Integrand size = 26, antiderivative size = 382 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {10516 b^2 d^3 x}{99225 c^2}+\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}+\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{945 c^3}-\frac {32 b d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{945 c}+\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \text {arcsinh}(c x))^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \]
[Out]
Time = 0.55 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.423, Rules used = {5808, 5776, 5812, 5798, 8, 30, 272, 45, 5804, 12, 380} \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {32 b d^3 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{945 c}+\frac {1}{9} d^3 x^3 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {2}{21} d^3 x^3 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {8}{105} d^3 x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 b d^3 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{81 c^3}+\frac {2 b d^3 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{441 c^3}+\frac {4 b d^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{525 c^3}+\frac {16 b d^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{315 c^3}+\frac {64 b d^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{945 c^3}+\frac {16}{315} d^3 x^3 (a+b \text {arcsinh}(c x))^2+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {4198 b^2 c^2 d^3 x^5}{165375}-\frac {10516 b^2 d^3 x}{99225 c^2}+\frac {5258 b^2 d^3 x^3}{297675} \]
[In]
[Out]
Rule 8
Rule 12
Rule 30
Rule 45
Rule 272
Rule 380
Rule 5776
Rule 5798
Rule 5804
Rule 5808
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{3} (2 d) \int x^2 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{9} \left (2 b c d^3\right ) \int x^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx \\ & = \frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{63 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{81 c^3}+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{21} \left (8 d^2\right ) \int x^2 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{21} \left (4 b c d^3\right ) \int x^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx+\frac {1}{9} \left (2 b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx \\ & = \frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{105 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{81 c^3}+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{105} \left (16 d^3\right ) \int x^2 (a+b \text {arcsinh}(c x))^2 \, dx+\frac {\left (2 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right ) \, dx}{567 c^2}-\frac {1}{105} \left (16 b c d^3\right ) \int x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx+\frac {1}{21} \left (4 b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right )}{35 c^4} \, dx \\ & = \frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \text {arcsinh}(c x))^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {\left (2 b^2 d^3\right ) \int \left (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{567 c^2}+\frac {\left (4 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \left (-2+5 c^2 x^2\right ) \, dx}{735 c^2}-\frac {1}{315} \left (32 b c d^3\right ) \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{105} \left (16 b^2 c^2 d^3\right ) \int \frac {-2+c^2 x^2+3 c^4 x^4}{15 c^4} \, dx \\ & = -\frac {4 b^2 d^3 x}{567 c^2}+\frac {2 b^2 d^3 x^3}{1701}+\frac {2}{189} b^2 c^2 d^3 x^5+\frac {38 b^2 c^4 d^3 x^7}{3969}+\frac {2}{729} b^2 c^6 d^3 x^9-\frac {32 b d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{945 c}+\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \text {arcsinh}(c x))^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{945} \left (32 b^2 d^3\right ) \int x^2 \, dx+\frac {\left (4 b^2 d^3\right ) \int \left (-2+c^2 x^2+8 c^4 x^4+5 c^6 x^6\right ) \, dx}{735 c^2}+\frac {\left (16 b^2 d^3\right ) \int \left (-2+c^2 x^2+3 c^4 x^4\right ) \, dx}{1575 c^2}+\frac {\left (64 b d^3\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{945 c} \\ & = -\frac {3796 b^2 d^3 x}{99225 c^2}+\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}+\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{945 c^3}-\frac {32 b d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{945 c}+\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \text {arcsinh}(c x))^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {\left (64 b^2 d^3\right ) \int 1 \, dx}{945 c^2} \\ & = -\frac {10516 b^2 d^3 x}{99225 c^2}+\frac {5258 b^2 d^3 x^3}{297675}+\frac {4198 b^2 c^2 d^3 x^5}{165375}+\frac {374 b^2 c^4 d^3 x^7}{27783}+\frac {2}{729} b^2 c^6 d^3 x^9+\frac {64 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{945 c^3}-\frac {32 b d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{945 c}+\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{315 c^3}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{525 c^3}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{441 c^3}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{81 c^3}+\frac {16}{315} d^3 x^3 (a+b \text {arcsinh}(c x))^2+\frac {8}{105} d^3 x^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{21} d^3 x^3 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{9} d^3 x^3 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \\ \end{align*}
Time = 1.29 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.72 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^3 \left (99225 a^2 c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )-630 a b \sqrt {1+c^2 x^2} \left (-5258+2629 c^2 x^2+6297 c^4 x^4+4675 c^6 x^6+1225 c^8 x^8\right )+b^2 \left (-3312540 c x+552090 c^3 x^3+793422 c^5 x^5+420750 c^7 x^7+85750 c^9 x^9\right )-630 b \left (-315 a c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right )+b \sqrt {1+c^2 x^2} \left (-5258+2629 c^2 x^2+6297 c^4 x^4+4675 c^6 x^6+1225 c^8 x^8\right )\right ) \text {arcsinh}(c x)+99225 b^2 c^3 x^3 \left (105+189 c^2 x^2+135 c^4 x^4+35 c^6 x^6\right ) \text {arcsinh}(c x)^2\right )}{31255875 c^3} \]
[In]
[Out]
Time = 0.23 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.14
method | result | size |
parts | \(d^{3} a^{2} \left (\frac {1}{9} c^{6} x^{9}+\frac {3}{7} c^{4} x^{7}+\frac {3}{5} c^{2} x^{5}+\frac {1}{3} x^{3}\right )+\frac {d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{4}}{9}-\frac {16 \operatorname {arcsinh}\left (c x \right )^{2} x c}{315}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{63}-\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{105}-\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{81}-\frac {3406208 c x}{31255875}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {622 c x \left (c^{2} x^{2}+1\right )^{3}}{250047}+\frac {15224 c x \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {115504 c x \left (c^{2} x^{2}+1\right )}{31255875}+\frac {32 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{315}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{441}+\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}+\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )}{c^{3}}+\frac {2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{9} x^{9}}{9}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {2099 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(437\) |
derivativedivides | \(\frac {d^{3} a^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{4}}{9}-\frac {16 \operatorname {arcsinh}\left (c x \right )^{2} x c}{315}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{63}-\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{105}-\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{81}-\frac {3406208 c x}{31255875}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {622 c x \left (c^{2} x^{2}+1\right )^{3}}{250047}+\frac {15224 c x \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {115504 c x \left (c^{2} x^{2}+1\right )}{31255875}+\frac {32 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{315}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{441}+\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}+\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{9} x^{9}}{9}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {2099 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(438\) |
default | \(\frac {d^{3} a^{2} \left (\frac {1}{9} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {3}{5} c^{5} x^{5}+\frac {1}{3} c^{3} x^{3}\right )+d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{4}}{9}-\frac {16 \operatorname {arcsinh}\left (c x \right )^{2} x c}{315}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{63}-\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{105}-\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{315}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{81}-\frac {3406208 c x}{31255875}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{4}}{729}+\frac {622 c x \left (c^{2} x^{2}+1\right )^{3}}{250047}+\frac {15224 c x \left (c^{2} x^{2}+1\right )^{2}}{10418625}-\frac {115504 c x \left (c^{2} x^{2}+1\right )}{31255875}+\frac {32 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{315}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{441}+\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{525}+\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{945}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{9} x^{9}}{9}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{81}-\frac {187 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{3969}-\frac {2099 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{33075}-\frac {2629 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{99225}+\frac {5258 \sqrt {c^{2} x^{2}+1}}{99225}\right )}{c^{3}}\) | \(438\) |
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.05 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {42875 \, {\left (81 \, a^{2} + 2 \, b^{2}\right )} c^{9} d^{3} x^{9} + 1125 \, {\left (11907 \, a^{2} + 374 \, b^{2}\right )} c^{7} d^{3} x^{7} + 189 \, {\left (99225 \, a^{2} + 4198 \, b^{2}\right )} c^{5} d^{3} x^{5} + 105 \, {\left (99225 \, a^{2} + 5258 \, b^{2}\right )} c^{3} d^{3} x^{3} - 3312540 \, b^{2} c d^{3} x + 99225 \, {\left (35 \, b^{2} c^{9} d^{3} x^{9} + 135 \, b^{2} c^{7} d^{3} x^{7} + 189 \, b^{2} c^{5} d^{3} x^{5} + 105 \, b^{2} c^{3} d^{3} x^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 630 \, {\left (11025 \, a b c^{9} d^{3} x^{9} + 42525 \, a b c^{7} d^{3} x^{7} + 59535 \, a b c^{5} d^{3} x^{5} + 33075 \, a b c^{3} d^{3} x^{3} - {\left (1225 \, b^{2} c^{8} d^{3} x^{8} + 4675 \, b^{2} c^{6} d^{3} x^{6} + 6297 \, b^{2} c^{4} d^{3} x^{4} + 2629 \, b^{2} c^{2} d^{3} x^{2} - 5258 \, b^{2} d^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 630 \, {\left (1225 \, a b c^{8} d^{3} x^{8} + 4675 \, a b c^{6} d^{3} x^{6} + 6297 \, a b c^{4} d^{3} x^{4} + 2629 \, a b c^{2} d^{3} x^{2} - 5258 \, a b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{31255875 \, c^{3}} \]
[In]
[Out]
Time = 1.81 (sec) , antiderivative size = 626, normalized size of antiderivative = 1.64 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{6} d^{3} x^{9}}{9} + \frac {3 a^{2} c^{4} d^{3} x^{7}}{7} + \frac {3 a^{2} c^{2} d^{3} x^{5}}{5} + \frac {a^{2} d^{3} x^{3}}{3} + \frac {2 a b c^{6} d^{3} x^{9} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {2 a b c^{5} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1}}{81} + \frac {6 a b c^{4} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {374 a b c^{3} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{3969} + \frac {6 a b c^{2} d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {4198 a b c d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{33075} + \frac {2 a b d^{3} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {5258 a b d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{99225 c} + \frac {10516 a b d^{3} \sqrt {c^{2} x^{2} + 1}}{99225 c^{3}} + \frac {b^{2} c^{6} d^{3} x^{9} \operatorname {asinh}^{2}{\left (c x \right )}}{9} + \frac {2 b^{2} c^{6} d^{3} x^{9}}{729} - \frac {2 b^{2} c^{5} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{81} + \frac {3 b^{2} c^{4} d^{3} x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {374 b^{2} c^{4} d^{3} x^{7}}{27783} - \frac {374 b^{2} c^{3} d^{3} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3969} + \frac {3 b^{2} c^{2} d^{3} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {4198 b^{2} c^{2} d^{3} x^{5}}{165375} - \frac {4198 b^{2} c d^{3} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{33075} + \frac {b^{2} d^{3} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {5258 b^{2} d^{3} x^{3}}{297675} - \frac {5258 b^{2} d^{3} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{99225 c} - \frac {10516 b^{2} d^{3} x}{99225 c^{2}} + \frac {10516 b^{2} d^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{99225 c^{3}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{3}}{3} & \text {otherwise} \end {cases} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 922 vs. \(2 (340) = 680\).
Time = 0.25 (sec) , antiderivative size = 922, normalized size of antiderivative = 2.41 \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Too large to display} \]
[In]
[Out]
Exception generated. \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
[In]
[Out]
Timed out. \[ \int x^2 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \]
[In]
[Out]