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

Optimal. Leaf size=110 \[ -\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}-\frac{b c x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}+\frac{b x \sqrt{d-c^2 d x^2}}{3 c \sqrt{1-c^2 x^2}} \]

[Out]

(b*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[1 - c^2*x^2]) - (b*c*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) - ((d
- c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*c^2*d)

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Rubi [A]  time = 0.0680963, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {4677} \[ -\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}-\frac{b c x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}+\frac{b x \sqrt{d-c^2 d x^2}}{3 c \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x*Sqrt[d - c^2*d*x^2])/(3*c*Sqrt[1 - c^2*x^2]) - (b*c*x^3*Sqrt[d - c^2*d*x^2])/(9*Sqrt[1 - c^2*x^2]) - ((d
- c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/(3*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]

Rubi steps

\begin{align*} \int x \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right ) \, dx &=-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}+\frac{\left (b \sqrt{d-c^2 d x^2}\right ) \int \left (1-c^2 x^2\right ) \, dx}{3 c \sqrt{1-c^2 x^2}}\\ &=\frac{b x \sqrt{d-c^2 d x^2}}{3 c \sqrt{1-c^2 x^2}}-\frac{b c x^3 \sqrt{d-c^2 d x^2}}{9 \sqrt{1-c^2 x^2}}-\frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0841051, size = 70, normalized size = 0.64 \[ \frac{\sqrt{d-c^2 d x^2} \left (\left (c^2 x^2-1\right ) \left (a+b \sin ^{-1}(c x)\right )+\frac{b c \left (x-\frac{c^2 x^3}{3}\right )}{\sqrt{1-c^2 x^2}}\right )}{3 c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [C]  time = 0.137, size = 343, normalized size = 3.1 \begin{align*} -{\frac{a}{3\,{c}^{2}d} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{{\frac{3}{2}}}}+b \left ({\frac{i+3\,\arcsin \left ( cx \right ) }{72\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( 4\,{c}^{4}{x}^{4}-5\,{c}^{2}{x}^{2}-4\,i\sqrt{-{c}^{2}{x}^{2}+1}{x}^{3}{c}^{3}+3\,i\sqrt{-{c}^{2}{x}^{2}+1}xc+1 \right ) }-{\frac{\arcsin \left ( cx \right ) +i}{8\,{c}^{2} \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{\arcsin \left ( cx \right ) -i}{8\,{c}^{2} \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 ) }+{\frac{-i+3\,\arcsin \left ( cx \right ) }{72\,{c}^{2} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) } \left ( 4\,i\sqrt{-{c}^{2}{x}^{2}+1}{x}^{3}{c}^{3}+4\,{c}^{4}{x}^{4}-3\,i\sqrt{-{c}^{2}{x}^{2}+1}xc-5\,{c}^{2}{x}^{2}+1 \right ) } \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*a/c^2/d*(-c^2*d*x^2+d)^(3/2)+b*(1/72*(-d*(c^2*x^2-1))^(1/2)*(4*c^4*x^4-5*c^2*x^2-4*I*(-c^2*x^2+1)^(1/2)*x
^3*c^3+3*I*(-c^2*x^2+1)^(1/2)*x*c+1)*(I+3*arcsin(c*x))/c^2/(c^2*x^2-1)-1/8*(-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/(c^2*x^2-1)-1/8*(-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/(c^2*x^2-1)+1/72*(-d*(c^2*x^2-1))^(1/2)*(4*I*(-c^2*x^2+1)^(1/2)*x^3*c^3+4*c^4*x
^4-3*I*(-c^2*x^2+1)^(1/2)*x*c-5*c^2*x^2+1)*(-I+3*arcsin(c*x))/c^2/(c^2*x^2-1))

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Maxima [A]  time = 1.67976, size = 101, normalized size = 0.92 \begin{align*} -\frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} b \arcsin \left (c x\right )}{3 \, c^{2} d} - \frac{{\left (c^{2} d^{\frac{3}{2}} x^{3} - 3 \, d^{\frac{3}{2}} x\right )} b}{9 \, c d} - \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}} a}{3 \, c^{2} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(-c^2*d*x^2 + d)^(3/2)*b*arcsin(c*x)/(c^2*d) - 1/9*(c^2*d^(3/2)*x^3 - 3*d^(3/2)*x)*b/(c*d) - 1/3*(-c^2*d*
x^2 + d)^(3/2)*a/(c^2*d)

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Fricas [A]  time = 2.32224, size = 248, normalized size = 2.25 \begin{align*} \frac{{\left (b c^{3} x^{3} - 3 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} + 3 \,{\left (a c^{4} x^{4} - 2 \, a c^{2} x^{2} +{\left (b c^{4} x^{4} - 2 \, b c^{2} x^{2} + b\right )} \arcsin \left (c x\right ) + a\right )} \sqrt{-c^{2} d x^{2} + d}}{9 \,{\left (c^{4} x^{2} - c^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*((b*c^3*x^3 - 3*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 3*(a*c^4*x^4 - 2*a*c^2*x^2 + (b*c^4*x^4 -
2*b*c^2*x^2 + b)*arcsin(c*x) + a)*sqrt(-c^2*d*x^2 + d))/(c^4*x^2 - c^2)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError