Integrand size = 26, antiderivative size = 130 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c^3 d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c^3 d \sqrt {d+c^2 d x^2}} \]
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Time = 0.10 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {5810, 5783, 266} \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 b c^3 d \sqrt {c^2 d x^2+d}}+\frac {b \sqrt {c^2 x^2+1} \log \left (c^2 x^2+1\right )}{2 c^3 d \sqrt {c^2 d x^2+d}} \]
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Rule 266
Rule 5783
Rule 5810
Rubi steps \begin{align*} \text {integral}& = -\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {d+c^2 d x^2}} \, dx}{c^2 d}+\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int \frac {x}{1+c^2 x^2} \, dx}{c d \sqrt {d+c^2 d x^2}} \\ & = -\frac {x (a+b \text {arcsinh}(c x))}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c^3 d \sqrt {d+c^2 d x^2}}+\frac {b \sqrt {1+c^2 x^2} \log \left (1+c^2 x^2\right )}{2 c^3 d \sqrt {d+c^2 d x^2}} \\ \end{align*}
Time = 0.33 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.12 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=-\frac {a x \sqrt {d \left (1+c^2 x^2\right )}}{c^2 d^2 \left (1+c^2 x^2\right )}+\frac {b \left (-2 c x \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \left (\text {arcsinh}(c x)^2+2 \log \left (\sqrt {1+c^2 x^2}\right )\right )\right )}{2 c^3 d \sqrt {d \left (1+c^2 x^2\right )}}+\frac {a \log \left (c d x+\sqrt {d} \sqrt {d \left (1+c^2 x^2\right )}\right )}{c^3 d^{3/2}} \]
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Time = 0.20 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.78
method | result | size |
default | \(-\frac {a x}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}\) | \(232\) |
parts | \(-\frac {a x}{c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {a \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{2} d \sqrt {c^{2} d}}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{2 \sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}-\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right ) x}{c^{2} d^{2} \left (c^{2} x^{2}+1\right )}+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (1+\left (c x +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{\sqrt {c^{2} x^{2}+1}\, c^{3} d^{2}}\) | \(232\) |
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \]
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