Integrand size = 21, antiderivative size = 509 \[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=-\frac {865 a c^2 x^2 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {65 a^3 c^2 x^4 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {245}{384} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {115 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{768 a \sqrt {1+a^2 x^2}}-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{64 a \sqrt {1+a^2 x^2}} \]
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Time = 0.42 (sec) , antiderivative size = 509, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5786, 5785, 5783, 5776, 5812, 30, 5798, 14, 267} \[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=-\frac {15 a c^2 x^2 \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{32 \sqrt {a^2 x^2+1}}+\frac {5}{16} c^2 x \text {arcsinh}(a x)^3 \sqrt {a^2 c x^2+c}+\frac {245}{384} c^2 x \text {arcsinh}(a x) \sqrt {a^2 c x^2+c}+\frac {1}{36} c^2 x \left (a^2 x^2+1\right )^2 \text {arcsinh}(a x) \sqrt {a^2 c x^2+c}+\frac {65}{576} c^2 x \left (a^2 x^2+1\right ) \text {arcsinh}(a x) \sqrt {a^2 c x^2+c}+\frac {5 c^2 \text {arcsinh}(a x)^4 \sqrt {a^2 c x^2+c}}{64 a \sqrt {a^2 x^2+1}}-\frac {c^2 \left (a^2 x^2+1\right )^{5/2} \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{12 a}-\frac {5 c^2 \left (a^2 x^2+1\right )^{3/2} \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{32 a}-\frac {115 c^2 \text {arcsinh}(a x)^2 \sqrt {a^2 c x^2+c}}{768 a \sqrt {a^2 x^2+1}}+\frac {1}{6} x \text {arcsinh}(a x)^3 \left (a^2 c x^2+c\right )^{5/2}+\frac {5}{24} c x \text {arcsinh}(a x)^3 \left (a^2 c x^2+c\right )^{3/2}-\frac {865 a c^2 x^2 \sqrt {a^2 c x^2+c}}{2304 \sqrt {a^2 x^2+1}}-\frac {c^2 \left (a^2 x^2+1\right )^{5/2} \sqrt {a^2 c x^2+c}}{216 a}-\frac {65 a^3 c^2 x^4 \sqrt {a^2 c x^2+c}}{2304 \sqrt {a^2 x^2+1}} \]
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Rule 14
Rule 30
Rule 267
Rule 5776
Rule 5783
Rule 5785
Rule 5786
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {1}{6} (5 c) \int \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3 \, dx-\frac {\left (a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right )^2 \text {arcsinh}(a x)^2 \, dx}{2 \sqrt {1+a^2 x^2}} \\ & = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {1}{8} \left (5 c^2\right ) \int \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3 \, dx+\frac {\left (c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{5/2} \text {arcsinh}(a x) \, dx}{6 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \text {arcsinh}(a x)^2 \, dx}{8 \sqrt {1+a^2 x^2}} \\ & = \frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x) \, dx}{36 \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (1+a^2 x^2\right )^{3/2} \text {arcsinh}(a x) \, dx}{16 \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)^3}{\sqrt {1+a^2 x^2}} \, dx}{16 \sqrt {1+a^2 x^2}}-\frac {\left (a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right )^2 \, dx}{36 \sqrt {1+a^2 x^2}}-\frac {\left (15 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \text {arcsinh}(a x)^2 \, dx}{16 \sqrt {1+a^2 x^2}} \\ & = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \, dx}{48 \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \int \sqrt {1+a^2 x^2} \text {arcsinh}(a x) \, dx}{64 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \, dx}{144 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \left (1+a^2 x^2\right ) \, dx}{64 \sqrt {1+a^2 x^2}}+\frac {\left (15 a^2 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {x^2 \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{16 \sqrt {1+a^2 x^2}} \\ & = -\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {245}{384} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{64 a \sqrt {1+a^2 x^2}}+\frac {\left (5 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{96 \sqrt {1+a^2 x^2}}+\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{128 \sqrt {1+a^2 x^2}}-\frac {\left (15 c^2 \sqrt {c+a^2 c x^2}\right ) \int \frac {\text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{32 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (x+a^2 x^3\right ) \, dx}{144 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{96 \sqrt {1+a^2 x^2}}-\frac {\left (5 a c^2 \sqrt {c+a^2 c x^2}\right ) \int \left (x+a^2 x^3\right ) \, dx}{64 \sqrt {1+a^2 x^2}}-\frac {\left (15 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{128 \sqrt {1+a^2 x^2}}-\frac {\left (15 a c^2 \sqrt {c+a^2 c x^2}\right ) \int x \, dx}{32 \sqrt {1+a^2 x^2}} \\ & = -\frac {865 a c^2 x^2 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {65 a^3 c^2 x^4 \sqrt {c+a^2 c x^2}}{2304 \sqrt {1+a^2 x^2}}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2}}{216 a}+\frac {245}{384} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {65}{576} c^2 x \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)+\frac {1}{36} c^2 x \left (1+a^2 x^2\right )^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)-\frac {115 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{768 a \sqrt {1+a^2 x^2}}-\frac {15 a c^2 x^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 \sqrt {1+a^2 x^2}}-\frac {5 c^2 \left (1+a^2 x^2\right )^{3/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{32 a}-\frac {c^2 \left (1+a^2 x^2\right )^{5/2} \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^2}{12 a}+\frac {5}{16} c^2 x \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^3+\frac {5}{24} c x \left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^3+\frac {1}{6} x \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3+\frac {5 c^2 \sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^4}{64 a \sqrt {1+a^2 x^2}} \\ \end{align*}
Time = 0.60 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.35 \[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=\frac {c^2 \sqrt {c+a^2 c x^2} \left (4320 \text {arcsinh}(a x)^4-9720 \cosh (2 \text {arcsinh}(a x))-243 \cosh (4 \text {arcsinh}(a x))-8 \cosh (6 \text {arcsinh}(a x))-72 \text {arcsinh}(a x)^2 (270 \cosh (2 \text {arcsinh}(a x))+27 \cosh (4 \text {arcsinh}(a x))+2 \cosh (6 \text {arcsinh}(a x)))+288 \text {arcsinh}(a x)^3 (45 \sinh (2 \text {arcsinh}(a x))+9 \sinh (4 \text {arcsinh}(a x))+\sinh (6 \text {arcsinh}(a x)))+12 \text {arcsinh}(a x) (1620 \sinh (2 \text {arcsinh}(a x))+81 \sinh (4 \text {arcsinh}(a x))+4 \sinh (6 \text {arcsinh}(a x)))\right )}{55296 a \sqrt {1+a^2 x^2}} \]
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Time = 0.24 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.58
method | result | size |
default | \(\frac {5 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right )^{4} c^{2}}{64 \sqrt {a^{2} x^{2}+1}\, a}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (32 a^{7} x^{7}+32 x^{6} a^{6} \sqrt {a^{2} x^{2}+1}+64 a^{5} x^{5}+48 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+38 a^{3} x^{3}+18 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+6 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (36 \operatorname {arcsinh}\left (a x \right )^{3}-18 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )-1\right ) c^{2}}{13824 a \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}+8 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+12 a^{3} x^{3}+8 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+4 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (32 \operatorname {arcsinh}\left (a x \right )^{3}-24 \operatorname {arcsinh}\left (a x \right )^{2}+12 \,\operatorname {arcsinh}\left (a x \right )-3\right ) c^{2}}{4096 a \left (a^{2} x^{2}+1\right )}+\frac {15 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}+2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+2 a x +\sqrt {a^{2} x^{2}+1}\right ) \left (4 \operatorname {arcsinh}\left (a x \right )^{3}-6 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )-3\right ) c^{2}}{512 a \left (a^{2} x^{2}+1\right )}+\frac {15 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (2 a^{3} x^{3}-2 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+2 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (4 \operatorname {arcsinh}\left (a x \right )^{3}+6 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )+3\right ) c^{2}}{512 a \left (a^{2} x^{2}+1\right )}+\frac {3 \sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (8 a^{5} x^{5}-8 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+12 a^{3} x^{3}-8 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+4 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (32 \operatorname {arcsinh}\left (a x \right )^{3}+24 \operatorname {arcsinh}\left (a x \right )^{2}+12 \,\operatorname {arcsinh}\left (a x \right )+3\right ) c^{2}}{4096 a \left (a^{2} x^{2}+1\right )}+\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \left (32 a^{7} x^{7}-32 x^{6} a^{6} \sqrt {a^{2} x^{2}+1}+64 a^{5} x^{5}-48 a^{4} x^{4} \sqrt {a^{2} x^{2}+1}+38 a^{3} x^{3}-18 a^{2} x^{2} \sqrt {a^{2} x^{2}+1}+6 a x -\sqrt {a^{2} x^{2}+1}\right ) \left (36 \operatorname {arcsinh}\left (a x \right )^{3}+18 \operatorname {arcsinh}\left (a x \right )^{2}+6 \,\operatorname {arcsinh}\left (a x \right )+1\right ) c^{2}}{13824 a \left (a^{2} x^{2}+1\right )}\) | \(802\) |
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\[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=\int { {\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]
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Timed out. \[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=\text {Timed out} \]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: RuntimeError} \]
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Exception generated. \[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (c+a^2 c x^2\right )^{5/2} \text {arcsinh}(a x)^3 \, dx=\int {\mathrm {asinh}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2} \,d x \]
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