3.3.68 \(\int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{3/2}} \, dx\) [268]

3.3.68.1 Optimal result

Integrand size = 23, antiderivative size = 144 \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=-\frac {2 e (c+d x) \sqrt {1-(c+d x)^2}}{b d \sqrt {a+b \arcsin (c+d x)}}+\frac {2 e \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{b^{3/2} d}+\frac {2 e \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{b^{3/2} d} \]

output

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

3.3.68.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.17 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.17 \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\frac {i e e^{-\frac {2 i a}{b}} \left (-\sqrt {2} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )+\sqrt {2} e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {1}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )+2 i e^{\frac {2 i a}{b}} \sin (2 \arcsin (c+d x))\right )}{2 b d \sqrt {a+b \arcsin (c+d x)}} \]

input

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

output

((I/2)*e*(-(Sqrt[2]*Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[1/2, ((-2 
*I)*(a + b*ArcSin[c + d*x]))/b]) + Sqrt[2]*E^(((4*I)*a)/b)*Sqrt[(I*(a + b* 
ArcSin[c + d*x]))/b]*Gamma[1/2, ((2*I)*(a + b*ArcSin[c + d*x]))/b] + (2*I) 
*E^(((2*I)*a)/b)*Sin[2*ArcSin[c + d*x]]))/(b*d*E^(((2*I)*a)/b)*Sqrt[a + b* 
ArcSin[c + d*x]])
 

3.3.68.3 Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.98, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {5304, 27, 5142, 3042, 3787, 25, 3042, 3785, 3786, 3832, 3833}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 5304

\(\displaystyle \frac {\int \frac {e (c+d x)}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e \int \frac {c+d x}{(a+b \arcsin (c+d x))^{3/2}}d(c+d x)}{d}\)

\(\Big \downarrow \) 5142

\(\displaystyle \frac {e \left (\frac {2 \int \frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e \left (\frac {2 \int \frac {\sin \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\)

\(\Big \downarrow \) 3787

\(\displaystyle \frac {e \left (\frac {2 \left (\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))-\sin \left (\frac {2 a}{b}\right ) \int -\frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e \left (\frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\cos \left (\frac {2 a}{b}\right ) \int \frac {\cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e \left (\frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\cos \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\)

\(\Big \downarrow \) 3785

\(\displaystyle \frac {e \left (\frac {2 \left (\sin \left (\frac {2 a}{b}\right ) \int \frac {\sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\)

\(\Big \downarrow \) 3786

\(\displaystyle \frac {e \left (\frac {2 \left (2 \sin \left (\frac {2 a}{b}\right ) \int \sin \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}+2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\)

\(\Big \downarrow \) 3832

\(\displaystyle \frac {e \left (\frac {2 \left (2 \cos \left (\frac {2 a}{b}\right ) \int \cos \left (\frac {2 (a+b \arcsin (c+d x))}{b}\right )d\sqrt {a+b \arcsin (c+d x)}+\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\)

\(\Big \downarrow \) 3833

\(\displaystyle \frac {e \left (\frac {2 \left (\sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )\right )}{b^2}-\frac {2 (c+d x) \sqrt {1-(c+d x)^2}}{b \sqrt {a+b \arcsin (c+d x)}}\right )}{d}\)

input

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

output

(e*((-2*(c + d*x)*Sqrt[1 - (c + d*x)^2])/(b*Sqrt[a + b*ArcSin[c + d*x]]) + 
 (2*(Sqrt[b]*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]] 
)/(Sqrt[b]*Sqrt[Pi])] + Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + 
 d*x]])/(Sqrt[b]*Sqrt[Pi])]*Sin[(2*a)/b]))/b^2))/d
 

3.3.68.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3785

Int[sin[Pi/2 + (e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> S 
imp[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 3786

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[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 3787

Int[sin[(e_.) + (f_.)*(x_)]/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Cos 
[(d*e - c*f)/d]   Int[Sin[c*(f/d) + f*x]/Sqrt[c + d*x], x], x] + Simp[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 3832

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 3833

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 5142

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] - Simp 
[1/(b^2*c^(m + 1)*(n + 1))   Subst[Int[ExpandTrigReduce[x^(n + 1), Sin[-a/b 
 + x/b]^(m - 1)*(m - (m + 1)*Sin[-a/b + x/b]^2), x], x], x, a + b*ArcSin[c* 
x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]
 

rule 5304

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

3.3.68.4 Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.17

method	result	size
default	\(\frac {e \left (2 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )-2 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right )+\sin \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right )\right )}{d b \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(169\)

input

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

output

e/d/b*(2*(-1/b)^(1/2)*Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)*cos(2*a/b)*Fresne 
lC(2*2^(1/2)/Pi^(1/2)/(-2/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)-2*(-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)+sin(-2*(a+b*arcsin(d*x+c))/ 
b+2*a/b))/(a+b*arcsin(d*x+c))^(1/2)
 

3.3.68.5 Fricas [F(-2)]

Exception generated. \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]

input

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

output

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

3.3.68.6 Sympy [F]

\[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=e \left (\int \frac {c}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx + \int \frac {d x}{a \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}}\, dx\right ) \]

input

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

output

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

3.3.68.7 Maxima [F]

\[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int { \frac {d e x + c e}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

input

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

output

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

3.3.68.8 Giac [F]

\[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int { \frac {d e x + c e}{{\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

input

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

output

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

3.3.68.9 Mupad [F(-1)]

Timed out. \[ \int \frac {c e+d e x}{(a+b \arcsin (c+d x))^{3/2}} \, dx=\int \frac {c\,e+d\,e\,x}{{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{3/2}} \,d x \]

input

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

output

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