3.6.42 \(\int \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2 \, dx\) [542]

Optimal. Leaf size=222 \[ -\frac {1}{4} b^2 x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {b c x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^3}{6 b c \sqrt {1-c^2 x^2}} \]

[Out]

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

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Rubi [A]
time = 0.21, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {4763, 4741, 4737, 4723, 327, 222} \begin {gather*} \frac {\sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^3}{6 b c \sqrt {1-c^2 x^2}}-\frac {b c x^2 \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {c d x+d} \sqrt {e-c e x} (a+b \text {ArcSin}(c x))^2+\frac {b^2 \text {ArcSin}(c x) \sqrt {c d x+d} \sqrt {e-c e x}}{4 c \sqrt {1-c^2 x^2}}-\frac {1}{4} b^2 x \sqrt {c d x+d} \sqrt {e-c e x} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 4723

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

Rule 4737

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

Rule 4741

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

Rule 4763

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]

Rubi steps

\begin {align*} \int \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}-\frac {\left (b c \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=-\frac {b c x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {b c x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b^2 \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {b c x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}+\frac {1}{2} x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.58, size = 288, normalized size = 1.30 \begin {gather*} \frac {4 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3-12 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+6 b \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x) (b \cos (2 \text {ArcSin}(c x))+2 a \sin (2 \text {ArcSin}(c x)))+6 b \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^2 (2 a+b \sin (2 \text {ArcSin}(c x)))+3 \sqrt {d+c d x} \sqrt {e-c e x} \left (4 a^2 c x \sqrt {1-c^2 x^2}+2 a b \cos (2 \text {ArcSin}(c x))-b^2 \sin (2 \text {ArcSin}(c x))\right )}{24 c \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 12*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sq
rt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[
c*x]*(b*Cos[2*ArcSin[c*x]] + 2*a*Sin[2*ArcSin[c*x]]) + 6*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(2*a
+ b*Sin[2*ArcSin[c*x]]) + 3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(4*a^2*c*x*Sqrt[1 - c^2*x^2] + 2*a*b*Cos[2*ArcSin[
c*x]] - b^2*Sin[2*ArcSin[c*x]]))/(24*c*Sqrt[1 - c^2*x^2])

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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \sqrt {c d x +d}\, \sqrt {-c e x +e}\, \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(sqrt(-c^2*d*x^2*e + d*e)*x + sqrt(d)*arcsin(c*x)*e^(1/2)/c)*a^2 + sqrt(d)*e^(1/2)*integrate((b^2*arctan2(
c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c
*x + 1), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {d \left (c x + 1\right )} \sqrt {- e \left (c x - 1\right )} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x}\,\sqrt {e-c\,e\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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