\(\int (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx\) [81]

Optimal result

Integrand size = 32, antiderivative size = 564 \[ \int (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=-\frac {245 b^2 d^2 e^2 x \sqrt {d+c d x} \sqrt {e-c e x}}{1152}-\frac {65 b^2 d^2 e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )}{1728}-\frac {1}{108} b^2 d^2 e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2+\frac {115 b^2 d^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)}{1152 c \sqrt {1-c^2 x^2}}-\frac {5 b c d^2 e^2 x^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))}{16 \sqrt {1-c^2 x^2}}+\frac {5 b d^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))}{48 c}+\frac {b d^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))}{18 c}+\frac {5}{16} d^2 e^2 x \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^2+\frac {5}{24} d^2 e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right ) (a+b \arcsin (c x))^2+\frac {1}{6} d^2 e^2 x \sqrt {d+c d x} \sqrt {e-c e x} \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))^2+\frac {5 d^2 e^2 \sqrt {d+c d x} \sqrt {e-c e x} (a+b \arcsin (c x))^3}{48 b c \sqrt {1-c^2 x^2}} \] Output:

-245/1152*b^2*d^2*e^2*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)-65/1728*b^2*d^2*e 
^2*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)-1/108*b^2*d^2*e^2*x*(c* 
d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)^2+115/1152*b^2*d^2*e^2*(c*d*x+d 
)^(1/2)*(-c*e*x+e)^(1/2)*arcsin(c*x)/c/(-c^2*x^2+1)^(1/2)-5/16*b*c*d^2*e^2 
*x^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2) 
+5/48*b*d^2*e^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1)^(3/2)*(a+b*a 
rcsin(c*x))/c+1/18*b*d^2*e^2*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2*x^2+1) 
^(5/2)*(a+b*arcsin(c*x))/c+5/16*d^2*e^2*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2) 
*(a+b*arcsin(c*x))^2+5/24*d^2*e^2*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2)*(-c^2 
*x^2+1)*(a+b*arcsin(c*x))^2+1/6*d^2*e^2*x*(c*d*x+d)^(1/2)*(-c*e*x+e)^(1/2) 
*(-c^2*x^2+1)^2*(a+b*arcsin(c*x))^2+5/48*d^2*e^2*(c*d*x+d)^(1/2)*(-c*e*x+e 
)^(1/2)*(a+b*arcsin(c*x))^3/b/c/(-c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 3.46 (sec) , antiderivative size = 450, normalized size of antiderivative = 0.80 \[ \int (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {d^2 e^2 \left (1440 b^2 \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^3-4320 a^2 \sqrt {d} \sqrt {e} \sqrt {1-c^2 x^2} \arctan \left (\frac {c x \sqrt {d+c d x} \sqrt {e-c e x}}{\sqrt {d} \sqrt {e} \left (-1+c^2 x^2\right )}\right )+12 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x) (270 b \cos (2 \arcsin (c x))+27 b \cos (4 \arcsin (c x))+2 b \cos (6 \arcsin (c x))+540 a \sin (2 \arcsin (c x))+108 a \sin (4 \arcsin (c x))+12 a \sin (6 \arcsin (c x)))+72 b \sqrt {d+c d x} \sqrt {e-c e x} \arcsin (c x)^2 (60 a+45 b \sin (2 \arcsin (c x))+9 b \sin (4 \arcsin (c x))+b \sin (6 \arcsin (c x)))+\sqrt {d+c d x} \sqrt {e-c e x} \left (9504 a^2 c x \sqrt {1-c^2 x^2}-7488 a^2 c^3 x^3 \sqrt {1-c^2 x^2}+2304 a^2 c^5 x^5 \sqrt {1-c^2 x^2}+3240 a b \cos (2 \arcsin (c x))+324 a b \cos (4 \arcsin (c x))+24 a b \cos (6 \arcsin (c x))-1620 b^2 \sin (2 \arcsin (c x))-81 b^2 \sin (4 \arcsin (c x))-4 b^2 \sin (6 \arcsin (c x))\right )\right )}{13824 c \sqrt {1-c^2 x^2}} \] Input:

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

Output:

(d^2*e^2*(1440*b^2*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[c*x]^3 - 4320*a^ 
2*Sqrt[d]*Sqrt[e]*Sqrt[1 - c^2*x^2]*ArcTan[(c*x*Sqrt[d + c*d*x]*Sqrt[e - c 
*e*x])/(Sqrt[d]*Sqrt[e]*(-1 + c^2*x^2))] + 12*b*Sqrt[d + c*d*x]*Sqrt[e - c 
*e*x]*ArcSin[c*x]*(270*b*Cos[2*ArcSin[c*x]] + 27*b*Cos[4*ArcSin[c*x]] + 2* 
b*Cos[6*ArcSin[c*x]] + 540*a*Sin[2*ArcSin[c*x]] + 108*a*Sin[4*ArcSin[c*x]] 
 + 12*a*Sin[6*ArcSin[c*x]]) + 72*b*Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*ArcSin[ 
c*x]^2*(60*a + 45*b*Sin[2*ArcSin[c*x]] + 9*b*Sin[4*ArcSin[c*x]] + b*Sin[6* 
ArcSin[c*x]]) + Sqrt[d + c*d*x]*Sqrt[e - c*e*x]*(9504*a^2*c*x*Sqrt[1 - c^2 
*x^2] - 7488*a^2*c^3*x^3*Sqrt[1 - c^2*x^2] + 2304*a^2*c^5*x^5*Sqrt[1 - c^2 
*x^2] + 3240*a*b*Cos[2*ArcSin[c*x]] + 324*a*b*Cos[4*ArcSin[c*x]] + 24*a*b* 
Cos[6*ArcSin[c*x]] - 1620*b^2*Sin[2*ArcSin[c*x]] - 81*b^2*Sin[4*ArcSin[c*x 
]] - 4*b^2*Sin[6*ArcSin[c*x]])))/(13824*c*Sqrt[1 - c^2*x^2])
 

Rubi [A] (verified)

Time = 1.82 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.77, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.406, Rules used = {5178, 5158, 5158, 5156, 5138, 262, 223, 5152, 5182, 211, 211, 211, 223}

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.

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

\(\Big \downarrow \) 5178

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

\(\Big \downarrow \) 5158

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx+\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 5158

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {3}{4} \int \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 5156

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx-b c \int x (a+b \arcsin (c x))dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 5138

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {3}{4} \left (-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {1-c^2 x^2}}dx\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {3}{4} \left (-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\int \frac {1}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {3}{4} \left (\frac {1}{2} \int \frac {(a+b \arcsin (c x))^2}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \int x \left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))dx+\frac {5}{6} \left (-\frac {1}{2} b c \int x \left (1-c^2 x^2\right ) (a+b \arcsin (c x))dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 5182

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \left (\frac {b \int \left (1-c^2 x^2\right )^{5/2}dx}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2}\right )+\frac {5}{6} \left (-\frac {1}{2} b c \left (\frac {b \int \left (1-c^2 x^2\right )^{3/2}dx}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \left (\frac {b \left (\frac {5}{6} \int \left (1-c^2 x^2\right )^{3/2}dx+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2}\right )+\frac {5}{6} \left (-\frac {1}{2} b c \left (\frac {b \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \left (\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \int \sqrt {1-c^2 x^2}dx+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2}\right )+\frac {5}{6} \left (-\frac {1}{2} b c \left (\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (-\frac {1}{3} b c \left (\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2}\right )+\frac {5}{6} \left (-\frac {1}{2} b c \left (\frac {b \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {1-c^2 x^2}}dx+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

\(\Big \downarrow \) 223

\(\displaystyle \frac {(c d x+d)^{5/2} (e-c e x)^{5/2} \left (\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2} (a+b \arcsin (c x))^2-\frac {1}{3} b c \left (\frac {b \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )+\frac {1}{6} x \left (1-c^2 x^2\right )^{5/2}\right )}{6 c}-\frac {\left (1-c^2 x^2\right )^3 (a+b \arcsin (c x))}{6 c^2}\right )+\frac {5}{6} \left (\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2} (a+b \arcsin (c x))^2-\frac {1}{2} b c \left (\frac {b \left (\frac {3}{4} \left (\frac {\arcsin (c x)}{2 c}+\frac {1}{2} x \sqrt {1-c^2 x^2}\right )+\frac {1}{4} x \left (1-c^2 x^2\right )^{3/2}\right )}{4 c}-\frac {\left (1-c^2 x^2\right )^2 (a+b \arcsin (c x))}{4 c^2}\right )+\frac {3}{4} \left (\frac {1}{2} x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2-b c \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )+\frac {(a+b \arcsin (c x))^3}{6 b c}\right )\right )\right )}{\left (1-c^2 x^2\right )^{5/2}}\)

Input:

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

Output:

((d + c*d*x)^(5/2)*(e - c*e*x)^(5/2)*((x*(1 - c^2*x^2)^(5/2)*(a + b*ArcSin 
[c*x])^2)/6 - (b*c*(-1/6*((1 - c^2*x^2)^3*(a + b*ArcSin[c*x]))/c^2 + (b*(( 
x*(1 - c^2*x^2)^(5/2))/6 + (5*((x*(1 - c^2*x^2)^(3/2))/4 + (3*((x*Sqrt[1 - 
 c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/6))/(6*c)))/3 + (5*((x*(1 - c^2*x^2 
)^(3/2)*(a + b*ArcSin[c*x])^2)/4 + (3*((x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[ 
c*x])^2)/2 + (a + b*ArcSin[c*x])^3/(6*b*c) - b*c*((x^2*(a + b*ArcSin[c*x]) 
)/2 - (b*c*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^2 + ArcSin[c*x]/(2*c^3)))/2)))/4 
- (b*c*(-1/4*((1 - c^2*x^2)^2*(a + b*ArcSin[c*x]))/c^2 + (b*((x*(1 - c^2*x 
^2)^(3/2))/4 + (3*((x*Sqrt[1 - c^2*x^2])/2 + ArcSin[c*x]/(2*c)))/4))/(4*c) 
))/2))/6))/(1 - c^2*x^2)^(5/2)
 

Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt 
[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 5138

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

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 5156

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

rule 5158

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x 
_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSin[c*x])^n/(2*p + 1)), x] + (S 
imp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], 
x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   Int[x*(1 
- c^2*x^2)^(p - 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c 
, d, e}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0]
 

rule 5178

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_) 
 + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(d + e*x)^q*((f + g*x)^q/(1 - c^2*x^ 
2)^q)   Int[(d + e*x)^(p - q)*(1 - c^2*x^2)^q*(a + b*ArcSin[c*x])^n, x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[e*f + d*g, 0] && EqQ[c^2*d^2 
- e^2, 0] && HalfIntegerQ[p, q] && GeQ[p - q, 0]
 

rule 5182

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_ 
.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSin[c*x])^n/(2*e*(p + 
1))), x] + Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 - c^2*x^2)^p]   I 
nt[(1 - c^2*x^2)^(p + 1/2)*(a + b*ArcSin[c*x])^(n - 1), x], x] /; FreeQ[{a, 
 b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 4.38 (sec) , antiderivative size = 1617, normalized size of antiderivative = 2.87

method	result	size
default	\(\text {Expression too large to display}\)	\(1617\)
parts	\(\text {Expression too large to display}\)	\(1617\)

Input:

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

Output:

-1/6*a^2/c/e*(c*d*x+d)^(5/2)*(-c*e*x+e)^(7/2)-1/6*a^2*d/c/e*(c*d*x+d)^(3/2 
)*(-c*e*x+e)^(7/2)-1/8*a^2*d^2/c/e*(c*d*x+d)^(1/2)*(-c*e*x+e)^(7/2)+1/24*a 
^2*d^2/c*(-c*e*x+e)^(5/2)*(c*d*x+d)^(1/2)+5/48*a^2*d^2*e/c*(-c*e*x+e)^(3/2 
)*(c*d*x+d)^(1/2)+5/16*a^2*d^2*e^2/c*(-c*e*x+e)^(1/2)*(c*d*x+d)^(1/2)+5/16 
*a^2*d^3*e^3*((c*d*x+d)*(-c*e*x+e))^(1/2)/(-c*e*x+e)^(1/2)/(c*d*x+d)^(1/2) 
/(c^2*d*e)^(1/2)*arctan((c^2*d*e)^(1/2)*x/(-c^2*d*e*x^2+d*e)^(1/2))+b^2*(- 
5/48*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/c/(c^2*x^2-1) 
*arcsin(c*x)^3*d^2*e^2+1/6912*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(-32*I* 
(-c^2*x^2+1)^(1/2)*x^6*c^6+32*c^7*x^7+48*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-64*c 
^5*x^5-18*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+38*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-6*c 
*x)*(6*I*arcsin(c*x)+18*arcsin(c*x)^2-1)*d^2*e^2/c/(c^2*x^2-1)+15/256*(-e* 
(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(2*I*(-c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3 
-I*(-c^2*x^2+1)^(1/2)-2*c*x)*(2*arcsin(c*x)^2-1-2*I*arcsin(c*x))*d^2*e^2/c 
/(c^2*x^2-1)-1/27648*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)*(I*c^2*x^2-c*x*( 
-c^2*x^2+1)^(1/2)-I)*(348*I*arcsin(c*x)+576*arcsin(c*x)^2-77)*cos(5*arcsin 
(c*x))*d^2*e^2/c/(c^2*x^2-1)+5/27648*(-e*(c*x-1))^(1/2)*(d*(c*x+1))^(1/2)* 
(I*(-c^2*x^2+1)^(1/2)*c*x+c^2*x^2-1)*(60*I*arcsin(c*x)+144*arcsin(c*x)^2-1 
7)*sin(5*arcsin(c*x))*d^2*e^2/c/(c^2*x^2-1)-3/1024*(-e*(c*x-1))^(1/2)*(d*( 
c*x+1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(44*I*arcsin(c*x)+32*ar 
csin(c*x)^2-19)*cos(3*arcsin(c*x))*d^2*e^2/c/(c^2*x^2-1)+9/1024*(-e*(c*...
 

Fricas [F]

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

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

Output:

integral((a^2*c^4*d^2*e^2*x^4 - 2*a^2*c^2*d^2*e^2*x^2 + a^2*d^2*e^2 + (b^2 
*c^4*d^2*e^2*x^4 - 2*b^2*c^2*d^2*e^2*x^2 + b^2*d^2*e^2)*arcsin(c*x)^2 + 2* 
(a*b*c^4*d^2*e^2*x^4 - 2*a*b*c^2*d^2*e^2*x^2 + a*b*d^2*e^2)*arcsin(c*x))*s 
qrt(c*d*x + d)*sqrt(-c*e*x + e), x)
 

Sympy [F(-1)]

Timed out. \[ \int (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=\text {Timed out} \] Input:

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int (d+c d x)^{5/2} (e-c e x)^{5/2} (a+b \arcsin (c x))^2 \, dx=\frac {\sqrt {e}\, \sqrt {d}\, d^{2} e^{2} \left (-30 \mathit {asin} \left (\frac {\sqrt {-c x +1}}{\sqrt {2}}\right ) a^{2}+8 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{5} x^{5}-26 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c^{3} x^{3}+33 \sqrt {c x +1}\, \sqrt {-c x +1}\, a^{2} c x +96 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x^{4}d x \right ) a b \,c^{5}-192 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+96 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )d x \right ) a b c +48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}-96 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}+48 \left (\int \sqrt {c x +1}\, \sqrt {-c x +1}\, \mathit {asin} \left (c x \right )^{2}d x \right ) b^{2} c \right )}{48 c} \] Input:

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

Output:

(sqrt(e)*sqrt(d)*d**2*e**2*( - 30*asin(sqrt( - c*x + 1)/sqrt(2))*a**2 + 8* 
sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2*c**5*x**5 - 26*sqrt(c*x + 1)*sqrt( - c 
*x + 1)*a**2*c**3*x**3 + 33*sqrt(c*x + 1)*sqrt( - c*x + 1)*a**2*c*x + 96*i 
nt(sqrt(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x**4,x)*a*b*c**5 - 192*int(sqr 
t(c*x + 1)*sqrt( - c*x + 1)*asin(c*x)*x**2,x)*a*b*c**3 + 96*int(sqrt(c*x + 
 1)*sqrt( - c*x + 1)*asin(c*x),x)*a*b*c + 48*int(sqrt(c*x + 1)*sqrt( - c*x 
 + 1)*asin(c*x)**2*x**4,x)*b**2*c**5 - 96*int(sqrt(c*x + 1)*sqrt( - c*x + 
1)*asin(c*x)**2*x**2,x)*b**2*c**3 + 48*int(sqrt(c*x + 1)*sqrt( - c*x + 1)* 
asin(c*x)**2,x)*b**2*c))/(48*c)