Integrand size = 26, antiderivative size = 149 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=-\frac {8 b x}{15 c^5 \sqrt {\pi }}+\frac {4 b x^3}{45 c^3 \sqrt {\pi }}-\frac {b x^5}{25 c \sqrt {\pi }}+\frac {8 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{15 c^6 \pi }-\frac {4 x^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{15 c^4 \pi }+\frac {x^4 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{5 c^2 \pi } \]
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Time = 0.19 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {5812, 5798, 8, 30} \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {x^4 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{5 \pi c^2}+\frac {8 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{15 \pi c^6}-\frac {4 x^2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))}{15 \pi c^4}-\frac {8 b x}{15 \sqrt {\pi } c^5}+\frac {4 b x^3}{45 \sqrt {\pi } c^3}-\frac {b x^5}{25 \sqrt {\pi } c} \]
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Rule 8
Rule 30
Rule 5798
Rule 5812
Rubi steps \begin{align*} \text {integral}& = \frac {x^4 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{5 c^2 \pi }-\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{5 c^2}-\frac {b \int x^4 \, dx}{5 c \sqrt {\pi }} \\ & = -\frac {b x^5}{25 c \sqrt {\pi }}-\frac {4 x^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{15 c^4 \pi }+\frac {x^4 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{5 c^2 \pi }+\frac {8 \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx}{15 c^4}+\frac {(4 b) \int x^2 \, dx}{15 c^3 \sqrt {\pi }} \\ & = \frac {4 b x^3}{45 c^3 \sqrt {\pi }}-\frac {b x^5}{25 c \sqrt {\pi }}+\frac {8 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{15 c^6 \pi }-\frac {4 x^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{15 c^4 \pi }+\frac {x^4 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{5 c^2 \pi }-\frac {(8 b) \int 1 \, dx}{15 c^5 \sqrt {\pi }} \\ & = -\frac {8 b x}{15 c^5 \sqrt {\pi }}+\frac {4 b x^3}{45 c^3 \sqrt {\pi }}-\frac {b x^5}{25 c \sqrt {\pi }}+\frac {8 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{15 c^6 \pi }-\frac {4 x^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{15 c^4 \pi }+\frac {x^4 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{5 c^2 \pi } \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.72 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {15 a \sqrt {1+c^2 x^2} \left (8-4 c^2 x^2+3 c^4 x^4\right )+b \left (-120 c x+20 c^3 x^3-9 c^5 x^5\right )+15 b \sqrt {1+c^2 x^2} \left (8-4 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)}{225 c^6 \sqrt {\pi }} \]
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Time = 0.20 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.30
method | result | size |
default | \(a \left (\frac {x^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}{5 \pi \,c^{2}}-\frac {4 \left (\frac {x^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{2}}-\frac {2 \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{4}}\right )}{5 c^{2}}\right )+\frac {b \left (45 \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}-15 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-9 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+60 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+120 \,\operatorname {arcsinh}\left (c x \right )-120 c x \sqrt {c^{2} x^{2}+1}\right )}{225 c^{6} \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}}\) | \(193\) |
parts | \(a \left (\frac {x^{4} \sqrt {\pi \,c^{2} x^{2}+\pi }}{5 \pi \,c^{2}}-\frac {4 \left (\frac {x^{2} \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{2}}-\frac {2 \sqrt {\pi \,c^{2} x^{2}+\pi }}{3 \pi \,c^{4}}\right )}{5 c^{2}}\right )+\frac {b \left (45 \,\operatorname {arcsinh}\left (c x \right ) c^{6} x^{6}-15 \,\operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}-9 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}+60 \,\operatorname {arcsinh}\left (c x \right ) c^{2} x^{2}+20 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+120 \,\operatorname {arcsinh}\left (c x \right )-120 c x \sqrt {c^{2} x^{2}+1}\right )}{225 c^{6} \sqrt {\pi }\, \sqrt {c^{2} x^{2}+1}}\) | \(193\) |
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Time = 0.28 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.08 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {15 \, \sqrt {\pi + \pi c^{2} x^{2}} {\left (3 \, b c^{6} x^{6} - b c^{4} x^{4} + 4 \, b c^{2} x^{2} + 8 \, b\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + \sqrt {\pi + \pi c^{2} x^{2}} {\left (45 \, a c^{6} x^{6} - 15 \, a c^{4} x^{4} + 60 \, a c^{2} x^{2} - {\left (9 \, b c^{5} x^{5} - 20 \, b c^{3} x^{3} + 120 \, b c x\right )} \sqrt {c^{2} x^{2} + 1} + 120 \, a\right )}}{225 \, {\left (\pi c^{8} x^{2} + \pi c^{6}\right )}} \]
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Time = 3.15 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.23 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {a \left (\begin {cases} \frac {x^{4} \sqrt {c^{2} x^{2} + 1}}{5 c^{2}} - \frac {4 x^{2} \sqrt {c^{2} x^{2} + 1}}{15 c^{4}} + \frac {8 \sqrt {c^{2} x^{2} + 1}}{15 c^{6}} & \text {for}\: c^{2} \neq 0 \\\frac {x^{6}}{6} & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} + \frac {b \left (\begin {cases} - \frac {x^{5}}{25 c} + \frac {x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{5 c^{2}} + \frac {4 x^{3}}{45 c^{3}} - \frac {4 x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{15 c^{4}} - \frac {8 x}{15 c^{5}} + \frac {8 \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{15 c^{6}} & \text {for}\: c \neq 0 \\0 & \text {otherwise} \end {cases}\right )}{\sqrt {\pi }} \]
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Time = 0.20 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.17 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\frac {1}{15} \, {\left (\frac {3 \, \sqrt {\pi + \pi c^{2} x^{2}} x^{4}}{\pi c^{2}} - \frac {4 \, \sqrt {\pi + \pi c^{2} x^{2}} x^{2}}{\pi c^{4}} + \frac {8 \, \sqrt {\pi + \pi c^{2} x^{2}}}{\pi c^{6}}\right )} b \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, {\left (\frac {3 \, \sqrt {\pi + \pi c^{2} x^{2}} x^{4}}{\pi c^{2}} - \frac {4 \, \sqrt {\pi + \pi c^{2} x^{2}} x^{2}}{\pi c^{4}} + \frac {8 \, \sqrt {\pi + \pi c^{2} x^{2}}}{\pi c^{6}}\right )} a - \frac {{\left (9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x\right )} b}{225 \, \sqrt {\pi } c^{5}} \]
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Exception generated. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))}{\sqrt {\pi +c^2 \pi x^2}} \, dx=\int \frac {x^5\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}{\sqrt {\Pi \,c^2\,x^2+\Pi }} \,d x \]
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