\(\int (c e+d e x)^2 \sqrt {a+b \arcsin (c+d x)} \, dx\) [120]

Optimal result

Integrand size = 25, antiderivative size = 274 \[ \int (c e+d e x)^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\frac {e^2 (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}}{3 d}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4 d}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{6}} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{12 d}+\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{2}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{4 d}-\frac {\sqrt {b} e^2 \sqrt {\frac {\pi }{6}} \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{12 d} \] Output:

1/3*e^2*(d*x+c)^3*(a+b*arcsin(d*x+c))^(1/2)/d-1/8*b^(1/2)*e^2*2^(1/2)*Pi^( 
1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2)) 
/d+1/72*b^(1/2)*e^2*6^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelS(6^(1/2)/Pi^(1/2)* 
(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))/d+1/8*b^(1/2)*e^2*2^(1/2)*Pi^(1/2)*Fres 
nelC(2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(a/b)/d-1/72*b 
^(1/2)*e^2*6^(1/2)*Pi^(1/2)*FresnelC(6^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^ 
(1/2)/b^(1/2))*sin(3*a/b)/d
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.91 \[ \int (c e+d e x)^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\frac {b e^2 e^{-\frac {3 i a}{b}} \left (9 e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {i (a+b \arcsin (c+d x))}{b}\right )+9 e^{\frac {4 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {i (a+b \arcsin (c+d x))}{b}\right )-\sqrt {3} \left (\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},-\frac {3 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {3}{2},\frac {3 i (a+b \arcsin (c+d x))}{b}\right )\right )\right )}{72 d \sqrt {a+b \arcsin (c+d x)}} \] Input:

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

Output:

(b*e^2*(9*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] + 9*E^(((4*I)*a)/b)*Sqrt[(I*(a + b*Arc 
Sin[c + d*x]))/b]*Gamma[3/2, (I*(a + b*ArcSin[c + d*x]))/b] - Sqrt[3]*(Sqr 
t[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, ((-3*I)*(a + b*ArcSin[c + d 
*x]))/b] + E^(((6*I)*a)/b)*Sqrt[(I*(a + b*ArcSin[c + d*x]))/b]*Gamma[3/2, 
((3*I)*(a + b*ArcSin[c + d*x]))/b])))/(72*d*E^(((3*I)*a)/b)*Sqrt[a + b*Arc 
Sin[c + d*x]])
 

Rubi [A] (verified)

Time = 0.88 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5304, 27, 5140, 5224, 25, 3042, 3793, 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.

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

\(\Big \downarrow \) 5304

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 5140

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{6} b \int \frac {(c+d x)^3}{\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}d(c+d x)\right )}{d}\)

\(\Big \downarrow \) 5224

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{6} \int -\frac {\sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 \left (\frac {1}{6} \int \frac {\sin ^3\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\frac {1}{3} (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e^2 \left (\frac {1}{6} \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )^3}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))+\frac {1}{3} (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}\right )}{d}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {e^2 \left (\frac {1}{6} \int \left (\frac {3 \sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{4 \sqrt {a+b \arcsin (c+d x)}}-\frac {\sin \left (\frac {3 a}{b}-\frac {3 (a+b \arcsin (c+d x))}{b}\right )}{4 \sqrt {a+b \arcsin (c+d x)}}\right )d(a+b \arcsin (c+d x))+\frac {1}{3} (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \left (\frac {1}{6} \left (\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {3}{2} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\frac {\pi }{6}} \sqrt {b} \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}\right )}{d}\)

Input:

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

Output:

(e^2*(((c + d*x)^3*Sqrt[a + b*ArcSin[c + d*x]])/3 + ((-3*Sqrt[b]*Sqrt[Pi/2 
]*Cos[a/b]*FresnelS[(Sqrt[2/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]])/2 + 
 (Sqrt[b]*Sqrt[Pi/6]*Cos[(3*a)/b]*FresnelS[(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c 
 + d*x]])/Sqrt[b]])/2 + (3*Sqrt[b]*Sqrt[Pi/2]*FresnelC[(Sqrt[2/Pi]*Sqrt[a 
+ b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[a/b])/2 - (Sqrt[b]*Sqrt[Pi/6]*FresnelC[ 
(Sqrt[6/Pi]*Sqrt[a + b*ArcSin[c + d*x]])/Sqrt[b]]*Sin[(3*a)/b])/2)/6))/d
 

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 3042

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

rule 3793

Int[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In 
t[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 5140

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x 
^(m + 1)*((a + b*ArcSin[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 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] && GtQ[n, 0]
 

rule 5224

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^ 
2)^(p_.), x_Symbol] :> Simp[(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 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]
 

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.44

method	result	size
default	\(-\frac {e^{2} \left (-9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \cos \left (\frac {a}{b}\right ) \operatorname {FresnelS}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b -9 \sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, \sqrt {a +b \arcsin \left (d x +c \right )}\, \sin \left (\frac {a}{b}\right ) \operatorname {FresnelC}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) b +\sqrt {-\frac {3}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \sin \left (\frac {3 a}{b}\right ) \operatorname {FresnelC}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, b +\sqrt {-\frac {3}{b}}\, \sqrt {2}\, \sqrt {\pi }\, \cos \left (\frac {3 a}{b}\right ) \operatorname {FresnelS}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (d x +c \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {a +b \arcsin \left (d x +c \right )}\, b +18 \arcsin \left (d x +c \right ) \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) b +18 \sin \left (-\frac {a +b \arcsin \left (d x +c \right )}{b}+\frac {a}{b}\right ) a -6 \arcsin \left (d x +c \right ) \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) b -6 \sin \left (-\frac {3 \left (a +b \arcsin \left (d x +c \right )\right )}{b}+\frac {3 a}{b}\right ) a \right )}{72 d \sqrt {a +b \arcsin \left (d x +c \right )}}\)	\(394\)

Input:

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

Output:

-1/72/d*e^2/(a+b*arcsin(d*x+c))^(1/2)*(-9*2^(1/2)*Pi^(1/2)*(-1/b)^(1/2)*(a 
+b*arcsin(d*x+c))^(1/2)*cos(a/b)*FresnelS(2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a 
+b*arcsin(d*x+c))^(1/2)/b)*b-9*2^(1/2)*Pi^(1/2)*(-1/b)^(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+(-3/b)^(1/2)*2^(1/2)*Pi^(1/2)*sin(3*a/b)*FresnelC(3*2^(1 
/2)/Pi^(1/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b*arcsin(d*x+c)) 
^(1/2)*b+(-3/b)^(1/2)*2^(1/2)*Pi^(1/2)*cos(3*a/b)*FresnelS(3*2^(1/2)/Pi^(1 
/2)/(-3/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b*arcsin(d*x+c))^(1/2)*b+ 
18*arcsin(d*x+c)*sin(-(a+b*arcsin(d*x+c))/b+a/b)*b+18*sin(-(a+b*arcsin(d*x 
+c))/b+a/b)*a-6*arcsin(d*x+c)*sin(-3*(a+b*arcsin(d*x+c))/b+3*a/b)*b-6*sin( 
-3*(a+b*arcsin(d*x+c))/b+3*a/b)*a)
 

Fricas [F(-2)]

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

integrate((d*e*x+c*e)^2*(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)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.30 (sec) , antiderivative size = 1169, normalized size of antiderivative = 4.27 \[ \int (c e+d e x)^2 \sqrt {a+b \arcsin (c+d x)} \, dx=\text {Too large to display} \] Input:

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

Output:

1/8*sqrt(2)*sqrt(pi)*a*b*e^2*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^2/sqrt(abs(b)) + b*sqrt(abs(b)))*d) + 1/16*I*sqrt(2)*sqrt(p 
i)*b^2*e^2*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^2/s 
qrt(abs(b)) + b*sqrt(abs(b)))*d) + 1/8*sqrt(2)*sqrt(pi)*a*b*e^2*erf(1/2*I* 
sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(abs(b)) - 1/2*sqrt(2)*sqrt(b*arcs 
in(d*x + c) + a)*sqrt(abs(b))/b)*e^(-I*a/b)/((-I*b^2/sqrt(abs(b)) + b*sqrt 
(abs(b)))*d) - 1/16*I*sqrt(2)*sqrt(pi)*b^2*e^2*erf(1/2*I*sqrt(2)*sqrt(b*ar 
csin(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^2/sqrt(abs(b)) + b*sqrt(abs(b)))*d) - 1/ 
4*sqrt(pi)*a*sqrt(b)*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt 
(b) - 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b 
)/((sqrt(6)*b + I*sqrt(6)*b^2/abs(b))*d) - 1/24*I*sqrt(pi)*b^(3/2)*e^2*erf 
(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - 1/2*I*sqrt(6)*sqrt(b*a 
rcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(3*I*a/b)/((sqrt(6)*b + I*sqrt(6)*b^ 
2/abs(b))*d) - 1/4*sqrt(pi)*a*sqrt(b)*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d 
*x + c) + a)/sqrt(b) + 1/2*I*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/a 
bs(b))*e^(-3*I*a/b)/((sqrt(6)*b - I*sqrt(6)*b^2/abs(b))*d) + 1/24*I*sqrt(p 
i)*b^(3/2)*e^2*erf(-1/2*sqrt(6)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + 1...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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