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

Optimal. Leaf size=171 \[ \frac{f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}}-\frac{g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b g x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{c \sqrt{d-c^2 d x^2}}+\frac{2 b^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt{d-c^2 d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.368733, antiderivative size = 207, normalized size of antiderivative = 1.21, number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.194, Rules used = {4777, 4763, 4641, 4677, 4619, 261} \[ \frac{f \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d-c^2 d x^2}}+\frac{2 a b g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}-\frac{g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 g \left (1-c^2 x^2\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b^2 g x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4777

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

Rule 4763

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSin[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g},
x] && EqQ[c^2*d + e, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n, 0] && (m == 1 || p > 0 ||
(n == 1 && p > -1) || (m == 2 && p < -2))

Rule 4641

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

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

Mathematica [A]  time = 0.596177, size = 291, normalized size = 1.7 \[ \frac{3 \sqrt{d} g \left (c^2 x^2-1\right ) \left (a^2 \sqrt{1-c^2 x^2}-2 a b c x-2 b^2 \sqrt{1-c^2 x^2}\right )-3 a^2 c f \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )+3 b \sqrt{d} \left (c^2 x^2-1\right ) \sin ^{-1}(c x)^2 \left (b g \sqrt{1-c^2 x^2}-a c f\right )-6 b \sqrt{d} g \left (c^2 x^2-1\right ) \sin ^{-1}(c x) \left (b c x-a \sqrt{1-c^2 x^2}\right )-b^2 c \sqrt{d} f \left (c^2 x^2-1\right ) \sin ^{-1}(c x)^3}{3 c^2 \sqrt{d} \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.303, size = 513, normalized size = 3. \begin{align*} -{\frac{{a}^{2}g}{{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}+{{a}^{2}f\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}-{\frac{{b}^{2} \left ( \arcsin \left ( cx \right ) \right ) ^{3}f}{3\,dc \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{{b}^{2}g \left ( \arcsin \left ( cx \right ) \right ) ^{2}{x}^{2}}{d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-2\,{\frac{{b}^{2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }g\arcsin \left ( cx \right ) \sqrt{-{c}^{2}{x}^{2}+1}x}{dc \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{{b}^{2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }g{x}^{2}}{d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{{b}^{2}g \left ( \arcsin \left ( cx \right ) \right ) ^{2}}{{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}-2\,{\frac{{b}^{2}\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }g}{{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{ab \left ( \arcsin \left ( cx \right ) \right ) ^{2}f}{dc \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-2\,{\frac{ab\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }g\arcsin \left ( cx \right ){x}^{2}}{d \left ({c}^{2}{x}^{2}-1 \right ) }}-2\,{\frac{ab\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }g\sqrt{-{c}^{2}{x}^{2}+1}x}{dc \left ({c}^{2}{x}^{2}-1 \right ) }}+2\,{\frac{ab\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }g\arcsin \left ( cx \right ) }{{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Exception raised: ValueError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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((g*x+f)*(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 (g x + f\right )}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt{-c^{2} d x^{2} + d}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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