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

Optimal. Leaf size=230 \[ -\frac{2 a b e x \sqrt{1-c^2 x^2}}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b^2 e \left (1-c^2 x^2\right )}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b^2 e x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt{c d x+d} \sqrt{e-c e x}} \]

[Out]

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

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Rubi [A]  time = 0.44103, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {4673, 4763, 4641, 4677, 4619, 261} \[ -\frac{2 a b e x \sqrt{1-c^2 x^2}}{\sqrt{c d x+d} \sqrt{e-c e x}}+\frac{e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b^2 e \left (1-c^2 x^2\right )}{c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b^2 e x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt{c d x+d} \sqrt{e-c e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 4673

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(q_), x_Symbol] :> D
ist[((d + e*x)^q*(f + g*x)^q)/(1 - c^2*x^2)^q, Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x]
, x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 - e^2, 0] && HalfIntegerQ[p, q]
&& GeQ[p - q, 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{\sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{d+c d x}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(e-c e x) \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{\sqrt{1-c^2 x^2} \int \left (\frac{e \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}-\frac{c e x \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{1-c^2 x^2}}\right ) \, dx}{\sqrt{d+c d x} \sqrt{e-c e x}}\\ &=\frac{\left (e \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 d x} \sqrt{e-c e x}}-\frac{\left (c e \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 d x} \sqrt{e-c e x}}\\ &=\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{\left (2 b e \sqrt{1-c^2 x^2}\right ) \int \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt{d+c d x} \sqrt{e-c e x}}\\ &=-\frac{2 a b e x \sqrt{1-c^2 x^2}}{\sqrt{d+c d x} \sqrt{e-c e x}}+\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{\left (2 b^2 e \sqrt{1-c^2 x^2}\right ) \int \sin ^{-1}(c x) \, dx}{\sqrt{d+c d x} \sqrt{e-c e x}}\\ &=-\frac{2 a b e x \sqrt{1-c^2 x^2}}{\sqrt{d+c d x} \sqrt{e-c e x}}-\frac{2 b^2 e x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt{d+c d x} \sqrt{e-c e x}}+\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{\left (2 b^2 c e \sqrt{1-c^2 x^2}\right ) \int \frac{x}{\sqrt{1-c^2 x^2}} \, dx}{\sqrt{d+c d x} \sqrt{e-c e x}}\\ &=-\frac{2 a b e x \sqrt{1-c^2 x^2}}{\sqrt{d+c d x} \sqrt{e-c e x}}-\frac{2 b^2 e \left (1-c^2 x^2\right )}{c \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{2 b^2 e x \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{\sqrt{d+c d x} \sqrt{e-c e x}}+\frac{e \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{e \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{3 b c \sqrt{d+c d x} \sqrt{e-c e x}}\\ \end{align*}

Mathematica [A]  time = 1.14844, size = 296, normalized size = 1.29 \[ \frac{3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a^2 \sqrt{1-c^2 x^2}-2 a b c x-2 b^2 \sqrt{1-c^2 x^2}\right )-3 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )+3 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (a+b \sqrt{1-c^2 x^2}\right )-6 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (b c x-a \sqrt{1-c^2 x^2}\right )+b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{3 c d \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.271, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\sqrt{-cex+e}{\frac{1}{\sqrt{cdx+d}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+d)^(1/2),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((a+b*arcsin(c*x))^2*(-c*e*x+e)^(1/2)/(c*d*x+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{{\left (b^{2} \arcsin \left (c x\right )^{2} + 2 \, a b \arcsin \left (c x\right ) + a^{2}\right )} \sqrt{-c e x + e}}{\sqrt{c d x + d}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{- e \left (c x - 1\right )} \left (a + b \operatorname{asin}{\left (c x \right )}\right )^{2}}{\sqrt{d \left (c x + 1\right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(-e*(c*x - 1))*(a + b*asin(c*x))**2/sqrt(d*(c*x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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