} + {\left (b d x^{2} + b c\right )} \arcsin \left (d x^{2} + c\right ) + \sqrt {-d^{2} x^{4} - 2 \, c d x^{2} - c^{2} + 1} b}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.11, size = 76, normalized size = 1.33 \begin {gather*} \begin {cases} \frac {a x^{2}}{2} + \frac {b c \operatorname {asin}{\left (c + d x^{2} \right )}}{2 d} + \frac {b x^{2} \operatorname {asin}{\left (c + d x^{2} \right )}}{2} + \frac {b \sqrt {- c^{2} - 2 c d x^{2} - d^{2} x^{4} + 1}}{2 d} & \text {for}\: d \neq 0 \\\frac {x^{2} \left (a + b \operatorname {asin}{\left (c \right )}\right )}{2} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.41, size = 49, normalized size = 0.86 \begin {gather*} \frac {{\left (d x^{2} + c\right )} a + {\left ({\left (d x^{2} + c\right )} \arcsin \left (d x^{2} + c\right ) + \sqrt {-{\left (d x^{2} + c\right )}^{2} + 1}\right )} b}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 0.73, size = 108, normalized size = 1.89 \begin {gather*} \frac {a\,x^2}{2}+\frac {b\,x^2\,\mathrm {asin}\left (d\,x^2+c\right )}{2}+\frac {b\,\sqrt {-c^2-2\,c\,d\,x^2-d^2\,x^4+1}}{2\,d}+\frac {b\,c\,\ln \left (\sqrt {-c^2-2\,c\,d\,x^2-d^2\,x^4+1}-\frac {d^2\,x^2+c\,d}{\sqrt {-d^2}}\right )}{2\,\sqrt {-d^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a + b*asin(c + d*x^2)),x)

[Out]

(a*x^2)/2 + (b*x^2*asin(c + d*x^2))/2 + (b*(1 - d^2*x^4 - 2*c*d*x^2 - c^2)^(1/2))/(2*d) + (b*c*log((1 - d^2*x^
4 - 2*c*d*x^2 - c^2)^(1/2) - (c*d + d^2*x^2)/(-d^2)^(1/2)))/(2*(-d^2)^(1/2))

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