Integrand size = 25, antiderivative size = 275 \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {x \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}+\frac {25 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^2}-\frac {27 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {25 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2} \]
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Time = 0.57 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.00, number of steps used = 28, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {5814, 5791, 3393, 3384, 3379, 3382, 5819, 5556} \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}+\frac {25 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^2}-\frac {27 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {25 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {x \left (c^2 x^2+1\right )^3}{b c (a+b \text {arcsinh}(c x))} \]
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Rule 3379
Rule 3382
Rule 3384
Rule 3393
Rule 5556
Rule 5791
Rule 5814
Rule 5819
Rubi steps \begin{align*} \text {integral}& = -\frac {x \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {\int \frac {\left (1+c^2 x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx}{b c}+\frac {(7 c) \int \frac {x^2 \left (1+c^2 x^2\right )^2}{a+b \text {arcsinh}(c x)} \, dx}{b} \\ & = -\frac {x \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {\text {Subst}\left (\int \frac {\cosh ^5\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2}+\frac {7 \text {Subst}\left (\int \frac {\cosh ^5\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2} \\ & = -\frac {x \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {\text {Subst}\left (\int \left (\frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 x}+\frac {5 \cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 x}+\frac {5 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2}+\frac {7 \text {Subst}\left (\int \left (\frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{64 x}+\frac {3 \cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{64 x}+\frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{64 x}-\frac {5 \cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{64 x}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^2} \\ & = -\frac {x \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {\text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^2}+\frac {7 \text {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}-\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}+\frac {7 \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}+\frac {5 \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^2}+\frac {21 \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}-\frac {35 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}+\frac {5 \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^2} \\ & = -\frac {x \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}-\frac {\left (35 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}+\frac {\left (5 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^2}+\frac {\left (7 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}+\frac {\left (5 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^2}+\frac {\cosh \left (\frac {5 a}{b}\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^2}+\frac {\left (21 \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}+\frac {\left (7 \cosh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}+\frac {\left (35 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}-\frac {\left (5 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^2}-\frac {\left (7 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}-\frac {\left (5 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^2}-\frac {\sinh \left (\frac {5 a}{b}\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^2}-\frac {\left (21 \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2}-\frac {\left (7 \sinh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {7 x}{b}\right )}{x} \, dx,x,a+b \text {arcsinh}(c x)\right )}{64 b^2 c^2} \\ & = -\frac {x \left (1+c^2 x^2\right )^3}{b c (a+b \text {arcsinh}(c x))}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^2}+\frac {27 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}+\frac {25 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}+\frac {7 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{64 b^2 c^2}-\frac {27 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {25 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2}-\frac {7 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 (a+b \text {arcsinh}(c x))}{b}\right )}{64 b^2 c^2} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.47 \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {64 b c x+192 b c^3 x^3+192 b c^5 x^5+64 b c^7 x^7-5 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )-27 (a+b \text {arcsinh}(c x)) \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-25 a \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-25 b \text {arcsinh}(c x) \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-7 a \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-7 b \text {arcsinh}(c x) \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+5 a \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+5 b \text {arcsinh}(c x) \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+27 a \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+27 b \text {arcsinh}(c x) \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+25 a \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+25 b \text {arcsinh}(c x) \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+7 a \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+7 b \text {arcsinh}(c x) \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{64 b^2 c^2 (a+b \text {arcsinh}(c x))} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(957\) vs. \(2(259)=518\).
Time = 0.28 (sec) , antiderivative size = 958, normalized size of antiderivative = 3.48
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\[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \]
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\[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} x}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]
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Exception generated. \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\text {Exception raised: RuntimeError} \]
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Timed out. \[ \int \frac {x \left (1+c^2 x^2\right )^{5/2}}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {x\,{\left (c^2\,x^2+1\right )}^{5/2}}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \]
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