3.111 \(\int \frac{x^3 (a+b \sin ^{-1}(c x))}{\sqrt{d-c^2 d x^2}} \, dx\)

Optimal. Leaf size=148 \[ -\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d}+\frac{b x^3 \sqrt{1-c^2 x^2}}{9 c \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2}}{3 c^3 \sqrt{d-c^2 d x^2}} \]

[Out]

(2*b*x*Sqrt[1 - c^2*x^2])/(3*c^3*Sqrt[d - c^2*d*x^2]) + (b*x^3*Sqrt[1 - c^2*x^2])/(9*c*Sqrt[d - c^2*d*x^2]) -
(2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^4*d) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^2*d
)

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Rubi [A]  time = 0.159491, antiderivative size = 148, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {4707, 4677, 8, 30} \[ -\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d}+\frac{b x^3 \sqrt{1-c^2 x^2}}{9 c \sqrt{d-c^2 d x^2}}+\frac{2 b x \sqrt{1-c^2 x^2}}{3 c^3 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*x*Sqrt[1 - c^2*x^2])/(3*c^3*Sqrt[d - c^2*d*x^2]) + (b*x^3*Sqrt[1 - c^2*x^2])/(9*c*Sqrt[d - c^2*d*x^2]) -
(2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^4*d) - (x^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcSin[c*x]))/(3*c^2*d
)

Rule 4707

Int[(((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[
(f*(f*x)^(m - 1)*Sqrt[d + e*x^2]*(a + b*ArcSin[c*x])^n)/(e*m), x] + (Dist[(f^2*(m - 1))/(c^2*m), Int[((f*x)^(m
- 2)*(a + b*ArcSin[c*x])^n)/Sqrt[d + e*x^2], x], x] + Dist[(b*f*n*Sqrt[1 - c^2*x^2])/(c*m*Sqrt[d + e*x^2]), I
nt[(f*x)^(m - 1)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[c^2*d + e, 0] &&
GtQ[n, 0] && GtQ[m, 1] && IntegerQ[m]

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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{x^3 \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx &=-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}+\frac{2 \int \frac{x \left (a+b \sin ^{-1}(c x)\right )}{\sqrt{d-c^2 d x^2}} \, dx}{3 c^2}+\frac{\left (b \sqrt{1-c^2 x^2}\right ) \int x^2 \, dx}{3 c \sqrt{d-c^2 d x^2}}\\ &=\frac{b x^3 \sqrt{1-c^2 x^2}}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}+\frac{\left (2 b \sqrt{1-c^2 x^2}\right ) \int 1 \, dx}{3 c^3 \sqrt{d-c^2 d x^2}}\\ &=\frac{2 b x \sqrt{1-c^2 x^2}}{3 c^3 \sqrt{d-c^2 d x^2}}+\frac{b x^3 \sqrt{1-c^2 x^2}}{9 c \sqrt{d-c^2 d x^2}}-\frac{2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^4 d}-\frac{x^2 \sqrt{d-c^2 d x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 c^2 d}\\ \end{align*}

Mathematica [A]  time = 0.0523435, size = 92, normalized size = 0.62 \[ \frac{3 a \left (c^4 x^4+c^2 x^2-2\right )+b c x \sqrt{1-c^2 x^2} \left (c^2 x^2+6\right )+3 b \left (c^4 x^4+c^2 x^2-2\right ) \sin ^{-1}(c x)}{9 c^4 \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*c*x*Sqrt[1 - c^2*x^2]*(6 + c^2*x^2) + 3*a*(-2 + c^2*x^2 + c^4*x^4) + 3*b*(-2 + c^2*x^2 + c^4*x^4)*ArcSin[c*
x])/(9*c^4*Sqrt[d - c^2*d*x^2])

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Maple [C]  time = 0.24, size = 381, normalized size = 2.6 \begin{align*} a \left ( -{\frac{{x}^{2}}{3\,{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}-{\frac{2}{3\,d{c}^{4}}\sqrt{-{c}^{2}d{x}^{2}+d}} \right ) +b \left ( -{\frac{i+3\,\arcsin \left ( cx \right ) }{72\,d{c}^{4} \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{3\,\arcsin \left ( cx \right ) +3\,i}{8\,d{c}^{4} \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{3\,\arcsin \left ( cx \right ) -3\,i}{8\,d{c}^{4} \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\,d{c}^{4} \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^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(1/2),x)

[Out]

a*(-1/3*x^2/c^2/d*(-c^2*d*x^2+d)^(1/2)-2/3/d/c^4*(-c^2*d*x^2+d)^(1/2))+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^4/d/(c^2*x^2-1)
-3/8*(-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^4/d/(c^2*x^2-1)-3/8*(-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^4/d/(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^4/
d/(c^2*x^2-1))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.8392, size = 255, normalized size = 1.72 \begin{align*} -\frac{{\left (b c^{3} x^{3} + 6 \, b c x\right )} \sqrt{-c^{2} d x^{2} + d} \sqrt{-c^{2} x^{2} + 1} + 3 \,{\left (a c^{4} x^{4} + a c^{2} x^{2} +{\left (b c^{4} x^{4} + b c^{2} x^{2} - 2 \, b\right )} \arcsin \left (c x\right ) - 2 \, a\right )} \sqrt{-c^{2} d x^{2} + d}}{9 \,{\left (c^{6} d x^{2} - c^{4} d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

-1/9*((b*c^3*x^3 + 6*b*c*x)*sqrt(-c^2*d*x^2 + d)*sqrt(-c^2*x^2 + 1) + 3*(a*c^4*x^4 + a*c^2*x^2 + (b*c^4*x^4 +
b*c^2*x^2 - 2*b)*arcsin(c*x) - 2*a)*sqrt(-c^2*d*x^2 + d))/(c^6*d*x^2 - c^4*d)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \left (a + b \operatorname{asin}{\left (c x \right )}\right )}{\sqrt{- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(a+b*asin(c*x))/(-c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x**3*(a + b*asin(c*x))/sqrt(-d*(c*x - 1)*(c*x + 1)), x)

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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 )} x^{3}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsin(c*x) + a)*x^3/sqrt(-c^2*d*x^2 + d), x)