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

Optimal. Leaf size=455 \[ -\frac {b c e x^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {2 b e x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {e \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {2 b c^2 e x^3 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} e x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 b^2 e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x}}{27 c}+\frac {b^2 e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {1}{4} b^2 e x \sqrt {c d x+d} \sqrt {e-c e x}-\frac {4 b^2 e \sqrt {c d x+d} \sqrt {e-c e x}}{9 c} \]

[Out]

-4/9*b^2*e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c-1/4*b^2*e*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)-2/27*b^2*e*(-c^2*x^
2+1)*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c+1/2*e*x*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+1/3*e*(-c
^2*x^2+1)*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c+1/4*b^2*e*arcsin(c*x)*(c*d*x+d)^(1/2)*(-c*e*x
+e)^(1/2)/c/(-c^2*x^2+1)^(1/2)-2/3*b*e*x*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)
-1/2*b*c*e*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)+2/9*b*c^2*e*x^3*(a+b*arcs
in(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)+1/6*e*(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 = 0.59, antiderivative size = 455, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.344, Rules used = {4673, 4763, 4647, 4641, 4627, 321, 216, 4677, 4645, 444, 43} \[ \frac {2 b c^2 e x^3 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}-\frac {b c e x^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {1-c^2 x^2}}-\frac {2 b e x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {e \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {1-c^2 x^2}}+\frac {e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {1}{2} e x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 b^2 e \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x}}{27 c}+\frac {b^2 e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {1}{4} b^2 e x \sqrt {c d x+d} \sqrt {e-c e x}-\frac {4 b^2 e \sqrt {c d x+d} \sqrt {e-c e x}}{9 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*b^2*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(9*c) - (b^2*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/4 - (2*b^2*e*Sqrt
[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(27*c) + (b^2*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(4*c*S
qrt[1 - c^2*x^2]) - (2*b*e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) - (b*c
*e*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(2*Sqrt[1 - c^2*x^2]) + (2*b*c^2*e*x^3*Sqrt[d + c*
d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) + (e*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*
ArcSin[c*x])^2)/2 + (e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c) + (e*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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 216

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 321

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 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

Rule 4627

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 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 4645

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2)
^p, x]}, Dist[a + b*ArcSin[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 - c^2*x^2], x], x], x]] /; F
reeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[p, 0]

Rule 4647

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[Sqrt[d + e*x^2]/(2*Sqrt[1 - c^2*x^2]), Int[(a + b*ArcSin[c*x])^n/Sqrt[1 -
c^2*x^2], x], x] - Dist[(b*c*n*Sqrt[d + e*x^2])/(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 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 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 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))

Rubi steps

\begin {align*} \int \sqrt {d+c d x} (e-c e x)^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int (e-c e x) \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (\sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \left (e \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2-c e x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2\right ) \, dx}{\sqrt {1-c^2 x^2}}\\ &=\frac {\left (e \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 {\left (c e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \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} e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {\left (e \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 (2 b e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \sqrt {1-c^2 x^2}}-\frac {\left (b c e \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 {2 b e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {b c e 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 {2 b c^2 e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {e \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 (2 b^2 c e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx}{3 \sqrt {1-c^2 x^2}}+\frac {\left (b^2 c^2 e \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 e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 b e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {b c e 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 {2 b c^2 e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {e \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 e \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 {\left (b^2 c e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {c^2 x}{3}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 e x \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b^2 e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {2 b e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {b c e 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 {2 b c^2 e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {e \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 e \sqrt {d+c d x} \sqrt {e-c e x}\right ) \operatorname {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-c^2 x}}+\frac {1}{3} \sqrt {1-c^2 x}\right ) \, dx,x,x^2\right )}{3 \sqrt {1-c^2 x^2}}\\ &=-\frac {4 b^2 e \sqrt {d+c d x} \sqrt {e-c e x}}{9 c}-\frac {1}{4} b^2 e x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {2 b^2 e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{27 c}+\frac {b^2 e \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}-\frac {2 b e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}-\frac {b c e 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 {2 b c^2 e x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}+\frac {1}{2} e x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {e \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {e \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 = 2.06, size = 440, normalized size = 0.97 \[ \frac {e \sqrt {c d x+d} \sqrt {e-c e x} \left (-3 \left (4 \left (3 a^2 \sqrt {1-c^2 x^2} \left (2 c^2 x^2-3 c x-2\right )-4 a b c x \left (c^2 x^2-3\right )+9 b^2 \sqrt {1-c^2 x^2}\right )+9 b^2 \sin \left (2 \sin ^{-1}(c x)\right )\right )+54 a b \cos \left (2 \sin ^{-1}(c x)\right )-4 b^2 \cos \left (3 \sin ^{-1}(c x)\right )\right )-108 a^2 \sqrt {d} e^{3/2} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )+18 b e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^2 \left (6 a+3 b \sqrt {1-c^2 x^2}+3 b \sin \left (2 \sin ^{-1}(c x)\right )+b \cos \left (3 \sin ^{-1}(c x)\right )\right )-6 b e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x) \left (2 \left (12 a c^2 x^2 \sqrt {1-c^2 x^2}-12 a \sqrt {1-c^2 x^2}-9 a \sin \left (2 \sin ^{-1}(c x)\right )+9 b c x+b \sin \left (3 \sin ^{-1}(c x)\right )\right )-9 b \cos \left (2 \sin ^{-1}(c x)\right )\right )+36 b^2 e \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^3}{216 c \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(36*b^2*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 108*a^2*Sqrt[d]*e^(3/2)*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))] + 18*b*e*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*
ArcSin[c*x]^2*(6*a + 3*b*Sqrt[1 - c^2*x^2] + b*Cos[3*ArcSin[c*x]] + 3*b*Sin[2*ArcSin[c*x]]) + e*Sqrt[d + c*d*x
]*Sqrt[e - c*e*x]*(54*a*b*Cos[2*ArcSin[c*x]] - 4*b^2*Cos[3*ArcSin[c*x]] - 3*(4*(9*b^2*Sqrt[1 - c^2*x^2] - 4*a*
b*c*x*(-3 + c^2*x^2) + 3*a^2*Sqrt[1 - c^2*x^2]*(-2 - 3*c*x + 2*c^2*x^2)) + 9*b^2*Sin[2*ArcSin[c*x]])) - 6*b*e*
Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]*(-9*b*Cos[2*ArcSin[c*x]] + 2*(9*b*c*x - 12*a*Sqrt[1 - c^2*x^2] + 1
2*a*c^2*x^2*Sqrt[1 - c^2*x^2] - 9*a*Sin[2*ArcSin[c*x]] + b*Sin[3*ArcSin[c*x]])))/(216*c*Sqrt[1 - c^2*x^2])

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fricas [F]  time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-{\left (a^{2} c e x - a^{2} e + {\left (b^{2} c e x - b^{2} e\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c e x - a b e\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c e x + e}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Warning, integrat
ion of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]S
implification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep
near 0Simplification assuming t_nostep near 0Warning, integration of abs or sign assumes constant sign by inte
rvals (correct if the argument is real):Check [abs(t_nostep)]Warning, integration of abs or sign assumes const
ant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Simplification assuming t_nostep
near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_
nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assu
ming t_nostep near 0Simplification assuming t_nostep near 0Warning, integration of abs or sign assumes constan
t sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Simplification assuming t_nostep ne
ar 0Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming t_no
step near 0Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assumi
ng t_nostep near 0Simplification assuming t_nostep near 0Warning, integration of abs or sign assumes constant
sign by intervals (correct if the argument is real):Check [abs(t_nostep)]Warning, integration of abs or sign a
ssumes constant sign by intervals (correct if the argument is real):Check [abs(t_nostep)]sym2poly/r2sym(const
gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \[ \int \sqrt {c d x +d}\, \left (-c e x +e \right )^{\frac {3}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, {\left (3 \, \sqrt {-c^{2} d e x^{2} + d e} e x + \frac {3 \, d e^{2} \arcsin \left (c x\right )}{\sqrt {d e} c} + \frac {2 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}}}{c d}\right )} a^{2} + \sqrt {d} \sqrt {e} \int -{\left ({\left (b^{2} c e x - b^{2} e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c e x - a b e\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(3*sqrt(-c^2*d*e*x^2 + d*e)*e*x + 3*d*e^2*arcsin(c*x)/(sqrt(d*e)*c) + 2*(-c^2*d*e*x^2 + d*e)^(3/2)/(c*d))*
a^2 + sqrt(d)*sqrt(e)*integrate(-((b^2*c*e*x - b^2*e)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c*
e*x - a*b*e)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,\sqrt {d+c\,d\,x}\,{\left (e-c\,e\,x\right )}^{3/2} \,d x \]

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)^(3/2),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {d \left (c x + 1\right )} \left (- e \left (c x - 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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