Integrand size = 26, antiderivative size = 141 \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {2 b \pi ^{5/2} x}{63 c^3}-\frac {b \pi ^{5/2} x^3}{189 c}-\frac {1}{21} b c \pi ^{5/2} x^5-\frac {19}{441} b c^3 \pi ^{5/2} x^7-\frac {1}{81} b c^5 \pi ^{5/2} x^9-\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 \pi }+\frac {\left (\pi +c^2 \pi x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 \pi ^2} \]
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Time = 0.12 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {272, 45, 5804, 12, 380} \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\left (\pi c^2 x^2+\pi \right )^{9/2} (a+b \text {arcsinh}(c x))}{9 \pi ^2 c^4}-\frac {\left (\pi c^2 x^2+\pi \right )^{7/2} (a+b \text {arcsinh}(c x))}{7 \pi c^4}-\frac {1}{81} \pi ^{5/2} b c^5 x^9-\frac {19}{441} \pi ^{5/2} b c^3 x^7+\frac {2 \pi ^{5/2} b x}{63 c^3}-\frac {1}{21} \pi ^{5/2} b c x^5-\frac {\pi ^{5/2} b x^3}{189 c} \]
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Rule 12
Rule 45
Rule 272
Rule 380
Rule 5804
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 \pi }+\frac {\left (\pi +c^2 \pi x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 \pi ^2}-\left (b c \sqrt {\pi }\right ) \int \frac {\pi ^2 \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right )}{63 c^4} \, dx \\ & = -\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 \pi }+\frac {\left (\pi +c^2 \pi x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 \pi ^2}-\frac {\left (b \pi ^{5/2}\right ) \int \left (1+c^2 x^2\right )^3 \left (-2+7 c^2 x^2\right ) \, dx}{63 c^3} \\ & = -\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 \pi }+\frac {\left (\pi +c^2 \pi x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 \pi ^2}-\frac {\left (b \pi ^{5/2}\right ) \int \left (-2+c^2 x^2+15 c^4 x^4+19 c^6 x^6+7 c^8 x^8\right ) \, dx}{63 c^3} \\ & = \frac {2 b \pi ^{5/2} x}{63 c^3}-\frac {b \pi ^{5/2} x^3}{189 c}-\frac {1}{21} b c \pi ^{5/2} x^5-\frac {19}{441} b c^3 \pi ^{5/2} x^7-\frac {1}{81} b c^5 \pi ^{5/2} x^9-\frac {\left (\pi +c^2 \pi x^2\right )^{7/2} (a+b \text {arcsinh}(c x))}{7 c^4 \pi }+\frac {\left (\pi +c^2 \pi x^2\right )^{9/2} (a+b \text {arcsinh}(c x))}{9 c^4 \pi ^2} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.77 \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {\pi ^{5/2} \left (63 a \left (1+c^2 x^2\right )^{7/2} \left (-2+7 c^2 x^2\right )-b c x \left (-126+21 c^2 x^2+189 c^4 x^4+171 c^6 x^6+49 c^8 x^8\right )+63 b \left (1+c^2 x^2\right )^{7/2} \left (-2+7 c^2 x^2\right ) \text {arcsinh}(c x)\right )}{3969 c^4} \]
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Time = 0.25 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.60
method | result | size |
default | \(a \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{9 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{63 \pi \,c^{4}}\right )+\frac {b \,\pi ^{\frac {5}{2}} \left (441 \,\operatorname {arcsinh}\left (c x \right ) c^{10} x^{10}+1638 \,\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}-49 c^{9} x^{9} \sqrt {c^{2} x^{2}+1}+2142 \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}-171 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+1008 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-189 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}-63 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-21 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-126 \,\operatorname {arcsinh}\left (c x \right )+126 c x \sqrt {c^{2} x^{2}+1}\right )}{3969 c^{4} \sqrt {c^{2} x^{2}+1}}\) | \(226\) |
parts | \(a \left (\frac {x^{2} \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{9 \pi \,c^{2}}-\frac {2 \left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {7}{2}}}{63 \pi \,c^{4}}\right )+\frac {b \,\pi ^{\frac {5}{2}} \left (441 \,\operatorname {arcsinh}\left (c x \right ) c^{10} x^{10}+1638 \,\operatorname {arcsinh}\left (c x \right ) c^{8} x^{8}-49 c^{9} x^{9} \sqrt {c^{2} x^{2}+1}+2142 \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}-171 c^{7} x^{7} \sqrt {c^{2} x^{2}+1}+1008 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-189 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}-63 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}-21 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}-126 \,\operatorname {arcsinh}\left (c x \right )+126 c x \sqrt {c^{2} x^{2}+1}\right )}{3969 c^{4} \sqrt {c^{2} x^{2}+1}}\) | \(226\) |
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Leaf count of result is larger than twice the leaf count of optimal. 263 vs. \(2 (113) = 226\).
Time = 0.30 (sec) , antiderivative size = 263, normalized size of antiderivative = 1.87 \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {63 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (7 \, \pi ^{2} b c^{10} x^{10} + 26 \, \pi ^{2} b c^{8} x^{8} + 34 \, \pi ^{2} b c^{6} x^{6} + 16 \, \pi ^{2} b c^{4} x^{4} - \pi ^{2} b c^{2} x^{2} - 2 \, \pi ^{2} b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (441 \, \pi ^{2} a c^{10} x^{10} + 1638 \, \pi ^{2} a c^{8} x^{8} + 2142 \, \pi ^{2} a c^{6} x^{6} + 1008 \, \pi ^{2} a c^{4} x^{4} - 63 \, \pi ^{2} a c^{2} x^{2} - 126 \, \pi ^{2} a - {\left (49 \, \pi ^{2} b c^{9} x^{9} + 171 \, \pi ^{2} b c^{7} x^{7} + 189 \, \pi ^{2} b c^{5} x^{5} + 21 \, \pi ^{2} b c^{3} x^{3} - 126 \, \pi ^{2} b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )}}{3969 \, {\left (c^{6} x^{2} + c^{4}\right )}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 379 vs. \(2 (133) = 266\).
Time = 90.31 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.69 \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\begin {cases} \frac {\pi ^{\frac {5}{2}} a c^{4} x^{8} \sqrt {c^{2} x^{2} + 1}}{9} + \frac {19 \pi ^{\frac {5}{2}} a c^{2} x^{6} \sqrt {c^{2} x^{2} + 1}}{63} + \frac {5 \pi ^{\frac {5}{2}} a x^{4} \sqrt {c^{2} x^{2} + 1}}{21} + \frac {\pi ^{\frac {5}{2}} a x^{2} \sqrt {c^{2} x^{2} + 1}}{63 c^{2}} - \frac {2 \pi ^{\frac {5}{2}} a \sqrt {c^{2} x^{2} + 1}}{63 c^{4}} - \frac {\pi ^{\frac {5}{2}} b c^{5} x^{9}}{81} + \frac {\pi ^{\frac {5}{2}} b c^{4} x^{8} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9} - \frac {19 \pi ^{\frac {5}{2}} b c^{3} x^{7}}{441} + \frac {19 \pi ^{\frac {5}{2}} b c^{2} x^{6} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{63} - \frac {\pi ^{\frac {5}{2}} b c x^{5}}{21} + \frac {5 \pi ^{\frac {5}{2}} b x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{21} - \frac {\pi ^{\frac {5}{2}} b x^{3}}{189 c} + \frac {\pi ^{\frac {5}{2}} b x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{63 c^{2}} + \frac {2 \pi ^{\frac {5}{2}} b x}{63 c^{3}} - \frac {2 \pi ^{\frac {5}{2}} b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{63 c^{4}} & \text {for}\: c \neq 0 \\\frac {\pi ^{\frac {5}{2}} a x^{4}}{4} & \text {otherwise} \end {cases} \]
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Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.11 \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{63} \, {\left (\frac {7 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} x^{2}}{\pi c^{2}} - \frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}}}{\pi c^{4}}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{63} \, {\left (\frac {7 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}} x^{2}}{\pi c^{2}} - \frac {2 \, {\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {7}{2}}}{\pi c^{4}}\right )} a - \frac {{\left (49 \, \pi ^{\frac {5}{2}} c^{8} x^{9} + 171 \, \pi ^{\frac {5}{2}} c^{6} x^{7} + 189 \, \pi ^{\frac {5}{2}} c^{4} x^{5} + 21 \, \pi ^{\frac {5}{2}} c^{2} x^{3} - 126 \, \pi ^{\frac {5}{2}} x\right )} b}{3969 \, c^{3}} \]
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Exception generated. \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int x^3 \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x)) \, dx=\int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2} \,d x \]
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