Integrand size = 26, antiderivative size = 465 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {100976 b^2 d^3 x}{4002075 c^4}-\frac {50488 b^2 d^3 x^3}{12006225 c^2}+\frac {12622 b^2 d^3 x^5}{6670125}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}-\frac {256 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{17325 c^5}+\frac {128 b d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{17325 c^3}-\frac {32 b d^3 x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{5775 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} (a+b \text {arcsinh}(c x))}{121 c^5}+\frac {16 d^3 x^5 (a+b \text {arcsinh}(c x))^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \]
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Time = 0.69 (sec) , antiderivative size = 465, normalized size of antiderivative = 1.00, number of steps used = 21, 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, 1167} \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{11} d^3 x^5 \left (c^2 x^2+1\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {2}{33} d^3 x^5 \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {8}{231} d^3 x^5 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {32 b d^3 x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{5775 c}-\frac {2 b d^3 \left (c^2 x^2+1\right )^{11/2} (a+b \text {arcsinh}(c x))}{121 c^5}+\frac {8 b d^3 \left (c^2 x^2+1\right )^{9/2} (a+b \text {arcsinh}(c x))}{297 c^5}-\frac {2 b d^3 \left (c^2 x^2+1\right )^{7/2} (a+b \text {arcsinh}(c x))}{1617 c^5}+\frac {4 b d^3 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{1155 c^5}-\frac {16 b d^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{693 c^5}-\frac {256 b d^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{17325 c^5}+\frac {128 b d^3 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{17325 c^3}+\frac {16 d^3 x^5 (a+b \text {arcsinh}(c x))^2}{1155}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {100976 b^2 d^3 x}{4002075 c^4}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}-\frac {50488 b^2 d^3 x^3}{12006225 c^2}+\frac {12622 b^2 d^3 x^5}{6670125} \]
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Rule 8
Rule 12
Rule 30
Rule 45
Rule 272
Rule 1167
Rule 5776
Rule 5798
Rule 5804
Rule 5808
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{11} (6 d) \int x^4 \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{11} \left (2 b c d^3\right ) \int x^5 \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))}{77 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{99 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} (a+b \text {arcsinh}(c x))}{121 c^5}+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{33} \left (8 d^2\right ) \int x^4 \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{33} \left (4 b c d^3\right ) \int x^5 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx+\frac {1}{11} \left (2 b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^3 \left (8-28 c^2 x^2+63 c^4 x^4\right )}{693 c^6} \, dx \\ & = -\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{165 c^5}+\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{231 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} (a+b \text {arcsinh}(c x))}{121 c^5}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{231} \left (16 d^3\right ) \int x^4 (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 (8-28 c^2 x^2+63 c^4 x^4\right ) \, dx}{7623 c^4}-\frac {1}{231} \left (16 b c d^3\right ) \int x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx+\frac {1}{33} \left (4 b^2 c^2 d^3\right ) \int \frac {\left (1+c^2 x^2\right )^2 \left (8-20 c^2 x^2+35 c^4 x^4\right )}{315 c^6} \, dx \\ & = -\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} (a+b \text {arcsinh}(c x))}{121 c^5}+\frac {16 d^3 x^5 (a+b \text {arcsinh}(c x))^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{11} d^3 x^5 \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 (8-4 c^2 x^2+3 c^4 x^4+113 c^6 x^6+161 c^8 x^8+63 c^{10} x^{10}\right ) \, dx}{7623 c^4}+\frac {\left (4 b^2 d^3\right ) \int \left (1+c^2 x^2\right )^2 \left (8-20 c^2 x^2+35 c^4 x^4\right ) \, dx}{10395 c^4}-\frac {\left (32 b c d^3\right ) \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{1155}+\frac {1}{231} \left (16 b^2 c^2 d^3\right ) \int \frac {8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6}{105 c^6} \, dx \\ & = \frac {16 b^2 d^3 x}{7623 c^4}-\frac {8 b^2 d^3 x^3}{22869 c^2}+\frac {2 b^2 d^3 x^5}{12705}+\frac {226 b^2 c^2 d^3 x^7}{53361}+\frac {46 b^2 c^4 d^3 x^9}{9801}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}-\frac {32 b d^3 x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{5775 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} (a+b \text {arcsinh}(c x))}{121 c^5}+\frac {16 d^3 x^5 (a+b \text {arcsinh}(c x))^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {\left (32 b^2 d^3\right ) \int x^4 \, dx}{5775}+\frac {\left (4 b^2 d^3\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+50 c^6 x^6+35 c^8 x^8\right ) \, dx}{10395 c^4}+\frac {\left (16 b^2 d^3\right ) \int \left (8-4 c^2 x^2+3 c^4 x^4+15 c^6 x^6\right ) \, dx}{24255 c^4}+\frac {\left (128 b d^3\right ) \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{5775 c} \\ & = \frac {8368 b^2 d^3 x}{800415 c^4}-\frac {4184 b^2 d^3 x^3}{2401245 c^2}+\frac {12622 b^2 d^3 x^5}{6670125}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}+\frac {128 b d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{17325 c^3}-\frac {32 b d^3 x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{5775 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} (a+b \text {arcsinh}(c x))}{121 c^5}+\frac {16 d^3 x^5 (a+b \text {arcsinh}(c x))^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2-\frac {\left (256 b d^3\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{17325 c^3}-\frac {\left (128 b^2 d^3\right ) \int x^2 \, dx}{17325 c^2} \\ & = \frac {8368 b^2 d^3 x}{800415 c^4}-\frac {50488 b^2 d^3 x^3}{12006225 c^2}+\frac {12622 b^2 d^3 x^5}{6670125}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}-\frac {256 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{17325 c^5}+\frac {128 b d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{17325 c^3}-\frac {32 b d^3 x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{5775 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} (a+b \text {arcsinh}(c x))}{121 c^5}+\frac {16 d^3 x^5 (a+b \text {arcsinh}(c x))^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2+\frac {\left (256 b^2 d^3\right ) \int 1 \, dx}{17325 c^4} \\ & = \frac {100976 b^2 d^3 x}{4002075 c^4}-\frac {50488 b^2 d^3 x^3}{12006225 c^2}+\frac {12622 b^2 d^3 x^5}{6670125}+\frac {9410 b^2 c^2 d^3 x^7}{1120581}+\frac {182 b^2 c^4 d^3 x^9}{29403}+\frac {2 b^2 c^6 d^3 x^{11}}{1331}-\frac {256 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{17325 c^5}+\frac {128 b d^3 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{17325 c^3}-\frac {32 b d^3 x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{5775 c}-\frac {16 b d^3 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{693 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{1155 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{1617 c^5}+\frac {8 b d^3 \left (1+c^2 x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{297 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{11/2} (a+b \text {arcsinh}(c x))}{121 c^5}+\frac {16 d^3 x^5 (a+b \text {arcsinh}(c x))^2}{1155}+\frac {8}{231} d^3 x^5 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {2}{33} d^3 x^5 \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{11} d^3 x^5 \left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \\ \end{align*}
Time = 0.28 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.64 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^3 \left (12006225 a^2 c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )-6930 a b \sqrt {1+c^2 x^2} \left (50488-25244 c^2 x^2+18933 c^4 x^4+117625 c^6 x^6+111475 c^8 x^8+33075 c^{10} x^{10}\right )+2 b^2 c x \left (174940920-29156820 c^2 x^2+13120569 c^4 x^4+58224375 c^6 x^6+42917875 c^8 x^8+10418625 c^{10} x^{10}\right )-6930 b \left (-3465 a c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )+b \sqrt {1+c^2 x^2} \left (50488-25244 c^2 x^2+18933 c^4 x^4+117625 c^6 x^6+111475 c^8 x^8+33075 c^{10} x^{10}\right )\right ) \text {arcsinh}(c x)+12006225 b^2 c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right ) \text {arcsinh}(c x)^2\right )}{13867189875 c^5} \]
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Time = 0.31 (sec) , antiderivative size = 519, normalized size of antiderivative = 1.12
method | result | size |
parts | \(d^{3} a^{2} \left (\frac {1}{11} c^{6} x^{11}+\frac {1}{3} c^{4} x^{9}+\frac {3}{7} c^{2} x^{7}+\frac {1}{5} x^{5}\right )+\frac {d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3} \left (c^{2} x^{2}+1\right )^{4}}{11}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{4}}{33}+\frac {16 \operatorname {arcsinh}\left (c x \right )^{2} x c}{1155}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{231}+\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{385}+\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{1155}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1617}-\frac {428 c x \left (c^{2} x^{2}+1\right )^{4}}{323433}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3465}-\frac {606416 c x \left (c^{2} x^{2}+1\right )}{13867189875}-\frac {5487704 c x \left (c^{2} x^{2}+1\right )^{2}}{4622396625}-\frac {148174 c x \left (c^{2} x^{2}+1\right )^{3}}{110937519}-\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1925}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{121}+\frac {382986368 c x}{13867189875}-\frac {32 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{1155}+\frac {34 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{3267}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{5}}{1331}\right )}{c^{5}}+\frac {2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{11} x^{11}}{11}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {91 c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {4705 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {c^{10} x^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(519\) |
derivativedivides | \(\frac {d^{3} a^{2} \left (\frac {1}{11} c^{11} x^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3} \left (c^{2} x^{2}+1\right )^{4}}{11}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{4}}{33}+\frac {16 \operatorname {arcsinh}\left (c x \right )^{2} x c}{1155}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{231}+\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{385}+\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{1155}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1617}-\frac {428 c x \left (c^{2} x^{2}+1\right )^{4}}{323433}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3465}-\frac {606416 c x \left (c^{2} x^{2}+1\right )}{13867189875}-\frac {5487704 c x \left (c^{2} x^{2}+1\right )^{2}}{4622396625}-\frac {148174 c x \left (c^{2} x^{2}+1\right )^{3}}{110937519}-\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1925}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{121}+\frac {382986368 c x}{13867189875}-\frac {32 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{1155}+\frac {34 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{3267}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{5}}{1331}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{11} x^{11}}{11}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {91 c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {4705 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {c^{10} x^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(520\) |
default | \(\frac {d^{3} a^{2} \left (\frac {1}{11} c^{11} x^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{3} b^{2} \left (\frac {\operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3} \left (c^{2} x^{2}+1\right )^{4}}{11}-\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{4}}{33}+\frac {16 \operatorname {arcsinh}\left (c x \right )^{2} x c}{1155}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{3}}{231}+\frac {2 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{385}+\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{1155}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{1617}-\frac {428 c x \left (c^{2} x^{2}+1\right )^{4}}{323433}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3465}-\frac {606416 c x \left (c^{2} x^{2}+1\right )}{13867189875}-\frac {5487704 c x \left (c^{2} x^{2}+1\right )^{2}}{4622396625}-\frac {148174 c x \left (c^{2} x^{2}+1\right )^{3}}{110937519}-\frac {4 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1925}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{121}+\frac {382986368 c x}{13867189875}-\frac {32 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{1155}+\frac {34 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {9}{2}}}{3267}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{5}}{1331}\right )+2 d^{3} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{11} x^{11}}{11}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{7} x^{7}}{7}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}-\frac {91 c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {4705 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {c^{10} x^{10} \sqrt {c^{2} x^{2}+1}}{121}\right )}{c^{5}}\) | \(520\) |
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Time = 0.28 (sec) , antiderivative size = 444, normalized size of antiderivative = 0.95 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {10418625 \, {\left (121 \, a^{2} + 2 \, b^{2}\right )} c^{11} d^{3} x^{11} + 471625 \, {\left (9801 \, a^{2} + 182 \, b^{2}\right )} c^{9} d^{3} x^{9} + 12375 \, {\left (480249 \, a^{2} + 9410 \, b^{2}\right )} c^{7} d^{3} x^{7} + 2079 \, {\left (1334025 \, a^{2} + 12622 \, b^{2}\right )} c^{5} d^{3} x^{5} - 58313640 \, b^{2} c^{3} d^{3} x^{3} + 349881840 \, b^{2} c d^{3} x + 12006225 \, {\left (105 \, b^{2} c^{11} d^{3} x^{11} + 385 \, b^{2} c^{9} d^{3} x^{9} + 495 \, b^{2} c^{7} d^{3} x^{7} + 231 \, b^{2} c^{5} d^{3} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6930 \, {\left (363825 \, a b c^{11} d^{3} x^{11} + 1334025 \, a b c^{9} d^{3} x^{9} + 1715175 \, a b c^{7} d^{3} x^{7} + 800415 \, a b c^{5} d^{3} x^{5} - {\left (33075 \, b^{2} c^{10} d^{3} x^{10} + 111475 \, b^{2} c^{8} d^{3} x^{8} + 117625 \, b^{2} c^{6} d^{3} x^{6} + 18933 \, b^{2} c^{4} d^{3} x^{4} - 25244 \, b^{2} c^{2} d^{3} x^{2} + 50488 \, b^{2} d^{3}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6930 \, {\left (33075 \, a b c^{10} d^{3} x^{10} + 111475 \, a b c^{8} d^{3} x^{8} + 117625 \, a b c^{6} d^{3} x^{6} + 18933 \, a b c^{4} d^{3} x^{4} - 25244 \, a b c^{2} d^{3} x^{2} + 50488 \, a b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{13867189875 \, c^{5}} \]
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Time = 3.48 (sec) , antiderivative size = 702, normalized size of antiderivative = 1.51 \[ \int x^4 \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^{11}}{11} + \frac {a^{2} c^{4} d^{3} x^{9}}{3} + \frac {3 a^{2} c^{2} d^{3} x^{7}}{7} + \frac {a^{2} d^{3} x^{5}}{5} + \frac {2 a b c^{6} d^{3} x^{11} \operatorname {asinh}{\left (c x \right )}}{11} - \frac {2 a b c^{5} d^{3} x^{10} \sqrt {c^{2} x^{2} + 1}}{121} + \frac {2 a b c^{4} d^{3} x^{9} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {182 a b c^{3} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1}}{3267} + \frac {6 a b c^{2} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {9410 a b c d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{160083} + \frac {2 a b d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {12622 a b d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{1334025 c} + \frac {50488 a b d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{4002075 c^{3}} - \frac {100976 a b d^{3} \sqrt {c^{2} x^{2} + 1}}{4002075 c^{5}} + \frac {b^{2} c^{6} d^{3} x^{11} \operatorname {asinh}^{2}{\left (c x \right )}}{11} + \frac {2 b^{2} c^{6} d^{3} x^{11}}{1331} - \frac {2 b^{2} c^{5} d^{3} x^{10} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{121} + \frac {b^{2} c^{4} d^{3} x^{9} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {182 b^{2} c^{4} d^{3} x^{9}}{29403} - \frac {182 b^{2} c^{3} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3267} + \frac {3 b^{2} c^{2} d^{3} x^{7} \operatorname {asinh}^{2}{\left (c x \right )}}{7} + \frac {9410 b^{2} c^{2} d^{3} x^{7}}{1120581} - \frac {9410 b^{2} c d^{3} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{160083} + \frac {b^{2} d^{3} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {12622 b^{2} d^{3} x^{5}}{6670125} - \frac {12622 b^{2} d^{3} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{1334025 c} - \frac {50488 b^{2} d^{3} x^{3}}{12006225 c^{2}} + \frac {50488 b^{2} d^{3} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4002075 c^{3}} + \frac {100976 b^{2} d^{3} x}{4002075 c^{4}} - \frac {100976 b^{2} d^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{4002075 c^{5}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{5}}{5} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1109 vs. \(2 (413) = 826\).
Time = 0.23 (sec) , antiderivative size = 1109, normalized size of antiderivative = 2.38 \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Too large to display} \]
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Exception generated. \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int x^4 \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \]
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