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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.659786, antiderivative size = 559, normalized size of antiderivative = 1., 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} \[ \frac{5 d^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{c d^3 x^2 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{3 d^3 x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{2 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{11 d^3 \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right )^2}{3 c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{2 b c^2 d^3 x^3 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{3 b c d^3 x^2 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{22 b d^3 x \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{2 b^2 d^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{3 b^2 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt{c d x+d} \sqrt{e-c e x}}+\frac{68 b^2 d^3 \left (1-c^2 x^2\right )}{9 c \sqrt{c d x+d} \sqrt{e-c e x}}-\frac{3 b^2 d^3 \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{4 c \sqrt{c d x+d} \sqrt{e-c e x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 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 2638

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

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 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 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 8

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

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]

Rubi steps

\begin{align*} \int \frac{(d+c d x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt{e-c e x}} \, dx &=\frac{\sqrt{1-c^2 x^2} \int \frac{(d+c d 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 d+c d \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 d^3 (a+b x)^2+3 c^3 d^3 (a+b x)^2 \sin (x)+3 c^3 d^3 (a+b x)^2 \sin ^2(x)+c^3 d^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{d^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 (d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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{d^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 d^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 d^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 d^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 d^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 d^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 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{6 b d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^3 \left (1-c^2 x^2\right )}{9 c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{3 b^2 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{2 b^2 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^3 \left (1-c^2 x^2\right )}{9 c \sqrt{d+c d x} \sqrt{e-c e x}}+\frac{3 b^2 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt{d+c d x} \sqrt{e-c e x}}-\frac{2 b^2 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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 d^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*}

Mathematica [A]  time = 3.5625, size = 434, normalized size = 0.78 \[ -\frac{d^2 \left (\sqrt{c d x+d} \sqrt{e-c e x} \left (6 \left (6 a^2 \sqrt{1-c^2 x^2} \left (2 c^2 x^2+9 c x+22\right )-8 a b c x \left (c^2 x^2+33\right )-27 b^2 (c x+10) \sqrt{1-c^2 x^2}\right )+162 a b \cos \left (2 \sin ^{-1}(c x)\right )+4 b^2 \cos \left (3 \sin ^{-1}(c x)\right )\right )+540 a^2 \sqrt{d} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )+18 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (-30 a+9 b (2 c x+5) \sqrt{1-c^2 x^2}-b \cos \left (3 \sin ^{-1}(c x)\right )\right )-6 b \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (-108 a c x \sqrt{1-c^2 x^2}-270 a \sqrt{1-c^2 x^2}+6 a \cos \left (3 \sin ^{-1}(c x)\right )+8 b c^3 x^3+36 b c^2 x^2+264 b c x-9 b \cos \left (2 \sin ^{-1}(c x)\right )-18 b\right )-180 b^2 \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3\right )}{216 c e \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d^2*(-180*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 + 540*a^2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcT
an[(c*x*Sqrt[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]*(-18*b + 264*b*c*x + 36*b*c^2*x^2 + 8*b*c^3*x^3 - 270*a*Sqrt[1 - c^2*x^2] - 108*a*c*x*Sqrt[1 -
c^2*x^2] - 9*b*Cos[2*ArcSin[c*x]] + 6*a*Cos[3*ArcSin[c*x]]) + 18*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]
^2*(-30*a + 9*b*(5 + 2*c*x)*Sqrt[1 - c^2*x^2] - b*Cos[3*ArcSin[c*x]]) + Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(6*(-2
7*b^2*(10 + c*x)*Sqrt[1 - c^2*x^2] - 8*a*b*c*x*(33 + c^2*x^2) + 6*a^2*Sqrt[1 - c^2*x^2]*(22 + 9*c*x + 2*c^2*x^
2)) + 162*a*b*Cos[2*ArcSin[c*x]] + 4*b^2*Cos[3*ArcSin[c*x]])))/(216*c*e*Sqrt[1 - c^2*x^2])

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Maple [F]  time = 0.263, size = 0, normalized size = 0. \begin{align*} \int{ \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2} \left ( cdx+d \right ) ^{{\frac{5}{2}}}{\frac{1}{\sqrt{-cex+e}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c*d*x+d)^(5/2)*(a+b*arcsin(c*x))^2/(-c*e*x+e)^(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((c*d*x+d)^(5/2)*(a+b*arcsin(c*x))^2/(-c*e*x+e)^(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 (a^{2} c^{2} d^{2} x^{2} + 2 \, a^{2} c d^{2} x + a^{2} d^{2} +{\left (b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c^{2} d^{2} x^{2} + 2 \, a b c d^{2} x + a b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c e x + e}}{c e x - e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (c d x + d\right )}^{\frac{5}{2}}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt{-c e x + e}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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