Integrand size = 28, antiderivative size = 405 \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {7 b^2 d x \sqrt {d+c^2 d x^2}}{1152 c^2}+\frac {43 b^2 d x^3 \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 c^2 d x^5 \sqrt {d+c^2 d x^2}+\frac {7 b^2 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{1152 c^3 \sqrt {1+c^2 x^2}}-\frac {b d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c \sqrt {1+c^2 x^2}}-\frac {7 b c d x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{48 \sqrt {1+c^2 x^2}}-\frac {b c^3 d x^6 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{18 \sqrt {1+c^2 x^2}}+\frac {d x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 c^2}+\frac {1}{8} d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{6} x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{48 b c^3 \sqrt {1+c^2 x^2}} \]
1/6*x^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2-7/1152*b^2*d*x*(c^2*d*x^2 +d)^(1/2)/c^2+43/1728*b^2*d*x^3*(c^2*d*x^2+d)^(1/2)+1/108*b^2*c^2*d*x^5*(c ^2*d*x^2+d)^(1/2)+1/16*d*x*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^2+1/ 8*d*x^3*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)+7/1152*b^2*d*arcsinh(c*x) *(c^2*d*x^2+d)^(1/2)/c^3/(c^2*x^2+1)^(1/2)-1/16*b*d*x^2*(a+b*arcsinh(c*x)) *(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-7/48*b*c*d*x^4*(a+b*arcsinh(c*x)) *(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/18*b*c^3*d*x^6*(a+b*arcsinh(c*x)) *(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/48*d*(a+b*arcsinh(c*x))^3*(c^2*d* x^2+d)^(1/2)/b/c^3/(c^2*x^2+1)^(1/2)
Time = 2.11 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.25 \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {864 a^2 c d x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+4032 a^2 c^3 d x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+2304 a^2 c^5 d x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-288 b^2 d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^3+216 a b d \sqrt {d+c^2 d x^2} \cosh (2 \text {arcsinh}(c x))-108 a b d \sqrt {d+c^2 d x^2} \cosh (4 \text {arcsinh}(c x))-24 a b d \sqrt {d+c^2 d x^2} \cosh (6 \text {arcsinh}(c x))-864 a^2 d^{3/2} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-108 b^2 d \sqrt {d+c^2 d x^2} \sinh (2 \text {arcsinh}(c x))+27 b^2 d \sqrt {d+c^2 d x^2} \sinh (4 \text {arcsinh}(c x))+4 b^2 d \sqrt {d+c^2 d x^2} \sinh (6 \text {arcsinh}(c x))+12 b d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) (18 b \cosh (2 \text {arcsinh}(c x))-9 b \cosh (4 \text {arcsinh}(c x))-2 b \cosh (6 \text {arcsinh}(c x))-36 a \sinh (2 \text {arcsinh}(c x))+36 a \sinh (4 \text {arcsinh}(c x))+12 a \sinh (6 \text {arcsinh}(c x)))+72 b d \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2 (-12 a-3 b \sinh (2 \text {arcsinh}(c x))+3 b \sinh (4 \text {arcsinh}(c x))+b \sinh (6 \text {arcsinh}(c x)))}{13824 c^3 \sqrt {1+c^2 x^2}} \]
(864*a^2*c*d*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 4032*a^2*c^3*d*x^3* Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 2304*a^2*c^5*d*x^5*Sqrt[1 + c^2*x^ 2]*Sqrt[d + c^2*d*x^2] - 288*b^2*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 + 21 6*a*b*d*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 108*a*b*d*Sqrt[d + c^2* d*x^2]*Cosh[4*ArcSinh[c*x]] - 24*a*b*d*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[ c*x]] - 864*a^2*d^(3/2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2 *d*x^2]] - 108*b^2*d*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + 27*b^2*d*S qrt[d + c^2*d*x^2]*Sinh[4*ArcSinh[c*x]] + 4*b^2*d*Sqrt[d + c^2*d*x^2]*Sinh [6*ArcSinh[c*x]] + 12*b*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(18*b*Cosh[2*Ar cSinh[c*x]] - 9*b*Cosh[4*ArcSinh[c*x]] - 2*b*Cosh[6*ArcSinh[c*x]] - 36*a*S inh[2*ArcSinh[c*x]] + 36*a*Sinh[4*ArcSinh[c*x]] + 12*a*Sinh[6*ArcSinh[c*x] ]) + 72*b*d*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(-12*a - 3*b*Sinh[2*ArcSinh [c*x]] + 3*b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]]))/(13824*c^3*Sq rt[1 + c^2*x^2])
Time = 2.31 (sec) , antiderivative size = 495, normalized size of antiderivative = 1.22, number of steps used = 17, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.607, Rules used = {6223, 6218, 27, 363, 262, 262, 222, 6221, 6191, 262, 262, 222, 6227, 6191, 262, 222, 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.
| \(\displaystyle \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \int x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 6218 |
| \(\displaystyle \frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x^4 \left (2 c^2 x^2+3\right )}{12 \sqrt {c^2 x^2+1}}dx+\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{12} b c \int \frac {x^4 \left (2 c^2 x^2+3\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 363 |
| \(\displaystyle \frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{12} b c \left (\frac {4}{3} \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )+\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )+\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )+\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \int x^3 (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int x (a+b \text {arcsinh}(c x))dx}{c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx\right )}{c}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 262 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )\right )}{c}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{c}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2+\frac {\sqrt {c^2 d x^2+d} \left (-\frac {(a+b \text {arcsinh}(c x))^3}{6 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{c}\right )}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\) |
(x^3*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/6 - (b*c*d*Sqrt[d + c^2 *d*x^2]*((x^4*(a + b*ArcSinh[c*x]))/4 + (c^2*x^6*(a + b*ArcSinh[c*x]))/6 - (b*c*((x^5*Sqrt[1 + c^2*x^2])/3 + (4*((x^3*Sqrt[1 + c^2*x^2])/(4*c^2) - ( 3*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)))/(4*c^2)))/3))/12 ))/(3*Sqrt[1 + c^2*x^2]) + (d*((x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x ])^2)/4 - (b*c*Sqrt[d + c^2*d*x^2]*((x^4*(a + b*ArcSinh[c*x]))/4 - (b*c*(( x^3*Sqrt[1 + c^2*x^2])/(4*c^2) - (3*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSi nh[c*x]/(2*c^3)))/(4*c^2)))/4))/(2*Sqrt[1 + c^2*x^2]) + (Sqrt[d + c^2*d*x^ 2]*((x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*c^2) - (a + b*ArcSinh[ c*x])^3/(6*b*c^3) - (b*((x^2*(a + b*ArcSinh[c*x]))/2 - (b*c*((x*Sqrt[1 + c ^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)))/2))/c))/(4*Sqrt[1 + c^2*x^2])))/ 2
3.3.67.3.1 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[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x _Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3)) Int[(e*x)^ m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d , 0] && NeQ[m + 2*p + 3, 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_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1551\) vs. \(2(351)=702\).
Time = 0.36 (sec) , antiderivative size = 1552, normalized size of antiderivative = 3.83
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1552\) |
| parts | \(\text {Expression too large to display}\) | \(1552\) |
1/6*a^2*x*(c^2*d*x^2+d)^(5/2)/c^2/d-1/24*a^2/c^2*x*(c^2*d*x^2+d)^(3/2)-1/1 6*a^2/c^2*d*x*(c^2*d*x^2+d)^(1/2)-1/16*a^2/c^2*d^2*ln(c^2*d*x/(c^2*d)^(1/2 )+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b^2*(-1/48*(d*(c^2*x^2+1))^(1/2)/(c^2 *x^2+1)^(1/2)/c^3*arcsinh(c*x)^3*d+1/6912*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^ 7+32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5+48*c^4*x^4*(c^2*x^2+1)^(1/2)+38* c^3*x^3+18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x+(c^2*x^2+1)^(1/2))*(18*arcsinh( c*x)^2-6*arcsinh(c*x)+1)*d/c^3/(c^2*x^2+1)+1/1024*(d*(c^2*x^2+1))^(1/2)*(8 *c^5*x^5+8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3+8*c^2*x^2*(c^2*x^2+1)^(1/2 )+4*c*x+(c^2*x^2+1)^(1/2))*(8*arcsinh(c*x)^2-4*arcsinh(c*x)+1)*d/c^3/(c^2* x^2+1)-1/256*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+ 2*c*x+(c^2*x^2+1)^(1/2))*(2*arcsinh(c*x)^2-2*arcsinh(c*x)+1)*d/c^3/(c^2*x^ 2+1)-1/256*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*x^2+1)^(1/2)+2* c*x-(c^2*x^2+1)^(1/2))*(2*arcsinh(c*x)^2+2*arcsinh(c*x)+1)*d/c^3/(c^2*x^2+ 1)+1/1024*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12* c^3*x^3-8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*x^2+1)^(1/2))*(8*arcsinh(c* x)^2+4*arcsinh(c*x)+1)*d/c^3/(c^2*x^2+1)+1/6912*(d*(c^2*x^2+1))^(1/2)*(32* c^7*x^7-32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5-48*c^4*x^4*(c^2*x^2+1)^(1/ 2)+38*c^3*x^3-18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x-(c^2*x^2+1)^(1/2))*(18*ar csinh(c*x)^2+6*arcsinh(c*x)+1)*d/c^3/(c^2*x^2+1))+2*a*b*(-1/32*(d*(c^2*x^2 +1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(c*x)^2*d+1/2304*(d*(c^2*x^2+1)...
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
integral((a^2*c^2*d*x^4 + a^2*d*x^2 + (b^2*c^2*d*x^4 + b^2*d*x^2)*arcsinh( c*x)^2 + 2*(a*b*c^2*d*x^4 + a*b*d*x^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^{2} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]
Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
\[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{2} \,d x } \]
Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]