Integrand size = 26, antiderivative size = 75 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {b x^2}{4 c \sqrt {\pi }}+\frac {x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{2 c^2 \pi }-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3 \sqrt {\pi }} \]
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Time = 0.09 (sec) , antiderivative size = 75, 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 = {5812, 5783, 30} \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {(a+b \text {arcsinh}(c x))^2}{4 \sqrt {\pi } b c^3}+\frac {x \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{2 \pi c^2}-\frac {b x^2}{4 \sqrt {\pi } c} \]
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Rule 30
Rule 5783
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{2 c^2 \pi }-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{2 c^2}-\frac {b \int x \, dx}{2 c \sqrt {\pi }} \\ & = -\frac {b x^2}{4 c \sqrt {\pi }}+\frac {x \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{2 c^2 \pi }-\frac {(a+b \text {arcsinh}(c x))^2}{4 b c^3 \sqrt {\pi }} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.92 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {4 a c x \sqrt {1+c^2 x^2}-2 b \text {arcsinh}(c x)^2-b \cosh (2 \text {arcsinh}(c x))+\text {arcsinh}(c x) (-4 a+2 b \sinh (2 \text {arcsinh}(c x)))}{8 c^3 \sqrt {\pi }} \]
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Time = 0.16 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.43
method | result | size |
default | \(\frac {a x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2 \pi \,c^{2}}-\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \left (-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )^{2}+1\right )}{4 \sqrt {\pi }\, c^{3}}\) | \(107\) |
parts | \(\frac {a x \sqrt {\pi \,c^{2} x^{2}+\pi }}{2 \pi \,c^{2}}-\frac {a \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{2 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \left (-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+c^{2} x^{2}+\operatorname {arcsinh}\left (c x \right )^{2}+1\right )}{4 \sqrt {\pi }\, c^{3}}\) | \(107\) |
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{\sqrt {\pi + \pi c^{2} x^{2}}} \,d x } \]
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Time = 1.03 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.63 \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {a \left (\begin {cases} \frac {x \sqrt {c^{2} x^{2} + 1}}{2 c^{2}} - \frac {\log {\left (2 c^{2} x + 2 \sqrt {c^{2} x^{2} + 1} \sqrt {c^{2}} \right )}}{2 c^{2} \sqrt {c^{2}}} & \text {for}\: c^{2} \neq 0 \\\frac {x^{3}}{3} & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} + \frac {b \left (\begin {cases} - \frac {x^{2}}{4 c} + \frac {x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} - \frac {\operatorname {asinh}^{2}{\left (c x \right )}}{4 c^{3}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} \]
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Exception generated. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}}{\sqrt {\pi + \pi c^{2} x^{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int \frac {x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \]
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