\(\int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx\) [252]

Optimal result

Integrand size = 23, antiderivative size = 256 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {15 b^{5/2} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 b^{5/2} e \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{128 d} \]

[Out]

-1/4*e*(a+b*arcsin(d*x+c))^(5/2)/d+1/2*e*(d*x+c)^2*(a+b*arcsin(d*x+c))^(5/2)/d-15/128*b^(5/2)*e*cos(2*a/b)*Fre
snelC(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/d-15/128*b^(5/2)*e*FresnelS(2*(a+b*arcsin(d*x+c))
^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/d+5/8*b*e*(d*x+c)*(a+b*arcsin(d*x+c))^(3/2)*(1-(d*x+c)^2)^(1/2)/d
+15/64*b^2*e*(a+b*arcsin(d*x+c))^(1/2)/d-15/32*b^2*e*(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4889, 12, 4725, 4795, 4737, 4809, 3393, 3387, 3386, 3432, 3385, 3433} \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=-\frac {15 \sqrt {\pi } b^{5/2} e \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 \sqrt {\pi } b^{5/2} e \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d} \]

[In]

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

[Out]

(15*b^2*e*Sqrt[a + b*ArcSin[c + d*x]])/(64*d) - (15*b^2*e*(c + d*x)^2*Sqrt[a + b*ArcSin[c + d*x]])/(32*d) + (5
*b*e*(c + d*x)*Sqrt[1 - (c + d*x)^2]*(a + b*ArcSin[c + d*x])^(3/2))/(8*d) - (e*(a + b*ArcSin[c + d*x])^(5/2))/
(4*d) + (e*(c + d*x)^2*(a + b*ArcSin[c + d*x])^(5/2))/(2*d) - (15*b^(5/2)*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*
Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(128*d) - (15*b^(5/2)*e*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcS
in[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(128*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*} \text {integral}& = \frac {\text {Subst}\left (\int e x (a+b \arcsin (x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int x (a+b \arcsin (x))^{5/2} \, dx,x,c+d x\right )}{d} \\ & = \frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {(5 b e) \text {Subst}\left (\int \frac {x^2 (a+b \arcsin (x))^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{4 d} \\ & = \frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {(5 b e) \text {Subst}\left (\int \frac {(a+b \arcsin (x))^{3/2}}{\sqrt {1-x^2}} \, dx,x,c+d x\right )}{8 d}-\frac {\left (15 b^2 e\right ) \text {Subst}\left (\int x \sqrt {a+b \arcsin (x)} \, dx,x,c+d x\right )}{16 d} \\ & = -\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}+\frac {\left (15 b^3 e\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} \sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{64 d} \\ & = -\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}+\frac {\left (15 b^2 e\right ) \text {Subst}\left (\int \frac {\sin ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{64 d} \\ & = -\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}+\frac {\left (15 b^2 e\right ) \text {Subst}\left (\int \left (\frac {1}{2 \sqrt {x}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}}\right ) \, dx,x,a+b \arcsin (c+d x)\right )}{64 d} \\ & = \frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{128 d} \\ & = \frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e \cos \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{128 d}-\frac {\left (15 b^2 e \sin \left (\frac {2 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{128 d} \\ & = \frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {\left (15 b^2 e \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 \arcsin (c+d x)}\right )}{64 d}-\frac {\left (15 b^2 e \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 \arcsin (c+d x)}\right )}{64 d} \\ & = \frac {15 b^2 e \sqrt {a+b \arcsin (c+d x)}}{64 d}-\frac {15 b^2 e (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{32 d}+\frac {5 b e (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{8 d}-\frac {e (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {e (c+d x)^2 (a+b \arcsin (c+d x))^{5/2}}{2 d}-\frac {15 b^{5/2} e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{128 d}-\frac {15 b^{5/2} e \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{128 d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.08 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.55 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {i b^3 e e^{-\frac {2 i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )}{32 \sqrt {2} d \sqrt {a+b \arcsin (c+d x)}} \]

[In]

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

[Out]

((I/32)*b^3*e*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] - E^(((4*
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[2]*d*E^(((2
*I)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(448\) vs. \(2(210)=420\).

Time = 0.85 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.75

method	result	size
default	\(-\frac {e \left (15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{3}-15 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b^{3}+32 \arcsin \left (d x +c \right )^{3} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+96 \arcsin \left (d x +c \right )^{2} \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+40 \arcsin \left (d x +c \right )^{2} \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+96 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b -30 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b^{3}+80 \arcsin \left (d x +c \right ) \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+32 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{3}-30 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \,b^{2}+40 \sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a^{2} b \right )}{128 d \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(449\)

[In]

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

[Out]

-1/128*e/d/(a+b*arcsin(d*x+c))^(1/2)*(15*(-1/b)^(1/2)*Pi^(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^3-15*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(
1/2)*sin(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*b^3+32*arcsin(d*x+c)^3*c
os(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3+96*arcsin(d*x+c)^2*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b^2+40*arcsin(
d*x+c)^2*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3+96*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2*b-30
*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3+80*arcsin(d*x+c)*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*
b^2+32*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^3-30*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b^2+40*sin(-2*(a+b*arc
sin(d*x+c))/b+2*a/b)*a^2*b)

Fricas [F(-2)]

Exception generated. \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*e*x+c*e)*(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)

Sympy [F]

\[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=e \left (\int a^{2} c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int b^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]

[In]

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

[Out]

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

Maxima [F]

\[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

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

[Out]

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.13 (sec) , antiderivative size = 1449, normalized size of antiderivative = 5.66 \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/4*I*sqrt(pi)*a^3*b^(3/2)*e*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^2 + I*b^3/abs(b))*d) - 3/8*sqrt(pi)*a^2*b^(5/2)*e*erf(-sqrt(b*arcsin(d*x + c) + a)/sqr
t(b) - I*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(2*I*a/b)/((b^2 + I*b^3/abs(b))*d) - 1/4*I*sqrt(pi)*a^3
*b^(3/2)*e*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^2 - I*b^3/abs(b))*d) - 3/8*sqrt(pi)*a^2*b^(5/2)*e*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^2 - I*b^3/abs(b))*d) - 1/8*sqrt(b*arcsin(d*x + c) + a)*b^
2*e*arcsin(d*x + c)^2*e^(2*I*arcsin(d*x + c))/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*b^2*e*arcsin(d*x + c)^2*e^(-
2*I*arcsin(d*x + c))/d + 3/8*sqrt(pi)*a^2*b^2*e*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))*d) - 9/64*I*sqrt(pi)*a*b^3*e*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))*d) + 1/4*I*sqrt(pi)*a^3*b*e*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))*d) + 3/8*sqrt(pi)*a^2*b^2*e*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))*d
) + 9/64*I*sqrt(pi)*a*b^3*e*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)/((b^(3/2) - I*b^(5/2)/abs(b))*d) - 1/4*I*sqrt(pi)*a^3*sqrt(b)*e*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))*d) + 9/64*I*sqr
t(pi)*a*b^(5/2)*e*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))*d) + 15/256*sqrt(pi)*b^(7/2)*e*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))*d) - 9/64*I*sqrt(pi)*a*b^(5/2)*e*erf(-s
qrt(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/ab
s(b))*d) + 15/256*sqrt(pi)*b^(7/2)*e*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))*d) - 1/4*sqrt(b*arcsin(d*x + c) + a)*a*b*e*arcsin(d*x + c)*e^
(2*I*arcsin(d*x + c))/d - 5/32*I*sqrt(b*arcsin(d*x + c) + a)*b^2*e*arcsin(d*x + c)*e^(2*I*arcsin(d*x + c))/d -
1/4*sqrt(b*arcsin(d*x + c) + a)*a*b*e*arcsin(d*x + c)*e^(-2*I*arcsin(d*x + c))/d + 5/32*I*sqrt(b*arcsin(d*x +
c) + a)*b^2*e*arcsin(d*x + c)*e^(-2*I*arcsin(d*x + c))/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*a^2*e*e^(2*I*arcsi
n(d*x + c))/d - 5/32*I*sqrt(b*arcsin(d*x + c) + a)*a*b*e*e^(2*I*arcsin(d*x + c))/d + 15/128*sqrt(b*arcsin(d*x
+ c) + a)*b^2*e*e^(2*I*arcsin(d*x + c))/d - 1/8*sqrt(b*arcsin(d*x + c) + a)*a^2*e*e^(-2*I*arcsin(d*x + c))/d +
5/32*I*sqrt(b*arcsin(d*x + c) + a)*a*b*e*e^(-2*I*arcsin(d*x + c))/d + 15/128*sqrt(b*arcsin(d*x + c) + a)*b^2*
e*e^(-2*I*arcsin(d*x + c))/d

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x) (a+b \arcsin (c+d x))^{5/2} \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2} \,d x \]

[In]

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

[Out]

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