3.3.51 \(\int (c e+d e x)^2 (a+b \text {ArcSin}(c+d x))^{5/2} \, dx\) [251]

Optimal. Leaf size=427 \[ -\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \text {ArcSin}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \text {ArcSin}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{5/2}}{3 d}+\frac {15 b^{5/2} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {5 b^{5/2} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{144 d}-\frac {15 b^{5/2} e^2 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 d}+\frac {5 b^{5/2} e^2 \sqrt {\frac {\pi }{6}} \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{144 d} \]

[Out]

1/3*e^2*(d*x+c)^3*(a+b*arcsin(d*x+c))^(5/2)/d-5/864*b^(5/2)*e^2*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)*(a+b*arcs
in(d*x+c))^(1/2)/b^(1/2))*6^(1/2)*Pi^(1/2)/d+5/864*b^(5/2)*e^2*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(
1/2)/b^(1/2))*sin(3*a/b)*6^(1/2)*Pi^(1/2)/d+15/32*b^(5/2)*e^2*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d
*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d-15/32*b^(5/2)*e^2*FresnelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)
/b^(1/2))*sin(a/b)*2^(1/2)*Pi^(1/2)/d+5/9*b*e^2*(a+b*arcsin(d*x+c))^(3/2)*(1-(d*x+c)^2)^(1/2)/d+5/18*b*e^2*(d*
x+c)^2*(a+b*arcsin(d*x+c))^(3/2)*(1-(d*x+c)^2)^(1/2)/d-5/6*b^2*e^2*(d*x+c)*(a+b*arcsin(d*x+c))^(1/2)/d-5/36*b^
2*e^2*(d*x+c)^3*(a+b*arcsin(d*x+c))^(1/2)/d

________________________________________________________________________________________

Rubi [A]
time = 0.88, antiderivative size = 427, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 13, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {4889, 12, 4725, 4795, 4767, 4715, 4809, 3387, 3386, 3432, 3385, 3433, 3393} \begin {gather*} -\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^2 \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{16 d}+\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} e^2 \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{144 d}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {5 \sqrt {\frac {\pi }{6}} b^{5/2} e^2 \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{144 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \text {ArcSin}(c+d x)}}{36 d}-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \text {ArcSin}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{5/2}}{3 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} (c+d x)^2 (a+b \text {ArcSin}(c+d x))^{3/2}}{18 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{3/2}}{9 d} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-5*b^2*e^2*(c + d*x)*Sqrt[a + b*ArcSin[c + d*x]])/(6*d) - (5*b^2*e^2*(c + d*x)^3*Sqrt[a + b*ArcSin[c + d*x]])
/(36*d) + (5*b*e^2*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(9*d) + (5*b*e^2*(c + d*x)^2*Sqrt[1 -
(c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(18*d) + (e^2*(c + d*x)^3*(a + b*ArcSin[c + d*x])^(5/2))/(3*d) + (
15*b^(5/2)*e^2*Sqrt[Pi/2]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(16*d) - (5*b^(
5/2)*e^2*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(144*d) - (15*b^(
5/2)*e^2*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/(16*d) + (5*b^(5/2)*e
^2*Sqrt[Pi/6]*FresnelC[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/(144*d)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 3385

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Cos[f*(x^2/d)],
x], x, Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3386

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[Sin[f*(x^2/d)], x], x,
Sqrt[c + d*x]], x] /; FreeQ[{c, d, e, f}, x] && ComplexFreeQ[f] && EqQ[d*e - c*f, 0]

Rule 3387

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[Cos[(d*e - c*f)/d], Int[Sin[c*(f/d) +
f*x]/Sqrt[c + d*x], x], x] + Dist[Sin[(d*e - c*f)/d], Int[Cos[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] /; FreeQ[{c
, d, e, f}, x] && ComplexFreeQ[f] && NeQ[d*e - c*f, 0]

Rule 3393

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sin
[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f, m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1])
)

Rule 3432

Int[Sin[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelS[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 3433

Int[Cos[(d_.)*((e_.) + (f_.)*(x_))^2], x_Symbol] :> Simp[(Sqrt[Pi/2]/(f*Rt[d, 2]))*FresnelC[Sqrt[2/Pi]*Rt[d, 2
]*(e + f*x)], x] /; FreeQ[{d, e, f}, x]

Rule 4715

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSin[c*x])^n, x] - Dist[b*c*n, Int[
x*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 4725

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)*((a + b*ArcSin[c*x])^n/(m
+ 1)), x] - Dist[b*c*(n/(m + 1)), Int[x^(m + 1)*((a + b*ArcSin[c*x])^(n - 1)/Sqrt[1 - c^2*x^2]), x], x] /; Fre
eQ[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 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]

Rule 4795

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

Rule 4809

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

Rule 4889

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

Rubi steps

\begin {align*} \int (c e+d e x)^2 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac {\text {Subst}\left (\int e^2 x^2 \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int x^2 \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b e^2\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{6 d}\\ &=\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b e^2\right ) \text {Subst}\left (\int \frac {x \left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{9 d}-\frac {\left (5 b^2 e^2\right ) \text {Subst}\left (\int x^2 \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{12 d}\\ &=-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b^2 e^2\right ) \text {Subst}\left (\int \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{6 d}+\frac {\left (5 b^3 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{72 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {\left (5 b^3 e^2\right ) \text {Subst}\left (\int \frac {\sin ^3(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{72 d}+\frac {\left (5 b^3 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{12 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {\left (5 b^3 e^2\right ) \text {Subst}\left (\int \left (\frac {3 \sin (x)}{4 \sqrt {a+b x}}-\frac {\sin (3 x)}{4 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{72 d}+\frac {\left (5 b^3 e^2\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{12 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}-\frac {\left (5 b^3 e^2\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{288 d}+\frac {\left (5 b^3 e^2\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 d}+\frac {\left (5 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{12 d}-\frac {\left (5 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{12 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {\left (5 b^2 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{6 d}+\frac {\left (5 b^3 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 d}-\frac {\left (5 b^3 e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{288 d}-\frac {\left (5 b^2 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{6 d}-\frac {\left (5 b^3 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{96 d}+\frac {\left (5 b^3 e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{288 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {5 b^{5/2} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{6 d}-\frac {5 b^{5/2} e^2 \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{6 d}+\frac {\left (5 b^2 e^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{48 d}-\frac {\left (5 b^2 e^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{144 d}-\frac {\left (5 b^2 e^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{48 d}+\frac {\left (5 b^2 e^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{144 d}\\ &=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{6 d}-\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{9 d}+\frac {5 b e^2 (c+d x)^2 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{3 d}+\frac {15 b^{5/2} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{16 d}-\frac {5 b^{5/2} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{144 d}-\frac {15 b^{5/2} e^2 \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{16 d}+\frac {5 b^{5/2} e^2 \sqrt {\frac {\pi }{6}} C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{144 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.17, size = 249, normalized size = 0.58 \begin {gather*} \frac {b^3 e^2 e^{-\frac {3 i a}{b}} \left (-81 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {7}{2},-\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )-81 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {7}{2},\frac {i (a+b \text {ArcSin}(c+d x))}{b}\right )+\sqrt {3} \left (\sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {7}{2},-\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {7}{2},\frac {3 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )\right )}{648 d \sqrt {a+b \text {ArcSin}(c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(b^3*e^2*(-81*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-I)*(a + b*ArcSin[c + d*x]))
/b] - 81*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, (I*(a + b*ArcSin[c + d*x]))/b] + Sqrt[
3]*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-3*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((6*I)*a)/b)*Sq
rt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((3*I)*(a + b*ArcSin[c + d*x]))/b])))/(648*d*E^(((3*I)*a)/b)*Sqrt
[a + b*ArcSin[c + d*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(878\) vs. \(2(347)=694\).
time = 0.53, size = 879, normalized size = 2.06

method	result	size
default	\(-\frac {e^{2} \left (-5 \sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b^{3}-5 \sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) b^{3}+405 \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{3}+405 \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, b^{3}+216 \arcsin \left (d x +c \right )^{3} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{3}-72 \arcsin \left (d x +c \right )^{3} \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) b^{3}+648 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{2}-540 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{3}-216 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a \,b^{2}+60 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) b^{3}+648 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b -810 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b^{3}-1080 \arcsin \left (d x +c \right ) \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{2}-216 \arcsin \left (d x +c \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a^{2} b +30 \arcsin \left (d x +c \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) b^{3}+120 \arcsin \left (d x +c \right ) \cos \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a \,b^{2}+216 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{3}-810 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a \,b^{2}-540 \cos \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a^{2} b -72 \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a^{3}+30 \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a \,b^{2}+60 \cos \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a^{2} b \right )}{864 d \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(879\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/864/d*e^2*(-5*(-3/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1/2
)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3-5*(-3/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(
3*a/b)*FresnelC(3*2^(1/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3+405*(a+b*arcsin(d*x+c))^(1/2)
*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*b^
3+405*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(a/b)*FresnelC(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d
*x+c))^(1/2)/b)*(-1/b)^(1/2)*b^3+216*arcsin(d*x+c)^3*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^3-72*arcsin(d*x+c)^3*si
n(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*b^3+648*arcsin(d*x+c)^2*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a*b^2-540*arcsin(d*x
+c)^2*cos(-(a+b*arcsin(d*x+c))/b+a/b)*b^3-216*arcsin(d*x+c)^2*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a*b^2+60*arc
sin(d*x+c)^2*cos(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*b^3+648*arcsin(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*b-8
10*arcsin(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b^3-1080*arcsin(d*x+c)*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a*b^2-
216*arcsin(d*x+c)*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a^2*b+30*arcsin(d*x+c)*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/
b)*b^3+120*arcsin(d*x+c)*cos(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a*b^2+216*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a^3-810
*sin(-(a+b*arcsin(d*x+c))/b+a/b)*a*b^2-540*cos(-(a+b*arcsin(d*x+c))/b+a/b)*a^2*b-72*sin(-3*(a+b*arcsin(d*x+c))
/b+3*a/b)*a^3+30*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a*b^2+60*cos(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a^2*b)/(a+b*
arcsin(d*x+c))^(1/2)

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} e^{2} \left (\int a^{2} c^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} c^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 2 a^{2} c d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 2 b^{2} c d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 4 a b c d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e**2*(Integral(a**2*c**2*sqrt(a + b*asin(c + d*x)), x) + Integral(a**2*d**2*x**2*sqrt(a + b*asin(c + d*x)), x)
+ Integral(b**2*c**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2, x) + Integral(2*a*b*c**2*sqrt(a + b*asin(c +
d*x))*asin(c + d*x), x) + Integral(2*a**2*c*d*x*sqrt(a + b*asin(c + d*x)), x) + Integral(b**2*d**2*x**2*sqrt(
a + b*asin(c + d*x))*asin(c + d*x)**2, x) + Integral(2*a*b*d**2*x**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x),
x) + Integral(2*b**2*c*d*x*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2, x) + Integral(4*a*b*c*d*x*sqrt(a + b*as
in(c + d*x))*asin(c + d*x), x))

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Giac [C] Result contains complex when optimal does not.
time = 2.04, size = 2826, normalized size = 6.62 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/576*(72*sqrt(2)*sqrt(pi)*a^3*b^2*e^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(
2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b))) + 72*sqrt(2)*
sqrt(pi)*a^3*b^2*e^2*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*
x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b))) + 216*I*sqrt(2)*sqrt(pi)*a^2*b
^2*e^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*s
qrt(abs(b))/b)*e^(I*a/b)/(I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) - 216*I*sqrt(2)*sqrt(pi)*a^2*b^2*e^2*erf(1/2*I*
sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(
-I*a/b)/(-I*b^2/sqrt(abs(b)) + b*sqrt(abs(b))) + 24*I*sqrt(b*arcsin(d*x + c) + a)*b^2*e^2*arcsin(d*x + c)^2*e^
(3*I*arcsin(d*x + c)) - 72*I*sqrt(b*arcsin(d*x + c) + a)*b^2*e^2*arcsin(d*x + c)^2*e^(I*arcsin(d*x + c)) + 72*
I*sqrt(b*arcsin(d*x + c) + a)*b^2*e^2*arcsin(d*x + c)^2*e^(-I*arcsin(d*x + c)) - 24*I*sqrt(b*arcsin(d*x + c) +
a)*b^2*e^2*arcsin(d*x + c)^2*e^(-3*I*arcsin(d*x + c)) - 144*sqrt(pi)*a^3*sqrt(b)*e^2*erf(-1/2*sqrt(6)*sqrt(b*
arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*
b + I*sqrt(6)*b^2/abs(b)) - 144*I*sqrt(pi)*a^2*b^(3/2)*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b
) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*b + I*sqrt(6)*b^2/abs(b)) -
216*I*sqrt(2)*sqrt(pi)*a^2*b*e^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sq
rt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b))) - 135*I*sqrt(2)*sqrt(pi)
*b^3*e^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)
*sqrt(abs(b))/b)*e^(I*a/b)/(I*b/sqrt(abs(b)) + sqrt(abs(b))) + 216*I*sqrt(2)*sqrt(pi)*a^2*b*e^2*erf(1/2*I*sqrt
(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a
/b)/(-I*b/sqrt(abs(b)) + sqrt(abs(b))) + 135*I*sqrt(2)*sqrt(pi)*b^3*e^2*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x +
c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/(-I*b/sqrt(abs(b)) +
sqrt(abs(b))) - 144*sqrt(pi)*a^3*sqrt(b)*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + 1/2*I*sqr
t(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*b - I*sqrt(6)*b^2/abs(b)) + 144*I*sqrt(
pi)*a^2*b^(3/2)*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c
) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*b - I*sqrt(6)*b^2/abs(b)) + 48*I*sqrt(b*arcsin(d*x + c) + a)*a*b*
e^2*arcsin(d*x + c)*e^(3*I*arcsin(d*x + c)) - 20*sqrt(b*arcsin(d*x + c) + a)*b^2*e^2*arcsin(d*x + c)*e^(3*I*ar
csin(d*x + c)) - 144*I*sqrt(b*arcsin(d*x + c) + a)*a*b*e^2*arcsin(d*x + c)*e^(I*arcsin(d*x + c)) + 180*sqrt(b*
arcsin(d*x + c) + a)*b^2*e^2*arcsin(d*x + c)*e^(I*arcsin(d*x + c)) + 144*I*sqrt(b*arcsin(d*x + c) + a)*a*b*e^2
*arcsin(d*x + c)*e^(-I*arcsin(d*x + c)) + 180*sqrt(b*arcsin(d*x + c) + a)*b^2*e^2*arcsin(d*x + c)*e^(-I*arcsin
(d*x + c)) - 48*I*sqrt(b*arcsin(d*x + c) + a)*a*b*e^2*arcsin(d*x + c)*e^(-3*I*arcsin(d*x + c)) - 20*sqrt(b*arc
sin(d*x + c) + a)*b^2*e^2*arcsin(d*x + c)*e^(-3*I*arcsin(d*x + c)) + 144*sqrt(pi)*a^3*e^2*erf(-1/2*sqrt(6)*sqr
t(b*arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt
(6)*sqrt(b) + I*sqrt(6)*b^(3/2)/abs(b)) + 144*I*sqrt(pi)*a^2*b*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a
)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*sqrt(b) + I*sqrt(6)
*b^(3/2)/abs(b)) + 36*sqrt(pi)*a*b^2*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(6)*
sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/(sqrt(6)*sqrt(b) + I*sqrt(6)*b^(3/2)/abs(b)) - 144*sqr
t(pi)*a^3*e^2*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c)
+ a)*sqrt(abs(b))/b)*e^(I*a/b)/(I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b))) - 144*sqrt(pi)*a^3*e^2*erf(1
/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b
)*e^(-I*a/b)/(-I*sqrt(2)*b/sqrt(abs(b)) + sqrt(2)*sqrt(abs(b))) + 144*sqrt(pi)*a^3*e^2*erf(-1/2*sqrt(6)*sqrt(b
*arcsin(d*x + c) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6
)*sqrt(b) - I*sqrt(6)*b^(3/2)/abs(b)) - 144*I*sqrt(pi)*a^2*b*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/
sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*sqrt(b) - I*sqrt(6)*
b^(3/2)/abs(b)) + 36*sqrt(pi)*a*b^2*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + 1/2*I*sqrt(6)*s
qrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-3*I*a/b)/(sqrt(6)*sqrt(b) - I*sqrt(6)*b^(3/2)/abs(b)) - 36*sqrt
(pi)*a*b^(3/2)*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcs...

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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