Integrand size = 26, antiderivative size = 179 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=-\frac {23}{15} b c \pi ^{5/2} x-\frac {11}{45} b c^3 \pi ^{5/2} x^3-\frac {1}{25} b c^5 \pi ^{5/2} x^5+\pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-2 \pi ^{5/2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-b \pi ^{5/2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+b \pi ^{5/2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {5808, 5806, 5816, 4267, 2317, 2438, 8, 200} \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=-2 \pi ^{5/2} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {1}{5} \left (\pi c^2 x^2+\pi \right )^{5/2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \pi \left (\pi c^2 x^2+\pi \right )^{3/2} (a+b \text {arcsinh}(c x))+\pi ^2 \sqrt {\pi c^2 x^2+\pi } (a+b \text {arcsinh}(c x))-\pi ^{5/2} b \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+\pi ^{5/2} b \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )-\frac {1}{25} \pi ^{5/2} b c^5 x^5-\frac {11}{45} \pi ^{5/2} b c^3 x^3-\frac {23}{15} \pi ^{5/2} b c x \]
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Rule 8
Rule 200
Rule 2317
Rule 2438
Rule 4267
Rule 5806
Rule 5808
Rule 5816
Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\pi \int \frac {\left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx-\frac {1}{5} \left (b c \pi ^{5/2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx \\ & = \frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\pi ^2 \int \frac {\sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))}{x} \, dx-\frac {1}{5} \left (b c \pi ^{5/2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx-\frac {1}{3} \left (b c \pi ^{5/2}\right ) \int \left (1+c^2 x^2\right ) \, dx \\ & = -\frac {8}{15} b c \pi ^{5/2} x-\frac {11}{45} b c^3 \pi ^{5/2} x^3-\frac {1}{25} b c^5 \pi ^{5/2} x^5+\pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\pi ^{5/2} \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx-\left (b c \pi ^{5/2}\right ) \int 1 \, dx \\ & = -\frac {23}{15} b c \pi ^{5/2} x-\frac {11}{45} b c^3 \pi ^{5/2} x^3-\frac {1}{25} b c^5 \pi ^{5/2} x^5+\pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))+\pi ^{5/2} \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x)) \\ & = -\frac {23}{15} b c \pi ^{5/2} x-\frac {11}{45} b c^3 \pi ^{5/2} x^3-\frac {1}{25} b c^5 \pi ^{5/2} x^5+\pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-2 \pi ^{5/2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\left (b \pi ^{5/2}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )+\left (b \pi ^{5/2}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right ) \\ & = -\frac {23}{15} b c \pi ^{5/2} x-\frac {11}{45} b c^3 \pi ^{5/2} x^3-\frac {1}{25} b c^5 \pi ^{5/2} x^5+\pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-2 \pi ^{5/2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-\left (b \pi ^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )+\left (b \pi ^{5/2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right ) \\ & = -\frac {23}{15} b c \pi ^{5/2} x-\frac {11}{45} b c^3 \pi ^{5/2} x^3-\frac {1}{25} b c^5 \pi ^{5/2} x^5+\pi ^2 \sqrt {\pi +c^2 \pi x^2} (a+b \text {arcsinh}(c x))+\frac {1}{3} \pi \left (\pi +c^2 \pi x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{5} \left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))-2 \pi ^{5/2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )-b \pi ^{5/2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )+b \pi ^{5/2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) \\ \end{align*}
Time = 0.45 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.44 \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\frac {1}{225} \pi ^{5/2} \left (-345 b c x-55 b c^3 x^3-9 b c^5 x^5+345 a \sqrt {1+c^2 x^2}+165 a c^2 x^2 \sqrt {1+c^2 x^2}+45 a c^4 x^4 \sqrt {1+c^2 x^2}+345 b \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+165 b c^2 x^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+45 b c^4 x^4 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+225 b \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-225 b \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+225 a \log (x)-225 a \log \left (\pi \left (1+\sqrt {1+c^2 x^2}\right )\right )+225 b \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-225 b \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right ) \]
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Time = 0.15 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.59
method | result | size |
default | \(a \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{5}+\pi \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\pi \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )\right )\right )-b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{5}}{25}-\frac {11 b \,c^{3} \pi ^{\frac {5}{2}} x^{3}}{45}+\frac {23 b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}}}{15}-\frac {23 b c \,\pi ^{\frac {5}{2}} x}{15}+b \,\pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-b \,\pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}} x^{4} c^{4}}{5}+\frac {11 b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}} x^{2} c^{2}}{15}\) | \(284\) |
parts | \(a \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {5}{2}}}{5}+\pi \left (\frac {\left (\pi \,c^{2} x^{2}+\pi \right )^{\frac {3}{2}}}{3}+\pi \left (\sqrt {\pi \,c^{2} x^{2}+\pi }-\sqrt {\pi }\, \operatorname {arctanh}\left (\frac {\sqrt {\pi }}{\sqrt {\pi \,c^{2} x^{2}+\pi }}\right )\right )\right )\right )-b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right )+b \,\pi ^{\frac {5}{2}} \operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right )-\frac {b \,c^{5} \pi ^{\frac {5}{2}} x^{5}}{25}-\frac {11 b \,c^{3} \pi ^{\frac {5}{2}} x^{3}}{45}+\frac {23 b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}}}{15}-\frac {23 b c \,\pi ^{\frac {5}{2}} x}{15}+b \,\pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right )-b \,\pi ^{\frac {5}{2}} \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right )+\frac {b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}} x^{4} c^{4}}{5}+\frac {11 b \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (c x \right ) \pi ^{\frac {5}{2}} x^{2} c^{2}}{15}\) | \(284\) |
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\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\pi ^{\frac {5}{2}} \left (\int \frac {a \sqrt {c^{2} x^{2} + 1}}{x}\, dx + \int 2 a c^{2} x \sqrt {c^{2} x^{2} + 1}\, dx + \int a c^{4} x^{3} \sqrt {c^{2} x^{2} + 1}\, dx + \int \frac {b \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{x}\, dx + \int 2 b c^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx + \int b c^{4} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}\, dx\right ) \]
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\[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int { \frac {{\left (\pi + \pi c^{2} x^{2}\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x} \,d x } \]
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Exception generated. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (\pi +c^2 \pi x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (\Pi \,c^2\,x^2+\Pi \right )}^{5/2}}{x} \,d x \]
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