Integrand size = 26, antiderivative size = 165 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b \pi ^{3/2} x^2}{32 c}-\frac {7}{96} b c \pi ^{3/2} x^4-\frac {1}{36} b c^3 \pi ^{3/2} x^6+\frac {\pi ^{3/2} x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{32 b c^3} \]
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Time = 0.22 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {5808, 5806, 5812, 5783, 30, 14} \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{32 b c^3}+\frac {\pi ^{3/2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{6} x^3 \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} \pi x^3 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\frac {1}{36} \pi ^{3/2} b c^3 x^6-\frac {7}{96} \pi ^{3/2} b c x^4-\frac {\pi ^{3/2} b x^2}{32 c} \]
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Rule 14
Rule 30
Rule 5783
Rule 5806
Rule 5808
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{2} \pi \int x^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x)) \, dx-\frac {1}{6} \left (b c \pi ^{3/2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx \\ & = \frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} \pi ^{3/2} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{8} \left (b c \pi ^{3/2}\right ) \int x^3 \, dx-\frac {1}{6} \left (b c \pi ^{3/2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx \\ & = -\frac {7}{96} b c \pi ^{3/2} x^4-\frac {1}{36} b c^3 \pi ^{3/2} x^6+\frac {\pi ^{3/2} x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\pi ^{3/2} \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 c^2}-\frac {\left (b \pi ^{3/2}\right ) \int x \, dx}{16 c} \\ & = -\frac {b \pi ^{3/2} x^2}{32 c}-\frac {7}{96} b c \pi ^{3/2} x^4-\frac {1}{36} b c^3 \pi ^{3/2} x^6+\frac {\pi ^{3/2} x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} \pi x^3 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} x^3 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\pi ^{3/2} (a+b \text {arcsinh}(c x))^2}{32 b c^3} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.93 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{3/2} \left (144 a c x \sqrt {1+c^2 x^2}+672 a c^3 x^3 \sqrt {1+c^2 x^2}+384 a c^5 x^5 \sqrt {1+c^2 x^2}-72 b \text {arcsinh}(c x)^2+18 b \cosh (2 \text {arcsinh}(c x))-9 b \cosh (4 \text {arcsinh}(c x))-2 b \cosh (6 \text {arcsinh}(c x))-12 \text {arcsinh}(c x) (12 a+3 b \sinh (2 \text {arcsinh}(c x))-3 b \sinh (4 \text {arcsinh}(c x))-b \sinh (6 \text {arcsinh}(c x)))\right )}{2304 c^3} \]
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Time = 0.21 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.30
method | result | size |
default | \(\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{6 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{16 c^{2}}-\frac {a \,\pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{16 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \,\pi ^{\frac {3}{2}} \left (-48 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+8 c^{6} x^{6}-84 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+21 c^{4} x^{4}-18 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+9 c^{2} x^{2}+9 \operatorname {arcsinh}\left (c x \right )^{2}-4\right )}{288 c^{3}}\) | \(214\) |
parts | \(\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{6 \pi \,c^{2}}-\frac {a x \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{24 c^{2}}-\frac {a \pi x \sqrt {\pi \,c^{2} x^{2}+\pi }}{16 c^{2}}-\frac {a \,\pi ^{2} \ln \left (\frac {\pi \,c^{2} x}{\sqrt {\pi \,c^{2}}}+\sqrt {\pi \,c^{2} x^{2}+\pi }\right )}{16 c^{2} \sqrt {\pi \,c^{2}}}-\frac {b \,\pi ^{\frac {3}{2}} \left (-48 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+8 c^{6} x^{6}-84 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+21 c^{4} x^{4}-18 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+9 c^{2} x^{2}+9 \operatorname {arcsinh}\left (c x \right )^{2}-4\right )}{288 c^{3}}\) | \(214\) |
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\[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2} \,d x } \]
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Time = 5.49 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.59 \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\pi ^{\frac {3}{2}} a c^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{6} + \frac {7 \pi ^{\frac {3}{2}} a x^{3} \sqrt {c^{2} x^{2} + 1}}{24} + \frac {\pi ^{\frac {3}{2}} a x \sqrt {c^{2} x^{2} + 1}}{16 c^{2}} - \frac {\pi ^{\frac {3}{2}} a \operatorname {asinh}{\left (c x \right )}}{16 c^{3}} - \frac {\pi ^{\frac {3}{2}} b c^{3} x^{6}}{36} + \frac {\pi ^{\frac {3}{2}} b c^{2} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{6} - \frac {7 \pi ^{\frac {3}{2}} b c x^{4}}{96} + \frac {7 \pi ^{\frac {3}{2}} b x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{24} - \frac {\pi ^{\frac {3}{2}} b x^{2}}{32 c} + \frac {\pi ^{\frac {3}{2}} b x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16 c^{2}} - \frac {\pi ^{\frac {3}{2}} b \operatorname {asinh}^{2}{\left (c x \right )}}{32 c^{3}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {3}{2}} a x^{3}}{3} & \text {otherwise} \end {cases} \]
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Exception generated. \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2} \,d x } \]
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Timed out. \[ \int x^2 \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{3/2} \,d x \]
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