3.1.84 \(\int x^4 (d-c^2 d x^2)^{5/2} (a+b \arcsin (c x)) \, dx\) [84]

3.1.84.1 Optimal result

Integrand size = 27, antiderivative size = 430 \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {3 b d^2 x^2 \sqrt {d-c^2 d x^2}}{512 c^3 \sqrt {1-c^2 x^2}}+\frac {b d^2 x^4 \sqrt {d-c^2 d x^2}}{512 c \sqrt {1-c^2 x^2}}-\frac {31 b c d^2 x^6 \sqrt {d-c^2 d x^2}}{960 \sqrt {1-c^2 x^2}}+\frac {21 b c^3 d^2 x^8 \sqrt {d-c^2 d x^2}}{640 \sqrt {1-c^2 x^2}}-\frac {b c^5 d^2 x^{10} \sqrt {d-c^2 d x^2}}{100 \sqrt {1-c^2 x^2}}-\frac {3 d^2 x \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{256 c^4}-\frac {d^2 x^3 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))}{128 c^2}+\frac {1}{32} d^2 x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {1}{16} d x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {3 d^2 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))^2}{512 b c^5 \sqrt {1-c^2 x^2}} \]

output

1/16*d*x^5*(-c^2*d*x^2+d)^(3/2)*(a+b*arcsin(c*x))+1/10*x^5*(-c^2*d*x^2+d)^ 
(5/2)*(a+b*arcsin(c*x))-3/256*d^2*x*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2) 
/c^4-1/128*d^2*x^3*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)/c^2+1/32*d^2*x^5 
*(a+b*arcsin(c*x))*(-c^2*d*x^2+d)^(1/2)+3/512*b*d^2*x^2*(-c^2*d*x^2+d)^(1/ 
2)/c^3/(-c^2*x^2+1)^(1/2)+1/512*b*d^2*x^4*(-c^2*d*x^2+d)^(1/2)/c/(-c^2*x^2 
+1)^(1/2)-31/960*b*c*d^2*x^6*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+21/64 
0*b*c^3*d^2*x^8*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)-1/100*b*c^5*d^2*x^ 
10*(-c^2*d*x^2+d)^(1/2)/(-c^2*x^2+1)^(1/2)+3/512*d^2*(a+b*arcsin(c*x))^2*( 
-c^2*d*x^2+d)^(1/2)/b/c^5/(-c^2*x^2+1)^(1/2)
 

3.1.84.2 Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.51 \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\frac {d^2 \sqrt {d-c^2 d x^2} \left (225 a^2+b^2 c^2 x^2 \left (225+75 c^2 x^2-1240 c^4 x^4+1260 c^6 x^6-384 c^8 x^8\right )+30 a b c x \sqrt {1-c^2 x^2} \left (-15-10 c^2 x^2+248 c^4 x^4-336 c^6 x^6+128 c^8 x^8\right )+30 b \left (15 a+b c x \sqrt {1-c^2 x^2} \left (-15-10 c^2 x^2+248 c^4 x^4-336 c^6 x^6+128 c^8 x^8\right )\right ) \arcsin (c x)+225 b^2 \arcsin (c x)^2\right )}{38400 b c^5 \sqrt {1-c^2 x^2}} \]

input

Integrate[x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]
 

output

(d^2*Sqrt[d - c^2*d*x^2]*(225*a^2 + b^2*c^2*x^2*(225 + 75*c^2*x^2 - 1240*c 
^4*x^4 + 1260*c^6*x^6 - 384*c^8*x^8) + 30*a*b*c*x*Sqrt[1 - c^2*x^2]*(-15 - 
 10*c^2*x^2 + 248*c^4*x^4 - 336*c^6*x^6 + 128*c^8*x^8) + 30*b*(15*a + b*c* 
x*Sqrt[1 - c^2*x^2]*(-15 - 10*c^2*x^2 + 248*c^4*x^4 - 336*c^6*x^6 + 128*c^ 
8*x^8))*ArcSin[c*x] + 225*b^2*ArcSin[c*x]^2))/(38400*b*c^5*Sqrt[1 - c^2*x^ 
2])
 

3.1.84.3 Rubi [A] (verified)

Time = 1.42 (sec) , antiderivative size = 407, normalized size of antiderivative = 0.95, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {5202, 243, 49, 2009, 5202, 244, 2009, 5198, 15, 5210, 15, 5210, 15, 5152}

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.

\(\displaystyle \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx\)

\(\Big \downarrow \) 5202

\(\displaystyle \frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^5 \left (1-c^2 x^2\right )^2dx}{10 \sqrt {1-c^2 x^2}}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int x^4 \left (1-c^2 x^2\right )^2dx^2}{20 \sqrt {1-c^2 x^2}}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))\)

\(\Big \downarrow \) 49

\(\displaystyle \frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx-\frac {b c d^2 \sqrt {d-c^2 d x^2} \int \left (c^4 x^8-2 c^2 x^6+x^4\right )dx^2}{20 \sqrt {1-c^2 x^2}}+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} d \int x^4 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))dx+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5202

\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int x^5 \left (1-c^2 x^2\right )dx}{8 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 244

\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx-\frac {b c d \sqrt {d-c^2 d x^2} \int \left (x^5-c^2 x^7\right )dx}{8 \sqrt {1-c^2 x^2}}+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \int x^4 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))dx+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {1-c^2 x^2}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5198

\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{6 \sqrt {1-c^2 x^2}}-\frac {b c \sqrt {d-c^2 d x^2} \int x^5dx}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {1-c^2 x^2}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (\frac {\sqrt {d-c^2 d x^2} \int \frac {x^4 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {1-c^2 x^2}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}+\frac {b \int x^3dx}{4 c}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}\right )}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {1-c^2 x^2}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \int \frac {x^2 (a+b \arcsin (c x))}{\sqrt {1-c^2 x^2}}dx}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {1-c^2 x^2}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5210

\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}+\frac {b \int xdx}{2 c}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {1-c^2 x^2}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 15

\(\displaystyle \frac {1}{2} d \left (\frac {3}{8} d \left (\frac {\sqrt {d-c^2 d x^2} \left (\frac {3 \left (\frac {\int \frac {a+b \arcsin (c x)}{\sqrt {1-c^2 x^2}}dx}{2 c^2}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {b x^4}{16 c}\right )}{6 \sqrt {1-c^2 x^2}}+\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}\right )+\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {1-c^2 x^2}}\right )+\frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

\(\Big \downarrow \) 5152

\(\displaystyle \frac {1}{10} x^5 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x))+\frac {1}{2} d \left (\frac {1}{8} x^5 \left (d-c^2 d x^2\right )^{3/2} (a+b \arcsin (c x))+\frac {3}{8} d \left (\frac {1}{6} x^5 \sqrt {d-c^2 d x^2} (a+b \arcsin (c x))+\frac {\sqrt {d-c^2 d x^2} \left (-\frac {x^3 \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{4 c^2}+\frac {3 \left (\frac {(a+b \arcsin (c x))^2}{4 b c^3}-\frac {x \sqrt {1-c^2 x^2} (a+b \arcsin (c x))}{2 c^2}+\frac {b x^2}{4 c}\right )}{4 c^2}+\frac {b x^4}{16 c}\right )}{6 \sqrt {1-c^2 x^2}}-\frac {b c x^6 \sqrt {d-c^2 d x^2}}{36 \sqrt {1-c^2 x^2}}\right )-\frac {b c d \left (\frac {x^6}{6}-\frac {c^2 x^8}{8}\right ) \sqrt {d-c^2 d x^2}}{8 \sqrt {1-c^2 x^2}}\right )-\frac {b c d^2 \left (\frac {c^4 x^{10}}{5}-\frac {c^2 x^8}{2}+\frac {x^6}{3}\right ) \sqrt {d-c^2 d x^2}}{20 \sqrt {1-c^2 x^2}}\)

input

Int[x^4*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]),x]
 

output

-1/20*(b*c*d^2*Sqrt[d - c^2*d*x^2]*(x^6/3 - (c^2*x^8)/2 + (c^4*x^10)/5))/S 
qrt[1 - c^2*x^2] + (x^5*(d - c^2*d*x^2)^(5/2)*(a + b*ArcSin[c*x]))/10 + (d 
*(-1/8*(b*c*d*Sqrt[d - c^2*d*x^2]*(x^6/6 - (c^2*x^8)/8))/Sqrt[1 - c^2*x^2] 
 + (x^5*(d - c^2*d*x^2)^(3/2)*(a + b*ArcSin[c*x]))/8 + (3*d*(-1/36*(b*c*x^ 
6*Sqrt[d - c^2*d*x^2])/Sqrt[1 - c^2*x^2] + (x^5*Sqrt[d - c^2*d*x^2]*(a + b 
*ArcSin[c*x]))/6 + (Sqrt[d - c^2*d*x^2]*((b*x^4)/(16*c) - (x^3*Sqrt[1 - c^ 
2*x^2]*(a + b*ArcSin[c*x]))/(4*c^2) + (3*((b*x^2)/(4*c) - (x*Sqrt[1 - c^2* 
x^2]*(a + b*ArcSin[c*x]))/(2*c^2) + (a + b*ArcSin[c*x])^2/(4*b*c^3)))/(4*c 
^2)))/(6*Sqrt[1 - c^2*x^2])))/8))/2
 

3.1.84.3.1 Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 49

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

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 244

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand 
Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p 
, 0]
 

rule 2009

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

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 5198

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

rule 5202

Int[((a_.) + ArcSin[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_. 
)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*ArcS 
in[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1))   Int[(f*x) 
^m*(d + e*x^2)^(p - 1)*(a + b*ArcSin[c*x])^n, x], x] - Simp[b*c*(n/(f*(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, m}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && GtQ[p, 0] &&  !LtQ[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]
 

3.1.84.4 Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.20 (sec) , antiderivative size = 1106, normalized size of antiderivative = 2.57

method	result	size
default	\(\text {Expression too large to display}\)	\(1106\)
parts	\(\text {Expression too large to display}\)	\(1106\)

input

int(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x,method=_RETURNVERBOSE)
 

output

-1/10*a*x^3*(-c^2*d*x^2+d)^(7/2)/c^2/d-3/80*a/c^4*x*(-c^2*d*x^2+d)^(7/2)/d 
+1/160*a/c^4*x*(-c^2*d*x^2+d)^(5/2)+1/128*a/c^4*d*x*(-c^2*d*x^2+d)^(3/2)+3 
/256*a/c^4*d^2*x*(-c^2*d*x^2+d)^(1/2)+3/256*a/c^4*d^3/(c^2*d)^(1/2)*arctan 
((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))+b*(-3/512*(-d*(c^2*x^2-1))^(1/2)*(- 
c^2*x^2+1)^(1/2)/c^5/(c^2*x^2-1)*arcsin(c*x)^2*d^2+1/102400*(-d*(c^2*x^2-1 
))^(1/2)*(-512*I*(-c^2*x^2+1)^(1/2)*x^10*c^10+512*c^11*x^11+1280*I*(-c^2*x 
^2+1)^(1/2)*x^8*c^8-1536*c^9*x^9-1120*I*(-c^2*x^2+1)^(1/2)*x^6*c^6+1696*c^ 
7*x^7+400*I*(-c^2*x^2+1)^(1/2)*x^4*c^4-832*c^5*x^5-50*I*(-c^2*x^2+1)^(1/2) 
*x^2*c^2+170*c^3*x^3+I*(-c^2*x^2+1)^(1/2)-10*c*x)*(I+10*arcsin(c*x))*d^2/c 
^5/(c^2*x^2-1)+1/2048*(-d*(c^2*x^2-1))^(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)*(-I+2*arcsin(c*x))*d^2/c^5/(c^2*x 
^2-1)-3/819200*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I) 
*(11*I+40*arcsin(c*x))*cos(9*arcsin(c*x))*d^2/c^5/(c^2*x^2-1)+1/819200*(-d 
*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1/2)*x*c+c^2*x^2-1)*(17*I+280*arcsin( 
c*x))*sin(9*arcsin(c*x))*d^2/c^5/(c^2*x^2-1)+1/98304*(-d*(c^2*x^2-1))^(1/2 
)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)*(5*I+72*arcsin(c*x))*cos(7*arcsin(c 
*x))*d^2/c^5/(c^2*x^2-1)-1/98304*(-d*(c^2*x^2-1))^(1/2)*(I*(-c^2*x^2+1)^(1 
/2)*x*c+c^2*x^2-1)*(11*I+24*arcsin(c*x))*sin(7*arcsin(c*x))*d^2/c^5/(c^2*x 
^2-1)+1/12288*(-d*(c^2*x^2-1))^(1/2)*(I*c^2*x^2-c*x*(-c^2*x^2+1)^(1/2)-I)* 
(7*I+18*arcsin(c*x))*cos(5*arcsin(c*x))*d^2/c^5/(c^2*x^2-1)-5/12288*(-d...
 

3.1.84.5 Fricas [F]

\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{4} \,d x } \]

input

integrate(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="fricas" 
)
 

output

integral((a*c^4*d^2*x^8 - 2*a*c^2*d^2*x^6 + a*d^2*x^4 + (b*c^4*d^2*x^8 - 2 
*b*c^2*d^2*x^6 + b*d^2*x^4)*arcsin(c*x))*sqrt(-c^2*d*x^2 + d), x)
 

3.1.84.6 Sympy [F(-1)]

Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\text {Timed out} \]

input

integrate(x**4*(-c**2*d*x**2+d)**(5/2)*(a+b*asin(c*x)),x)
 

output

Timed out
 

3.1.84.7 Maxima [F]

\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{4} \,d x } \]

input

integrate(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="maxima" 
)
 

output

b*sqrt(d)*integrate((c^4*d^2*x^8 - 2*c^2*d^2*x^6 + d^2*x^4)*sqrt(c*x + 1)* 
sqrt(-c*x + 1)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)), x) - 1/1280*(12 
8*(-c^2*d*x^2 + d)^(7/2)*x^3/(c^2*d) - 8*(-c^2*d*x^2 + d)^(5/2)*x/c^4 + 48 
*(-c^2*d*x^2 + d)^(7/2)*x/(c^4*d) - 10*(-c^2*d*x^2 + d)^(3/2)*d*x/c^4 - 15 
*sqrt(-c^2*d*x^2 + d)*d^2*x/c^4 - 15*d^(5/2)*arcsin(c*x)/c^5)*a
 

3.1.84.8 Giac [F]

\[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int { {\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \arcsin \left (c x\right ) + a\right )} x^{4} \,d x } \]

input

integrate(x^4*(-c^2*d*x^2+d)^(5/2)*(a+b*arcsin(c*x)),x, algorithm="giac")
 

output

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

3.1.84.9 Mupad [F(-1)]

Timed out. \[ \int x^4 \left (d-c^2 d x^2\right )^{5/2} (a+b \arcsin (c x)) \, dx=\int x^4\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2} \,d x \]

input

int(x^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2),x)
 

output

int(x^4*(a + b*asin(c*x))*(d - c^2*d*x^2)^(5/2), x)