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

Optimal. Leaf size=270 \[ \frac{f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b f g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}+\frac{b g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}} \]

[Out]

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

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Rubi [A]  time = 0.433264, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 7, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.226, Rules used = {4777, 4763, 4641, 4677, 8, 4707, 30} \[ \frac{f^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{2 b c \sqrt{d-c^2 d x^2}}-\frac{2 f g \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{c^2 \sqrt{d-c^2 d x^2}}+\frac{g^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2}{4 b c^3 \sqrt{d-c^2 d x^2}}-\frac{g^2 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )}{2 c^2 \sqrt{d-c^2 d x^2}}+\frac{2 b f g x \sqrt{1-c^2 x^2}}{c \sqrt{d-c^2 d x^2}}+\frac{b g^2 x^2 \sqrt{1-c^2 x^2}}{4 c \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*b*f*g*x*Sqrt[1 - c^2*x^2])/(c*Sqrt[d - c^2*d*x^2]) + (b*g^2*x^2*Sqrt[1 - c^2*x^2])/(4*c*Sqrt[d - c^2*d*x^2]
) - (2*f*g*(1 - c^2*x^2)*(a + b*ArcSin[c*x]))/(c^2*Sqrt[d - c^2*d*x^2]) - (g^2*x*(1 - c^2*x^2)*(a + b*ArcSin[c
*x]))/(2*c^2*Sqrt[d - c^2*d*x^2]) + (f^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(2*b*c*Sqrt[d - c^2*d*x^2])
+ (g^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^2)/(4*b*c^3*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 8

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

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

Mathematica [A]  time = 0.692357, size = 266, normalized size = 0.99 \[ \frac{\sqrt{d} g \left (c^2 x^2-1\right ) \left (4 c \left (a \sqrt{1-c^2 x^2} (4 f+g x)-4 b c f x\right )+b g \cos \left (2 \sin ^{-1}(c x)\right )\right )-4 a \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2} \left (2 c^2 f^2+g^2\right ) \tan ^{-1}\left (\frac{c x \sqrt{d-c^2 d x^2}}{\sqrt{d} \left (c^2 x^2-1\right )}\right )-2 b \sqrt{d} \left (c^2 x^2-1\right ) \left (2 c^2 f^2+g^2\right ) \sin ^{-1}(c x)^2+2 b \sqrt{d} g \left (c^2 x^2-1\right ) \sin ^{-1}(c x) \left (8 c f \sqrt{1-c^2 x^2}+g \sin \left (2 \sin ^{-1}(c x)\right )\right )}{8 c^3 \sqrt{d} \sqrt{1-c^2 x^2} \sqrt{d-c^2 d x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*b*Sqrt[d]*(2*c^2*f^2 + g^2)*(-1 + c^2*x^2)*ArcSin[c*x]^2 - 4*a*(2*c^2*f^2 + g^2)*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))] + Sqrt[d]*g*(-1 + c^2*x^2)*(4*c*(-4*b*
c*f*x + a*(4*f + g*x)*Sqrt[1 - c^2*x^2]) + b*g*Cos[2*ArcSin[c*x]]) + 2*b*Sqrt[d]*g*(-1 + c^2*x^2)*ArcSin[c*x]*
(8*c*f*Sqrt[1 - c^2*x^2] + g*Sin[2*ArcSin[c*x]]))/(8*c^3*Sqrt[d]*Sqrt[1 - c^2*x^2]*Sqrt[d - c^2*d*x^2])

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Maple [B]  time = 0.37, size = 549, normalized size = 2. \begin{align*} -{\frac{a{g}^{2}x}{2\,{c}^{2}d}\sqrt{-{c}^{2}d{x}^{2}+d}}+{\frac{a{g}^{2}}{2\,{c}^{2}}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}-2\,{\frac{afg\sqrt{-{c}^{2}d{x}^{2}+d}}{{c}^{2}d}}+{a{f}^{2}\arctan \left ({x\sqrt{{c}^{2}d}{\frac{1}{\sqrt{-{c}^{2}d{x}^{2}+d}}}} \right ){\frac{1}{\sqrt{{c}^{2}d}}}}+2\,{\frac{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }fg\arcsin \left ( cx \right ) }{{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b{g}^{2}{x}^{2}}{4\,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{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }fg\arcsin \left ( cx \right ){x}^{2}}{d \left ({c}^{2}{x}^{2}-1 \right ) }}-2\,{\frac{b\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }fg\sqrt{-{c}^{2}{x}^{2}+1}x}{dc \left ({c}^{2}{x}^{2}-1 \right ) }}-{\frac{b \left ( \arcsin \left ( cx \right ) \right ) ^{2}{f}^{2}}{2\,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 \left ( \arcsin \left ( cx \right ) \right ) ^{2}{g}^{2}}{4\,d{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}}-{\frac{b{g}^{2}\arcsin \left ( cx \right ){x}^{3}}{2\,d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{g}^{2}\arcsin \left ( cx \right ) x}{2\,{c}^{2}d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }}+{\frac{b{g}^{2}}{8\,d{c}^{3} \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-d \left ({c}^{2}{x}^{2}-1 \right ) }\sqrt{-{c}^{2}{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*a*g^2*x/c^2/d*(-c^2*d*x^2+d)^(1/2)+1/2*a*g^2/c^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2
))-2*a*f*g/c^2/d*(-c^2*d*x^2+d)^(1/2)+a*f^2/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+2*b*(-d
*(c^2*x^2-1))^(1/2)*f*g/c^2/d/(c^2*x^2-1)*arcsin(c*x)-1/4*b*(-d*(c^2*x^2-1))^(1/2)*g^2/c/d/(c^2*x^2-1)*(-c^2*x
^2+1)^(1/2)*x^2-2*b*(-d*(c^2*x^2-1))^(1/2)*f*g/d/(c^2*x^2-1)*arcsin(c*x)*x^2-2*b*(-d*(c^2*x^2-1))^(1/2)*f*g/c/
d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)*x-1/2*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^2-1/4*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^3/d/(c^2*x^2-1)*arcsin(c*x)^2*g^2-1/2*b*(-d*(c^2*x^2-
1))^(1/2)*g^2/d/(c^2*x^2-1)*arcsin(c*x)*x^3+1/2*b*(-d*(c^2*x^2-1))^(1/2)*g^2/c^2/d/(c^2*x^2-1)*arcsin(c*x)*x+1
/8*b*(-d*(c^2*x^2-1))^(1/2)*g^2/c^3/d/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)

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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)^2*(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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{-c^{2} d x^{2} + d}{\left (a g^{2} x^{2} + 2 \, a f g x + a f^{2} +{\left (b g^{2} x^{2} + 2 \, b f g x + b f^{2}\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)^2*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-c^2*d*x^2 + d)*(a*g^2*x^2 + 2*a*f*g*x + a*f^2 + (b*g^2*x^2 + 2*b*f*g*x + b*f^2)*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)**2*(a+b*asin(c*x))/(-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 )}^{2}{\left (b \arcsin \left (c x\right ) + a\right )}}{\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)^2*(a+b*arcsin(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

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