\(\int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{5/2}} \, dx\) [273]

Optimal result

Integrand size = 23, antiderivative size = 207 \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=-\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {8 e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d}+\frac {8 e \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} d} \]

[Out]

-8/3*e*cos(2*a/b)*FresnelS(2*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/b^(5/2)/d+8/3*e*FresnelC(2*(
a+b*arcsin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/b^(5/2)/d-2/3*e*(d*x+c)*(1-(d*x+c)^2)^(1/2)/b/d
/(a+b*arcsin(d*x+c))^(3/2)-4/3*e/b^2/d/(a+b*arcsin(d*x+c))^(1/2)+8/3*e*(d*x+c)^2/b^2/d/(a+b*arcsin(d*x+c))^(1/
2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {4889, 12, 4729, 4807, 4731, 4491, 3387, 3386, 3432, 3385, 3433, 4737} \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=\frac {8 \sqrt {\pi } e \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d}-\frac {8 \sqrt {\pi } e \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d}+\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {4 e}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {2 e \sqrt {1-(c+d x)^2} (c+d x)}{3 b d (a+b \arcsin (c+d x))^{3/2}} \]

[In]

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

[Out]

(-2*e*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(3*b*d*(a + b*ArcSin[c + d*x])^(3/2)) - (4*e)/(3*b^2*d*Sqrt[a + b*ArcSi
n[c + d*x]]) + (8*e*(c + d*x)^2)/(3*b^2*d*Sqrt[a + b*ArcSin[c + d*x]]) - (8*e*Sqrt[Pi]*Cos[(2*a)/b]*FresnelS[(
2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])])/(3*b^(5/2)*d) + (8*e*Sqrt[Pi]*FresnelC[(2*Sqrt[a + b*ArcSi
n[c + d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(3*b^(5/2)*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 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 4491

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 4729

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

Rule 4731

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Dist[1/(b*c^(m + 1)), Subst[Int[x^n*Sin[-
a/b + x/b]^m*Cos[-a/b + x/b], x], x, a + b*ArcSin[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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 4807

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

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 \frac {e x}{(a+b \arcsin (x))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e \text {Subst}\left (\int \frac {x}{(a+b \arcsin (x))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}+\frac {(2 e) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} (a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{3 b d}-\frac {(4 e) \text {Subst}\left (\int \frac {x^2}{\sqrt {1-x^2} (a+b \arcsin (x))^{3/2}} \, dx,x,c+d x\right )}{3 b d} \\ & = -\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {(16 e) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \arcsin (x)}} \, dx,x,c+d x\right )}{3 b^2 d} \\ & = -\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {(16 e) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}-\frac {x}{b}\right ) \sin \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {(16 e) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {(8 e) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \arcsin (c+d x)\right )}{3 b^3 d} \\ & = -\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {\left (8 e \cos \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 )}{3 b^3 d}+\frac {\left (8 e \sin \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 )}{3 b^3 d} \\ & = -\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {\left (16 e \cos \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 )}{3 b^3 d}+\frac {\left (16 e \sin \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 )}{3 b^3 d} \\ & = -\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{3 b d (a+b \arcsin (c+d x))^{3/2}}-\frac {4 e}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}+\frac {8 e (c+d x)^2}{3 b^2 d \sqrt {a+b \arcsin (c+d x)}}-\frac {8 e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{3 b^{5/2} d}+\frac {8 e \sqrt {\pi } \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{3 b^{5/2} d} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 1.22 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.93 \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{5/2}} \, dx=-\frac {e \left (2 (a+b \arcsin (c+d x)) \left (e^{-2 i \arcsin (c+d x)}+e^{2 i \arcsin (c+d x)}-\sqrt {2} e^{-\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-\sqrt {2} e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )\right )+b \sin (2 \arcsin (c+d x))\right )}{3 b^2 d (a+b \arcsin (c+d x))^{3/2}} \]

[In]

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

[Out]

-1/3*(e*(2*(a + b*ArcSin[c + d*x])*(E^((-2*I)*ArcSin[c + d*x]) + E^((2*I)*ArcSin[c + d*x]) - (Sqrt[2]*Sqrt[((-
I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b])/E^(((2*I)*a)/b) - Sqrt[2]*E^(((
2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b]) + b*Sin[2*ArcSin
[c + d*x]]))/(b^2*d*(a + b*ArcSin[c + d*x])^(3/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(369\) vs. \(2(169)=338\).

Time = 0.98 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.79

method	result	size
default	\(\frac {e \left (8 \arcsin \left (d x +c \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +8 \arcsin \left (d x +c \right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) a +8 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) a -4 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b +\sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b -4 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a \right )}{3 d \,b^{2} \left (a +b \arcsin \left (d x +c \right )\right )^{\frac {3}{2}}}\)	\(370\)

[In]

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

[Out]

1/3*e/d/b^2*(8*arcsin(d*x+c)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(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+8*arcsin(d*x+c)*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1
/2)*sin(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+8*(-1/b)^(1/2)*Pi^(1/2)
*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*FresnelS(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*a+
8*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*sin(2*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arc
sin(d*x+c))^(1/2)/b)*a-4*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b+sin(-2*(a+b*arcsin(d*x+c))/b+2*a/
b)*b-4*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a)/(a+b*arcsin(d*x+c))^(3/2)

Fricas [F(-2)]

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

[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]

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

[In]

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

[Out]

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

Maxima [F]

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

[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 [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[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)