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

Optimal. Leaf size=613 \[ \frac {1}{4} c^2 d^2 x^3 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {3 b c d^2 x^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {4 b d^2 x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {5 d^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt {1-c^2 x^2}}-\frac {2 d^2 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}-\frac {4 b c^2 d^2 x^3 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}-\frac {b c^3 d^2 x^4 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {c d x+d} \sqrt {e-c e x}+\frac {4 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x}}{27 c}+\frac {15 b^2 d^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}-\frac {15}{64} b^2 d^2 x \sqrt {c d x+d} \sqrt {e-c e x}+\frac {8 b^2 d^2 \sqrt {c d x+d} \sqrt {e-c e x}}{9 c} \]

[Out]

8/9*b^2*d^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c-15/64*b^2*d^2*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)-1/32*b^2*c^2*d
^2*x^3*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+4/27*b^2*d^2*(-c^2*x^2+1)*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c+3/8*d^2*x
*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)+1/4*c^2*d^2*x^3*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*
e*x+e)^(1/2)-2/3*d^2*(-c^2*x^2+1)*(a+b*arcsin(c*x))^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c+15/64*b^2*d^2*arcsin(
c*x)*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/c/(-c^2*x^2+1)^(1/2)+4/3*b*d^2*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)-3/8*b*c*d^2*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)-4/9*b*c^2*d^2*x^3*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)-1/8*b*c^3*d^2*x
^4*(a+b*arcsin(c*x))*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)/(-c^2*x^2+1)^(1/2)+5/24*d^2*(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)

________________________________________________________________________________________

Rubi [A]  time = 1.01, antiderivative size = 613, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 13, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {4673, 4763, 4647, 4641, 4627, 321, 216, 4677, 4645, 444, 43, 4697, 4707} \[ -\frac {b c^3 d^2 x^4 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {1}{4} c^2 d^2 x^3 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {4 b c^2 d^2 x^3 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt {1-c^2 x^2}}-\frac {3 b c d^2 x^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {4 b d^2 x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt {1-c^2 x^2}}+\frac {5 d^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt {1-c^2 x^2}}-\frac {2 d^2 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac {3}{8} d^2 x \sqrt {c d x+d} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {c d x+d} \sqrt {e-c e x}+\frac {4 b^2 d^2 \left (1-c^2 x^2\right ) \sqrt {c d x+d} \sqrt {e-c e x}}{27 c}+\frac {15 b^2 d^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}-\frac {15}{64} b^2 d^2 x \sqrt {c d x+d} \sqrt {e-c e x}+\frac {8 b^2 d^2 \sqrt {c d x+d} \sqrt {e-c e x}}{9 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8*b^2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(9*c) - (15*b^2*d^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/64 - (b^2*c
^2*d^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/32 + (4*b^2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(27
*c) + (15*b^2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(64*c*Sqrt[1 - c^2*x^2]) + (4*b*d^2*x*Sqrt[d +
c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(3*Sqrt[1 - c^2*x^2]) - (3*b*c*d^2*x^2*Sqrt[d + c*d*x]*Sqrt[e - c*
e*x]*(a + b*ArcSin[c*x]))/(8*Sqrt[1 - c^2*x^2]) - (4*b*c^2*d^2*x^3*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcS
in[c*x]))/(9*Sqrt[1 - c^2*x^2]) - (b*c^3*d^2*x^4*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(8*Sqrt[
1 - c^2*x^2]) + (3*d^2*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/8 + (c^2*d^2*x^3*Sqrt[d + c*d*
x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^2)/4 - (2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcS
in[c*x])^2)/(3*c) + (5*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x])^3)/(24*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 4697

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

Rule 4707

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

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 (d+c d x)^{5/2} \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 (d+c d x)^2 \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 (d^2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+2 c d^2 x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+c^2 d^2 x^2 \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 (d^2 \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 (2 c d^2 \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 {\left (c^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^2 \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} d^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d^2 \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 (d^2 \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 (4 b d^2 \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 d^2 \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 {\left (c^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (b c^3 d^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x^3 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{2 \sqrt {1-c^2 x^2}}\\ &=\frac {4 b d^2 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 d^2 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 {4 b c^2 d^2 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 {b c^3 d^2 x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d^2 \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 {d^2 \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 (d^2 \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}{8 \sqrt {1-c^2 x^2}}+\frac {\left (b c d^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{4 \sqrt {1-c^2 x^2}}-\frac {\left (4 b^2 c d^2 \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 d^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 {\left (b^2 c^4 d^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^4}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {1}{4} b^2 d^2 x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {4 b d^2 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 {3 b c d^2 x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {4 b c^2 d^2 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 {b c^3 d^2 x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d^2 \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 {5 d^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt {1-c^2 x^2}}+\frac {\left (b^2 d^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 {\left (2 b^2 c d^2 \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 {\left (3 b^2 c^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{32 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 c^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{8 \sqrt {1-c^2 x^2}}\\ &=-\frac {15}{64} b^2 d^2 x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {b^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{4 c \sqrt {1-c^2 x^2}}+\frac {4 b d^2 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 {3 b c d^2 x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {4 b c^2 d^2 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 {b c^3 d^2 x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d^2 \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 {5 d^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt {1-c^2 x^2}}+\frac {\left (3 b^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{64 \sqrt {1-c^2 x^2}}-\frac {\left (b^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{16 \sqrt {1-c^2 x^2}}-\frac {\left (2 b^2 c d^2 \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 {8 b^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x}}{9 c}-\frac {15}{64} b^2 d^2 x \sqrt {d+c d x} \sqrt {e-c e x}-\frac {1}{32} b^2 c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x}+\frac {4 b^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{27 c}+\frac {15 b^2 d^2 \sqrt {d+c d x} \sqrt {e-c e x} \sin ^{-1}(c x)}{64 c \sqrt {1-c^2 x^2}}+\frac {4 b d^2 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 {3 b c d^2 x^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}-\frac {4 b c^2 d^2 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 {b c^3 d^2 x^4 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{8 \sqrt {1-c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {1}{4} c^2 d^2 x^3 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac {2 d^2 \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 {5 d^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{24 b c \sqrt {1-c^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 2.60, size = 555, normalized size = 0.91 \[ \frac {d^2 \sqrt {c d x+d} \sqrt {e-c e x} \left (3 \left (1536 a^2 c^2 x^2 \sqrt {1-c^2 x^2}+864 a^2 c x \sqrt {1-c^2 x^2}-1536 a^2 \sqrt {1-c^2 x^2}+576 a^2 c^3 x^3 \sqrt {1-c^2 x^2}-1024 a b c^3 x^3+3072 a b c x-36 a b \cos \left (4 \sin ^{-1}(c x)\right )+2304 b^2 \sqrt {1-c^2 x^2}-288 b^2 \sin \left (2 \sin ^{-1}(c x)\right )+9 b^2 \sin \left (4 \sin ^{-1}(c x)\right )\right )+1728 a b \cos \left (2 \sin ^{-1}(c x)\right )+256 b^2 \cos \left (3 \sin ^{-1}(c x)\right )\right )-4320 a^2 d^{5/2} \sqrt {e} \sqrt {1-c^2 x^2} \tan ^{-1}\left (\frac {c x \sqrt {c d x+d} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (c^2 x^2-1\right )}\right )-72 b d^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^2 \left (-60 a+48 b \sqrt {1-c^2 x^2}-24 b \sin \left (2 \sin ^{-1}(c x)\right )+3 b \sin \left (4 \sin ^{-1}(c x)\right )+16 b \cos \left (3 \sin ^{-1}(c x)\right )\right )+12 b d^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x) \left (768 a c^2 x^2 \sqrt {1-c^2 x^2}-768 a \sqrt {1-c^2 x^2}+288 a \sin \left (2 \sin ^{-1}(c x)\right )-36 a \sin \left (4 \sin ^{-1}(c x)\right )+576 b c x+64 b \sin \left (3 \sin ^{-1}(c x)\right )+144 b \cos \left (2 \sin ^{-1}(c x)\right )-9 b \cos \left (4 \sin ^{-1}(c x)\right )\right )+1440 b^2 d^2 \sqrt {c d x+d} \sqrt {e-c e x} \sin ^{-1}(c x)^3}{6912 c \sqrt {1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(1440*b^2*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 4320*a^2*d^(5/2)*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTa
n[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 12*b*d^2*Sqrt[d + c*d*x]*Sqrt[e -
c*e*x]*ArcSin[c*x]*(576*b*c*x - 768*a*Sqrt[1 - c^2*x^2] + 768*a*c^2*x^2*Sqrt[1 - c^2*x^2] + 144*b*Cos[2*ArcSin
[c*x]] - 9*b*Cos[4*ArcSin[c*x]] + 288*a*Sin[2*ArcSin[c*x]] + 64*b*Sin[3*ArcSin[c*x]] - 36*a*Sin[4*ArcSin[c*x]]
) - 72*b*d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^2*(-60*a + 48*b*Sqrt[1 - c^2*x^2] + 16*b*Cos[3*ArcSin
[c*x]] - 24*b*Sin[2*ArcSin[c*x]] + 3*b*Sin[4*ArcSin[c*x]]) + d^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1728*a*b*Cos
[2*ArcSin[c*x]] + 256*b^2*Cos[3*ArcSin[c*x]] + 3*(3072*a*b*c*x - 1024*a*b*c^3*x^3 - 1536*a^2*Sqrt[1 - c^2*x^2]
+ 2304*b^2*Sqrt[1 - c^2*x^2] + 864*a^2*c*x*Sqrt[1 - c^2*x^2] + 1536*a^2*c^2*x^2*Sqrt[1 - c^2*x^2] + 576*a^2*c
^3*x^3*Sqrt[1 - c^2*x^2] - 36*a*b*Cos[4*ArcSin[c*x]] - 288*b^2*Sin[2*ArcSin[c*x]] + 9*b^2*Sin[4*ArcSin[c*x]]))
)/(6912*c*Sqrt[1 - c^2*x^2])

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fricas [F]  time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a^{2} c^{2} d^{2} x^{2} + 2 \, a^{2} c d^{2} x + a^{2} d^{2} + {\left (b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \arcsin \left (c x\right )^{2} + 2 \, {\left (a b c^{2} d^{2} x^{2} + 2 \, a b c d^{2} x + a b d^{2}\right )} \arcsin \left (c x\right )\right )} \sqrt {c d x + d} \sqrt {-c e x + e}, x\right ) \]

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), 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)^(5/2)*(a+b*arcsin(c*x))^2*(-c*e*x+e)^(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 = 0.37, size = 0, normalized size = 0.00 \[ \int \left (c d x +d \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2} \sqrt {-c e x +e}\, dx \]

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]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{24} \, {\left (15 \, \sqrt {-c^{2} d e x^{2} + d e} d^{2} x + \frac {15 \, d^{3} e \arcsin \left (c x\right )}{\sqrt {d e} c} - \frac {6 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}} d x}{e} - \frac {16 \, {\left (-c^{2} d e x^{2} + d e\right )}^{\frac {3}{2}} d}{c e}\right )} a^{2} + \sqrt {d} \sqrt {e} \int {\left ({\left (b^{2} c^{2} d^{2} x^{2} + 2 \, b^{2} c d^{2} x + b^{2} d^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} + 2 \, {\left (a b c^{2} d^{2} x^{2} + 2 \, a b c d^{2} x + a b d^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )\right )} \sqrt {c x + 1} \sqrt {-c x + 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/24*(15*sqrt(-c^2*d*e*x^2 + d*e)*d^2*x + 15*d^3*e*arcsin(c*x)/(sqrt(d*e)*c) - 6*(-c^2*d*e*x^2 + d*e)^(3/2)*d*
x/e - 16*(-c^2*d*e*x^2 + d*e)^(3/2)*d/(c*e))*a^2 + sqrt(d)*sqrt(e)*integrate(((b^2*c^2*d^2*x^2 + 2*b^2*c*d^2*x
+ b^2*d^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^2*d^2*x^2 + 2*a*b*c*d^2*x + a*b*d^2)*arcta
n2(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 \[ \int {\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\right )}^{5/2}\,\sqrt {e-c\,e\,x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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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*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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