\(\int \frac {x^4 (a+b \arcsin (c x))^2}{(d-c^2 d x^2)^{3/2}} \, dx\) [242]

Optimal result

Integrand size = 29, antiderivative size = 414 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {b^2 x \sqrt {d-c^2 d x^2}}{4 c^4 d^2}+\frac {b^2 \sqrt {1-c^2 x^2} \arcsin (c x)}{4 c^5 d \sqrt {d-c^2 d x^2}}-\frac {b x^2 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^3 d \sqrt {d-c^2 d x^2}}+\frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {i \sqrt {1-c^2 x^2} (a+b \arcsin (c x))^2}{c^5 d \sqrt {d-c^2 d x^2}}+\frac {3 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^4 d^2}-\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{2 b c^5 d \sqrt {d-c^2 d x^2}}+\frac {2 b \sqrt {1-c^2 x^2} (a+b \arcsin (c x)) \log \left (1+e^{2 i \arcsin (c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}}-\frac {i b^2 \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )}{c^5 d \sqrt {d-c^2 d x^2}} \] Output:

-1/4*b^2*x*(-c^2*d*x^2+d)^(1/2)/c^4/d^2+1/4*b^2*(-c^2*x^2+1)^(1/2)*arcsin( 
c*x)/c^5/d/(-c^2*d*x^2+d)^(1/2)-1/2*b*x^2*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c 
*x))/c^3/d/(-c^2*d*x^2+d)^(1/2)+x^3*(a+b*arcsin(c*x))^2/c^2/d/(-c^2*d*x^2+ 
d)^(1/2)-I*(-c^2*x^2+1)^(1/2)*(a+b*arcsin(c*x))^2/c^5/d/(-c^2*d*x^2+d)^(1/ 
2)+3/2*x*(-c^2*d*x^2+d)^(1/2)*(a+b*arcsin(c*x))^2/c^4/d^2-1/2*(-c^2*x^2+1) 
^(1/2)*(a+b*arcsin(c*x))^3/b/c^5/d/(-c^2*d*x^2+d)^(1/2)+2*b*(-c^2*x^2+1)^( 
1/2)*(a+b*arcsin(c*x))*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)/c^5/d/(-c^2*d*x^ 
2+d)^(1/2)-I*b^2*(-c^2*x^2+1)^(1/2)*polylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^ 
2)/c^5/d/(-c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.74 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.75 \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-4 a^2 c \sqrt {d} x \left (-3+c^2 x^2\right )+12 a^2 \sqrt {d-c^2 d x^2} \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )+2 a b \sqrt {d} \left (8 c x \arcsin (c x)+\sqrt {1-c^2 x^2} \left (-6 \arcsin (c x)^2+\cos (2 \arcsin (c x))+4 \log \left (1-c^2 x^2\right )+2 \arcsin (c x) \sin (2 \arcsin (c x))\right )\right )+b^2 \sqrt {d} \left (8 c x \arcsin (c x)^2-8 i \sqrt {1-c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )+\sqrt {1-c^2 x^2} \left (-4 \arcsin (c x)^3+2 \arcsin (c x) \left (\cos (2 \arcsin (c x))+8 \log \left (1+e^{2 i \arcsin (c x)}\right )\right )-\sin (2 \arcsin (c x))+2 \arcsin (c x)^2 (-4 i+\sin (2 \arcsin (c x)))\right )\right )}{8 c^5 d^{3/2} \sqrt {d-c^2 d x^2}} \] Input:

Integrate[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]
 

Output:

(-4*a^2*c*Sqrt[d]*x*(-3 + c^2*x^2) + 12*a^2*Sqrt[d - c^2*d*x^2]*ArcTan[(c* 
x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))] + 2*a*b*Sqrt[d]*(8*c*x*Ar 
cSin[c*x] + Sqrt[1 - c^2*x^2]*(-6*ArcSin[c*x]^2 + Cos[2*ArcSin[c*x]] + 4*L 
og[1 - c^2*x^2] + 2*ArcSin[c*x]*Sin[2*ArcSin[c*x]])) + b^2*Sqrt[d]*(8*c*x* 
ArcSin[c*x]^2 - (8*I)*Sqrt[1 - c^2*x^2]*PolyLog[2, -E^((2*I)*ArcSin[c*x])] 
 + Sqrt[1 - c^2*x^2]*(-4*ArcSin[c*x]^3 + 2*ArcSin[c*x]*(Cos[2*ArcSin[c*x]] 
 + 8*Log[1 + E^((2*I)*ArcSin[c*x])]) - Sin[2*ArcSin[c*x]] + 2*ArcSin[c*x]^ 
2*(-4*I + Sin[2*ArcSin[c*x]]))))/(8*c^5*d^(3/2)*Sqrt[d - c^2*d*x^2])
 

Rubi [A] (verified)

Time = 2.27 (sec) , antiderivative size = 394, normalized size of antiderivative = 0.95, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.483, Rules used = {5206, 5210, 262, 223, 5138, 262, 223, 5152, 5180, 3042, 4202, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 5206

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

\(\Big \downarrow \) 5210

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5138

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 223

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

\(\Big \downarrow \) 5152

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

\(\Big \downarrow \) 5180

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4202

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle \frac {x^3 (a+b \arcsin (c x))^2}{c^2 d \sqrt {d-c^2 d x^2}}-\frac {3 \left (-\frac {x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{2 c^2 d}+\frac {\sqrt {1-c^2 x^2} (a+b \arcsin (c x))^3}{6 b c^3 \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {1-c^2 x^2} \left (\frac {1}{2} x^2 (a+b \arcsin (c x))-\frac {1}{2} b c \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )\right )}{c \sqrt {d-c^2 d x^2}}\right )}{c^2 d}-\frac {2 b \sqrt {1-c^2 x^2} \left (\frac {\frac {i (a+b \arcsin (c x))^2}{2 b}-2 i \left (-\frac {1}{2} i \log \left (1+e^{2 i \arcsin (c x)}\right ) (a+b \arcsin (c x))-\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 i \arcsin (c x)}\right )\right )}{c^4}-\frac {x^2 (a+b \arcsin (c x))}{2 c^2}+\frac {b \left (\frac {\arcsin (c x)}{2 c^3}-\frac {x \sqrt {1-c^2 x^2}}{2 c^2}\right )}{2 c}\right )}{c d \sqrt {d-c^2 d x^2}}\)

Input:

Int[(x^4*(a + b*ArcSin[c*x])^2)/(d - c^2*d*x^2)^(3/2),x]
 

Output:

(x^3*(a + b*ArcSin[c*x])^2)/(c^2*d*Sqrt[d - c^2*d*x^2]) - (3*(-1/2*(x*Sqrt 
[d - c^2*d*x^2]*(a + b*ArcSin[c*x])^2)/(c^2*d) + (Sqrt[1 - c^2*x^2]*(a + b 
*ArcSin[c*x])^3)/(6*b*c^3*Sqrt[d - c^2*d*x^2]) + (b*Sqrt[1 - c^2*x^2]*((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))/(c*Sqrt[d - c^2*d*x^2])))/(c^2*d) - (2*b*Sqrt[1 - c^2*x^ 
2]*(-1/2*(x^2*(a + b*ArcSin[c*x]))/c^2 + (b*(-1/2*(x*Sqrt[1 - c^2*x^2])/c^ 
2 + ArcSin[c*x]/(2*c^3)))/(2*c) + (((I/2)*(a + b*ArcSin[c*x])^2)/b - (2*I) 
*((-1/2*I)*(a + b*ArcSin[c*x])*Log[1 + E^((2*I)*ArcSin[c*x])] - (b*PolyLog 
[2, -E^((2*I)*ArcSin[c*x])])/4))/c^4))/(c*d*Sqrt[d - c^2*d*x^2])
 

Defintions of rubi rules used

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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

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

rule 4202

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I 
*((c + d*x)^(m + 1)/(d*(m + 1))), x] - Simp[2*I   Int[(c + d*x)^m*(E^(2*I*( 
e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] && IGt 
Q[m, 0]
 

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 5180

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

rule 5206

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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
 Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSin[c*x])^n, x], x] + Simp 
[b*f*(n/(2*c*(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}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[m, 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]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 806 vs. \(2 (392 ) = 784\).

Time = 0.74 (sec) , antiderivative size = 807, normalized size of antiderivative = 1.95

method	result	size
default	\(-\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{2 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (-i+8 \arcsin \left (c x \right )\right )}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (18 \arcsin \left (c x \right )^{2}-1\right ) x}{16 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{16 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}\right )+\frac {3 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{2 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {9 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x}{4 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (3 \arcsin \left (c x \right )\right )}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{4 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}\)	\(807\)
parts	\(-\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {-c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {-c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{2 c^{4} d \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{3}}{2 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (c x \right ) \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (c x \right )^{2}+\operatorname {polylog}\left (2, -\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )\right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {i \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \left (-i+8 \arcsin \left (c x \right )\right )}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (18 \arcsin \left (c x \right )^{2}-1\right ) x}{16 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \cos \left (3 \arcsin \left (c x \right )\right )}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (2 \arcsin \left (c x \right )^{2}-1\right ) \sin \left (3 \arcsin \left (c x \right )\right )}{16 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}\right )+\frac {3 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right )^{2}}{2 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}+\frac {2 i a b \sqrt {-c^{2} x^{2}+1}\, \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {2 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}\, \ln \left (1+\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )^{2}\right )}{d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {-c^{2} x^{2}+1}}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {9 a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) x}{4 d^{2} c^{4} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \cos \left (3 \arcsin \left (c x \right )\right )}{8 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}-\frac {a b \sqrt {-d \left (c^{2} x^{2}-1\right )}\, \arcsin \left (c x \right ) \sin \left (3 \arcsin \left (c x \right )\right )}{4 d^{2} c^{5} \left (c^{2} x^{2}-1\right )}\)	\(807\)

Input:

int(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-1/2*a^2*x^3/c^2/d/(-c^2*d*x^2+d)^(1/2)+3/2*a^2/c^4*x/d/(-c^2*d*x^2+d)^(1/ 
2)-3/2*a^2/c^4/d/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2) 
)+b^2*(1/2*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^5/(c^2*x^2-1)*a 
rcsin(c*x)^3+I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^5/(c^2*x^2- 
1)*(2*I*arcsin(c*x)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)+2*arcsin(c*x)^2+pol 
ylog(2,-(I*c*x+(-c^2*x^2+1)^(1/2))^2))-1/8*I*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x 
^2-1))^(1/2)/d^2/c^5/(c^2*x^2-1)*arcsin(c*x)*(-I+8*arcsin(c*x))-1/16*(-d*( 
c^2*x^2-1))^(1/2)/d^2/c^4/(c^2*x^2-1)*(18*arcsin(c*x)^2-1)*x-1/8*(-d*(c^2* 
x^2-1))^(1/2)/d^2/c^5/(c^2*x^2-1)*arcsin(c*x)*cos(3*arcsin(c*x))-1/16*(-d* 
(c^2*x^2-1))^(1/2)/d^2/c^5/(c^2*x^2-1)*(2*arcsin(c*x)^2-1)*sin(3*arcsin(c* 
x)))+3/2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2/c^5/(c^2*x^2-1) 
*arcsin(c*x)^2+2*I*a*b*(-c^2*x^2+1)^(1/2)*(-d*(c^2*x^2-1))^(1/2)/d^2/c^5/( 
c^2*x^2-1)*arcsin(c*x)-2*a*b*(-d*(c^2*x^2-1))^(1/2)*(-c^2*x^2+1)^(1/2)/d^2 
/c^5/(c^2*x^2-1)*ln(1+(I*c*x+(-c^2*x^2+1)^(1/2))^2)-1/8*a*b*(-d*(c^2*x^2-1 
))^(1/2)/d^2/c^5/(c^2*x^2-1)*(-c^2*x^2+1)^(1/2)-9/4*a*b*(-d*(c^2*x^2-1))^( 
1/2)/d^2/c^4/(c^2*x^2-1)*arcsin(c*x)*x-1/8*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/ 
c^5/(c^2*x^2-1)*cos(3*arcsin(c*x))-1/4*a*b*(-d*(c^2*x^2-1))^(1/2)/d^2/c^5/ 
(c^2*x^2-1)*arcsin(c*x)*sin(3*arcsin(c*x))
 

Fricas [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x**4*(a+b*asin(c*x))**2/(-c**2*d*x**2+d)**(3/2),x)
 

Output:

Integral(x**4*(a + b*asin(c*x))**2/(-d*(c*x - 1)*(c*x + 1))**(3/2), x)
 

Maxima [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2} x^{4}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="maxim 
a")
 

Output:

-1/2*a^2*(x^3/(sqrt(-c^2*d*x^2 + d)*c^2*d) - 3*x/(sqrt(-c^2*d*x^2 + d)*c^4 
*d) + 3*arcsin(c*x)/(c^5*d^(3/2))) + sqrt(d)*integrate((b^2*x^4*arctan2(c* 
x, sqrt(c*x + 1)*sqrt(-c*x + 1))^2 + 2*a*b*x^4*arctan2(c*x, sqrt(c*x + 1)* 
sqrt(-c*x + 1)))*sqrt(c*x + 1)*sqrt(-c*x + 1)/(c^4*d^2*x^4 - 2*c^2*d^2*x^2 
 + d^2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4*(a+b*arcsin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x, algorithm="giac" 
)
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \] Input:

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(3/2),x)
 

Output:

int((x^4*(a + b*asin(c*x))^2)/(d - c^2*d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {-2 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{3} b^{2}-6 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right )^{2} a b -2 \mathit {asin} \left (c x \right )^{2} b^{2} c^{3} x^{3}+2 \mathit {asin} \left (c x \right )^{2} b^{2} c x -6 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) a^{2}-2 \sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b^{2} c^{2} x^{2}+\sqrt {-c^{2} x^{2}+1}\, \mathit {asin} \left (c x \right ) b^{2}-4 \mathit {asin} \left (c x \right ) a b \,c^{3} x^{3}+4 \mathit {asin} \left (c x \right ) a b c x -8 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) a b c -4 \sqrt {-c^{2} x^{2}+1}\, \left (\int \frac {\mathit {asin} \left (c x \right )^{2}}{\sqrt {-c^{2} x^{2}+1}\, c^{2} x^{2}-\sqrt {-c^{2} x^{2}+1}}d x \right ) b^{2} c -2 \sqrt {-c^{2} x^{2}+1}\, a b \,c^{2} x^{2}+2 \sqrt {-c^{2} x^{2}+1}\, a b -2 a^{2} c^{3} x^{3}+6 a^{2} c x +b^{2} c^{3} x^{3}-b^{2} c x}{4 \sqrt {d}\, \sqrt {-c^{2} x^{2}+1}\, c^{5} d} \] Input:

int(x^4*(a+b*asin(c*x))^2/(-c^2*d*x^2+d)^(3/2),x)
 

Output:

( - 2*sqrt( - c**2*x**2 + 1)*asin(c*x)**3*b**2 - 6*sqrt( - c**2*x**2 + 1)* 
asin(c*x)**2*a*b - 2*asin(c*x)**2*b**2*c**3*x**3 + 2*asin(c*x)**2*b**2*c*x 
 - 6*sqrt( - c**2*x**2 + 1)*asin(c*x)*a**2 - 2*sqrt( - c**2*x**2 + 1)*asin 
(c*x)*b**2*c**2*x**2 + sqrt( - c**2*x**2 + 1)*asin(c*x)*b**2 - 4*asin(c*x) 
*a*b*c**3*x**3 + 4*asin(c*x)*a*b*c*x - 8*sqrt( - c**2*x**2 + 1)*int(asin(c 
*x)/(sqrt( - c**2*x**2 + 1)*c**2*x**2 - sqrt( - c**2*x**2 + 1)),x)*a*b*c - 
 4*sqrt( - c**2*x**2 + 1)*int(asin(c*x)**2/(sqrt( - c**2*x**2 + 1)*c**2*x* 
*2 - sqrt( - c**2*x**2 + 1)),x)*b**2*c - 2*sqrt( - c**2*x**2 + 1)*a*b*c**2 
*x**2 + 2*sqrt( - c**2*x**2 + 1)*a*b - 2*a**2*c**3*x**3 + 6*a**2*c*x + b** 
2*c**3*x**3 - b**2*c*x)/(4*sqrt(d)*sqrt( - c**2*x**2 + 1)*c**5*d)