Integrand size = 22, antiderivative size = 145 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=-\frac {35 b d^3 x \sqrt {1+c^2 x^2}}{1024 c}-\frac {35 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{1536 c}-\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{384 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2}}{64 c}-\frac {35 b d^3 \text {arcsinh}(c x)}{1024 c^2}+\frac {d^3 \left (1+c^2 x^2\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2} \]
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Time = 0.05 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5798, 201, 221} \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^3 \left (c^2 x^2+1\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {35 b d^3 \text {arcsinh}(c x)}{1024 c^2}-\frac {b d^3 x \left (c^2 x^2+1\right )^{7/2}}{64 c}-\frac {7 b d^3 x \left (c^2 x^2+1\right )^{5/2}}{384 c}-\frac {35 b d^3 x \left (c^2 x^2+1\right )^{3/2}}{1536 c}-\frac {35 b d^3 x \sqrt {c^2 x^2+1}}{1024 c} \]
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Rule 201
Rule 221
Rule 5798
Rubi steps \begin{align*} \text {integral}& = \frac {d^3 \left (1+c^2 x^2\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {\left (b d^3\right ) \int \left (1+c^2 x^2\right )^{7/2} \, dx}{8 c} \\ & = -\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2}}{64 c}+\frac {d^3 \left (1+c^2 x^2\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {\left (7 b d^3\right ) \int \left (1+c^2 x^2\right )^{5/2} \, dx}{64 c} \\ & = -\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{384 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2}}{64 c}+\frac {d^3 \left (1+c^2 x^2\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {\left (35 b d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{384 c} \\ & = -\frac {35 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{1536 c}-\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{384 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2}}{64 c}+\frac {d^3 \left (1+c^2 x^2\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {\left (35 b d^3\right ) \int \sqrt {1+c^2 x^2} \, dx}{512 c} \\ & = -\frac {35 b d^3 x \sqrt {1+c^2 x^2}}{1024 c}-\frac {35 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{1536 c}-\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{384 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2}}{64 c}+\frac {d^3 \left (1+c^2 x^2\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2}-\frac {\left (35 b d^3\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{1024 c} \\ & = -\frac {35 b d^3 x \sqrt {1+c^2 x^2}}{1024 c}-\frac {35 b d^3 x \left (1+c^2 x^2\right )^{3/2}}{1536 c}-\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2}}{384 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2}}{64 c}-\frac {35 b d^3 \text {arcsinh}(c x)}{1024 c^2}+\frac {d^3 \left (1+c^2 x^2\right )^4 (a+b \text {arcsinh}(c x))}{8 c^2} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {d^3 \left (c x \left (384 a c x \left (4+6 c^2 x^2+4 c^4 x^4+c^6 x^6\right )-b \sqrt {1+c^2 x^2} \left (279+326 c^2 x^2+200 c^4 x^4+48 c^6 x^6\right )\right )+3 b \left (93+512 c^2 x^2+768 c^4 x^4+512 c^6 x^6+128 c^8 x^8\right ) \text {arcsinh}(c x)\right )}{3072 c^2} \]
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Time = 0.17 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.99
method | result | size |
derivativedivides | \(\frac {\frac {d^{3} a \left (c^{2} x^{2}+1\right )^{4}}{8}+d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {93 \,\operatorname {arcsinh}\left (c x \right )}{1024}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}-\frac {7 c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{384}-\frac {35 c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{1536}-\frac {35 c x \sqrt {c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(143\) |
default | \(\frac {\frac {d^{3} a \left (c^{2} x^{2}+1\right )^{4}}{8}+d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {93 \,\operatorname {arcsinh}\left (c x \right )}{1024}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}-\frac {7 c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{384}-\frac {35 c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{1536}-\frac {35 c x \sqrt {c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(143\) |
parts | \(\frac {d^{3} a \left (c^{2} x^{2}+1\right )^{4}}{8 c^{2}}+\frac {d^{3} b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}}{8}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}}{2}+\frac {3 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}+\frac {\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}}{2}+\frac {93 \,\operatorname {arcsinh}\left (c x \right )}{1024}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}-\frac {7 c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{384}-\frac {35 c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{1536}-\frac {35 c x \sqrt {c^{2} x^{2}+1}}{1024}\right )}{c^{2}}\) | \(145\) |
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Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.28 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {384 \, a c^{8} d^{3} x^{8} + 1536 \, a c^{6} d^{3} x^{6} + 2304 \, a c^{4} d^{3} x^{4} + 1536 \, a c^{2} d^{3} x^{2} + 3 \, {\left (128 \, b c^{8} d^{3} x^{8} + 512 \, b c^{6} d^{3} x^{6} + 768 \, b c^{4} d^{3} x^{4} + 512 \, b c^{2} d^{3} x^{2} + 93 \, b d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (48 \, b c^{7} d^{3} x^{7} + 200 \, b c^{5} d^{3} x^{5} + 326 \, b c^{3} d^{3} x^{3} + 279 \, b c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}}{3072 \, c^{2}} \]
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Time = 0.88 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.74 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {a c^{6} d^{3} x^{8}}{8} + \frac {a c^{4} d^{3} x^{6}}{2} + \frac {3 a c^{2} d^{3} x^{4}}{4} + \frac {a d^{3} x^{2}}{2} + \frac {b c^{6} d^{3} x^{8} \operatorname {asinh}{\left (c x \right )}}{8} - \frac {b c^{5} d^{3} x^{7} \sqrt {c^{2} x^{2} + 1}}{64} + \frac {b c^{4} d^{3} x^{6} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {25 b c^{3} d^{3} x^{5} \sqrt {c^{2} x^{2} + 1}}{384} + \frac {3 b c^{2} d^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {163 b c d^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{1536} + \frac {b d^{3} x^{2} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {93 b d^{3} x \sqrt {c^{2} x^{2} + 1}}{1024 c} + \frac {93 b d^{3} \operatorname {asinh}{\left (c x \right )}}{1024 c^{2}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{2}}{2} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 352 vs. \(2 (125) = 250\).
Time = 0.19 (sec) , antiderivative size = 352, normalized size of antiderivative = 2.43 \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{8} \, a c^{6} d^{3} x^{8} + \frac {1}{2} \, a c^{4} d^{3} x^{6} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{2}} - \frac {56 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{6}} - \frac {105 \, \sqrt {c^{2} x^{2} + 1} x}{c^{8}} + \frac {105 \, \operatorname {arsinh}\left (c x\right )}{c^{9}}\right )} c\right )} b c^{6} d^{3} + \frac {3}{4} \, a c^{2} d^{3} x^{4} + \frac {1}{96} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{4} d^{3} + \frac {3}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{2} \, a d^{3} x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} b d^{3} \]
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Exception generated. \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x \left (d+c^2 d x^2\right )^3 (a+b \text {arcsinh}(c x)) \, dx=\int x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \]
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