Integrand size = 19, antiderivative size = 265 \[ \int \left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^3 \, dx=-\frac {4144 c^2 \sqrt {1+a^2 x^2}}{1125 a}-\frac {272 c^2 \left (1+a^2 x^2\right )^{3/2}}{3375 a}-\frac {6 c^2 \left (1+a^2 x^2\right )^{5/2}}{625 a}+\frac {298}{75} c^2 x \text {arcsinh}(a x)+\frac {76}{225} a^2 c^2 x^3 \text {arcsinh}(a x)+\frac {6}{125} a^4 c^2 x^5 \text {arcsinh}(a x)-\frac {8 c^2 \sqrt {1+a^2 x^2} \text {arcsinh}(a x)^2}{5 a}-\frac {4 c^2 \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x)^2}{15 a}-\frac {3 c^2 \left (1+a^2 x^2\right )^{5/2} \text {arcsinh}(a x)^2}{25 a}+\frac {8}{15} c^2 x \text {arcsinh}(a x)^3+\frac {4}{15} c^2 x \left (1+a^2 x^2\right ) \text {arcsinh}(a x)^3+\frac {1}{5} c^2 x \left (1+a^2 x^2\right )^2 \text {arcsinh}(a x)^3 \]
-272/3375*c^2*(a^2*x^2+1)^(3/2)/a-6/625*c^2*(a^2*x^2+1)^(5/2)/a+298/75*c^2 *x*arcsinh(a*x)+76/225*a^2*c^2*x^3*arcsinh(a*x)+6/125*a^4*c^2*x^5*arcsinh( a*x)-4/15*c^2*(a^2*x^2+1)^(3/2)*arcsinh(a*x)^2/a-3/25*c^2*(a^2*x^2+1)^(5/2 )*arcsinh(a*x)^2/a+8/15*c^2*x*arcsinh(a*x)^3+4/15*c^2*x*(a^2*x^2+1)*arcsin h(a*x)^3+1/5*c^2*x*(a^2*x^2+1)^2*arcsinh(a*x)^3-4144/1125*c^2*(a^2*x^2+1)^ (1/2)/a-8/5*c^2*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)/a
Time = 0.10 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.52 \[ \int \left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^3 \, dx=\frac {c^2 \left (-2 \sqrt {1+a^2 x^2} \left (31841+842 a^2 x^2+81 a^4 x^4\right )+30 a x \left (2235+190 a^2 x^2+27 a^4 x^4\right ) \text {arcsinh}(a x)-225 \sqrt {1+a^2 x^2} \left (149+38 a^2 x^2+9 a^4 x^4\right ) \text {arcsinh}(a x)^2+1125 a x \left (15+10 a^2 x^2+3 a^4 x^4\right ) \text {arcsinh}(a x)^3\right )}{16875 a} \]
(c^2*(-2*Sqrt[1 + a^2*x^2]*(31841 + 842*a^2*x^2 + 81*a^4*x^4) + 30*a*x*(22 35 + 190*a^2*x^2 + 27*a^4*x^4)*ArcSinh[a*x] - 225*Sqrt[1 + a^2*x^2]*(149 + 38*a^2*x^2 + 9*a^4*x^4)*ArcSinh[a*x]^2 + 1125*a*x*(15 + 10*a^2*x^2 + 3*a^ 4*x^4)*ArcSinh[a*x]^3))/(16875*a)
Time = 1.43 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.38, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.737, Rules used = {6201, 27, 6201, 6187, 6213, 6187, 241, 6199, 27, 353, 53, 1576, 1140, 2009}
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.
| \(\displaystyle \int \text {arcsinh}(a x)^3 \left (a^2 c x^2+c\right )^2 \, dx\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {3}{5} a c^2 \int x \left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2dx+\frac {4}{5} c \int c \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3dx+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3}{5} a c^2 \int x \left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2dx+\frac {4}{5} c^2 \int \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3dx+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 6201 |
| \(\displaystyle -\frac {3}{5} a c^2 \int x \left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2dx+\frac {4}{5} c^2 \left (-a \int x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2dx+\frac {2}{3} \int \text {arcsinh}(a x)^3dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \int \frac {x \text {arcsinh}(a x)^2}{\sqrt {a^2 x^2+1}}dx\right )-a \int x \sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2dx+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3\right )-\frac {3}{5} a c^2 \int x \left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2dx+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \int \text {arcsinh}(a x)dx}{a}\right )\right )-a \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \int \left (a^2 x^2+1\right ) \text {arcsinh}(a x)dx}{3 a}\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3\right )-\frac {3}{5} a c^2 \left (\frac {\left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2}{5 a^2}-\frac {2 \int \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)dx}{5 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {4}{5} c^2 \left (\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-a \int \frac {x}{\sqrt {a^2 x^2+1}}dx\right )}{a}\right )\right )-a \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \int \left (a^2 x^2+1\right ) \text {arcsinh}(a x)dx}{3 a}\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3\right )-\frac {3}{5} a c^2 \left (\frac {\left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2}{5 a^2}-\frac {2 \int \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)dx}{5 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \int \left (a^2 x^2+1\right ) \text {arcsinh}(a x)dx}{3 a}\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {\left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2}{5 a^2}-\frac {2 \int \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)dx}{5 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 6199 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (-a \int \frac {x \left (a^2 x^2+3\right )}{3 \sqrt {a^2 x^2+1}}dx+\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{3 a}\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {\left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2}{5 a^2}-\frac {2 \left (-a \int \frac {x \left (3 a^4 x^4+10 a^2 x^2+15\right )}{15 \sqrt {a^2 x^2+1}}dx+\frac {1}{5} a^4 x^5 \text {arcsinh}(a x)+\frac {2}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{5 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{3} a \int \frac {x \left (a^2 x^2+3\right )}{\sqrt {a^2 x^2+1}}dx+\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{3 a}\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {\left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2}{5 a^2}-\frac {2 \left (-\frac {1}{15} a \int \frac {x \left (3 a^4 x^4+10 a^2 x^2+15\right )}{\sqrt {a^2 x^2+1}}dx+\frac {1}{5} a^4 x^5 \text {arcsinh}(a x)+\frac {2}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{5 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{6} a \int \frac {a^2 x^2+3}{\sqrt {a^2 x^2+1}}dx^2+\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{3 a}\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {\left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2}{5 a^2}-\frac {2 \left (-\frac {1}{15} a \int \frac {x \left (3 a^4 x^4+10 a^2 x^2+15\right )}{\sqrt {a^2 x^2+1}}dx+\frac {1}{5} a^4 x^5 \text {arcsinh}(a x)+\frac {2}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{5 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{6} a \int \left (\sqrt {a^2 x^2+1}+\frac {2}{\sqrt {a^2 x^2+1}}\right )dx^2+\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{3 a}\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {\left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2}{5 a^2}-\frac {2 \left (-\frac {1}{15} a \int \frac {x \left (3 a^4 x^4+10 a^2 x^2+15\right )}{\sqrt {a^2 x^2+1}}dx+\frac {1}{5} a^4 x^5 \text {arcsinh}(a x)+\frac {2}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{5 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 1576 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{6} a \int \left (\sqrt {a^2 x^2+1}+\frac {2}{\sqrt {a^2 x^2+1}}\right )dx^2+\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{3 a}\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {\left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2}{5 a^2}-\frac {2 \left (-\frac {1}{30} a \int \frac {3 a^4 x^4+10 a^2 x^2+15}{\sqrt {a^2 x^2+1}}dx^2+\frac {1}{5} a^4 x^5 \text {arcsinh}(a x)+\frac {2}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{5 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 1140 |
| \(\displaystyle \frac {4}{5} c^2 \left (-a \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (-\frac {1}{6} a \int \left (\sqrt {a^2 x^2+1}+\frac {2}{\sqrt {a^2 x^2+1}}\right )dx^2+\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{3 a}\right )+\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )\right )-\frac {3}{5} a c^2 \left (\frac {\left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2}{5 a^2}-\frac {2 \left (-\frac {1}{30} a \int \left (3 \left (a^2 x^2+1\right )^{3/2}+4 \sqrt {a^2 x^2+1}+\frac {8}{\sqrt {a^2 x^2+1}}\right )dx^2+\frac {1}{5} a^4 x^5 \text {arcsinh}(a x)+\frac {2}{3} a^2 x^3 \text {arcsinh}(a x)+x \text {arcsinh}(a x)\right )}{5 a}\right )+\frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{5} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x)^3+\frac {4}{5} c^2 \left (\frac {1}{3} x \left (a^2 x^2+1\right ) \text {arcsinh}(a x)^3+\frac {2}{3} \left (x \text {arcsinh}(a x)^3-3 a \left (\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^2}{a^2}-\frac {2 \left (x \text {arcsinh}(a x)-\frac {\sqrt {a^2 x^2+1}}{a}\right )}{a}\right )\right )-a \left (\frac {\left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2}{3 a^2}-\frac {2 \left (\frac {1}{3} a^2 x^3 \text {arcsinh}(a x)-\frac {1}{6} a \left (\frac {2 \left (a^2 x^2+1\right )^{3/2}}{3 a^2}+\frac {4 \sqrt {a^2 x^2+1}}{a^2}\right )+x \text {arcsinh}(a x)\right )}{3 a}\right )\right )-\frac {3}{5} a c^2 \left (\frac {\left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2}{5 a^2}-\frac {2 \left (\frac {1}{5} a^4 x^5 \text {arcsinh}(a x)+\frac {2}{3} a^2 x^3 \text {arcsinh}(a x)-\frac {1}{30} a \left (\frac {6 \left (a^2 x^2+1\right )^{5/2}}{5 a^2}+\frac {8 \left (a^2 x^2+1\right )^{3/2}}{3 a^2}+\frac {16 \sqrt {a^2 x^2+1}}{a^2}\right )+x \text {arcsinh}(a x)\right )}{5 a}\right )\) |
(c^2*x*(1 + a^2*x^2)^2*ArcSinh[a*x]^3)/5 - (3*a*c^2*(((1 + a^2*x^2)^(5/2)* ArcSinh[a*x]^2)/(5*a^2) - (2*(-1/30*(a*((16*Sqrt[1 + a^2*x^2])/a^2 + (8*(1 + a^2*x^2)^(3/2))/(3*a^2) + (6*(1 + a^2*x^2)^(5/2))/(5*a^2))) + x*ArcSinh [a*x] + (2*a^2*x^3*ArcSinh[a*x])/3 + (a^4*x^5*ArcSinh[a*x])/5))/(5*a)))/5 + (4*c^2*((x*(1 + a^2*x^2)*ArcSinh[a*x]^3)/3 - a*(((1 + a^2*x^2)^(3/2)*Arc Sinh[a*x]^2)/(3*a^2) - (2*(-1/6*(a*((4*Sqrt[1 + a^2*x^2])/a^2 + (2*(1 + a^ 2*x^2)^(3/2))/(3*a^2))) + x*ArcSinh[a*x] + (a^2*x^3*ArcSinh[a*x])/3))/(3*a )) + (2*(x*ArcSinh[a*x]^3 - 3*a*((Sqrt[1 + a^2*x^2]*ArcSinh[a*x]^2)/a^2 - (2*(-(Sqrt[1 + a^2*x^2]/a) + x*ArcSinh[a*x]))/a)))/3))/5
3.4.29.3.1 Defintions of rubi rules used
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, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x _Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( p_.), x_Symbol] :> Simp[1/2 Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] , x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSinh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Simp[2*d*(p/(2*p + 1)) Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x ], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x* (1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Time = 0.29 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.75
| method | result | size |
| derivativedivides | \(\frac {c^{2} \left (3375 a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{3}-2025 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}+11250 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3}+810 a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )-8550 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-162 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+16875 a x \operatorname {arcsinh}\left (a x \right )^{3}+5700 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )-33525 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-1684 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+67050 a x \,\operatorname {arcsinh}\left (a x \right )-63682 \sqrt {a^{2} x^{2}+1}\right )}{16875 a}\) | \(200\) |
| default | \(\frac {c^{2} \left (3375 a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )^{3}-2025 a^{4} x^{4} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}+11250 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )^{3}+810 a^{5} x^{5} \operatorname {arcsinh}\left (a x \right )-8550 a^{2} x^{2} \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-162 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+16875 a x \operatorname {arcsinh}\left (a x \right )^{3}+5700 a^{3} x^{3} \operatorname {arcsinh}\left (a x \right )-33525 \operatorname {arcsinh}\left (a x \right )^{2} \sqrt {a^{2} x^{2}+1}-1684 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+67050 a x \,\operatorname {arcsinh}\left (a x \right )-63682 \sqrt {a^{2} x^{2}+1}\right )}{16875 a}\) | \(200\) |
1/16875/a*c^2*(3375*a^5*x^5*arcsinh(a*x)^3-2025*a^4*x^4*arcsinh(a*x)^2*(a^ 2*x^2+1)^(1/2)+11250*a^3*x^3*arcsinh(a*x)^3+810*a^5*x^5*arcsinh(a*x)-8550* a^2*x^2*arcsinh(a*x)^2*(a^2*x^2+1)^(1/2)-162*a^4*x^4*(a^2*x^2+1)^(1/2)+168 75*a*x*arcsinh(a*x)^3+5700*a^3*x^3*arcsinh(a*x)-33525*arcsinh(a*x)^2*(a^2* x^2+1)^(1/2)-1684*a^2*x^2*(a^2*x^2+1)^(1/2)+67050*a*x*arcsinh(a*x)-63682*( a^2*x^2+1)^(1/2))
Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.77 \[ \int \left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^3 \, dx=\frac {1125 \, {\left (3 \, a^{5} c^{2} x^{5} + 10 \, a^{3} c^{2} x^{3} + 15 \, a c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{3} - 225 \, {\left (9 \, a^{4} c^{2} x^{4} + 38 \, a^{2} c^{2} x^{2} + 149 \, c^{2}\right )} \sqrt {a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (27 \, a^{5} c^{2} x^{5} + 190 \, a^{3} c^{2} x^{3} + 2235 \, a c^{2} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} + 1}\right ) - 2 \, {\left (81 \, a^{4} c^{2} x^{4} + 842 \, a^{2} c^{2} x^{2} + 31841 \, c^{2}\right )} \sqrt {a^{2} x^{2} + 1}}{16875 \, a} \]
1/16875*(1125*(3*a^5*c^2*x^5 + 10*a^3*c^2*x^3 + 15*a*c^2*x)*log(a*x + sqrt (a^2*x^2 + 1))^3 - 225*(9*a^4*c^2*x^4 + 38*a^2*c^2*x^2 + 149*c^2)*sqrt(a^2 *x^2 + 1)*log(a*x + sqrt(a^2*x^2 + 1))^2 + 30*(27*a^5*c^2*x^5 + 190*a^3*c^ 2*x^3 + 2235*a*c^2*x)*log(a*x + sqrt(a^2*x^2 + 1)) - 2*(81*a^4*c^2*x^4 + 8 42*a^2*c^2*x^2 + 31841*c^2)*sqrt(a^2*x^2 + 1))/a
Time = 0.57 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.99 \[ \int \left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^3 \, dx=\begin {cases} \frac {a^{4} c^{2} x^{5} \operatorname {asinh}^{3}{\left (a x \right )}}{5} + \frac {6 a^{4} c^{2} x^{5} \operatorname {asinh}{\left (a x \right )}}{125} - \frac {3 a^{3} c^{2} x^{4} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{25} - \frac {6 a^{3} c^{2} x^{4} \sqrt {a^{2} x^{2} + 1}}{625} + \frac {2 a^{2} c^{2} x^{3} \operatorname {asinh}^{3}{\left (a x \right )}}{3} + \frac {76 a^{2} c^{2} x^{3} \operatorname {asinh}{\left (a x \right )}}{225} - \frac {38 a c^{2} x^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{75} - \frac {1684 a c^{2} x^{2} \sqrt {a^{2} x^{2} + 1}}{16875} + c^{2} x \operatorname {asinh}^{3}{\left (a x \right )} + \frac {298 c^{2} x \operatorname {asinh}{\left (a x \right )}}{75} - \frac {149 c^{2} \sqrt {a^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a x \right )}}{75 a} - \frac {63682 c^{2} \sqrt {a^{2} x^{2} + 1}}{16875 a} & \text {for}\: a \neq 0 \\0 & \text {otherwise} \end {cases} \]
Piecewise((a**4*c**2*x**5*asinh(a*x)**3/5 + 6*a**4*c**2*x**5*asinh(a*x)/12 5 - 3*a**3*c**2*x**4*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/25 - 6*a**3*c**2*x* *4*sqrt(a**2*x**2 + 1)/625 + 2*a**2*c**2*x**3*asinh(a*x)**3/3 + 76*a**2*c* *2*x**3*asinh(a*x)/225 - 38*a*c**2*x**2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/ 75 - 1684*a*c**2*x**2*sqrt(a**2*x**2 + 1)/16875 + c**2*x*asinh(a*x)**3 + 2 98*c**2*x*asinh(a*x)/75 - 149*c**2*sqrt(a**2*x**2 + 1)*asinh(a*x)**2/(75*a ) - 63682*c**2*sqrt(a**2*x**2 + 1)/(16875*a), Ne(a, 0)), (0, True))
Time = 0.21 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.79 \[ \int \left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^3 \, dx=-\frac {1}{75} \, {\left (9 \, \sqrt {a^{2} x^{2} + 1} a^{2} c^{2} x^{4} + 38 \, \sqrt {a^{2} x^{2} + 1} c^{2} x^{2} + \frac {149 \, \sqrt {a^{2} x^{2} + 1} c^{2}}{a^{2}}\right )} a \operatorname {arsinh}\left (a x\right )^{2} + \frac {1}{15} \, {\left (3 \, a^{4} c^{2} x^{5} + 10 \, a^{2} c^{2} x^{3} + 15 \, c^{2} x\right )} \operatorname {arsinh}\left (a x\right )^{3} - \frac {2}{16875} \, {\left (81 \, \sqrt {a^{2} x^{2} + 1} a^{2} c^{2} x^{4} + 842 \, \sqrt {a^{2} x^{2} + 1} c^{2} x^{2} - \frac {15 \, {\left (27 \, a^{4} c^{2} x^{5} + 190 \, a^{2} c^{2} x^{3} + 2235 \, c^{2} x\right )} \operatorname {arsinh}\left (a x\right )}{a} + \frac {31841 \, \sqrt {a^{2} x^{2} + 1} c^{2}}{a^{2}}\right )} a \]
-1/75*(9*sqrt(a^2*x^2 + 1)*a^2*c^2*x^4 + 38*sqrt(a^2*x^2 + 1)*c^2*x^2 + 14 9*sqrt(a^2*x^2 + 1)*c^2/a^2)*a*arcsinh(a*x)^2 + 1/15*(3*a^4*c^2*x^5 + 10*a ^2*c^2*x^3 + 15*c^2*x)*arcsinh(a*x)^3 - 2/16875*(81*sqrt(a^2*x^2 + 1)*a^2* c^2*x^4 + 842*sqrt(a^2*x^2 + 1)*c^2*x^2 - 15*(27*a^4*c^2*x^5 + 190*a^2*c^2 *x^3 + 2235*c^2*x)*arcsinh(a*x)/a + 31841*sqrt(a^2*x^2 + 1)*c^2/a^2)*a
Exception generated. \[ \int \left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \left (c+a^2 c x^2\right )^2 \text {arcsinh}(a x)^3 \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^2 \,d x \]