3.6.52 \(\int (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^2 \, dx\) [552]

Optimal. Leaf size=502 \[ -\frac {1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}-\frac {245 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}-\frac {65 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}+\frac {115 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \text {ArcSin}(c x)}{1152 c \left (1-c^2 x^2\right )^{5/2}}-\frac {5 b c x^2 (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))}{48 c \sqrt {1-c^2 x^2}}+\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^2}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^2}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^3}{48 b c \left (1-c^2 x^2\right )^{5/2}} \]

[Out]

-1/108*b^2*x*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)-245/1152*b^2*x*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)/(-c^2*x^2+1)^2-6
5/1728*b^2*x*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)/(-c^2*x^2+1)+115/1152*b^2*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)*arcsi
n(c*x)/c/(-c^2*x^2+1)^(5/2)-5/16*b*c*x^2*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(5/2)
+1/6*x*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2+5/16*x*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)*(a+b*arcsi
n(c*x))^2/(-c^2*x^2+1)^2+5/24*x*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^2/(-c^2*x^2+1)+5/48*(c*d*x+
d)^(5/2)*(-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))^3/b/c/(-c^2*x^2+1)^(5/2)+5/48*b*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)*(
a+b*arcsin(c*x))/c/(-c^2*x^2+1)^(1/2)+1/18*b*(c*d*x+d)^(5/2)*(-c*e*x+e)^(5/2)*(a+b*arcsin(c*x))*(-c^2*x^2+1)^(
1/2)/c

________________________________________________________________________________________

Rubi [A]
time = 0.37, antiderivative size = 502, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.281, Rules used = {4763, 4743, 4741, 4737, 4723, 327, 222, 4767, 201} \begin {gather*} \frac {5 (c d x+d)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}+\frac {5 x (c d x+d)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^2}{24 \left (1-c^2 x^2\right )}+\frac {5 x (c d x+d)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^2}{16 \left (1-c^2 x^2\right )^2}+\frac {b \sqrt {1-c^2 x^2} (c d x+d)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))}{18 c}+\frac {5 b (c d x+d)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))}{48 c \sqrt {1-c^2 x^2}}-\frac {5 b c x^2 (c d x+d)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac {1}{6} x (c d x+d)^{5/2} (e-c e x)^{5/2} (a+b \text {ArcSin}(c x))^2+\frac {115 b^2 \text {ArcSin}(c x) (c d x+d)^{5/2} (e-c e x)^{5/2}}{1152 c \left (1-c^2 x^2\right )^{5/2}}-\frac {65 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac {245 b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}-\frac {1}{108} b^2 x (c d x+d)^{5/2} (e-c e x)^{5/2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/108*(b^2*x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)) - (245*b^2*x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))/(1152*(1
- c^2*x^2)^2) - (65*b^2*x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2))/(1728*(1 - c^2*x^2)) + (115*b^2*(d + c*d*x)^(5/
2)*(e - c*e*x)^(5/2)*ArcSin[c*x])/(1152*c*(1 - c^2*x^2)^(5/2)) - (5*b*c*x^2*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2
)*(a + b*ArcSin[c*x]))/(16*(1 - c^2*x^2)^(5/2)) + (5*b*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])
)/(48*c*Sqrt[1 - c^2*x^2]) + (b*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x]))/(18
*c) + (x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^2)/6 + (5*x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/
2)*(a + b*ArcSin[c*x])^2)/(16*(1 - c^2*x^2)^2) + (5*x*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^
2)/(24*(1 - c^2*x^2)) + (5*(d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*(a + b*ArcSin[c*x])^3)/(48*b*c*(1 - c^2*x^2)^(5
/2))

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 222

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 327

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 4723

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 4737

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*Si
mp[Sqrt[1 - c^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSin[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[c
^2*d + e, 0] && NeQ[n, -1]

Rule 4741

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[(1/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 - c^2*x^2]], Int[(a + b*ArcSin[c*x])^n/S
qrt[1 - c^2*x^2], x], x] - Dist[b*c*(n/2)*Simp[Sqrt[d + e*x^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 4743

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((
a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (Dist[2*d*(p/(2*p + 1)), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n,
x], x] - Dist[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p], Int[x*(1 - c^2*x^2)^(p - 1/2)*(a + b*ArcS
in[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 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 4767

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/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^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]

Rubi steps

\begin {align*} \int (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx &=\frac {\left ((d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{\left (1-c^2 x^2\right )^{5/2}}\\ &=\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {\left (5 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{6 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (b c (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int x \left (1-c^2 x^2\right )^2 \left (a+b \sin ^{-1}(c x)\right ) \, dx}{3 \left (1-c^2 x^2\right )^{5/2}}\\ &=\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {\left (5 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \, dx}{8 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{5/2} \, dx}{18 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int x \left (1-c^2 x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx}{12 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac {1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}+\frac {5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt {1-c^2 x^2}}+\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {\left (5 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{108 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \left (1-c^2 x^2\right )^{3/2} \, dx}{48 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b c (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int x \left (a+b \sin ^{-1}(c x)\right ) \, dx}{8 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac {1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}-\frac {65 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac {5 b c x^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt {1-c^2 x^2}}+\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{144 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \sqrt {1-c^2 x^2} \, dx}{64 \left (1-c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 c^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac {x^2}{\sqrt {1-c^2 x^2}} \, dx}{16 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac {1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}-\frac {245 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}-\frac {65 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}-\frac {5 b c x^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt {1-c^2 x^2}}+\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{288 \left (1-c^2 x^2\right )^{5/2}}-\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{128 \left (1-c^2 x^2\right )^{5/2}}+\frac {\left (5 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{32 \left (1-c^2 x^2\right )^{5/2}}\\ &=-\frac {1}{108} b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}-\frac {245 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1152 \left (1-c^2 x^2\right )^2}-\frac {65 b^2 x (d+c d x)^{5/2} (e-c e x)^{5/2}}{1728 \left (1-c^2 x^2\right )}+\frac {115 b^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \sin ^{-1}(c x)}{1152 c \left (1-c^2 x^2\right )^{5/2}}-\frac {5 b c x^2 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{16 \left (1-c^2 x^2\right )^{5/2}}+\frac {5 b (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )}{48 c \sqrt {1-c^2 x^2}}+\frac {b (d+c d x)^{5/2} (e-c e x)^{5/2} \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}{18 c}+\frac {1}{6} x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{16 \left (1-c^2 x^2\right )^2}+\frac {5 x (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^2}{24 \left (1-c^2 x^2\right )}+\frac {5 (d+c d x)^{5/2} (e-c e x)^{5/2} \left (a+b \sin ^{-1}(c x)\right )^3}{48 b c \left (1-c^2 x^2\right )^{5/2}}\\ \end {align*}

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Mathematica [A]
time = 1.70, size = 450, normalized size = 0.90 \begin {gather*} \frac {d^2 e^2 \left (1440 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^3-4320 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \text {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 )+12 b \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x) (270 b \cos (2 \text {ArcSin}(c x))+27 b \cos (4 \text {ArcSin}(c x))+2 b \cos (6 \text {ArcSin}(c x))+540 a \sin (2 \text {ArcSin}(c x))+108 a \sin (4 \text {ArcSin}(c x))+12 a \sin (6 \text {ArcSin}(c x)))+72 b \sqrt {d+c d x} \sqrt {e-c e x} \text {ArcSin}(c x)^2 (60 a+45 b \sin (2 \text {ArcSin}(c x))+9 b \sin (4 \text {ArcSin}(c x))+b \sin (6 \text {ArcSin}(c x)))+\sqrt {d+c d x} \sqrt {e-c e x} \left (9504 a^2 c x \sqrt {1-c^2 x^2}-7488 a^2 c^3 x^3 \sqrt {1-c^2 x^2}+2304 a^2 c^5 x^5 \sqrt {1-c^2 x^2}+3240 a b \cos (2 \text {ArcSin}(c x))+324 a b \cos (4 \text {ArcSin}(c x))+24 a b \cos (6 \text {ArcSin}(c x))-1620 b^2 \sin (2 \text {ArcSin}(c x))-81 b^2 \sin (4 \text {ArcSin}(c x))-4 b^2 \sin (6 \text {ArcSin}(c x))\right )\right )}{13824 c \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(d^2*e^2*(1440*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 4320*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))] + 12*b*Sqrt[d + c*d*x]*Sqrt[e -
c*e*x]*ArcSin[c*x]*(270*b*Cos[2*ArcSin[c*x]] + 27*b*Cos[4*ArcSin[c*x]] + 2*b*Cos[6*ArcSin[c*x]] + 540*a*Sin[2
*ArcSin[c*x]] + 108*a*Sin[4*ArcSin[c*x]] + 12*a*Sin[6*ArcSin[c*x]]) + 72*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*Arc
Sin[c*x]^2*(60*a + 45*b*Sin[2*ArcSin[c*x]] + 9*b*Sin[4*ArcSin[c*x]] + b*Sin[6*ArcSin[c*x]]) + Sqrt[d + c*d*x]*
Sqrt[e - c*e*x]*(9504*a^2*c*x*Sqrt[1 - c^2*x^2] - 7488*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] + 2304*a^2*c^5*x^5*Sqrt[1
- c^2*x^2] + 3240*a*b*Cos[2*ArcSin[c*x]] + 324*a*b*Cos[4*ArcSin[c*x]] + 24*a*b*Cos[6*ArcSin[c*x]] - 1620*b^2*
Sin[2*ArcSin[c*x]] - 81*b^2*Sin[4*ArcSin[c*x]] - 4*b^2*Sin[6*ArcSin[c*x]])))/(13824*c*Sqrt[1 - c^2*x^2])

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Maple [F]
time = 0.15, size = 0, normalized size = 0.00 \[\int \left (c d x +d \right )^{\frac {5}{2}} \left (-c e x +e \right )^{\frac {5}{2}} \left (a +b \arcsin \left (c x \right )\right )^{2}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*(15*sqrt(-c^2*d*x^2*e + d*e)*d^2*x*e^2 + 10*(-c^2*d*x^2*e + d*e)^(3/2)*d*x*e + 8*(-c^2*d*x^2*e + d*e)^(5/
2)*x + 15*d^(5/2)*arcsin(c*x)*e^(5/2)/c)*a^2 + sqrt(d)*e^(1/2)*integrate(((b^2*c^4*d^2*x^4*e^2 - 2*b^2*c^2*d^2
*x^2*e^2 + b^2*d^2*e^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*(a*b*c^4*d^2*x^4*e^2 - 2*a*b*c^2*d^2*
x^2*e^2 + a*b*d^2*e^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1), x)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(((b^2*c^4*d^2*x^4 - 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsin(c*x)^2*e^2 + 2*(a*b*c^4*d^2*x^4 - 2*a*b*c^2*d
^2*x^2 + a*b*d^2)*arcsin(c*x)*e^2 + (a^2*c^4*d^2*x^4 - 2*a^2*c^2*d^2*x^2 + a^2*d^2)*e^2)*sqrt(c*d*x + d)*sqrt(
-(c*x - 1)*e), x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)^(5/2),x)

[Out]

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

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