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

Optimal. Leaf size=380 \[ \frac {3 \sqrt {\pi } b^{3/2} e^3 \sin \left (\frac {2 a}{b}\right ) C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{64 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 \sin \left (\frac {4 a}{b}\right ) C\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{512 d}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 \cos \left (\frac {4 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {3 \sqrt {\pi } b^{3/2} e^3 \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{64 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {3 b e^3 \sqrt {1-(c+d x)^2} (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {9 b e^3 \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d} \]

[Out]

-3/32*e^3*(a+b*arcsin(d*x+c))^(3/2)/d+1/4*e^3*(d*x+c)^4*(a+b*arcsin(d*x+c))^(3/2)/d+3/1024*b^(3/2)*e^3*cos(4*a
/b)*FresnelS(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-3/1024*b^(3/2)*e^3*Fresn
elC(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-3/64*b^(3/2)*e^3*cos(2
*a/b)*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/d+3/64*b^(3/2)*e^3*FresnelC(2*(a+b*arcsi
n(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/d+9/64*b*e^3*(d*x+c)*(1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x
+c))^(1/2)/d+3/32*b*e^3*(d*x+c)^3*(1-(d*x+c)^2)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/d

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Rubi [A]  time = 1.12, antiderivative size = 380, normalized size of antiderivative = 1.00, number of steps used = 27, number of rules used = 12, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {4805, 12, 4629, 4707, 4641, 4635, 4406, 3306, 3305, 3351, 3304, 3352} \[ \frac {3 \sqrt {\pi } b^{3/2} e^3 \sin \left (\frac {2 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {\pi } \sqrt {b}}\right )}{64 d}-\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 \sin \left (\frac {4 a}{b}\right ) \text {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{512 d}+\frac {3 \sqrt {\frac {\pi }{2}} b^{3/2} e^3 \cos \left (\frac {4 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {3 \sqrt {\pi } b^{3/2} e^3 \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{64 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}+\frac {3 b e^3 \sqrt {1-(c+d x)^2} (c+d x)^3 \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {9 b e^3 \sqrt {1-(c+d x)^2} (c+d x) \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}-\frac {3 e^3 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{32 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9*b*e^3*(c + d*x)*Sqrt[1 - (c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(64*d) + (3*b*e^3*(c + d*x)^3*Sqrt[1 - (
c + d*x)^2]*Sqrt[a + b*ArcSin[c + d*x]])/(32*d) - (3*e^3*(a + b*ArcSin[c + d*x])^(3/2))/(32*d) + (e^3*(c + d*x
)^4*(a + b*ArcSin[c + d*x])^(3/2))/(4*d) + (3*b^(3/2)*e^3*Sqrt[Pi/2]*Cos[(4*a)/b]*FresnelS[(2*Sqrt[2/Pi]*Sqrt[
a + b*ArcSin[c + d*x]])/Sqrt[b]])/(512*d) - (3*b^(3/2)*e^3*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(2*Sqrt[a + b*ArcSin
[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(64*d) + (3*b^(3/2)*e^3*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sq
rt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(64*d) - (3*b^(3/2)*e^3*Sqrt[Pi/2]*FresnelC[(2*Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c +
d*x]])/Sqrt[b]]*Sin[(4*a)/b])/(512*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 3304

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 3305

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 3306

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 3351

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

Rule 3352

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

Rule 4406

Int[Cos[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sin[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[E
xpandTrigReduce[(c + d*x)^m, Sin[a + b*x]^n*Cos[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0]
&& IGtQ[p, 0]

Rule 4629

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 4635

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[(a + b*x)^n*S
in[x]^m*Cos[x], x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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 4707

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

Rule 4805

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 )^{3/2} \, dx &=\frac {\operatorname {Subst}\left (\int e^3 x^3 \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \operatorname {Subst}\left (\int x^3 \left (a+b \sin ^{-1}(x)\right )^{3/2} \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {\left (3 b e^3\right ) \operatorname {Subst}\left (\int \frac {x^4 \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}\\ &=\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {\left (9 b e^3\right ) \operatorname {Subst}\left (\int \frac {x^2 \sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{32 d}-\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{64 d}\\ &=\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \sin ^{-1}(c+d x)\right )^{3/2}}{4 d}-\frac {\left (9 b e^3\right ) \operatorname {Subst}\left (\int \frac {\sqrt {a+b \sin ^{-1}(x)}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{64 d}-\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin ^3(x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}-\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt {a+b \sin ^{-1}(x)}} \, dx,x,c+d x\right )}{128 d}\\ &=\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \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 )^{3/2}}{4 d}-\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \left (\frac {\sin (2 x)}{4 \sqrt {a+b x}}-\frac {\sin (4 x)}{8 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c+d x)\right )}{64 d}-\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\cos (x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{128 d}\\ &=\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \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 )^{3/2}}{4 d}+\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\sin (4 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{512 d}-\frac {\left (3 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{256 d}-\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{2 \sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{128 d}\\ &=\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \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 )^{3/2}}{4 d}-\frac {\left (9 b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {\sin (2 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c+d x)\right )}{256 d}-\frac {\left (3 b^2 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {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 )}{256 d}+\frac {\left (3 b^2 e^3 \cos \left (\frac {4 a}{b}\right )\right ) \operatorname {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 )}{512 d}+\frac {\left (3 b^2 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {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 )}{256 d}-\frac {\left (3 b^2 e^3 \sin \left (\frac {4 a}{b}\right )\right ) \operatorname {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 )}{512 d}\\ &=\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \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 )^{3/2}}{4 d}-\frac {\left (3 b e^3 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{128 d}-\frac {\left (9 b^2 e^3 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {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 )}{256 d}+\frac {\left (3 b e^3 \cos \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{256 d}+\frac {\left (3 b e^3 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{128 d}+\frac {\left (9 b^2 e^3 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {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 )}{256 d}-\frac {\left (3 b e^3 \sin \left (\frac {4 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {4 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{256 d}\\ &=\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \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 )^{3/2}}{4 d}+\frac {3 b^{3/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {3 b^{3/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{256 d}+\frac {3 b^{3/2} e^3 \sqrt {\pi } C\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 {3 b^{3/2} e^3 \sqrt {\frac {\pi }{2}} C\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 )}{512 d}-\frac {\left (9 b e^3 \cos \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \sin \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{128 d}+\frac {\left (9 b e^3 \sin \left (\frac {2 a}{b}\right )\right ) \operatorname {Subst}\left (\int \cos \left (\frac {2 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c+d x)}\right )}{128 d}\\ &=\frac {9 b e^3 (c+d x) \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{64 d}+\frac {3 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} \sqrt {a+b \sin ^{-1}(c+d x)}}{32 d}-\frac {3 e^3 \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 )^{3/2}}{4 d}+\frac {3 b^{3/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) S\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b}}\right )}{512 d}-\frac {3 b^{3/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) S\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{64 d}+\frac {3 b^{3/2} e^3 \sqrt {\pi } C\left (\frac {2 \sqrt {a+b \sin ^{-1}(c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{64 d}-\frac {3 b^{3/2} e^3 \sqrt {\frac {\pi }{2}} C\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 )}{512 d}\\ \end {align*}

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Mathematica [C]  time = 0.20, size = 273, normalized size = 0.72 \[ -\frac {i b e^3 e^{-\frac {4 i a}{b}} \sqrt {a+b \sin ^{-1}(c+d x)} \left (8 \sqrt {2} e^{\frac {2 i a}{b}} \sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {5}{2},-\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-8 \sqrt {2} e^{\frac {6 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {5}{2},\frac {2 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )-\sqrt {\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {5}{2},-\frac {4 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )+e^{\frac {8 i a}{b}} \sqrt {-\frac {i \left (a+b \sin ^{-1}(c+d x)\right )}{b}} \Gamma \left (\frac {5}{2},\frac {4 i \left (a+b \sin ^{-1}(c+d x)\right )}{b}\right )\right )}{512 d \sqrt {\frac {\left (a+b \sin ^{-1}(c+d x)\right )^2}{b^2}}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1/512*I)*b*e^3*Sqrt[a + b*ArcSin[c + d*x]]*(8*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*G
amma[5/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] - 8*Sqrt[2]*E^(((6*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/
b]*Gamma[5/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b] - Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((-4*I)*(a +
b*ArcSin[c + d*x]))/b] + E^(((8*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[5/2, ((4*I)*(a + b*ArcS
in[c + d*x]))/b]))/(d*E^(((4*I)*a)/b)*Sqrt[(a + b*ArcSin[c + d*x])^2/b^2])

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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [B]  time = 2.47, size = 2282, normalized size = 6.01 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*sqrt(pi)*a^2*b^2*i*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x
+ c) + a)/sqrt(b))*e^(4*a*i/b + 3)/((sqrt(2)*b^(7/2)*i/abs(b) + sqrt(2)*b^(5/2))*d) - 1/8*sqrt(pi)*a^2*b^2*i*e
rf(sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-4*a
*i/b + 3)/((sqrt(2)*b^(7/2)*i/abs(b) - sqrt(2)*b^(5/2))*d) - 1/8*sqrt(pi)*a^2*b^(3/2)*i*erf(-sqrt(2)*sqrt(b*ar
csin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(4*a*i/b + 3)/((sqrt(2)*b
^3*i/abs(b) + sqrt(2)*b^2)*d) + 1/8*sqrt(pi)*a^2*b^(3/2)*i*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) -
sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 3)/((b^3*i/abs(b) + b^2)*d) + 1/8*sqrt(pi)*a^2*b^(3/2)*i*er
f(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 3)/((b^3*i
/abs(b) - b^2)*d) + 1/16*sqrt(pi)*a^2*b^(3/2)*i*erf(sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqr
t(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-4*a*i/b + 3)/((sqrt(2)*b^3*i/abs(b) - sqrt(2)*b^2)*d) + 1/16*sqr
t(pi)*a^2*b*i*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/
sqrt(b))*e^(4*a*i/b + 3)/((sqrt(2)*b^(5/2)*i/abs(b) + sqrt(2)*b^(3/2))*d) - 1/8*sqrt(pi)*a^2*b*i*erf(sqrt(b*ar
csin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 3)/((b^(5/2)*i/abs(b)
- b^(3/2))*d) - 1/64*sqrt(pi)*a*b^3*erf(sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b
*arcsin(d*x + c) + a)/sqrt(b))*e^(-4*a*i/b + 3)/((sqrt(2)*b^(7/2)*i/abs(b) - sqrt(2)*b^(5/2))*d) + 1/16*sqrt(p
i)*a^2*b*i*erf(sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt
(b))*e^(-4*a*i/b + 3)/((sqrt(2)*b^(5/2)*i/abs(b) - sqrt(2)*b^(3/2))*d) - 3/1024*sqrt(pi)*b^3*i*erf(sqrt(2)*sqr
t(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-4*a*i/b + 3)/((sq
rt(2)*b^(5/2)*i/abs(b) - sqrt(2)*b^(3/2))*d) + 1/64*sqrt(pi)*a*b^(5/2)*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a
)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(4*a*i/b + 3)/((sqrt(2)*b^3*i/abs(b) + sqr
t(2)*b^2)*d) - 3/1024*sqrt(pi)*b^(5/2)*i*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*s
qrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(4*a*i/b + 3)/((sqrt(2)*b^2*i/abs(b) + sqrt(2)*b)*d) - 1/16*sqrt(pi)*a*b
^(5/2)*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 3
)/((b^3*i/abs(b) + b^2)*d) - 1/8*sqrt(pi)*a^2*sqrt(b)*i*erf(-sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sq
rt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 3)/((b^2*i/abs(b) + b)*d) + 3/128*sqrt(pi)*b^(5/2)*i*erf(-sqrt
(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 3)/((b^2*i/abs(b)
+ b)*d) + 1/16*sqrt(pi)*a*b^(5/2)*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) +
a)/sqrt(b))*e^(-2*a*i/b + 3)/((b^3*i/abs(b) - b^2)*d) + 3/128*sqrt(pi)*b^(5/2)*i*erf(sqrt(b*arcsin(d*x + c) +
a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-2*a*i/b + 3)/((b^2*i/abs(b) - b)*d) - 1/64*sqr
t(pi)*a*b^2*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sq
rt(b))*e^(4*a*i/b + 3)/((sqrt(2)*b^(5/2)*i/abs(b) + sqrt(2)*b^(3/2))*d) + 1/16*sqrt(pi)*a*b^2*erf(-sqrt(b*arcs
in(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(2*a*i/b + 3)/((b^(5/2)*i/abs(b) +
b^(3/2))*d) - 1/16*sqrt(pi)*a*b^2*erf(sqrt(b*arcsin(d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(b*arcsin(d*x + c) +
a)/sqrt(b))*e^(-2*a*i/b + 3)/((b^(5/2)*i/abs(b) - b^(3/2))*d) + 1/64*sqrt(pi)*a*b^2*erf(sqrt(2)*sqrt(b*arcsin(
d*x + c) + a)*sqrt(b)*i/abs(b) - sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b))*e^(-4*a*i/b + 3)/((sqrt(2)*b^(5/
2)*i/abs(b) - sqrt(2)*b^(3/2))*d) + 3/512*sqrt(b*arcsin(d*x + c) + a)*b*i*e^(4*i*arcsin(d*x + c) + 3)/d + 1/64
*sqrt(b*arcsin(d*x + c) + a)*b*arcsin(d*x + c)*e^(4*i*arcsin(d*x + c) + 3)/d - 3/64*sqrt(b*arcsin(d*x + c) + a
)*b*i*e^(2*i*arcsin(d*x + c) + 3)/d - 1/16*sqrt(b*arcsin(d*x + c) + a)*b*arcsin(d*x + c)*e^(2*i*arcsin(d*x + c
) + 3)/d + 3/64*sqrt(b*arcsin(d*x + c) + a)*b*i*e^(-2*i*arcsin(d*x + c) + 3)/d - 1/16*sqrt(b*arcsin(d*x + c) +
a)*b*arcsin(d*x + c)*e^(-2*i*arcsin(d*x + c) + 3)/d - 3/512*sqrt(b*arcsin(d*x + c) + a)*b*i*e^(-4*i*arcsin(d*
x + c) + 3)/d + 1/64*sqrt(b*arcsin(d*x + c) + a)*b*arcsin(d*x + c)*e^(-4*i*arcsin(d*x + c) + 3)/d + 1/64*sqrt(
b*arcsin(d*x + c) + a)*a*e^(4*i*arcsin(d*x + c) + 3)/d - 1/16*sqrt(b*arcsin(d*x + c) + a)*a*e^(2*i*arcsin(d*x
+ c) + 3)/d - 1/16*sqrt(b*arcsin(d*x + c) + a)*a*e^(-2*i*arcsin(d*x + c) + 3)/d + 1/64*sqrt(b*arcsin(d*x + c)
+ a)*a*e^(-4*i*arcsin(d*x + c) + 3)/d

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maple [A]  time = 0.34, size = 582, normalized size = 1.53 \[ -\frac {e^{3} \left (-3 \sqrt {2}\, \sqrt {\pi }\, \cos \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 ) \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, b^{2}+3 \sqrt {2}\, \sqrt {\pi }\, \sin \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 ) \sqrt {\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, b^{2}+48 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \mathrm {S}\left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}-48 \sqrt {\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \FresnelC \left (\frac {2 \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {\frac {1}{b}}\, b}\right ) b^{2}+128 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b^{2}-32 \arcsin \left (d x +c \right )^{2} \cos \left (\frac {4 a +4 b \arcsin \left (d x +c \right )}{b}-\frac {4 a}{b}\right ) b^{2}+256 \arcsin \left (d x +c \right ) \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a b -96 \arcsin \left (d x +c \right ) \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) b^{2}-64 \arcsin \left (d x +c \right ) \cos \left (\frac {4 a +4 b \arcsin \left (d x +c \right )}{b}-\frac {4 a}{b}\right ) a b +12 \arcsin \left (d x +c \right ) \sin \left (\frac {4 a +4 b \arcsin \left (d x +c \right )}{b}-\frac {4 a}{b}\right ) b^{2}+128 \cos \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a^{2}-96 \sin \left (\frac {2 a +2 b \arcsin \left (d x +c \right )}{b}-\frac {2 a}{b}\right ) a b -32 \cos \left (\frac {4 a +4 b \arcsin \left (d x +c \right )}{b}-\frac {4 a}{b}\right ) a^{2}+12 \sin \left (\frac {4 a +4 b \arcsin \left (d x +c \right )}{b}-\frac {4 a}{b}\right ) a b \right )}{1024 d \sqrt {a +b \arcsin \left (d x +c \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1024/d*e^3*(-3*2^(1/2)*Pi^(1/2)*cos(4*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2
)/b)*(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*b^2+3*2^(1/2)*Pi^(1/2)*sin(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(1/b)
^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*b^2+48*(1/b)^(1/2)*Pi^(1/2)*(a+b*arc
sin(d*x+c))^(1/2)*cos(2*a/b)*FresnelS(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2-48*(1/b)^(1/2)*P
i^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*FresnelC(2/Pi^(1/2)/(1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^2+
128*arcsin(d*x+c)^2*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*b^2-32*arcsin(d*x+c)^2*cos(4*(a+b*arcsin(d*x+c))/b-4*a/
b)*b^2+256*arcsin(d*x+c)*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a*b-96*arcsin(d*x+c)*sin(2*(a+b*arcsin(d*x+c))/b-2
*a/b)*b^2-64*arcsin(d*x+c)*cos(4*(a+b*arcsin(d*x+c))/b-4*a/b)*a*b+12*arcsin(d*x+c)*sin(4*(a+b*arcsin(d*x+c))/b
-4*a/b)*b^2+128*cos(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a^2-96*sin(2*(a+b*arcsin(d*x+c))/b-2*a/b)*a*b-32*cos(4*(a+b
*arcsin(d*x+c))/b-4*a/b)*a^2+12*sin(4*(a+b*arcsin(d*x+c))/b-4*a/b)*a*b)/(a+b*arcsin(d*x+c))^(1/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (d e x + c e\right )}^{3} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ e^{3} \left (\int a c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 a c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int 3 a c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 b c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 b c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*e*x+c*e)**3*(a+b*asin(d*x+c))**(3/2),x)

[Out]

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

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