Integrand size = 26, antiderivative size = 355 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=-\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {7 b c^3 d^2 x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}} \]
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Time = 0.33 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {5807, 5808, 5806, 5816, 4267, 2317, 2438, 8, 276} \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=-\frac {5 c^2 d^2 \sqrt {c^2 d x^2+d} \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}+\frac {5}{2} c^2 d^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 d \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {5 b c^2 d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {c^2 x^2+1}}+\frac {5 b c^2 d^2 \sqrt {c^2 d x^2+d} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {c^2 x^2+1}}-\frac {b c d^2 \sqrt {c^2 d x^2+d}}{2 x \sqrt {c^2 x^2+1}}-\frac {b c^5 d^2 x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}}-\frac {7 b c^3 d^2 x \sqrt {c^2 d x^2+d}}{3 \sqrt {c^2 x^2+1}} \]
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Rule 8
Rule 276
Rule 2317
Rule 2438
Rule 4267
Rule 5806
Rule 5807
Rule 5808
Rule 5816
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} \left (5 c^2 d\right ) \int \frac {\left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{x} \, dx+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {\left (1+c^2 x^2\right )^2}{x^2} \, dx}{2 \sqrt {1+c^2 x^2}} \\ & = \frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {1}{2} \left (5 c^2 d^2\right ) \int \frac {\sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x} \, dx+\frac {\left (b c d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (2 c^2+\frac {1}{x^2}+c^4 x^2\right ) \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{6 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2}}{6 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{x \sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (5 b c^3 d^2 \sqrt {d+c^2 d x^2}\right ) \int 1 \, dx}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {7 b c^3 d^2 x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}+\frac {\left (5 c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}(\int (a+b x) \text {csch}(x) \, dx,x,\text {arcsinh}(c x))}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {7 b c^3 d^2 x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1-e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \log \left (1+e^x\right ) \, dx,x,\text {arcsinh}(c x)\right )}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {7 b c^3 d^2 x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {\left (5 b c^2 d^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}} \\ & = -\frac {b c d^2 \sqrt {d+c^2 d x^2}}{2 x \sqrt {1+c^2 x^2}}-\frac {7 b c^3 d^2 x \sqrt {d+c^2 d x^2}}{3 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}}+\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{2 \sqrt {1+c^2 x^2}} \\ \end{align*}
Time = 6.41 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.19 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\frac {1}{72} d^2 \left (\frac {12 a \sqrt {d+c^2 d x^2} \left (-3+14 c^2 x^2+2 c^4 x^4\right )}{x^2}-\frac {8 b c^2 \sqrt {d+c^2 d x^2} \left (3 c x+c^3 x^3-3 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )}{\sqrt {1+c^2 x^2}}+180 a c^2 \sqrt {d} \log (x)-180 a c^2 \sqrt {d} \log \left (d+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+\frac {144 b c^2 \sqrt {d+c^2 d x^2} \left (-c x+\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+\text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+\operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )\right )}{\sqrt {1+c^2 x^2}}+\frac {9 b c^2 \sqrt {d+c^2 d x^2} \left (-2 \coth \left (\frac {1}{2} \text {arcsinh}(c x)\right )-\text {arcsinh}(c x) \text {csch}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+4 \text {arcsinh}(c x) \log \left (1-e^{-\text {arcsinh}(c x)}\right )-4 \text {arcsinh}(c x) \log \left (1+e^{-\text {arcsinh}(c x)}\right )+4 \operatorname {PolyLog}\left (2,-e^{-\text {arcsinh}(c x)}\right )-4 \operatorname {PolyLog}\left (2,e^{-\text {arcsinh}(c x)}\right )-\text {arcsinh}(c x) \text {sech}^2\left (\frac {1}{2} \text {arcsinh}(c x)\right )+2 \tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )}{\sqrt {1+c^2 x^2}}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.95
method | result | size |
default | \(a \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}+\frac {5 c^{2} \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5}+d \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )\right )\right )}{2}\right )+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-2 c^{5} x^{5}+42 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+45 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-45 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-42 c^{3} x^{3}+45 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-45 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-9 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-9 c x \right ) d^{2}}{18 \sqrt {c^{2} x^{2}+1}\, x^{2}}\) | \(339\) |
parts | \(a \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}+\frac {5 c^{2} \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5}+d \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )\right )\right )}{2}\right )+\frac {b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}-2 c^{5} x^{5}+42 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+45 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1-c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-45 \,\operatorname {arcsinh}\left (c x \right ) \ln \left (1+c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-42 c^{3} x^{3}+45 \operatorname {polylog}\left (2, c x +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-45 \operatorname {polylog}\left (2, -c x -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-9 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-9 c x \right ) d^{2}}{18 \sqrt {c^{2} x^{2}+1}\, x^{2}}\) | \(339\) |
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\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{x^{3}}\, dx \]
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\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{x^3} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^3} \,d x \]
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