3.2.56 \(\int x \sqrt {a+b \arcsin (c+d x)} \, dx\) [156]

3.2.56.1 Optimal result

Integrand size = 16, antiderivative size = 269 \[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=-\frac {c (c+d x) \sqrt {a+b \arcsin (c+d x)}}{d^2}-\frac {\sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))}{4 d^2}+\frac {\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{8 d^2}+\frac {\sqrt {b} c \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{d^2}-\frac {\sqrt {b} c \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{d^2}+\frac {\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 d^2} \]

output

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

3.2.56.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.26 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.92 \[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=\frac {-2 \sqrt {a+b \arcsin (c+d x)} \cos (2 \arcsin (c+d x))+\sqrt {b} \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {4 b c e^{-\frac {i a}{b}} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {2 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )\right )}{\sqrt {a+b \arcsin (c+d x)}}+\sqrt {b} \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{8 d^2} \]

input

Integrate[x*Sqrt[a + b*ArcSin[c + d*x]],x]
 

output

(-2*Sqrt[a + b*ArcSin[c + d*x]]*Cos[2*ArcSin[c + d*x]] + Sqrt[b]*Sqrt[Pi]* 
Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sqrt[Pi])] 
- (4*b*c*(Sqrt[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-I)*(a + b*A 
rcSin[c + d*x]))/b] + E^(((2*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]* 
Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b]))/(E^((I*a)/b)*Sqrt[a + b*ArcSin 
[c + d*x]]) + Sqrt[b]*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(S 
qrt[b]*Sqrt[Pi])]*Sin[(2*a)/b])/(8*d^2)
 

3.2.56.3 Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.11, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5304, 25, 27, 5246, 7267, 7292, 7293, 2009}

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.

\(\displaystyle \int x \sqrt {a+b \arcsin (c+d x)} \, dx\)

\(\Big \downarrow \) 5304

\(\displaystyle \frac {\int x \sqrt {a+b \arcsin (c+d x)}d(c+d x)}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\int -x \sqrt {a+b \arcsin (c+d x)}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int -d x \sqrt {a+b \arcsin (c+d x)}d(c+d x)}{d^2}\)

\(\Big \downarrow \) 5246

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

\(\Big \downarrow \) 7267

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

\(\Big \downarrow \) 7292

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

\(\Big \downarrow \) 7293

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \left (\frac {1}{2} \sqrt {\frac {\pi }{2}} b^{3/2} c \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{16} \sqrt {\pi } b^{3/2} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{16} \sqrt {\pi } b^{3/2} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{2} \sqrt {\frac {\pi }{2}} b^{3/2} c \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} b c \sqrt {a+b \arcsin (c+d x)} \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )+\frac {1}{8} b \sqrt {a+b \arcsin (c+d x)} \cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )\right )}{b d^2}\)

input

Int[x*Sqrt[a + b*ArcSin[c + d*x]],x]
 

output

(-2*((b*Sqrt[a + b*ArcSin[c + d*x]]*Cos[(2*a)/b - (2*(a + b*ArcSin[c + d*x 
]))/b])/8 - (b^(3/2)*Sqrt[Pi]*Cos[(2*a)/b]*FresnelC[(2*Sqrt[a + b*ArcSin[c 
 + d*x]])/(Sqrt[b]*Sqrt[Pi])])/16 - (b^(3/2)*c*Sqrt[Pi/2]*Cos[a/b]*Fresnel 
S[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/2 + (b^(3/2)*c*Sqrt[P 
i/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/ 
2 - (b^(3/2)*Sqrt[Pi]*FresnelS[(2*Sqrt[a + b*ArcSin[c + d*x]])/(Sqrt[b]*Sq 
rt[Pi])]*Sin[(2*a)/b])/16 - (b*c*Sqrt[a + b*ArcSin[c + d*x]]*Sin[a/b - (a 
+ b*ArcSin[c + d*x])/b])/2))/(b*d^2)
 

3.2.56.3.1 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 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 5246

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*((d_.) + (e_.)*(x_))^(m_.), x_S 
ymbol] :> Simp[1/c^(m + 1)   Subst[Int[(a + b*x)^n*Cos[x]*(c*d + e*Sin[x])^ 
m, x], x, ArcSin[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[m, 0]
 

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]
 

rule 7267

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfLinear[u, x]}, Si 
mp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/lst[[2]])], x 
] /;  !FalseQ[lst] && SubstForFractionalPowerQ[u, lst[[3]], x]]
 

rule 7292

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =! 
= u]
 

rule 7293

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v] 
]
 

3.2.56.4 Maple [A] (verified)

Time = 1.11 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.45

method	result	size
default	\(-\frac {4 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \cos \left (\frac {a}{b}\right ) b c +4 \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b c -\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b +\sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {2}{b}}\, b}\right ) b -8 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b c +2 \arcsin \left (d x +c \right ) \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) b -8 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a c +2 \cos \left (-\frac {2 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {2 a}{b}\right ) a}{8 d^{2} \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(390\)

input

int(x*(a+b*arcsin(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
 

output

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

3.2.56.5 Fricas [F(-2)]

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

input

integrate(x*(a+b*arcsin(d*x+c))^(1/2),x, algorithm="fricas")
 

output

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

3.2.56.6 Sympy [F]

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

input

integrate(x*(a+b*asin(d*x+c))**(1/2),x)
 

output

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

3.2.56.7 Maxima [F]

\[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=\int { \sqrt {b \arcsin \left (d x + c\right ) + a} x \,d x } \]

input

integrate(x*(a+b*arcsin(d*x+c))^(1/2),x, algorithm="maxima")
 

output

integrate(sqrt(b*arcsin(d*x + c) + a)*x, x)
 

3.2.56.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.00 (sec) , antiderivative size = 1079, normalized size of antiderivative = 4.01 \[ \int x \sqrt {a+b \arcsin (c+d x)} \, dx=\text {Too large to display} \]

input

integrate(x*(a+b*arcsin(d*x+c))^(1/2),x, algorithm="giac")
 

output

-1/2*sqrt(2)*sqrt(pi)*a*b^2*c*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + 
a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)* 
e^(I*a/b)/((I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d^2) - 1/4*I*sqrt(2)*sq 
rt(pi)*b^3*c*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 
 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a/b)/((I*b^3 
/sqrt(abs(b)) + b^2*sqrt(abs(b)))*d^2) - 1/2*sqrt(2)*sqrt(pi)*a*b^2*c*erf( 
1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt( 
b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + 
b^2*sqrt(abs(b)))*d^2) + 1/4*I*sqrt(2)*sqrt(pi)*b^3*c*erf(1/2*I*sqrt(2)*sq 
rt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c 
) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^3/sqrt(abs(b)) + b^2*sqrt(abs(b)) 
)*d^2) + sqrt(pi)*a*b*c*erf(-1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqr 
t(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(I*a 
/b)/((I*sqrt(2)*b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*d^2) + sqrt(pi) 
*a*b*c*erf(1/2*I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sq 
rt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*sqrt(2)* 
b^2/sqrt(abs(b)) + sqrt(2)*b*sqrt(abs(b)))*d^2) + 1/4*I*sqrt(pi)*a*b^(3/2) 
*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^2) - 1/16*sqrt(pi)*b^( 
5/2)*erf(-sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(b*arcsin(d*x + c...
 

3.2.56.9 Mupad [F(-1)]

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

input

int(x*(a + b*asin(c + d*x))^(1/2),x)
 

output

int(x*(a + b*asin(c + d*x))^(1/2), x)