3.3.50 \(\int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx\) [250]

3.3.50.1 Optimal result

Integrand size = 25, antiderivative size = 475 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {225 b^2 e^3 \sqrt {a+b \arcsin (c+d x)}}{2048 d}-\frac {45 b^2 e^3 (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}}{256 d}-\frac {15 b^2 e^3 (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}}{256 d}+\frac {15 b e^3 (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{64 d}+\frac {5 b e^3 (c+d x)^3 \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}}{32 d}-\frac {3 e^3 (a+b \arcsin (c+d x))^{5/2}}{32 d}+\frac {e^3 (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}}{4 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )}{4096 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )}{256 d}-\frac {15 b^{5/2} e^3 \sqrt {\pi } \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right ) \sin \left (\frac {2 a}{b}\right )}{256 d}+\frac {15 b^{5/2} e^3 \sqrt {\frac {\pi }{2}} \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right ) \sin \left (\frac {4 a}{b}\right )}{4096 d} \]

output

-3/32*e^3*(a+b*arcsin(d*x+c))^(5/2)/d+1/4*e^3*(d*x+c)^4*(a+b*arcsin(d*x+c) 
)^(5/2)/d+15/8192*b^(5/2)*e^3*cos(4*a/b)*FresnelC(2*2^(1/2)/Pi^(1/2)*(a+b* 
arcsin(d*x+c))^(1/2)/b^(1/2))*2^(1/2)*Pi^(1/2)/d+15/8192*b^(5/2)*e^3*Fresn 
elS(2*2^(1/2)/Pi^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b^(1/2))*sin(4*a/b)*2^(1/ 
2)*Pi^(1/2)/d-15/256*b^(5/2)*e^3*cos(2*a/b)*FresnelC(2*(a+b*arcsin(d*x+c)) 
^(1/2)/b^(1/2)/Pi^(1/2))*Pi^(1/2)/d-15/256*b^(5/2)*e^3*FresnelS(2*(a+b*arc 
sin(d*x+c))^(1/2)/b^(1/2)/Pi^(1/2))*sin(2*a/b)*Pi^(1/2)/d+15/64*b*e^3*(d*x 
+c)*(a+b*arcsin(d*x+c))^(3/2)*(1-(d*x+c)^2)^(1/2)/d+5/32*b*e^3*(d*x+c)^3*( 
a+b*arcsin(d*x+c))^(3/2)*(1-(d*x+c)^2)^(1/2)/d+225/2048*b^2*e^3*(a+b*arcsi 
n(d*x+c))^(1/2)/d-45/256*b^2*e^3*(d*x+c)^2*(a+b*arcsin(d*x+c))^(1/2)/d-15/ 
256*b^2*e^3*(d*x+c)^4*(a+b*arcsin(d*x+c))^(1/2)/d
 

3.3.50.2 Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 255, normalized size of antiderivative = 0.54 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\frac {i b^3 e^3 e^{-\frac {4 i a}{b}} \left (16 \sqrt {2} e^{\frac {2 i a}{b}} \sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},-\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-16 \sqrt {2} e^{\frac {6 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},\frac {2 i (a+b \arcsin (c+d x))}{b}\right )-\sqrt {-\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},-\frac {4 i (a+b \arcsin (c+d x))}{b}\right )+e^{\frac {8 i a}{b}} \sqrt {\frac {i (a+b \arcsin (c+d x))}{b}} \Gamma \left (\frac {7}{2},\frac {4 i (a+b \arcsin (c+d x))}{b}\right )\right )}{2048 d \sqrt {a+b \arcsin (c+d x)}} \]

input

Integrate[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^(5/2),x]
 

output

((I/2048)*b^3*e^3*(16*Sqrt[2]*E^(((2*I)*a)/b)*Sqrt[((-I)*(a + b*ArcSin[c + 
 d*x]))/b]*Gamma[7/2, ((-2*I)*(a + b*ArcSin[c + d*x]))/b] - 16*Sqrt[2]*E^( 
((6*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[((-I)*(a + b*ArcSin[c + d*x]))/b]*Gamma[7/2, ( 
(-4*I)*(a + b*ArcSin[c + d*x]))/b] + E^(((8*I)*a)/b)*Sqrt[(I*(a + b*ArcSin 
[c + d*x]))/b]*Gamma[7/2, ((4*I)*(a + b*ArcSin[c + d*x]))/b]))/(d*E^(((4*I 
)*a)/b)*Sqrt[a + b*ArcSin[c + d*x]])
 

3.3.50.3 Rubi [A] (verified)

Time = 2.53 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.16, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {5304, 27, 5140, 5210, 5140, 5210, 5140, 5152, 5224, 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.

\(\displaystyle \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 5304

\(\displaystyle \frac {\int e^3 (c+d x)^3 (a+b \arcsin (c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^3 \int (c+d x)^3 (a+b \arcsin (c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 5140

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}-\frac {5}{8} b \int \frac {(c+d x)^4 (a+b \arcsin (c+d x))^{3/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)\right )}{d}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}-\frac {5}{8} b \left (\frac {3}{8} b \int (c+d x)^3 \sqrt {a+b \arcsin (c+d x)}d(c+d x)+\frac {3}{4} \int \frac {(c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 5140

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}-\frac {5}{8} b \left (\frac {3}{8} b \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} b \int \frac {(c+d x)^4}{\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}d(c+d x)\right )+\frac {3}{4} \int \frac {(c+d x)^2 (a+b \arcsin (c+d x))^{3/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}-\frac {5}{8} b \left (\frac {3}{8} b \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} b \int \frac {(c+d x)^4}{\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}d(c+d x)\right )+\frac {3}{4} \left (\frac {3}{4} b \int (c+d x) \sqrt {a+b \arcsin (c+d x)}d(c+d x)+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^{3/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 5140

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}-\frac {5}{8} b \left (\frac {3}{8} b \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} b \int \frac {(c+d x)^4}{\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}d(c+d x)\right )+\frac {3}{4} \left (\frac {3}{4} b \left (\frac {1}{2} (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{4} b \int \frac {(c+d x)^2}{\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}d(c+d x)\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c+d x))^{3/2}}{\sqrt {1-(c+d x)^2}}d(c+d x)-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}-\frac {5}{8} b \left (\frac {3}{4} \left (\frac {3}{4} b \left (\frac {1}{2} (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{4} b \int \frac {(c+d x)^2}{\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}d(c+d x)\right )+\frac {(a+b \arcsin (c+d x))^{5/2}}{5 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}\right )+\frac {3}{8} b \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} b \int \frac {(c+d x)^4}{\sqrt {1-(c+d x)^2} \sqrt {a+b \arcsin (c+d x)}}d(c+d x)\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 5224

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}-\frac {5}{8} b \left (\frac {3}{8} b \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} \int \frac {\sin ^4\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )+\frac {3}{4} \left (\frac {3}{4} b \left (\frac {1}{2} (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{4} \int \frac {\sin ^2\left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )+\frac {(a+b \arcsin (c+d x))^{5/2}}{5 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}-\frac {5}{8} b \left (\frac {3}{4} \left (\frac {3}{4} b \left (\frac {1}{2} (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{4} \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )^2}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )+\frac {(a+b \arcsin (c+d x))^{5/2}}{5 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}\right )+\frac {3}{8} b \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} \int \frac {\sin \left (\frac {a}{b}-\frac {a+b \arcsin (c+d x)}{b}\right )^4}{\sqrt {a+b \arcsin (c+d x)}}d(a+b \arcsin (c+d x))\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}-\frac {5}{8} b \left (\frac {3}{4} \left (\frac {3}{4} b \left (\frac {1}{2} (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{4} \int \left (\frac {1}{2 \sqrt {a+b \arcsin (c+d x)}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 \sqrt {a+b \arcsin (c+d x)}}\right )d(a+b \arcsin (c+d x))\right )+\frac {(a+b \arcsin (c+d x))^{5/2}}{5 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}\right )+\frac {3}{8} b \left (\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}-\frac {1}{8} \int \left (\frac {\cos \left (\frac {4 a}{b}-\frac {4 (a+b \arcsin (c+d x))}{b}\right )}{8 \sqrt {a+b \arcsin (c+d x)}}-\frac {\cos \left (\frac {2 a}{b}-\frac {2 (a+b \arcsin (c+d x))}{b}\right )}{2 \sqrt {a+b \arcsin (c+d x)}}+\frac {3}{8 \sqrt {a+b \arcsin (c+d x)}}\right )d(a+b \arcsin (c+d x))\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \arcsin (c+d x))^{5/2}-\frac {5}{8} b \left (\frac {3}{4} \left (\frac {3}{4} b \left (\frac {1}{4} \left (\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\sqrt {a+b \arcsin (c+d x)}\right )+\frac {1}{2} (c+d x)^2 \sqrt {a+b \arcsin (c+d x)}\right )+\frac {(a+b \arcsin (c+d x))^{5/2}}{5 b}-\frac {1}{2} (c+d x) \sqrt {1-(c+d x)^2} (a+b \arcsin (c+d x))^{3/2}\right )+\frac {3}{8} b \left (\frac {1}{8} \left (-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \cos \left (\frac {4 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \cos \left (\frac {2 a}{b}\right ) \operatorname {FresnelC}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} \sin \left (\frac {2 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b} \sqrt {\pi }}\right )-\frac {1}{8} \sqrt {\frac {\pi }{2}} \sqrt {b} \sin \left (\frac {4 a}{b}\right ) \operatorname {FresnelS}\left (\frac {2 \sqrt {\frac {2}{\pi }} \sqrt {a+b \arcsin (c+d x)}}{\sqrt {b}}\right )-\frac {3}{4} \sqrt {a+b \arcsin (c+d x)}\right )+\frac {1}{4} (c+d x)^4 \sqrt {a+b \arcsin (c+d x)}\right )-\frac {1}{4} \sqrt {1-(c+d x)^2} (c+d x)^3 (a+b \arcsin (c+d x))^{3/2}\right )\right )}{d}\)

input

Int[(c*e + d*e*x)^3*(a + b*ArcSin[c + d*x])^(5/2),x]
 

output

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

3.3.50.3.1 Defintions of rubi rules used

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 5152

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_S 
ymbol] :> Simp[(1/(b*c*(n + 1)))*Simp[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 5210

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] + (Simp[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] + S 
imp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(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 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]
 

3.3.50.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(885\) vs. \(2(391)=782\).

Time = 1.23 (sec) , antiderivative size = 886, normalized size of antiderivative = 1.87

method	result	size
default	\(\text {Expression too large to display}\)	\(886\)

input

int((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^(5/2),x,method=_RETURNVERBOSE)
 

output

-1/8192*e^3/d/(a+b*arcsin(d*x+c))^(1/2)*(-15*Pi^(1/2)*2^(1/2)*cos(4*a/b)*F 
resnelC(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b* 
arcsin(d*x+c))^(1/2)*(-1/b)^(1/2)*b^3+15*Pi^(1/2)*2^(1/2)*sin(4*a/b)*Fresn 
elS(2*2^(1/2)/Pi^(1/2)/(-1/b)^(1/2)*(a+b*arcsin(d*x+c))^(1/2)/b)*(a+b*arcs 
in(d*x+c))^(1/2)*(-1/b)^(1/2)*b^3+480*(-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*arcsi 
n(d*x+c))^(1/2)/b)*b^3-480*(-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+1024*arcsin(d*x+c)^3*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3-2 
56*arcsin(d*x+c)^3*cos(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*b^3+3072*arcsin(d*x 
+c)^2*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b^2+1280*arcsin(d*x+c)^2*sin(- 
2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3-768*arcsin(d*x+c)^2*cos(-4*(a+b*arcsin( 
d*x+c))/b+4*a/b)*a*b^2-160*arcsin(d*x+c)^2*sin(-4*(a+b*arcsin(d*x+c))/b+4* 
a/b)*b^3+3072*arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2*b-960* 
arcsin(d*x+c)*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*b^3+2560*arcsin(d*x+c)*s 
in(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a*b^2-768*arcsin(d*x+c)*cos(-4*(a+b*arc 
sin(d*x+c))/b+4*a/b)*a^2*b+60*arcsin(d*x+c)*cos(-4*(a+b*arcsin(d*x+c))/b+4 
*a/b)*b^3-320*arcsin(d*x+c)*sin(-4*(a+b*arcsin(d*x+c))/b+4*a/b)*a*b^2+1024 
*cos(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^3-960*cos(-2*(a+b*arcsin(d*x+c))/b+ 
2*a/b)*a*b^2+1280*sin(-2*(a+b*arcsin(d*x+c))/b+2*a/b)*a^2*b-256*cos(-4*...
 

3.3.50.5 Fricas [F(-2)]

Exception generated. \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \]

input

integrate((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^(5/2),x, algorithm="fricas")
 

output

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

3.3.50.6 Sympy [F]

\[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=e^{3} \left (\int a^{2} c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int a^{2} d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 a^{2} c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int 3 a^{2} c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}}\, dx + \int b^{2} d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{3} x^{3} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 3 b^{2} c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 3 b^{2} c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}^{2}{\left (c + d x \right )}\, dx + \int 6 a b c d^{2} x^{2} \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx + \int 6 a b c^{2} d x \sqrt {a + b \operatorname {asin}{\left (c + d x \right )}} \operatorname {asin}{\left (c + d x \right )}\, dx\right ) \]

input

integrate((d*e*x+c*e)**3*(a+b*asin(d*x+c))**(5/2),x)
 

output

e**3*(Integral(a**2*c**3*sqrt(a + b*asin(c + d*x)), x) + Integral(a**2*d** 
3*x**3*sqrt(a + b*asin(c + d*x)), x) + Integral(b**2*c**3*sqrt(a + b*asin( 
c + d*x))*asin(c + d*x)**2, x) + Integral(2*a*b*c**3*sqrt(a + b*asin(c + d 
*x))*asin(c + d*x), x) + Integral(3*a**2*c*d**2*x**2*sqrt(a + b*asin(c + d 
*x)), x) + Integral(3*a**2*c**2*d*x*sqrt(a + b*asin(c + d*x)), x) + Integr 
al(b**2*d**3*x**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2, x) + Integra 
l(2*a*b*d**3*x**3*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x) + Integral(3 
*b**2*c*d**2*x**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2, x) + Integra 
l(3*b**2*c**2*d*x*sqrt(a + b*asin(c + d*x))*asin(c + d*x)**2, x) + Integra 
l(6*a*b*c*d**2*x**2*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x) + Integral 
(6*a*b*c**2*d*x*sqrt(a + b*asin(c + d*x))*asin(c + d*x), x))
 

3.3.50.7 Maxima [F]

\[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \arcsin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

input

integrate((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^(5/2),x, algorithm="maxima")
 

output

integrate((d*e*x + c*e)^3*(b*arcsin(d*x + c) + a)^(5/2), x)
 

3.3.50.8 Giac [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.85 (sec) , antiderivative size = 3408, normalized size of antiderivative = 7.17 \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\text {Too large to display} \]

input

integrate((d*e*x+c*e)^3*(a+b*arcsin(d*x+c))^(5/2),x, algorithm="giac")
 

output

-1/8192*(-512*I*sqrt(pi)*a^3*b^2*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + 
 a)/sqrt(b) + I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4* 
I*a/b)/(sqrt(2)*b^(5/2) - I*sqrt(2)*b^(7/2)/abs(b)) - 128*sqrt(b*arcsin(d* 
x + c) + a)*b^2*e^3*arcsin(d*x + c)^2*e^(4*I*arcsin(d*x + c)) + 512*sqrt(b 
*arcsin(d*x + c) + a)*b^2*e^3*arcsin(d*x + c)^2*e^(2*I*arcsin(d*x + c)) + 
512*sqrt(b*arcsin(d*x + c) + a)*b^2*e^3*arcsin(d*x + c)^2*e^(-2*I*arcsin(d 
*x + c)) - 128*sqrt(b*arcsin(d*x + c) + a)*b^2*e^3*arcsin(d*x + c)^2*e^(-4 
*I*arcsin(d*x + c)) - 1536*I*sqrt(pi)*a^3*b*e^3*erf(-sqrt(2)*sqrt(b*arcsin 
(d*x + c) + a)/sqrt(b) - I*sqrt(2)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs 
(b))*e^(4*I*a/b)/(sqrt(2)*b^(3/2) + I*sqrt(2)*b^(5/2)/abs(b)) - 192*sqrt(p 
i)*a^2*b^2*e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) - I*sqrt(2 
)*sqrt(b*arcsin(d*x + c) + a)*sqrt(b)/abs(b))*e^(4*I*a/b)/(sqrt(2)*b^(3/2) 
 + I*sqrt(2)*b^(5/2)/abs(b)) + 1024*I*sqrt(pi)*a^3*b*e^3*erf(-sqrt(b*arcsi 
n(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)) - 1024*I*sqrt(pi)*a^3*b*e^3*erf(-sq 
rt(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)) - 1024*I*sqrt(pi)*a^3*b* 
e^3*erf(-sqrt(2)*sqrt(b*arcsin(d*x + c) + a)/sqrt(b) + I*sqrt(2)*sqrt(b*ar 
csin(d*x + c) + a)*sqrt(b)/abs(b))*e^(-4*I*a/b)/(sqrt(2)*b^(3/2) - I*sqrt( 
2)*b^(5/2)/abs(b)) - 384*sqrt(pi)*a^2*b^2*e^3*erf(-sqrt(2)*sqrt(b*arcsi...
 

3.3.50.9 Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^3 (a+b \arcsin (c+d x))^{5/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^{5/2} \,d x \]

input

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^(5/2),x)
 

output

int((c*e + d*e*x)^3*(a + b*asin(c + d*x))^(5/2), x)