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

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

[Out]

(4*b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(9*c) - (b^2*d*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/4 + (2*b^2*d*Sqrt[
d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(27*c) + (b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(4*c*Sq
rt[1 - c^2*x^2]) + (2*b*d*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*
d*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*d*x^3*Sqrt[d + c*d
*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) + (d*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*A
rcSin[c*x])^2)/2 - (d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c) + (d*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])

________________________________________________________________________________________

Rubi [A]  time = 0.571039, antiderivative size = 455, normalized size of antiderivative = 1., 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} \[ -\frac{2 b c^2 d x^3 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{9 \sqrt{1-c^2 x^2}}-\frac{b c d x^2 \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{2 \sqrt{1-c^2 x^2}}+\frac{2 b d x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )}{3 \sqrt{1-c^2 x^2}}+\frac{d \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^3}{6 b c \sqrt{1-c^2 x^2}}-\frac{d \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2}{3 c}+\frac{1}{2} d x \sqrt{c d x+d} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2+\frac{2 b^2 d \left (1-c^2 x^2\right ) \sqrt{c d x+d} \sqrt{e-c e x}}{27 c}+\frac{b^2 d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)}{4 c \sqrt{1-c^2 x^2}}-\frac{1}{4} b^2 d x \sqrt{c d x+d} \sqrt{e-c e x}+\frac{4 b^2 d \sqrt{c d x+d} \sqrt{e-c e x}}{9 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/(9*c) - (b^2*d*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x])/4 + (2*b^2*d*Sqrt[
d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2))/(27*c) + (b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x])/(4*c*Sq
rt[1 - c^2*x^2]) + (2*b*d*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*
d*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*d*x^3*Sqrt[d + c*d
*x]*Sqrt[e - c*e*x]*(a + b*ArcSin[c*x]))/(9*Sqrt[1 - c^2*x^2]) + (d*x*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(a + b*A
rcSin[c*x])^2)/2 - (d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(1 - c^2*x^2)*(a + b*ArcSin[c*x])^2)/(3*c) + (d*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 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 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))

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 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 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 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 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 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 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 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 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])

Rubi steps

\begin{align*} \int (d+c d x)^{3/2} \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) \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 \sqrt{1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2+c d 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 (d \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 d \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} d x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d \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 \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 d \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 \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 d 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 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 d 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} d x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d \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 \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 d \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 \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 d x \sqrt{d+c d x} \sqrt{e-c e x}+\frac{2 b d 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 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 d 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} d x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d \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 \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 d \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 d \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 d x \sqrt{d+c d x} \sqrt{e-c e x}+\frac{b^2 d \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 d 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 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 d 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} d x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d \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 \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 d \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 d \sqrt{d+c d x} \sqrt{e-c e x}}{9 c}-\frac{1}{4} b^2 d x \sqrt{d+c d x} \sqrt{e-c e x}+\frac{2 b^2 d \sqrt{d+c d x} \sqrt{e-c e x} \left (1-c^2 x^2\right )}{27 c}+\frac{b^2 d \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 d 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 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 d 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} d x \sqrt{d+c d x} \sqrt{e-c e x} \left (a+b \sin ^{-1}(c x)\right )^2-\frac{d \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 \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*}

Mathematica [A]  time = 1.78636, size = 437, normalized size = 0.96 \[ \frac{d \sqrt{c d x+d} \sqrt{e-c e x} \left (12 \left (3 a^2 \sqrt{1-c^2 x^2} \left (2 c^2 x^2+3 c x-2\right )-4 a b c x \left (c^2 x^2-3\right )+9 b^2 \sqrt{1-c^2 x^2}\right )+54 a b \cos \left (2 \sin ^{-1}(c x)\right )-27 b^2 \sin \left (2 \sin ^{-1}(c x)\right )+4 b^2 \cos \left (3 \sin ^{-1}(c x)\right )\right )-108 a^2 d^{3/2} \sqrt{e} \sqrt{1-c^2 x^2} \tan ^{-1}\left (\frac{c x \sqrt{c d x+d} \sqrt{e-c e x}}{\sqrt{d} \sqrt{e} \left (c^2 x^2-1\right )}\right )-18 b d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^2 \left (-6 a+3 b \sqrt{1-c^2 x^2}-3 b \sin \left (2 \sin ^{-1}(c x)\right )+b \cos \left (3 \sin ^{-1}(c x)\right )\right )+6 b d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x) \left (2 \left (12 a c^2 x^2 \sqrt{1-c^2 x^2}-12 a \sqrt{1-c^2 x^2}+9 a \sin \left (2 \sin ^{-1}(c x)\right )+9 b c x+b \sin \left (3 \sin ^{-1}(c x)\right )\right )+9 b \cos \left (2 \sin ^{-1}(c x)\right )\right )+36 b^2 d \sqrt{c d x+d} \sqrt{e-c e x} \sin ^{-1}(c x)^3}{216 c \sqrt{1-c^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(36*b^2*d*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 108*a^2*d^(3/2)*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))] - 18*b*d*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]]) + d*Sqrt[d + c*d*
x]*Sqrt[e - c*e*x]*(12*(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)) + 54*a*b*Cos[2*ArcSin[c*x]] + 4*b^2*Cos[3*ArcSin[c*x]] - 27*b^2*Sin[2*ArcSin[c*x]]) + 6*b*d*S
qrt[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] + 12*
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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Maple [F]  time = 0.262, size = 0, normalized size = 0. \begin{align*} \int \left ( cdx+d \right ) ^{{\frac{3}{2}}} \left ( a+b\arcsin \left ( cx \right ) \right ) ^{2}\sqrt{-cex+e}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((c*d*x+d)^(3/2)*(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)^(3/2)*(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 ({\left (a^{2} c d x + a^{2} d +{\left (b^{2} c d x + b^{2} d\right )} \arcsin \left (c x\right )^{2} + 2 \,{\left (a b c d x + a b d\right )} \arcsin \left (c x\right )\right )} \sqrt{c d x + d} \sqrt{-c e x + e}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((a^2*c*d*x + a^2*d + (b^2*c*d*x + b^2*d)*arcsin(c*x)^2 + 2*(a*b*c*d*x + a*b*d)*arcsin(c*x))*sqrt(c*d*
x + d)*sqrt(-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)**(3/2)*(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{\left (c d x + d\right )}^{\frac{3}{2}} \sqrt{-c e x + e}{\left (b \arcsin \left (c x\right ) + a\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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