Integrand size = 24, antiderivative size = 114 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b x}{6 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {1+c^2 x^2} \arctan (c x)}{6 c^2 d^2 \sqrt {d+c^2 d x^2}} \]
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Time = 0.05 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {5798, 205, 209} \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {a+b \text {arcsinh}(c x)}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}+\frac {b \sqrt {c^2 x^2+1} \arctan (c x)}{6 c^2 d^2 \sqrt {c^2 d x^2+d}}+\frac {b x}{6 c d^2 \sqrt {c^2 x^2+1} \sqrt {c^2 d x^2+d}} \]
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Rule 205
Rule 209
Rule 5798
Rubi steps \begin{align*} \text {integral}& = -\frac {a+b \text {arcsinh}(c x)}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\left (1+c^2 x^2\right )^2} \, dx}{3 c d^2 \sqrt {d+c^2 d x^2}} \\ & = \frac {b x}{6 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {1}{1+c^2 x^2} \, dx}{6 c d^2 \sqrt {d+c^2 d x^2}} \\ & = \frac {b x}{6 c d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}+\frac {b \sqrt {1+c^2 x^2} \arctan (c x)}{6 c^2 d^2 \sqrt {d+c^2 d x^2}} \\ \end{align*}
Time = 0.29 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.90 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+c^2 d x^2} \left (b c x+b c^3 x^3-2 a \sqrt {1+c^2 x^2}-2 b \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+b \left (1+c^2 x^2\right )^2 \arctan (c x)\right )}{6 c^2 d^3 \left (1+c^2 x^2\right )^{5/2}} \]
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Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.74
method | result | size |
default | \(-\frac {a}{3 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{6 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3} c}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}+\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{6 \sqrt {c^{2} x^{2}+1}\, c^{2} d^{3}}-\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{6 \sqrt {c^{2} x^{2}+1}\, c^{2} d^{3}}\) | \(198\) |
parts | \(-\frac {a}{3 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{6 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3} c}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}+\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}+i\right )}{6 \sqrt {c^{2} x^{2}+1}\, c^{2} d^{3}}-\frac {i b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (c x +\sqrt {c^{2} x^{2}+1}-i\right )}{6 \sqrt {c^{2} x^{2}+1}\, c^{2} d^{3}}\) | \(198\) |
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Time = 0.29 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.46 \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {{\left (b c^{4} x^{4} + 2 \, b c^{2} x^{2} + b\right )} \sqrt {d} \arctan \left (\frac {2 \, \sqrt {c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} + 1} c \sqrt {d} x}{c^{4} d x^{4} - d}\right ) + 4 \, \sqrt {c^{2} d x^{2} + d} b \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, \sqrt {c^{2} d x^{2} + d} {\left (\sqrt {c^{2} x^{2} + 1} b c x - 2 \, a\right )}}{12 \, {\left (c^{6} d^{3} x^{4} + 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \]
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\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \]
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