Integrand size = 10, antiderivative size = 282 \[ \int x^5 \arcsin (a x)^4 \, dx=\frac {245 x^2}{1152 a^4}+\frac {65 x^4}{3456 a^2}+\frac {x^6}{324}-\frac {245 x \sqrt {1-a^2 x^2} \arcsin (a x)}{576 a^5}-\frac {65 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{864 a^3}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{54 a}+\frac {245 \arcsin (a x)^2}{1152 a^6}-\frac {5 x^2 \arcsin (a x)^2}{16 a^4}-\frac {5 x^4 \arcsin (a x)^2}{48 a^2}-\frac {1}{18} x^6 \arcsin (a x)^2+\frac {5 x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{24 a^5}+\frac {5 x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{36 a^3}+\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{9 a}-\frac {5 \arcsin (a x)^4}{96 a^6}+\frac {1}{6} x^6 \arcsin (a x)^4 \]
245/1152*x^2/a^4+65/3456*x^4/a^2+1/324*x^6+245/1152*arcsin(a*x)^2/a^6-5/16 *x^2*arcsin(a*x)^2/a^4-5/48*x^4*arcsin(a*x)^2/a^2-1/18*x^6*arcsin(a*x)^2-5 /96*arcsin(a*x)^4/a^6+1/6*x^6*arcsin(a*x)^4-245/576*x*arcsin(a*x)*(-a^2*x^ 2+1)^(1/2)/a^5-65/864*x^3*arcsin(a*x)*(-a^2*x^2+1)^(1/2)/a^3-1/54*x^5*arcs in(a*x)*(-a^2*x^2+1)^(1/2)/a+5/24*x*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/a^5+5 /36*x^3*arcsin(a*x)^3*(-a^2*x^2+1)^(1/2)/a^3+1/9*x^5*arcsin(a*x)^3*(-a^2*x ^2+1)^(1/2)/a
Time = 0.10 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.59 \[ \int x^5 \arcsin (a x)^4 \, dx=\frac {a^2 x^2 \left (2205+195 a^2 x^2+32 a^4 x^4\right )-6 a x \sqrt {1-a^2 x^2} \left (735+130 a^2 x^2+32 a^4 x^4\right ) \arcsin (a x)-9 \left (-245+360 a^2 x^2+120 a^4 x^4+64 a^6 x^6\right ) \arcsin (a x)^2+144 a x \sqrt {1-a^2 x^2} \left (15+10 a^2 x^2+8 a^4 x^4\right ) \arcsin (a x)^3+108 \left (-5+16 a^6 x^6\right ) \arcsin (a x)^4}{10368 a^6} \]
(a^2*x^2*(2205 + 195*a^2*x^2 + 32*a^4*x^4) - 6*a*x*Sqrt[1 - a^2*x^2]*(735 + 130*a^2*x^2 + 32*a^4*x^4)*ArcSin[a*x] - 9*(-245 + 360*a^2*x^2 + 120*a^4* x^4 + 64*a^6*x^6)*ArcSin[a*x]^2 + 144*a*x*Sqrt[1 - a^2*x^2]*(15 + 10*a^2*x ^2 + 8*a^4*x^4)*ArcSin[a*x]^3 + 108*(-5 + 16*a^6*x^6)*ArcSin[a*x]^4)/(1036 8*a^6)
Time = 2.39 (sec) , antiderivative size = 501, normalized size of antiderivative = 1.78, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.300, Rules used = {5138, 5210, 5138, 5210, 15, 5138, 5210, 15, 5138, 5152, 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.
| \(\displaystyle \int x^5 \arcsin (a x)^4 \, dx\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \int \frac {x^6 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {5 \int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{6 a^2}+\frac {\int x^5 \arcsin (a x)^2dx}{2 a}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \int \frac {x^6 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a}+\frac {5 \int \frac {x^4 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {5 \left (\frac {3 \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {3 \int x^3 \arcsin (a x)^2dx}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )}{6 a^2}+\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \left (\frac {5 \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{6 a^2}+\frac {\int x^5dx}{6 a}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^2}\right )}{2 a}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {5 \left (\frac {3 \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {3 \int x^3 \arcsin (a x)^2dx}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )}{6 a^2}+\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \left (\frac {5 \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^2}+\frac {x^6}{36 a}\right )}{2 a}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {5 \left (\frac {3 \int \frac {x^2 \arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )}{6 a^2}+\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \left (\frac {5 \int \frac {x^4 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^2}+\frac {x^6}{36 a}\right )}{2 a}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {3 \int x \arcsin (a x)^2dx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}+\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int x^3dx}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}\right )\right )}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )}{6 a^2}+\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \left (\frac {5 \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}+\frac {\int x^3dx}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^2}+\frac {x^6}{36 a}\right )}{2 a}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {3 \int x \arcsin (a x)^2dx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}+\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )}{6 a^2}+\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \left (\frac {5 \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^2}+\frac {x^6}{36 a}\right )}{2 a}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 5138 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {5 \left (\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 a}+\frac {\int \frac {\arcsin (a x)^3}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}+\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )}{6 a^2}+\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \left (\frac {5 \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^2}+\frac {x^6}{36 a}\right )}{2 a}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \left (\frac {5 \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^2}+\frac {x^6}{36 a}\right )}{2 a}+\frac {5 \left (\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \int \frac {x^2 \arcsin (a x)}{\sqrt {1-a^2 x^2}}dx\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 5210 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^2}+\frac {x^6}{36 a}\right )}{2 a}+\frac {5 \left (\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}+\frac {\int xdx}{2 a}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}\right )\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \left (\frac {5 \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^2}+\frac {x^6}{36 a}\right )}{2 a}+\frac {5 \left (\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (\frac {3 \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}+\frac {3 \left (\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\int \frac {\arcsin (a x)}{\sqrt {1-a^2 x^2}}dx}{2 a^2}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 a}+\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}\right )}{4 a^2}-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}\right )}{6 a^2}-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}\right )\) |
| \(\Big \downarrow \) 5152 |
| \(\displaystyle \frac {1}{6} x^6 \arcsin (a x)^4-\frac {2}{3} a \left (-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{6 a^2}+\frac {5 \left (-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)^3}{4 a^2}+\frac {3 \left (\frac {\arcsin (a x)^4}{8 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)^3}{2 a^2}+\frac {3 \left (\frac {1}{2} x^2 \arcsin (a x)^2-a \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )\right )}{2 a}\right )}{4 a^2}+\frac {3 \left (\frac {1}{4} x^4 \arcsin (a x)^2-\frac {1}{2} a \left (-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {3 \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )}{4 a^2}+\frac {x^4}{16 a}\right )\right )}{4 a}\right )}{6 a^2}+\frac {\frac {1}{6} x^6 \arcsin (a x)^2-\frac {1}{3} a \left (-\frac {x^5 \sqrt {1-a^2 x^2} \arcsin (a x)}{6 a^2}+\frac {5 \left (-\frac {x^3 \sqrt {1-a^2 x^2} \arcsin (a x)}{4 a^2}+\frac {3 \left (\frac {\arcsin (a x)^2}{4 a^3}-\frac {x \sqrt {1-a^2 x^2} \arcsin (a x)}{2 a^2}+\frac {x^2}{4 a}\right )}{4 a^2}+\frac {x^4}{16 a}\right )}{6 a^2}+\frac {x^6}{36 a}\right )}{2 a}\right )\) |
(x^6*ArcSin[a*x]^4)/6 - (2*a*(-1/6*(x^5*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a ^2 + ((x^6*ArcSin[a*x]^2)/6 - (a*(x^6/(36*a) - (x^5*Sqrt[1 - a^2*x^2]*ArcS in[a*x])/(6*a^2) + (5*(x^4/(16*a) - (x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(4 *a^2) + (3*(x^2/(4*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(2*a^2) + ArcSin [a*x]^2/(4*a^3)))/(4*a^2)))/(6*a^2)))/3)/(2*a) + (5*(-1/4*(x^3*Sqrt[1 - a^ 2*x^2]*ArcSin[a*x]^3)/a^2 + (3*((x^4*ArcSin[a*x]^2)/4 - (a*(x^4/(16*a) - ( x^3*Sqrt[1 - a^2*x^2]*ArcSin[a*x])/(4*a^2) + (3*(x^2/(4*a) - (x*Sqrt[1 - a ^2*x^2]*ArcSin[a*x])/(2*a^2) + ArcSin[a*x]^2/(4*a^3)))/(4*a^2)))/2))/(4*a) + (3*(-1/2*(x*Sqrt[1 - a^2*x^2]*ArcSin[a*x]^3)/a^2 + ArcSin[a*x]^4/(8*a^3 ) + (3*((x^2*ArcSin[a*x]^2)/2 - a*(x^2/(4*a) - (x*Sqrt[1 - a^2*x^2]*ArcSin [a*x])/(2*a^2) + ArcSin[a*x]^2/(4*a^3))))/(2*a)))/(4*a^2)))/(6*a^2)))/3
3.1.32.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
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]
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]
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]
Time = 0.11 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.22
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{6} x^{6} \arcsin \left (a x \right )^{4}}{6}-\frac {\arcsin \left (a x \right )^{3} \left (-8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-15 a x \sqrt {-a^{2} x^{2}+1}+15 \arcsin \left (a x \right )\right )}{72}-\frac {\arcsin \left (a x \right )^{2} a^{6} x^{6}}{18}+\frac {\arcsin \left (a x \right ) \left (-8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-15 a x \sqrt {-a^{2} x^{2}+1}+15 \arcsin \left (a x \right )\right )}{432}+\frac {115 \arcsin \left (a x \right )^{2}}{1152}+\frac {\left (a^{2} x^{2}-1\right )^{3}}{324}+\frac {13 \left (a^{2} x^{2}-1\right )^{2}}{864}+\frac {7 a^{2} x^{2}}{36}-\frac {11}{288}-\frac {5 a^{4} x^{4} \arcsin \left (a x \right )^{2}}{48}+\frac {5 \arcsin \left (a x \right ) \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{192}+\frac {5 \left (2 a^{2} x^{2}+3\right )^{2}}{1536}-\frac {5 \arcsin \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}{16}-\frac {5 \arcsin \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{16}+\frac {5 \arcsin \left (a x \right )^{4}}{32}}{a^{6}}\) | \(345\) |
| default | \(\frac {\frac {a^{6} x^{6} \arcsin \left (a x \right )^{4}}{6}-\frac {\arcsin \left (a x \right )^{3} \left (-8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-15 a x \sqrt {-a^{2} x^{2}+1}+15 \arcsin \left (a x \right )\right )}{72}-\frac {\arcsin \left (a x \right )^{2} a^{6} x^{6}}{18}+\frac {\arcsin \left (a x \right ) \left (-8 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-10 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-15 a x \sqrt {-a^{2} x^{2}+1}+15 \arcsin \left (a x \right )\right )}{432}+\frac {115 \arcsin \left (a x \right )^{2}}{1152}+\frac {\left (a^{2} x^{2}-1\right )^{3}}{324}+\frac {13 \left (a^{2} x^{2}-1\right )^{2}}{864}+\frac {7 a^{2} x^{2}}{36}-\frac {11}{288}-\frac {5 a^{4} x^{4} \arcsin \left (a x \right )^{2}}{48}+\frac {5 \arcsin \left (a x \right ) \left (-2 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-3 a x \sqrt {-a^{2} x^{2}+1}+3 \arcsin \left (a x \right )\right )}{192}+\frac {5 \left (2 a^{2} x^{2}+3\right )^{2}}{1536}-\frac {5 \arcsin \left (a x \right )^{2} \left (a^{2} x^{2}-1\right )}{16}-\frac {5 \arcsin \left (a x \right ) \left (a x \sqrt {-a^{2} x^{2}+1}+\arcsin \left (a x \right )\right )}{16}+\frac {5 \arcsin \left (a x \right )^{4}}{32}}{a^{6}}\) | \(345\) |
1/a^6*(1/6*a^6*x^6*arcsin(a*x)^4-1/72*arcsin(a*x)^3*(-8*(-a^2*x^2+1)^(1/2) *a^5*x^5-10*a^3*x^3*(-a^2*x^2+1)^(1/2)-15*a*x*(-a^2*x^2+1)^(1/2)+15*arcsin (a*x))-1/18*arcsin(a*x)^2*a^6*x^6+1/432*arcsin(a*x)*(-8*(-a^2*x^2+1)^(1/2) *a^5*x^5-10*a^3*x^3*(-a^2*x^2+1)^(1/2)-15*a*x*(-a^2*x^2+1)^(1/2)+15*arcsin (a*x))+115/1152*arcsin(a*x)^2+1/324*(a^2*x^2-1)^3+13/864*(a^2*x^2-1)^2+7/3 6*a^2*x^2-11/288-5/48*a^4*x^4*arcsin(a*x)^2+5/192*arcsin(a*x)*(-2*a^3*x^3* (-a^2*x^2+1)^(1/2)-3*a*x*(-a^2*x^2+1)^(1/2)+3*arcsin(a*x))+5/1536*(2*a^2*x ^2+3)^2-5/16*arcsin(a*x)^2*(a^2*x^2-1)-5/16*arcsin(a*x)*(a*x*(-a^2*x^2+1)^ (1/2)+arcsin(a*x))+5/32*arcsin(a*x)^4)
Time = 0.26 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.54 \[ \int x^5 \arcsin (a x)^4 \, dx=\frac {32 \, a^{6} x^{6} + 195 \, a^{4} x^{4} + 108 \, {\left (16 \, a^{6} x^{6} - 5\right )} \arcsin \left (a x\right )^{4} + 2205 \, a^{2} x^{2} - 9 \, {\left (64 \, a^{6} x^{6} + 120 \, a^{4} x^{4} + 360 \, a^{2} x^{2} - 245\right )} \arcsin \left (a x\right )^{2} + 6 \, \sqrt {-a^{2} x^{2} + 1} {\left (24 \, {\left (8 \, a^{5} x^{5} + 10 \, a^{3} x^{3} + 15 \, a x\right )} \arcsin \left (a x\right )^{3} - {\left (32 \, a^{5} x^{5} + 130 \, a^{3} x^{3} + 735 \, a x\right )} \arcsin \left (a x\right )\right )}}{10368 \, a^{6}} \]
1/10368*(32*a^6*x^6 + 195*a^4*x^4 + 108*(16*a^6*x^6 - 5)*arcsin(a*x)^4 + 2 205*a^2*x^2 - 9*(64*a^6*x^6 + 120*a^4*x^4 + 360*a^2*x^2 - 245)*arcsin(a*x) ^2 + 6*sqrt(-a^2*x^2 + 1)*(24*(8*a^5*x^5 + 10*a^3*x^3 + 15*a*x)*arcsin(a*x )^3 - (32*a^5*x^5 + 130*a^3*x^3 + 735*a*x)*arcsin(a*x)))/a^6
Time = 1.04 (sec) , antiderivative size = 269, normalized size of antiderivative = 0.95 \[ \int x^5 \arcsin (a x)^4 \, dx=\begin {cases} \frac {x^{6} \operatorname {asin}^{4}{\left (a x \right )}}{6} - \frac {x^{6} \operatorname {asin}^{2}{\left (a x \right )}}{18} + \frac {x^{6}}{324} + \frac {x^{5} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{9 a} - \frac {x^{5} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{54 a} - \frac {5 x^{4} \operatorname {asin}^{2}{\left (a x \right )}}{48 a^{2}} + \frac {65 x^{4}}{3456 a^{2}} + \frac {5 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{36 a^{3}} - \frac {65 x^{3} \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{864 a^{3}} - \frac {5 x^{2} \operatorname {asin}^{2}{\left (a x \right )}}{16 a^{4}} + \frac {245 x^{2}}{1152 a^{4}} + \frac {5 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}^{3}{\left (a x \right )}}{24 a^{5}} - \frac {245 x \sqrt {- a^{2} x^{2} + 1} \operatorname {asin}{\left (a x \right )}}{576 a^{5}} - \frac {5 \operatorname {asin}^{4}{\left (a x \right )}}{96 a^{6}} + \frac {245 \operatorname {asin}^{2}{\left (a x \right )}}{1152 a^{6}} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((x**6*asin(a*x)**4/6 - x**6*asin(a*x)**2/18 + x**6/324 + x**5*sq rt(-a**2*x**2 + 1)*asin(a*x)**3/(9*a) - x**5*sqrt(-a**2*x**2 + 1)*asin(a*x )/(54*a) - 5*x**4*asin(a*x)**2/(48*a**2) + 65*x**4/(3456*a**2) + 5*x**3*sq rt(-a**2*x**2 + 1)*asin(a*x)**3/(36*a**3) - 65*x**3*sqrt(-a**2*x**2 + 1)*a sin(a*x)/(864*a**3) - 5*x**2*asin(a*x)**2/(16*a**4) + 245*x**2/(1152*a**4) + 5*x*sqrt(-a**2*x**2 + 1)*asin(a*x)**3/(24*a**5) - 245*x*sqrt(-a**2*x**2 + 1)*asin(a*x)/(576*a**5) - 5*asin(a*x)**4/(96*a**6) + 245*asin(a*x)**2/( 1152*a**6), Ne(a, 0)), (0, True))
\[ \int x^5 \arcsin (a x)^4 \, dx=\int { x^{5} \arcsin \left (a x\right )^{4} \,d x } \]
1/6*x^6*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1))^4 + 2*a*integrate(1/3*s qrt(a*x + 1)*sqrt(-a*x + 1)*x^6*arctan2(a*x, sqrt(a*x + 1)*sqrt(-a*x + 1)) ^3/(a^2*x^2 - 1), x)
Time = 0.27 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.28 \[ \int x^5 \arcsin (a x)^4 \, dx=\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{9 \, a^{5}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{3} \arcsin \left (a x\right )^{4}}{6 \, a^{6}} - \frac {13 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )^{3}}{36 \, a^{5}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{4}}{2 \, a^{6}} - \frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{54 \, a^{5}} + \frac {11 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )^{3}}{24 \, a^{5}} - \frac {{\left (a^{2} x^{2} - 1\right )}^{3} \arcsin \left (a x\right )^{2}}{18 \, a^{6}} + \frac {{\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{4}}{2 \, a^{6}} + \frac {97 \, {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} x \arcsin \left (a x\right )}{864 \, a^{5}} - \frac {13 \, {\left (a^{2} x^{2} - 1\right )}^{2} \arcsin \left (a x\right )^{2}}{48 \, a^{6}} + \frac {11 \, \arcsin \left (a x\right )^{4}}{96 \, a^{6}} - \frac {299 \, \sqrt {-a^{2} x^{2} + 1} x \arcsin \left (a x\right )}{576 \, a^{5}} + \frac {{\left (a^{2} x^{2} - 1\right )}^{3}}{324 \, a^{6}} - \frac {11 \, {\left (a^{2} x^{2} - 1\right )} \arcsin \left (a x\right )^{2}}{16 \, a^{6}} + \frac {97 \, {\left (a^{2} x^{2} - 1\right )}^{2}}{3456 \, a^{6}} - \frac {299 \, \arcsin \left (a x\right )^{2}}{1152 \, a^{6}} + \frac {299 \, {\left (a^{2} x^{2} - 1\right )}}{1152 \, a^{6}} + \frac {9971}{82944 \, a^{6}} \]
1/9*(a^2*x^2 - 1)^2*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a^5 + 1/6*(a^2*x^2 - 1)^3*arcsin(a*x)^4/a^6 - 13/36*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)^3/a^5 + 1/2*(a^2*x^2 - 1)^2*arcsin(a*x)^4/a^6 - 1/54*(a^2*x^2 - 1)^2*sqrt(-a^2*x ^2 + 1)*x*arcsin(a*x)/a^5 + 11/24*sqrt(-a^2*x^2 + 1)*x*arcsin(a*x)^3/a^5 - 1/18*(a^2*x^2 - 1)^3*arcsin(a*x)^2/a^6 + 1/2*(a^2*x^2 - 1)*arcsin(a*x)^4/ a^6 + 97/864*(-a^2*x^2 + 1)^(3/2)*x*arcsin(a*x)/a^5 - 13/48*(a^2*x^2 - 1)^ 2*arcsin(a*x)^2/a^6 + 11/96*arcsin(a*x)^4/a^6 - 299/576*sqrt(-a^2*x^2 + 1) *x*arcsin(a*x)/a^5 + 1/324*(a^2*x^2 - 1)^3/a^6 - 11/16*(a^2*x^2 - 1)*arcsi n(a*x)^2/a^6 + 97/3456*(a^2*x^2 - 1)^2/a^6 - 299/1152*arcsin(a*x)^2/a^6 + 299/1152*(a^2*x^2 - 1)/a^6 + 9971/82944/a^6
Timed out. \[ \int x^5 \arcsin (a x)^4 \, dx=\int x^5\,{\mathrm {asin}\left (a\,x\right )}^4 \,d x \]