3.3.50 \(\int (c e+d e x)^3 (a+b \text {ArcSin}(c+d x))^{5/2} \, dx\) [250]

Optimal. Leaf size=475 \[ \frac {225 b^2 e^3 \sqrt {a+b \text {ArcSin}(c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {ArcSin}(c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {ArcSin}(c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \text {ArcSin}(c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \text {ArcSin}(c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {ArcSin}(c+d x))^{5/2}}{4 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{256 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{256 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{4096 d} \]

[Out]

-3/32*e^3*(a+b*arcsin(d*x+c))^(5/2)/d+1/4*e^3*(d*x+c)^4*(a+b*arcsin(d*x+c))^(5/2)/d+15/8192*b^(5/2)*e^3*cos(4*
a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d+15/8192*b^(5/2)*e^3*Fre
snelS(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(4*a/b)*2^(1/2)*Pi^(1/2)/d-15/256*b^(5/2)*e^3*c
os(2*a/b)*FresnelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/d-15/256*b^(5/2)*e^3*FresnelS(2*(a+b
*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/d+15/64*b*e^3*(d*x+c)*(a+b*arcsin(d*x+c))^(3/2)*(1
-(d*x+c)^2)^(1/2)/d+5/32*b*e^3*(d*x+c)^3*(a+b*arcsin(d*x+c))^(3/2)*(1-(d*x+c)^2)^(1/2)/d+225/2048*b^2*e^3*(a+b
*arcsin(d*x+c))^(1/2)/d-45/256*b^2*e^3*(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)/d-15/256*b^2*e^3*(d*x+c)^4*(a+b*arc
sin(d*x+c))^(1/2)/d

________________________________________________________________________________________

Rubi [A]
time = 0.95, antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {4889, 12, 4725, 4795, 4737, 4809, 3393, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^3 \cos \left (\frac {4 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 \sqrt {\pi } b^{5/2} e^3 \cos \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{256 d}-\frac {15 \sqrt {\pi } b^{5/2} e^3 \sin \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{256 d}+\frac {15 \sqrt {\frac {\pi }{2}} b^{5/2} e^3 \sin \left (\frac {4 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \text {ArcSin}(c+d x)}}{256 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \text {ArcSin}(c+d x)}}{256 d}+\frac {225 b^2 e^3 \sqrt {a+b \text {ArcSin}(c+d x)}}{2048 d}+\frac {e^3 (c+d x)^4 (a+b \text {ArcSin}(c+d x))^{5/2}}{4 d}+\frac {5 b e^3 \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \text {ArcSin}(c+d x))^{3/2}}{32 d}+\frac {15 b e^3 \sqrt {1-(c+d x)^2} (c+d x) (a+b \text {ArcSin}(c+d x))^{3/2}}{64 d}-\frac {3 e^3 (a+b \text {ArcSin}(c+d x))^{5/2}}{32 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^(5/2),x]

[Out]

(225*b^2*e^3*Sqrt[a + b*ArcSin[c + d*x]])/(2048*d) - (45*b^2*e^3*(c + d*x)^2*Sqrt[a + b*ArcSin[c + d*x]])/(256
*d) - (15*b^2*e^3*(c + d*x)^4*Sqrt[a + b*ArcSin[c + d*x]])/(256*d) + (15*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]
*(a + b*ArcSin[c + d*x])^(3/2))/(64*d) + (5*b*e^3*(c + d*x)^3*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3
/2))/(32*d) - (3*e^3*(a + b*ArcSin[c + d*x])^(5/2))/(32*d) + (e^3*(c + d*x)^4*(a + b*ArcSin[c + d*x])^(5/2))/(
4*d) + (15*b^(5/2)*e^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/(
4096*d) - (15*b^(5/2)*e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/
(256*d) - (15*b^(5/2)*e^3*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/
(256*d) + (15*b^(5/2)*e^3*Sqrt[Pi/2]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(4*a)/b]
)/(4096*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 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 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 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)^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2} \, dx &=\frac {\text {Subst}\left (\int e^3 x^3 \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int x^3 \left (a+b \sin ^{-1}(x)\right )^{5/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}-\frac {\left (5 b e^3\right ) \text {Subst}\left (\int \frac {x^4 \left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}\\ &=\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}-\frac {\left (15 b e^3\right ) \text {Subst}\left (\int \frac {x^2 \left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{32 d}-\frac {\left (15 b^2 e^3\right ) \text {Subst}\left (\int x^3 \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{64 d}\\ &=-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}-\frac {\left (15 b e^3\right ) \text {Subst}\left (\int \frac {\left (a+b \sin ^{-1}(x)\right )^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{64 d}-\frac {\left (45 b^2 e^3\right ) \text {Subst}\left (\int x \sqrt {a+b \sin ^{-1}(x)} \, dx,x,c+d x\right )}{128 d}+\frac {\left (15 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{512 d}\\ &=-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {\left (15 b^3 e^3\right ) \text {Subst}\left (\int \frac {\sin ^4(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{512 d}+\frac {\left (45 b^3 e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{512 d}\\ &=-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {\left (15 b^3 e^3\right ) \text {Subst}\left (\int \left (\frac {3}{8 \sqrt {a+b x}}-\frac {\cos (2 x)}{2 \sqrt {a+b x}}+\frac {\cos (4 x)}{8 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{512 d}+\frac {\left (45 b^3 e^3\right ) \text {Subst}\left (\int \frac {\sin ^2(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{512 d}\\ &=\frac {45 b^2 e^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {\left (15 b^3 e^3\right ) \text {Subst}\left (\int \frac {\cos (4 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4096 d}-\frac {\left (15 b^3 e^3\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1024 d}+\frac {\left (45 b^3 e^3\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {a+b x}}-\frac {\cos (2 x)}{2 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{512 d}\\ &=\frac {225 b^2 e^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}-\frac {\left (45 b^3 e^3\right ) \text {Subst}\left (\int \frac {\cos (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1024 d}-\frac {\left (15 b^3 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1024 d}+\frac {\left (15 b^3 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}+4 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4096 d}-\frac {\left (15 b^3 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1024 d}+\frac {\left (15 b^3 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}+4 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{4096 d}\\ &=\frac {225 b^2 e^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}-\frac {\left (15 b^2 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{512 d}-\frac {\left (45 b^3 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1024 d}+\frac {\left (15 b^2 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{2048 d}-\frac {\left (15 b^2 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{512 d}-\frac {\left (45 b^3 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{1024 d}+\frac {\left (15 b^2 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{2048 d}\\ &=\frac {225 b^2 e^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{1024 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{1024 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{4096 d}-\frac {\left (45 b^2 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{512 d}-\frac {\left (45 b^2 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{512 d}\\ &=\frac {225 b^2 e^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \sin ^{-1}(c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{5/2}}{4 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{256 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{256 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{4096 d}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 0.19, size = 269, normalized size = 0.57 \begin {gather*} -\frac {e^3 e^{-\frac {4 i a}{b}} (a+b \text {ArcSin}(c+d x))^{5/2} \left (-16 \sqrt {2} e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {7}{2},-\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )-16 \sqrt {2} e^{\frac {6 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {7}{2},\frac {2 i (a+b \text {ArcSin}(c+d x))}{b}\right )+\sqrt {\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {7}{2},-\frac {4 i (a+b \text {ArcSin}(c+d x))}{b}\right )+e^{\frac {8 i a}{b}} \sqrt {-\frac {i (a+b \text {ArcSin}(c+d x))}{b}} \text {Gamma}\left (\frac {7}{2},\frac {4 i (a+b \text {ArcSin}(c+d x))}{b}\right )\right )}{2048 d \left (\frac {(a+b \text {ArcSin}(c+d x))^2}{b^2}\right )^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^(5/2),x]

[Out]

-1/2048*(e^3*(a + b*ArcSin[c + d*x])^(5/2)*(-16*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Ga
mma[7/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] - 16*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/
b]*Gamma[7/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b] + Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-4*I)*(a +
b*ArcSin[c + d*x]))/b] + E^(((8*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((4*I)*(a + b*ArcS
in[c + d*x]))/b]))/(d*E^(((4*I)*a)/b)*((a + b*ArcSin[c + d*x])^2/b^2)^(3/2))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(891\) vs. \(2(391)=782\).
time = 0.52, size = 892, normalized size = 1.88

method	result	size
default	\(-\frac {e^{3} \left (-15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {4 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b^{3}+15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {4 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b^{3}+240 \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{3}-240 \sqrt {-\frac {2}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{3}+1024 \arcsin \left (d x +c \right )^{3} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}-256 \arcsin \left (d x +c \right )^{3} \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) b^{3}+3072 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+1280 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}-768 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a \,b^{2}-160 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) b^{3}+3072 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b -960 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+2560 \arcsin \left (d x +c \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}-768 \arcsin \left (d x +c \right ) \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a^{2} b +60 \arcsin \left (d x +c \right ) \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) b^{3}-320 \arcsin \left (d x +c \right ) \sin \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a \,b^{2}+1024 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{3}-960 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+1280 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b -256 \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a^{3}+60 \cos \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a \,b^{2}-160 \sin \left (-\frac {4 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {4 a}{b}\right ) a^{2} b \right )}{8192 d \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(892\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)

[Out]

-1/8192/d*e^3/(a+b*arcsin(d*x+c))^(1/2)*(-15*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(4*a/b
)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3+15*(-1/b)^(1/2)*Pi^(1/2)*2^(1/2)*(
a+b*arcsin(d*x+c))^(1/2)*sin(4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3+
240*(-2/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2
)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3-240*(-2/b)^(1/2)*Pi^(1/2)*2^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*Fres
nelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3+1024*arcsin(d*x+c)^3*cos(-2*(a+b*arcsin(
d*x+c))/b+2*a/b)*b^3-256*arcsin(d*x+c)^3*cos(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*b^3+3072*arcsin(d*x+c)^2*cos(-2*(
a+b*arcsin(d*x+c))/b+2*a/b)*a*b^2+1280*arcsin(d*x+c)^2*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3-768*arcsin(d*x+
c)^2*cos(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a*b^2-160*arcsin(d*x+c)^2*sin(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*b^3+307
2*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2*b-960*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b
)*b^3+2560*arcsin(d*x+c)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b^2-768*arcsin(d*x+c)*cos(-4*(a+b*arcsin(d*x+c)
)/b+4*a/b)*a^2*b+60*arcsin(d*x+c)*cos(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*b^3-320*arcsin(d*x+c)*sin(-4*(a+b*arcsin
(d*x+c))/b+4*a/b)*a*b^2+1024*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^3-960*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a
*b^2+1280*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2*b-256*cos(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a^3+60*cos(-4*(a+b
*arcsin(d*x+c))/b+4*a/b)*a*b^2-160*sin(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a^2*b)

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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)^3*(a+b*arcsin(d*x+c))^(5/2),x, algorithm="maxima")

[Out]

integrate((d*x*e + c*e)^3*(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)^3*(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^{3} \left (\int a^{2} c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a^{2} d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 a^{2} c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int 3 a^{2} c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 b^{2} c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 3 b^{2} c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 6 a b c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 6 a b c^{2} 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)**3*(a+b*asin(d*x+c))**(5/2),x)

[Out]

e**3*(Integral(a**2*c**3*sqrt(a + b*asin(c + d*x)), x) + Integral(a**2*d**3*x**3*sqrt(a + b*asin(c + d*x)), x)
+ Integral(b**2*c**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2, x) + Integral(2*a*b*c**3*sqrt(a + b*asin(c +
d*x))*asin(c + d*x), x) + Integral(3*a**2*c*d**2*x**2*sqrt(a + b*asin(c + d*x)), x) + Integral(3*a**2*c**2*d*
x*sqrt(a + b*asin(c + d*x)), x) + Integral(b**2*d**3*x**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2, x) + Int
egral(2*a*b*d**3*x**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x) + Integral(3*b**2*c*d**2*x**2*sqrt(a + b*asi
n(c + d*x))*asin(c + d*x)**2, x) + Integral(3*b**2*c**2*d*x*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2, x) + I
ntegral(6*a*b*c*d**2*x**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x) + Integral(6*a*b*c**2*d*x*sqrt(a + b*asi
n(c + d*x))*asin(c + d*x), x))

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Giac [C] Result contains complex when optimal does not.
time = 1.74, size = 3408, normalized size = 7.17 \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)^3*(a+b*arcsin(d*x+c))^(5/2),x, algorithm="giac")

[Out]

-1/8192*(-512*I*sqrt(pi)*a^3*b^2*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsi
n(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4*I*a/b)/(sqrt(2)*b^(5/2) - I*sqrt(2)*b^(7/2)/abs(b)) - 128*sqrt(b*arcsin(
d*x + c) + a)*b^2*e^3*arcsin(d*x + c)^2*e^(4*I*arcsin(d*x + c)) + 512*sqrt(b*arcsin(d*x + c) + a)*b^2*e^3*arcs
in(d*x + c)^2*e^(2*I*arcsin(d*x + c)) + 512*sqrt(b*arcsin(d*x + c) + a)*b^2*e^3*arcsin(d*x + c)^2*e^(-2*I*arcs
in(d*x + c)) - 128*sqrt(b*arcsin(d*x + c) + a)*b^2*e^3*arcsin(d*x + c)^2*e^(-4*I*arcsin(d*x + c)) - 1536*I*sqr
t(pi)*a^3*b*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(
b)/abs(b))*e^(4*I*a/b)/(sqrt(2)*b^(3/2) + I*sqrt(2)*b^(5/2)/abs(b)) - 192*sqrt(pi)*a^2*b^2*e^3*erf(-sqrt(2)*sq
rt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/(sqrt(2)
*b^(3/2) + I*sqrt(2)*b^(5/2)/abs(b)) + 1024*I*sqrt(pi)*a^3*b*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*
sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(b^(3/2) + I*b^(5/2)/abs(b)) - 1024*I*sqrt(pi)*a^3*b*e
^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(b^(3
/2) - I*b^(5/2)/abs(b)) - 1024*I*sqrt(pi)*a^3*b*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(
2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4*I*a/b)/(sqrt(2)*b^(3/2) - I*sqrt(2)*b^(5/2)/abs(b)) - 384
*sqrt(pi)*a^2*b^2*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)
*sqrt(b)/abs(b))*e^(-4*I*a/b)/(sqrt(2)*b^(3/2) - I*sqrt(2)*b^(5/2)/abs(b)) + 1536*I*sqrt(pi)*a^3*sqrt(b)*e^3*e
rf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I
*a/b)/(sqrt(2)*b + I*sqrt(2)*b^2/abs(b)) - 192*sqrt(pi)*a^2*b^(3/2)*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) +
a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/(sqrt(2)*b + I*sqrt(2)*b^2/abs(
b)) - 1024*I*sqrt(pi)*a^3*sqrt(b)*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)
*sqrt(b)/abs(b))*e^(2*I*a/b)/(b + I*b^2/abs(b)) + 1536*sqrt(pi)*a^2*b^(3/2)*e^3*erf(-sqrt(b*arcsin(d*x + c) +
a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(b + I*b^2/abs(b)) + 1024*I*sqrt(pi)*a^
3*sqrt(b)*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I
*a/b)/(b - I*b^2/abs(b)) + 1536*sqrt(pi)*a^2*b^(3/2)*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*a
rcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(b - I*b^2/abs(b)) + 1024*I*sqrt(pi)*a^3*sqrt(b)*e^3*erf(-sqr
t(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4*I*a/b)/
(sqrt(2)*b - I*sqrt(2)*b^2/abs(b)) - 256*sqrt(b*arcsin(d*x + c) + a)*a*b*e^3*arcsin(d*x + c)*e^(4*I*arcsin(d*x
+ c)) - 80*I*sqrt(b*arcsin(d*x + c) + a)*b^2*e^3*arcsin(d*x + c)*e^(4*I*arcsin(d*x + c)) + 1024*sqrt(b*arcsin
(d*x + c) + a)*a*b*e^3*arcsin(d*x + c)*e^(2*I*arcsin(d*x + c)) + 640*I*sqrt(b*arcsin(d*x + c) + a)*b^2*e^3*arc
sin(d*x + c)*e^(2*I*arcsin(d*x + c)) + 1024*sqrt(b*arcsin(d*x + c) + a)*a*b*e^3*arcsin(d*x + c)*e^(-2*I*arcsin
(d*x + c)) - 640*I*sqrt(b*arcsin(d*x + c) + a)*b^2*e^3*arcsin(d*x + c)*e^(-2*I*arcsin(d*x + c)) - 256*sqrt(b*a
rcsin(d*x + c) + a)*a*b*e^3*arcsin(d*x + c)*e^(-4*I*arcsin(d*x + c)) + 80*I*sqrt(b*arcsin(d*x + c) + a)*b^2*e^
3*arcsin(d*x + c)*e^(-4*I*arcsin(d*x + c)) - 512*I*sqrt(pi)*a^3*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/s
qrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/(sqrt(2)*sqrt(b) + I*sqrt(2)*b^(3/2
)/abs(b)) + 384*sqrt(pi)*a^2*b*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(
d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/(sqrt(2)*sqrt(b) + I*sqrt(2)*b^(3/2)/abs(b)) - 72*I*sqrt(pi)*a*b^2*e
^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^
(4*I*a/b)/(sqrt(2)*sqrt(b) + I*sqrt(2)*b^(3/2)/abs(b)) + 15*sqrt(pi)*b^3*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x +
c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/(sqrt(2)*sqrt(b) + I*sqrt(
2)*b^(3/2)/abs(b)) - 1536*sqrt(pi)*a^2*b*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x +
c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(sqrt(b) + I*b^(3/2)/abs(b)) + 576*I*sqrt(pi)*a*b^2*e^3*erf(-sqrt(b*arcsin
(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/(sqrt(b) + I*b^(3/2)/abs(b)
) - 1024*I*sqrt(pi)*a^3*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/a
bs(b))*e^(-2*I*a/b)/(sqrt(b) - I*b^(3/2)/abs(b)) - 1536*sqrt(pi)*a^2*b*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sq
rt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*a/b)/(sqrt(b) - I*b^(3/2)/abs(b)) - 576*I*sqrt(p
i)*a*b^2*e^3*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-2*I*
a/b)/(sqrt(b) - I*b^(3/2)/abs(b)) + 512*I*sqrt(...

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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 )}^3\,{\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)^3*(a + b*asin(c + d*x))^(5/2),x)

[Out]

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^(5/2), x)

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