\(\int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx\) [558]

Optimal result

Integrand size = 32, antiderivative size = 559 \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\frac {68 b^2 d^3 \left (1-c^2 x^2\right )}{9 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b^2 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {2 b^2 d^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 b^2 d^3 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {22 b d^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {2 b c^2 d^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {5 d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {d+c d x} \sqrt {e-c e x}} \]

[Out]

68/9*b^2*d^3*(-c^2*x^2+1)/c/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)+3/4*b^2*d^3*x*(-c^2*x^2+1)/(c*d*x+d)^(1/2)/(-c*e*
x+e)^(1/2)-2/27*b^2*d^3*(-c^2*x^2+1)^2/c/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)-11/3*d^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*d^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*d^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*d^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*d^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*d^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*d^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*d^3*(a+b*arcsi
n(c*x))^3*(-c^2*x^2+1)^(1/2)/b/c/(c*d*x+d)^(1/2)/(-c*e*x+e)^(1/2)

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 559, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {4763, 4857, 3398, 3377, 2718, 3392, 32, 2715, 8, 2713} \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\frac {5 d^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {c d^3 x^2 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{2 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{3 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {3 b c d^3 x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {22 b d^3 x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{3 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {2 b c^2 d^3 x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{9 \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {3 b^2 d^3 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c \sqrt {c d x+d} \sqrt {e-c e x}}-\frac {2 b^2 d^3 \left (1-c^2 x^2\right )^2}{27 c \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {3 b^2 d^3 x \left (1-c^2 x^2\right )}{4 \sqrt {c d x+d} \sqrt {e-c e x}}+\frac {68 b^2 d^3 \left (1-c^2 x^2\right )}{9 c \sqrt {c d x+d} \sqrt {e-c e x}} \]

[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 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 2713

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 2715

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] && Integ
erQ[2*n]

Rule 2718

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

Rule 3377

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 3392

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 3398

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 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]

Rule 4857

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*} \text {integral}& = \frac {\sqrt {1-c^2 x^2} \int \frac {(d+c d x)^3 (a+b \arcsin (c x))^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} \text {Subst}\left (\int (a+b x)^2 (c d+c d \sin (x))^3 \, dx,x,\arcsin (c x)\right )}{c^4 \sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {\sqrt {1-c^2 x^2} \text {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,\arcsin (c x)\right )}{c^4 \sqrt {d+c d x} \sqrt {e-c e x}} \\ & = \frac {d^3 \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}}+\frac {\left (d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin ^3(x) \, dx,x,\arcsin (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 ) \text {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\arcsin (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 ) \text {Subst}\left (\int (a+b x)^2 \sin ^2(x) \, dx,x,\arcsin (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} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{9 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2}{c \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {3 d^3 x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {d^3 \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}}+\frac {\left (2 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x)^2 \sin (x) \, dx,x,\arcsin (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 ) \text {Subst}\left (\int (a+b x)^2 \, dx,x,\arcsin (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 ) \text {Subst}(\int (a+b x) \cos (x) \, dx,x,\arcsin (c x))}{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 ) \text {Subst}\left (\int \sin ^3(x) \, dx,x,\arcsin (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 ) \text {Subst}\left (\int \sin ^2(x) \, dx,x,\arcsin (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} (a+b \arcsin (c x))}{\sqrt {d+c d x} \sqrt {e-c e x}}+\frac {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}}+\frac {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}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {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}}+\frac {\left (4 b d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int (a+b x) \cos (x) \, dx,x,\arcsin (c x))}{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 ) \text {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 ) \text {Subst}(\int 1 \, dx,x,\arcsin (c x))}{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 ) \text {Subst}(\int \sin (x) \, dx,x,\arcsin (c x))}{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} \arcsin (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} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{9 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {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}}-\frac {\left (4 b^2 d^3 \sqrt {1-c^2 x^2}\right ) \text {Subst}(\int \sin (x) \, dx,x,\arcsin (c x))}{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} \arcsin (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} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{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} (a+b \arcsin (c x))}{9 \sqrt {d+c d x} \sqrt {e-c e x}}-\frac {11 d^3 \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x))^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 ) (a+b \arcsin (c x))^2}{3 \sqrt {d+c d x} \sqrt {e-c e x}}+\frac {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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 9.93 (sec) , antiderivative size = 434, normalized size of antiderivative = 0.78 \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=-\frac {d^2 \left (-180 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3+540 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )-6 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) \left (-18 b+264 b c x+36 b c^2 x^2+8 b c^3 x^3-270 a \sqrt {1-c^2 x^2}-108 a c x \sqrt {1-c^2 x^2}-9 b \cos (2 \arcsin (c x))+6 a \cos (3 \arcsin (c x))\right )+18 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 \left (-30 a+9 b (5+2 c x) \sqrt {1-c^2 x^2}-b \cos (3 \arcsin (c x))\right )+\sqrt {d+c d x} \sqrt {e-c e x} \left (6 \left (-27 b^2 (10+c x) \sqrt {1-c^2 x^2}-8 a b c x \left (33+c^2 x^2\right )+6 a^2 \sqrt {1-c^2 x^2} \left (22+9 c x+2 c^2 x^2\right )\right )+162 a b \cos (2 \arcsin (c x))+4 b^2 \cos (3 \arcsin (c x))\right )\right )}{216 c e \sqrt {1-c^2 x^2}} \]

[In]

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

[Out]

-1/216*(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
]*ArcTan[(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*Sqr
t[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]*ArcSi
n[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*(-27*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]])))/(c*e*Sqrt[1 - c^2*x^2])

Maple [F]

\[\int \frac {\left (c d x +d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}}{\sqrt {-c e x +e}}d x\]

[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)

Fricas [F]

\[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c e x + e}} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\text {Exception raised: ValueError} \]

[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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\int { \frac {{\left (c d x + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{\sqrt {-c e x + e}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {(d+c d x)^{5/2} (a+b \arcsin (c x))^2}{\sqrt {e-c e x}} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{5/2}}{\sqrt {e-c\,e\,x}} \,d x \]

[In]

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

[Out]

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