Integrand size = 26, antiderivative size = 376 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {79 b^2 d^3 x^2}{5120 c^2}+\frac {79 b^2 d^3 x^4}{15360}+\frac {401 b^2 c^2 d^3 x^6}{28800}+\frac {57 b^2 c^4 d^3 x^8}{6400}+\frac {1}{500} b^2 c^6 d^3 x^{10}+\frac {79 b d^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2560 c^3}-\frac {79 b d^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3840 c}-\frac {31}{960} b c d^3 x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))-\frac {1}{32} b c d^3 x^5 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{50} b c d^3 x^5 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {79 d^3 (a+b \text {arcsinh}(c x))^2}{5120 c^4}+\frac {1}{40} d^3 x^4 (a+b \text {arcsinh}(c x))^2+\frac {1}{20} d^3 x^4 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {3}{40} d^3 x^4 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{10} d^3 x^4 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \] Output:
-79/5120*b^2*d^3*x^2/c^2+79/15360*b^2*d^3*x^4+401/28800*b^2*c^2*d^3*x^6+57 /6400*b^2*c^4*d^3*x^8+1/500*b^2*c^6*d^3*x^10+79/2560*b*d^3*x*(c^2*x^2+1)^( 1/2)*(a+b*arcsinh(c*x))/c^3-79/3840*b*d^3*x^3*(c^2*x^2+1)^(1/2)*(a+b*arcsi nh(c*x))/c-31/960*b*c*d^3*x^5*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))-1/32*b* c*d^3*x^5*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))-1/50*b*c*d^3*x^5*(c^2*x^2+1 )^(5/2)*(a+b*arcsinh(c*x))-79/5120*d^3*(a+b*arcsinh(c*x))^2/c^4+1/40*d^3*x ^4*(a+b*arcsinh(c*x))^2+1/20*d^3*x^4*(c^2*x^2+1)*(a+b*arcsinh(c*x))^2+3/40 *d^3*x^4*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2+1/10*d^3*x^4*(c^2*x^2+1)^3*(a+ b*arcsinh(c*x))^2
Time = 0.27 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.76 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^3 \left (c x \left (28800 a^2 c^3 x^3 \left (10+20 c^2 x^2+15 c^4 x^4+4 c^6 x^6\right )-30 a b \sqrt {1+c^2 x^2} \left (-1185+790 c^2 x^2+3208 c^4 x^4+2736 c^6 x^6+768 c^8 x^8\right )+b^2 c x \left (-17775+5925 c^2 x^2+16040 c^4 x^4+10260 c^6 x^6+2304 c^8 x^8\right )\right )+30 b \left (-b c x \sqrt {1+c^2 x^2} \left (-1185+790 c^2 x^2+3208 c^4 x^4+2736 c^6 x^6+768 c^8 x^8\right )+15 a \left (-79+1280 c^4 x^4+2560 c^6 x^6+1920 c^8 x^8+512 c^{10} x^{10}\right )\right ) \text {arcsinh}(c x)+225 b^2 \left (-79+1280 c^4 x^4+2560 c^6 x^6+1920 c^8 x^8+512 c^{10} x^{10}\right ) \text {arcsinh}(c x)^2\right )}{1152000 c^4} \] Input:
Integrate[x^3*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^3*(c*x*(28800*a^2*c^3*x^3*(10 + 20*c^2*x^2 + 15*c^4*x^4 + 4*c^6*x^6) - 30*a*b*Sqrt[1 + c^2*x^2]*(-1185 + 790*c^2*x^2 + 3208*c^4*x^4 + 2736*c^6*x^ 6 + 768*c^8*x^8) + b^2*c*x*(-17775 + 5925*c^2*x^2 + 16040*c^4*x^4 + 10260* c^6*x^6 + 2304*c^8*x^8)) + 30*b*(-(b*c*x*Sqrt[1 + c^2*x^2]*(-1185 + 790*c^ 2*x^2 + 3208*c^4*x^4 + 2736*c^6*x^6 + 768*c^8*x^8)) + 15*a*(-79 + 1280*c^4 *x^4 + 2560*c^6*x^6 + 1920*c^8*x^8 + 512*c^10*x^10))*ArcSinh[c*x] + 225*b^ 2*(-79 + 1280*c^4*x^4 + 2560*c^6*x^6 + 1920*c^8*x^8 + 512*c^10*x^10)*ArcSi nh[c*x]^2))/(1152000*c^4)
Leaf count is larger than twice the leaf count of optimal. \(899\) vs. \(2(376)=752\).
Time = 4.02 (sec) , antiderivative size = 899, normalized size of antiderivative = 2.39, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.654, Rules used = {6223, 27, 6223, 243, 49, 2009, 6223, 244, 2009, 6191, 6221, 15, 6227, 15, 6227, 15, 6198}
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.
| \(\displaystyle \int x^3 \left (c^2 d x^2+d\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {1}{5} b c d^3 \int x^4 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {3}{5} d \int d^2 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{5} b c d^3 \int x^4 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))dx+\frac {3}{5} d^3 \int x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle -\frac {1}{5} b c d^3 \left (\frac {1}{2} \int x^4 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {1}{10} b c \int x^5 \left (c^2 x^2+1\right )^2dx+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))\right )+\frac {3}{5} d^3 \left (-\frac {1}{4} b c \int x^4 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{2} \int x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {1}{5} b c d^3 \left (\frac {1}{2} \int x^4 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {1}{20} b c \int x^4 \left (c^2 x^2+1\right )^2dx^2+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))\right )+\frac {3}{5} d^3 \left (-\frac {1}{4} b c \int x^4 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{2} \int x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle \frac {3}{5} d^3 \left (-\frac {1}{4} b c \int x^4 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{2} \int x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \int x^4 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx-\frac {1}{20} b c \int \left (c^4 x^8+2 c^2 x^6+x^4\right )dx^2+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{5} d^3 \left (-\frac {1}{4} b c \int x^4 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{2} \int x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2dx+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \int x^4 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {3}{5} d^3 \left (-\frac {1}{4} b c \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{8} b c \int x^5 \left (c^2 x^2+1\right )dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{2} \left (-\frac {1}{3} b c \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} \int x^3 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{8} b c \int x^5 \left (c^2 x^2+1\right )dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (-\frac {1}{3} b c \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} \int x^3 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{8} b c \int \left (c^2 x^7+x^5\right )dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx-\frac {1}{8} b c \int \left (c^2 x^7+x^5\right )dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (-\frac {1}{3} b c \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{3} \int x^3 (a+b \text {arcsinh}(c x))^2dx+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {1}{3} b c \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \int x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {1}{3} b c \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx-\frac {1}{6} b c \int x^5dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx\right )-\frac {1}{3} b c \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {3}{5} d^3 \left (-\frac {1}{4} b c \left (\frac {3}{8} \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}-\frac {b \int x^3dx}{4 c}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {3}{5} d^3 \left (\frac {1}{2} \left (\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )\right )-\frac {1}{3} b c \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{6} x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{4} b c \left (\frac {3}{8} \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{8} x^4 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2\right )-\frac {1}{5} b c d^3 \left (\frac {1}{2} \left (\frac {3}{8} \left (\frac {1}{6} \left (-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{4 c^2}+\frac {x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {b x^4}{16 c}\right )+\frac {1}{6} x^5 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))-\frac {1}{36} b c x^6\right )+\frac {1}{8} x^5 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )\right )+\frac {1}{10} x^5 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )\right )+\frac {1}{10} d^3 x^4 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {1}{10} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2 x^4-\frac {1}{5} b c d^3 \left (\frac {1}{10} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x)) x^5-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )+\frac {1}{2} \left (\frac {1}{8} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) x^5-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^5+\frac {1}{6} \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b \int xdx}{2 c}-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )\right )+\frac {3}{5} d^3 \left (\frac {1}{8} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2 x^4-\frac {1}{4} b c \left (\frac {1}{8} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) x^5-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^5+\frac {1}{6} \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b \int xdx}{2 c}-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )+\frac {1}{2} \left (\frac {1}{6} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2 x^4-\frac {1}{3} b c \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^5+\frac {1}{6} \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b \int xdx}{2 c}-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )+\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b \int xdx}{2 c}-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {1}{10} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2 x^4-\frac {1}{5} b c d^3 \left (\frac {1}{10} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x)) x^5-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )+\frac {1}{2} \left (\frac {1}{8} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) x^5-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^5+\frac {1}{6} \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (-\frac {b x^2}{4 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x}{2 c^2}-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )\right )+\frac {3}{5} d^3 \left (\frac {1}{8} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2 x^4-\frac {1}{4} b c \left (\frac {1}{8} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) x^5-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^5+\frac {1}{6} \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (-\frac {b x^2}{4 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x}{2 c^2}-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )+\frac {1}{2} \left (\frac {1}{6} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2 x^4-\frac {1}{3} b c \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^5+\frac {1}{6} \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (-\frac {b x^2}{4 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x}{2 c^2}-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )+\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (-\frac {b x^2}{4 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x}{2 c^2}-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )\right )\right )\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {1}{10} d^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2 x^4-\frac {1}{5} b c d^3 \left (\frac {1}{10} \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x)) x^5-\frac {1}{20} b c \left (\frac {c^4 x^{10}}{5}+\frac {c^2 x^8}{2}+\frac {x^6}{3}\right )+\frac {1}{2} \left (\frac {1}{8} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) x^5-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^5+\frac {1}{6} \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (-\frac {b x^2}{4 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x}{2 c^2}-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}\right )}{4 c^2}\right )\right )\right )\right )+\frac {3}{5} d^3 \left (\frac {1}{8} \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2 x^4-\frac {1}{4} b c \left (\frac {1}{8} \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) x^5-\frac {1}{8} b c \left (\frac {c^2 x^8}{8}+\frac {x^6}{6}\right )+\frac {3}{8} \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^5+\frac {1}{6} \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (-\frac {b x^2}{4 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x}{2 c^2}-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}\right )}{4 c^2}\right )\right )\right )+\frac {1}{2} \left (\frac {1}{6} \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2 x^4-\frac {1}{3} b c \left (-\frac {1}{36} b c x^6+\frac {1}{6} \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^5+\frac {1}{6} \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (-\frac {b x^2}{4 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x}{2 c^2}-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}\right )}{4 c^2}\right )\right )+\frac {1}{3} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))^2-\frac {1}{2} b c \left (-\frac {b x^4}{16 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^3}{4 c^2}-\frac {3 \left (-\frac {b x^2}{4 c}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x}{2 c^2}-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3}\right )}{4 c^2}\right )\right )\right )\right )\) |
Input:
Int[x^3*(d + c^2*d*x^2)^3*(a + b*ArcSinh[c*x])^2,x]
Output:
(d^3*x^4*(1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])^2)/10 - (b*c*d^3*(-1/20*(b*c *(x^6/3 + (c^2*x^8)/2 + (c^4*x^10)/5)) + (x^5*(1 + c^2*x^2)^(5/2)*(a + b*A rcSinh[c*x]))/10 + (-1/8*(b*c*(x^6/6 + (c^2*x^8)/8)) + (x^5*(1 + c^2*x^2)^ (3/2)*(a + b*ArcSinh[c*x]))/8 + (3*(-1/36*(b*c*x^6) + (x^5*Sqrt[1 + c^2*x^ 2]*(a + b*ArcSinh[c*x]))/6 + (-1/16*(b*x^4)/c + (x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4*c^2) - (3*(-1/4*(b*x^2)/c + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/(4*b*c^3)))/(4*c^2))/6 ))/8)/2))/5 + (3*d^3*((x^4*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/8 - (b* c*(-1/8*(b*c*(x^6/6 + (c^2*x^8)/8)) + (x^5*(1 + c^2*x^2)^(3/2)*(a + b*ArcS inh[c*x]))/8 + (3*(-1/36*(b*c*x^6) + (x^5*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh [c*x]))/6 + (-1/16*(b*x^4)/c + (x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) )/(4*c^2) - (3*(-1/4*(b*x^2)/c + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]) )/(2*c^2) - (a + b*ArcSinh[c*x])^2/(4*b*c^3)))/(4*c^2))/6))/8))/4 + ((x^4* (1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/6 - (b*c*(-1/36*(b*c*x^6) + (x^5*Sqr t[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/6 + (-1/16*(b*x^4)/c + (x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4*c^2) - (3*(-1/4*(b*x^2)/c + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSinh[c*x])^2/(4*b*c^3)) )/(4*c^2))/6))/3 + ((x^4*(a + b*ArcSinh[c*x])^2)/4 - (b*c*(-1/16*(b*x^4)/c + (x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4*c^2) - (3*(-1/4*(b*x^2) /c + (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(2*c^2) - (a + b*ArcSin...
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcSinh[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_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c ^2*d] && NeQ[n, -1]
Int[((a_.) + ArcSinh[(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*Arc Sinh[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*ArcSinh[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]] I nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d , e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcSinh[(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*Arc Sinh[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*ArcSinh[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*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[((a_.) + ArcSinh[(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*ArcSinh[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*ArcSinh[c*x])^n, x], x] - Simp[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*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
Time = 1.71 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.05
| method | result | size |
| derivativedivides | \(\frac {d^{3} a^{2} \left (\frac {\left (c^{2} x^{2}+1\right )^{5}}{10}-\frac {\left (c^{2} x^{2}+1\right )^{4}}{8}\right )+d^{3} b^{2} \left (\frac {x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{4} \operatorname {arcsinh}\left (x c \right )^{2}}{10}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{4}}{40}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{50}+\frac {7 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{800}+\frac {49 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{4800}+\frac {49 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3840}+\frac {49 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{2560}+\frac {49 \operatorname {arcsinh}\left (x c \right )^{2}}{5120}+\frac {\left (c^{2} x^{2}+1\right )^{5}}{500}-\frac {7 \left (c^{2} x^{2}+1\right )^{4}}{6400}-\frac {49 \left (c^{2} x^{2}+1\right )^{3}}{28800}-\frac {49 \left (c^{2} x^{2}+1\right )^{2}}{15360}-\frac {49 c^{2} x^{2}}{5120}-\frac {49}{5120}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}}{10}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {79 \,\operatorname {arcsinh}\left (x c \right )}{5120}+\frac {7 x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1600}+\frac {49 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{9600}+\frac {49 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{7680}+\frac {49 \sqrt {c^{2} x^{2}+1}\, x c}{5120}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{100}\right )}{c^{4}}\) | \(393\) |
| default | \(\frac {d^{3} a^{2} \left (\frac {\left (c^{2} x^{2}+1\right )^{5}}{10}-\frac {\left (c^{2} x^{2}+1\right )^{4}}{8}\right )+d^{3} b^{2} \left (\frac {x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{4} \operatorname {arcsinh}\left (x c \right )^{2}}{10}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{4}}{40}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{50}+\frac {7 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{800}+\frac {49 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{4800}+\frac {49 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3840}+\frac {49 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{2560}+\frac {49 \operatorname {arcsinh}\left (x c \right )^{2}}{5120}+\frac {\left (c^{2} x^{2}+1\right )^{5}}{500}-\frac {7 \left (c^{2} x^{2}+1\right )^{4}}{6400}-\frac {49 \left (c^{2} x^{2}+1\right )^{3}}{28800}-\frac {49 \left (c^{2} x^{2}+1\right )^{2}}{15360}-\frac {49 c^{2} x^{2}}{5120}-\frac {49}{5120}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}}{10}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {79 \,\operatorname {arcsinh}\left (x c \right )}{5120}+\frac {7 x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1600}+\frac {49 x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{9600}+\frac {49 x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{7680}+\frac {49 \sqrt {c^{2} x^{2}+1}\, x c}{5120}-\frac {x c \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{100}\right )}{c^{4}}\) | \(393\) |
| parts | \(d^{3} a^{2} \left (\frac {1}{10} c^{6} x^{10}+\frac {3}{8} c^{4} x^{8}+\frac {1}{2} x^{6} c^{2}+\frac {1}{4} x^{4}\right )+\frac {d^{3} b^{2} \left (\frac {x^{2} c^{2} \left (c^{2} x^{2}+1\right )^{4} \operatorname {arcsinh}\left (x c \right )^{2}}{10}-\frac {\operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{4}}{40}-\frac {\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{50}+\frac {7 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{800}+\frac {49 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{4800}+\frac {49 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3840}+\frac {49 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c}{2560}+\frac {49 \operatorname {arcsinh}\left (x c \right )^{2}}{5120}+\frac {\left (c^{2} x^{2}+1\right )^{5}}{500}-\frac {7 \left (c^{2} x^{2}+1\right )^{4}}{6400}-\frac {49 \left (c^{2} x^{2}+1\right )^{3}}{28800}-\frac {49 \left (c^{2} x^{2}+1\right )^{2}}{15360}-\frac {49 c^{2} x^{2}}{5120}-\frac {49}{5120}\right )}{c^{4}}+\frac {2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{10} c^{10}}{10}+\frac {3 \,\operatorname {arcsinh}\left (x c \right ) x^{8} c^{8}}{8}+\frac {\operatorname {arcsinh}\left (x c \right ) x^{6} c^{6}}{2}+\frac {\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}}{4}-\frac {79 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}}{7680}+\frac {79 \sqrt {c^{2} x^{2}+1}\, x c}{5120}-\frac {79 \,\operatorname {arcsinh}\left (x c \right )}{5120}-\frac {401 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}}{9600}-\frac {57 x^{7} c^{7} \sqrt {c^{2} x^{2}+1}}{1600}-\frac {x^{9} c^{9} \sqrt {c^{2} x^{2}+1}}{100}\right )}{c^{4}}\) | \(414\) |
| orering | \(\frac {\left (208128 c^{12} x^{12}+1019388 c^{10} x^{10}+1928796 c^{8} x^{8}+1587835 c^{6} x^{6}-38650 c^{4} x^{4}-408825 c^{2} x^{2}-118500\right ) \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}}{768000 c^{4} \left (c^{2} x^{2}+1\right )^{4}}-\frac {\left (62208 c^{10} x^{10}+293364 c^{8} x^{8}+512560 c^{6} x^{6}+316905 c^{4} x^{4}-278475 c^{2} x^{2}-142200\right ) \left (3 x^{2} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+6 x^{4} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {2 x^{3} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}\right )}{2304000 x^{2} c^{4} \left (c^{2} x^{2}+1\right )^{3}}+\frac {\left (2304 c^{8} x^{8}+10260 c^{6} x^{6}+16040 c^{4} x^{4}+5925 c^{2} x^{2}-17775\right ) \left (6 x \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2}+42 x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{2} d +\frac {12 x^{2} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b c}{\sqrt {c^{2} x^{2}+1}}+24 x^{5} \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{4} d^{2}+\frac {24 x^{4} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{3} d b}{\sqrt {c^{2} x^{2}+1}}+\frac {2 x^{3} \left (c^{2} d \,x^{2}+d \right )^{3} b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {2 x^{4} \left (c^{2} d \,x^{2}+d \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{2304000 x \,c^{4} \left (c^{2} x^{2}+1\right )^{2}}\) | \(542\) |
Input:
int(x^3*(c^2*d*x^2+d)^3*(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/c^4*(d^3*a^2*(1/10*(c^2*x^2+1)^5-1/8*(c^2*x^2+1)^4)+d^3*b^2*(1/10*x^2*c^ 2*(c^2*x^2+1)^4*arcsinh(x*c)^2-1/40*arcsinh(x*c)^2*(c^2*x^2+1)^4-1/50*arcs inh(x*c)*x*c*(c^2*x^2+1)^(9/2)+7/800*arcsinh(x*c)*x*c*(c^2*x^2+1)^(7/2)+49 /4800*arcsinh(x*c)*x*c*(c^2*x^2+1)^(5/2)+49/3840*arcsinh(x*c)*x*c*(c^2*x^2 +1)^(3/2)+49/2560*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x*c+49/5120*arcsinh(x*c)^ 2+1/500*(c^2*x^2+1)^5-7/6400*(c^2*x^2+1)^4-49/28800*(c^2*x^2+1)^3-49/15360 *(c^2*x^2+1)^2-49/5120*c^2*x^2-49/5120)+2*d^3*a*b*(1/10*arcsinh(x*c)*x^10* c^10+3/8*arcsinh(x*c)*x^8*c^8+1/2*arcsinh(x*c)*x^6*c^6+1/4*arcsinh(x*c)*c^ 4*x^4-79/5120*arcsinh(x*c)+7/1600*x*c*(c^2*x^2+1)^(7/2)+49/9600*x*c*(c^2*x ^2+1)^(5/2)+49/7680*x*c*(c^2*x^2+1)^(3/2)+49/5120*(c^2*x^2+1)^(1/2)*x*c-1/ 100*x*c*(c^2*x^2+1)^(9/2)))
Time = 0.11 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.13 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {2304 \, {\left (50 \, a^{2} + b^{2}\right )} c^{10} d^{3} x^{10} + 540 \, {\left (800 \, a^{2} + 19 \, b^{2}\right )} c^{8} d^{3} x^{8} + 40 \, {\left (14400 \, a^{2} + 401 \, b^{2}\right )} c^{6} d^{3} x^{6} + 75 \, {\left (3840 \, a^{2} + 79 \, b^{2}\right )} c^{4} d^{3} x^{4} - 17775 \, b^{2} c^{2} d^{3} x^{2} + 225 \, {\left (512 \, b^{2} c^{10} d^{3} x^{10} + 1920 \, b^{2} c^{8} d^{3} x^{8} + 2560 \, b^{2} c^{6} d^{3} x^{6} + 1280 \, b^{2} c^{4} d^{3} x^{4} - 79 \, b^{2} d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (7680 \, a b c^{10} d^{3} x^{10} + 28800 \, a b c^{8} d^{3} x^{8} + 38400 \, a b c^{6} d^{3} x^{6} + 19200 \, a b c^{4} d^{3} x^{4} - 1185 \, a b d^{3} - {\left (768 \, b^{2} c^{9} d^{3} x^{9} + 2736 \, b^{2} c^{7} d^{3} x^{7} + 3208 \, b^{2} c^{5} d^{3} x^{5} + 790 \, b^{2} c^{3} d^{3} x^{3} - 1185 \, b^{2} c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 30 \, {\left (768 \, a b c^{9} d^{3} x^{9} + 2736 \, a b c^{7} d^{3} x^{7} + 3208 \, a b c^{5} d^{3} x^{5} + 790 \, a b c^{3} d^{3} x^{3} - 1185 \, a b c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}}{1152000 \, c^{4}} \] Input:
integrate(x^3*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
1/1152000*(2304*(50*a^2 + b^2)*c^10*d^3*x^10 + 540*(800*a^2 + 19*b^2)*c^8* d^3*x^8 + 40*(14400*a^2 + 401*b^2)*c^6*d^3*x^6 + 75*(3840*a^2 + 79*b^2)*c^ 4*d^3*x^4 - 17775*b^2*c^2*d^3*x^2 + 225*(512*b^2*c^10*d^3*x^10 + 1920*b^2* c^8*d^3*x^8 + 2560*b^2*c^6*d^3*x^6 + 1280*b^2*c^4*d^3*x^4 - 79*b^2*d^3)*lo g(c*x + sqrt(c^2*x^2 + 1))^2 + 30*(7680*a*b*c^10*d^3*x^10 + 28800*a*b*c^8* d^3*x^8 + 38400*a*b*c^6*d^3*x^6 + 19200*a*b*c^4*d^3*x^4 - 1185*a*b*d^3 - ( 768*b^2*c^9*d^3*x^9 + 2736*b^2*c^7*d^3*x^7 + 3208*b^2*c^5*d^3*x^5 + 790*b^ 2*c^3*d^3*x^3 - 1185*b^2*c*d^3*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^ 2 + 1)) - 30*(768*a*b*c^9*d^3*x^9 + 2736*a*b*c^7*d^3*x^7 + 3208*a*b*c^5*d^ 3*x^5 + 790*a*b*c^3*d^3*x^3 - 1185*a*b*c*d^3*x)*sqrt(c^2*x^2 + 1))/c^4
Time = 2.48 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.74 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx =\text {Too large to display} \] Input:
integrate(x**3*(c**2*d*x**2+d)**3*(a+b*asinh(c*x))**2,x)
Output:
Piecewise((a**2*c**6*d**3*x**10/10 + 3*a**2*c**4*d**3*x**8/8 + a**2*c**2*d **3*x**6/2 + a**2*d**3*x**4/4 + a*b*c**6*d**3*x**10*asinh(c*x)/5 - a*b*c** 5*d**3*x**9*sqrt(c**2*x**2 + 1)/50 + 3*a*b*c**4*d**3*x**8*asinh(c*x)/4 - 5 7*a*b*c**3*d**3*x**7*sqrt(c**2*x**2 + 1)/800 + a*b*c**2*d**3*x**6*asinh(c* x) - 401*a*b*c*d**3*x**5*sqrt(c**2*x**2 + 1)/4800 + a*b*d**3*x**4*asinh(c* x)/2 - 79*a*b*d**3*x**3*sqrt(c**2*x**2 + 1)/(3840*c) + 79*a*b*d**3*x*sqrt( c**2*x**2 + 1)/(2560*c**3) - 79*a*b*d**3*asinh(c*x)/(2560*c**4) + b**2*c** 6*d**3*x**10*asinh(c*x)**2/10 + b**2*c**6*d**3*x**10/500 - b**2*c**5*d**3* x**9*sqrt(c**2*x**2 + 1)*asinh(c*x)/50 + 3*b**2*c**4*d**3*x**8*asinh(c*x)* *2/8 + 57*b**2*c**4*d**3*x**8/6400 - 57*b**2*c**3*d**3*x**7*sqrt(c**2*x**2 + 1)*asinh(c*x)/800 + b**2*c**2*d**3*x**6*asinh(c*x)**2/2 + 401*b**2*c**2 *d**3*x**6/28800 - 401*b**2*c*d**3*x**5*sqrt(c**2*x**2 + 1)*asinh(c*x)/480 0 + b**2*d**3*x**4*asinh(c*x)**2/4 + 79*b**2*d**3*x**4/15360 - 79*b**2*d** 3*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(3840*c) - 79*b**2*d**3*x**2/(5120*c **2) + 79*b**2*d**3*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(2560*c**3) - 79*b**2 *d**3*asinh(c*x)**2/(5120*c**4), Ne(c, 0)), (a**2*d**3*x**4/4, True))
Leaf count of result is larger than twice the leaf count of optimal. 1112 vs. \(2 (336) = 672\).
Time = 0.09 (sec) , antiderivative size = 1112, normalized size of antiderivative = 2.96 \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Too large to display} \] Input:
integrate(x^3*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
1/10*b^2*c^6*d^3*x^10*arcsinh(c*x)^2 + 1/10*a^2*c^6*d^3*x^10 + 3/8*b^2*c^4 *d^3*x^8*arcsinh(c*x)^2 + 3/8*a^2*c^4*d^3*x^8 + 1/2*b^2*c^2*d^3*x^6*arcsin h(c*x)^2 + 1/2*a^2*c^2*d^3*x^6 + 1/6400*(1280*x^10*arcsinh(c*x) - (128*sqr t(c^2*x^2 + 1)*x^9/c^2 - 144*sqrt(c^2*x^2 + 1)*x^7/c^4 + 168*sqrt(c^2*x^2 + 1)*x^5/c^6 - 210*sqrt(c^2*x^2 + 1)*x^3/c^8 + 315*sqrt(c^2*x^2 + 1)*x/c^1 0 - 315*arcsinh(c*x)/c^11)*c)*a*b*c^6*d^3 + 1/64000*((128*x^10/c^2 - 180*x ^8/c^4 + 280*x^6/c^6 - 525*x^4/c^8 + 1575*x^2/c^10 - 1575*log(c*x + sqrt(c ^2*x^2 + 1))^2/c^12)*c^2 - 10*(128*sqrt(c^2*x^2 + 1)*x^9/c^2 - 144*sqrt(c^ 2*x^2 + 1)*x^7/c^4 + 168*sqrt(c^2*x^2 + 1)*x^5/c^6 - 210*sqrt(c^2*x^2 + 1) *x^3/c^8 + 315*sqrt(c^2*x^2 + 1)*x/c^10 - 315*arcsinh(c*x)/c^11)*c*arcsinh (c*x))*b^2*c^6*d^3 + 1/4*b^2*d^3*x^4*arcsinh(c*x)^2 + 1/512*(384*x^8*arcsi nh(c*x) - (48*sqrt(c^2*x^2 + 1)*x^7/c^2 - 56*sqrt(c^2*x^2 + 1)*x^5/c^4 + 7 0*sqrt(c^2*x^2 + 1)*x^3/c^6 - 105*sqrt(c^2*x^2 + 1)*x/c^8 + 105*arcsinh(c* x)/c^9)*c)*a*b*c^4*d^3 + 1/3072*((36*x^8/c^2 - 56*x^6/c^4 + 105*x^4/c^6 - 315*x^2/c^8 + 315*log(c*x + sqrt(c^2*x^2 + 1))^2/c^10)*c^2 - 6*(48*sqrt(c^ 2*x^2 + 1)*x^7/c^2 - 56*sqrt(c^2*x^2 + 1)*x^5/c^4 + 70*sqrt(c^2*x^2 + 1)*x ^3/c^6 - 105*sqrt(c^2*x^2 + 1)*x/c^8 + 105*arcsinh(c*x)/c^9)*c*arcsinh(c*x ))*b^2*c^4*d^3 + 1/4*a^2*d^3*x^4 + 1/48*(48*x^6*arcsinh(c*x) - (8*sqrt(c^2 *x^2 + 1)*x^5/c^2 - 10*sqrt(c^2*x^2 + 1)*x^3/c^4 + 15*sqrt(c^2*x^2 + 1)*x/ c^6 - 15*arcsinh(c*x)/c^7)*c)*a*b*c^2*d^3 + 1/288*((8*x^6/c^2 - 15*x^4/...
Exception generated. \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^3*(c^2*d*x^2+d)^3*(a+b*arcsinh(c*x))^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \] Input:
int(x^3*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3,x)
Output:
int(x^3*(a + b*asinh(c*x))^2*(d + c^2*d*x^2)^3, x)
\[ \int x^3 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^{3} \left (7680 \mathit {asinh} \left (c x \right ) a b \,c^{10} x^{10}+28800 \mathit {asinh} \left (c x \right ) a b \,c^{8} x^{8}+38400 \mathit {asinh} \left (c x \right ) a b \,c^{6} x^{6}+19200 \mathit {asinh} \left (c x \right ) a b \,c^{4} x^{4}-768 \sqrt {c^{2} x^{2}+1}\, a b \,c^{9} x^{9}-2736 \sqrt {c^{2} x^{2}+1}\, a b \,c^{7} x^{7}-3208 \sqrt {c^{2} x^{2}+1}\, a b \,c^{5} x^{5}-790 \sqrt {c^{2} x^{2}+1}\, a b \,c^{3} x^{3}+1185 \sqrt {c^{2} x^{2}+1}\, a b c x +38400 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{9}d x \right ) b^{2} c^{10}+115200 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{7}d x \right ) b^{2} c^{8}+115200 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{5}d x \right ) b^{2} c^{6}+38400 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{3}d x \right ) b^{2} c^{4}-1185 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a b +3840 a^{2} c^{10} x^{10}+14400 a^{2} c^{8} x^{8}+19200 a^{2} c^{6} x^{6}+9600 a^{2} c^{4} x^{4}\right )}{38400 c^{4}} \] Input:
int(x^3*(c^2*d*x^2+d)^3*(a+b*asinh(c*x))^2,x)
Output:
(d**3*(7680*asinh(c*x)*a*b*c**10*x**10 + 28800*asinh(c*x)*a*b*c**8*x**8 + 38400*asinh(c*x)*a*b*c**6*x**6 + 19200*asinh(c*x)*a*b*c**4*x**4 - 768*sqrt (c**2*x**2 + 1)*a*b*c**9*x**9 - 2736*sqrt(c**2*x**2 + 1)*a*b*c**7*x**7 - 3 208*sqrt(c**2*x**2 + 1)*a*b*c**5*x**5 - 790*sqrt(c**2*x**2 + 1)*a*b*c**3*x **3 + 1185*sqrt(c**2*x**2 + 1)*a*b*c*x + 38400*int(asinh(c*x)**2*x**9,x)*b **2*c**10 + 115200*int(asinh(c*x)**2*x**7,x)*b**2*c**8 + 115200*int(asinh( c*x)**2*x**5,x)*b**2*c**6 + 38400*int(asinh(c*x)**2*x**3,x)*b**2*c**4 - 11 85*log(sqrt(c**2*x**2 + 1) + c*x)*a*b + 3840*a**2*c**10*x**10 + 14400*a**2 *c**8*x**8 + 19200*a**2*c**6*x**6 + 9600*a**2*c**4*x**4))/(38400*c**4)