3.555 \(\int \frac {(e-c e x)^{5/2} (a+b \sin ^{-1}(c x))^2}{\sqrt {d+c d x}} \, dx\)

Optimal. Leaf size=559 \[ \frac {5 e^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {11 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {22 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {3 b^2 e^3 x \left (1-c^2 x^2\right )}{4 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {68 b^2 e^3 \left (1-c^2 x^2\right )}{9 c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {3 b^2 e^3 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt {c d x+d} \sqrt {e-c e x}} \]

[Out]

-68/9*b^2*e^3*(-c^2*x^2+1)/c/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)+3/4*b^2*e^3*x*(-c^2*x^2+1)/(c*d*x+d)^(1/2)/(-c*e
*x+e)^(1/2)+2/27*b^2*e^3*(-c^2*x^2+1)^2/c/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)+11/3*e^3*(-c^2*x^2+1)*(a+b*arcsin(c
*x))^2/c/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-3/2*e^3*x*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e
)^(1/2)+1/3*c*e^3*x^2*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-3/4*b^2*e^3*arcsin(c*x
)*(-c^2*x^2+1)^(1/2)/c/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-22/3*b*e^3*x*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(c*d
*x+d)^(1/2)/(-c*e*x+e)^(1/2)+3/2*b*c*e^3*x^2*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(c*d*x+d)^(1/2)/(-c*e*x+e)^(
1/2)-2/9*b*c^2*e^3*x^3*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(1/2)/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)+5/6*e^3*(a+b*arcs
in(c*x))^3*(-c^2*x^2+1)^(1/2)/b/c/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)

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Rubi [A]  time = 0.69, antiderivative size = 559, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4673, 4773, 3317, 3296, 2638, 3311, 32, 2635, 8, 2633} \[ \frac {5 e^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {11 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {22 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {3 b^2 e^3 x \left (1-c^2 x^2\right )}{4 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {68 b^2 e^3 \left (1-c^2 x^2\right )}{9 c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {3 b^2 e^3 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt {c d x+d} \sqrt {e-c e x}} \]

Antiderivative was successfully verified.

[In]

Int[((e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/Sqrt[d + c*d*x],x]

[Out]

(-68*b^2*e^3*(1 - c^2*x^2))/(9*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (3*b^2*e^3*x*(1 - c^2*x^2))/(4*Sqrt[d + c*
d*x]*Sqrt[e - c*e*x]) + (2*b^2*e^3*(1 - c^2*x^2)^2)/(27*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (3*b^2*e^3*Sqrt[1
- c^2*x^2]*ArcSin[c*x])/(4*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (22*b*e^3*x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c
*x]))/(3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (3*b*c*e^3*x^2*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(2*Sqrt[d +
c*d*x]*Sqrt[e - c*e*x]) - (2*b*c^2*e^3*x^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(9*Sqrt[d + c*d*x]*Sqrt[e -
c*e*x]) + (11*e^3*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) - (3*e^3*x*(1 - c
^2*x^2)*(a + b*ArcSin[c*x])^2)/(2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (c*e^3*x^2*(1 - c^2*x^2)*(a + b*ArcSin[c*
x])^2)/(3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]) + (5*e^3*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])^3)/(6*b*c*Sqrt[d + c
*d*x]*Sqrt[e - c*e*x])

Rule 8

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

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 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 4773

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

Rubi steps

\begin {align*} \int \frac {(e-c e x)^{5/2} \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)^3 \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} \operatorname {Subst}\left (\int (a+b x)^2 (c e-c e \sin (x))^3 \, dx,x,\sin ^{-1}(c x)\right )}{c^4 \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \left (c^3 e^3 (a+b x)^2-3 c^3 e^3 (a+b x)^2 \sin (x)+3 c^3 e^3 (a+b x)^2 \sin ^2(x)-c^3 e^3 (a+b x)^2 \sin ^3(x)\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^4 \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {e^3 \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 (e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin ^3(x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (3 e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (3 e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 e^3 \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 {3 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {e^3 \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 e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (3 e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^2 \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (6 b e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (2 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \sin ^3(x) \, dx,x,\sin ^{-1}(c x)\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (3 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \sin ^2(x) \, dx,x,\sin ^{-1}(c x)\right )}{2 c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=\frac {3 b^2 e^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {6 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {11 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 e^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (4 b e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x) \cos (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (2 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt {1-c^2 x^2}\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {\left (3 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int 1 \, dx,x,\sin ^{-1}(c x)\right )}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (6 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {56 b^2 e^3 \left (1-c^2 x^2\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b^2 e^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 b^2 e^3 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {22 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {11 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 e^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {\left (4 b^2 e^3 \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int \sin (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}\\ &=-\frac {68 b^2 e^3 \left (1-c^2 x^2\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b^2 e^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b^2 e^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 b^2 e^3 \sqrt {1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {22 b e^3 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c e^3 x^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b c^2 e^3 x^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {11 e^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 e^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {c e^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 e^3 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}}\\ \end {align*}

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Mathematica [A]  time = 3.62, size = 473, normalized size = 0.85 \[ \frac {e^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (72 a^2 c^2 x^2 \sqrt {1-c^2 x^2}-324 a^2 c x \sqrt {1-c^2 x^2}+792 a^2 \sqrt {1-c^2 x^2}-1620 a b c x+12 a b \sin \left (3 \sin ^{-1}(c x)\right )-162 a b \cos \left (2 \sin ^{-1}(c x)\right )-1620 b^2 \sqrt {1-c^2 x^2}+81 b^2 \sin \left (2 \sin ^{-1}(c x)\right )+4 b^2 \cos \left (3 \sin ^{-1}(c x)\right )\right )-540 a^2 \sqrt {d} e^{5/2} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )+18 b e^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^2 \left (30 a+45 b \sqrt {1-c^2 x^2}-9 b \sin \left (2 \sin ^{-1}(c x)\right )-b \cos \left (3 \sin ^{-1}(c x)\right )\right )-6 b e^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x) \left (108 a c x \sqrt {1-c^2 x^2}-270 a \sqrt {1-c^2 x^2}+6 a \cos \left (3 \sin ^{-1}(c x)\right )+8 b c^3 x^3+264 b c x+27 b \cos \left (2 \sin ^{-1}(c x)\right )\right )+180 b^2 e^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^3}{216 c d \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/Sqrt[d + c*d*x],x]

[Out]

(180*b^2*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 540*a^2*Sqrt[d]*e^(5/2)*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))] - 6*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e
*x]*ArcSin[c*x]*(264*b*c*x + 8*b*c^3*x^3 - 270*a*Sqrt[1 - c^2*x^2] + 108*a*c*x*Sqrt[1 - c^2*x^2] + 27*b*Cos[2*
ArcSin[c*x]] + 6*a*Cos[3*ArcSin[c*x]]) + 18*b*e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(30*a + 45*b*S
qrt[1 - c^2*x^2] - b*Cos[3*ArcSin[c*x]] - 9*b*Sin[2*ArcSin[c*x]]) + e^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(-1620
*a*b*c*x + 792*a^2*Sqrt[1 - c^2*x^2] - 1620*b^2*Sqrt[1 - c^2*x^2] - 324*a^2*c*x*Sqrt[1 - c^2*x^2] + 72*a^2*c^2
*x^2*Sqrt[1 - c^2*x^2] - 162*a*b*Cos[2*ArcSin[c*x]] + 4*b^2*Cos[3*ArcSin[c*x]] + 81*b^2*Sin[2*ArcSin[c*x]] + 1
2*a*b*Sin[3*ArcSin[c*x]]))/(216*c*d*Sqrt[1 - c^2*x^2])

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fricas [F]  time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c^{2} e^{2} x^{2} - 2 \, a^{2} c e^{2} x + a^{2} e^{2} + {\left (b^{2} c^{2} e^{2} x^{2} - 2 \, b^{2} c e^{2} x + b^{2} e^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} e^{2} x^{2} - 2 \, a b c e^{2} x + a b e^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {-c e x + e}}{\sqrt {c d x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2),x, algorithm="fricas")

[Out]

integral((a^2*c^2*e^2*x^2 - 2*a^2*c*e^2*x + a^2*e^2 + (b^2*c^2*e^2*x^2 - 2*b^2*c*e^2*x + b^2*e^2)*arcsin(c*x)^
2 + 2*(a*b*c^2*e^2*x^2 - 2*a*b*c*e^2*x + a*b*e^2)*arcsin(c*x))*sqrt(-c*e*x + e)/sqrt(c*d*x + d), 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*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2),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 = 0.36, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c e x +e \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {c d x +d}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2),x)

[Out]

int((-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{6} \, {\left (\frac {2 \, \sqrt {-c^{2} d e x^{2} + d e} c e^{2} x^{2}}{d} - \frac {9 \, \sqrt {-c^{2} d e x^{2} + d e} e^{2} x}{d} + \frac {15 \, e^{3} \arcsin \left (c x\right )}{\sqrt {d e} c} + \frac {22 \, \sqrt {-c^{2} d e x^{2} + d e} e^{2}}{c d}\right )} a^{2} + \sqrt {d} \sqrt {e} \int \frac {{\left ({\left (b^{2} c^{2} e^{2} x^{2} - 2 \, b^{2} c e^{2} x + b^{2} e^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} e^{2} x^{2} - 2 \, a b c e^{2} x + a b e^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}}{c d x + d}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2/(c*d*x+d)^(1/2),x, algorithm="maxima")

[Out]

1/6*(2*sqrt(-c^2*d*e*x^2 + d*e)*c*e^2*x^2/d - 9*sqrt(-c^2*d*e*x^2 + d*e)*e^2*x/d + 15*e^3*arcsin(c*x)/(sqrt(d*
e)*c) + 22*sqrt(-c^2*d*e*x^2 + d*e)*e^2/(c*d))*a^2 + sqrt(d)*sqrt(e)*integrate(((b^2*c^2*e^2*x^2 - 2*b^2*c*e^2
*x + b^2*e^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^2*e^2*x^2 - 2*a*b*c*e^2*x + a*b*e^2)*arc
tan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c*d*x + d), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (e-c\,e\,x\right )}^{5/2}}{\sqrt {d+c\,d\,x}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*asin(c*x))^2*(e - c*e*x)^(5/2))/(d + c*d*x)^(1/2),x)

[Out]

int(((a + b*asin(c*x))^2*(e - c*e*x)^(5/2))/(d + c*d*x)^(1/2), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c*e*x+e)**(5/2)*(a+b*asin(c*x))**2/(c*d*x+d)**(1/2),x)

[Out]

Timed out

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