Integrand size = 23, antiderivative size = 228 \[ \int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {3 d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {3 d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {d e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c} \]
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Time = 0.32 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5790, 5819, 5556, 3389, 2211, 2236, 2235} \[ \int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {3 \sqrt {\pi } d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {\sqrt {3 \pi } d e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {3 \sqrt {\pi } d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {\sqrt {3 \pi } d e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {2 d \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5790
Rule 5819
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {(6 c d) \int \frac {x \sqrt {1+c^2 x^2}}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b} \\ & = -\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {(6 d) \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c} \\ & = -\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {(6 d) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{4 \sqrt {x}}+\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{4 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c} \\ & = -\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {(3 d) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b^2 c}-\frac {(3 d) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{2 b^2 c} \\ & = -\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {(3 d) \text {Subst}\left (\int \frac {e^{-i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c}+\frac {(3 d) \text {Subst}\left (\int \frac {e^{i \left (\frac {i a}{b}-\frac {i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c}-\frac {(3 d) \text {Subst}\left (\int \frac {e^{-i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c}+\frac {(3 d) \text {Subst}\left (\int \frac {e^{i \left (\frac {3 i a}{b}-\frac {3 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{4 b^2 c} \\ & = -\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {(3 d) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b^2 c}-\frac {(3 d) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b^2 c}+\frac {(3 d) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b^2 c}+\frac {(3 d) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 b^2 c} \\ & = -\frac {2 d \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {3 d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {3 d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {d e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 295, normalized size of antiderivative = 1.29 \[ \int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {d e^{-3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \left (-e^{\frac {3 a}{b}}-3 e^{\frac {3 a}{b}+2 \text {arcsinh}(c x)}-3 e^{\frac {3 a}{b}+4 \text {arcsinh}(c x)}-e^{\frac {3 a}{b}+6 \text {arcsinh}(c x)}+3 e^{\frac {4 a}{b}+3 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} e^{3 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+3 e^{\frac {2 a}{b}+3 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}+3 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{4 b c \sqrt {a+b \text {arcsinh}(c x)}} \]
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\[\int \frac {c^{2} d \,x^{2}+d}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d \left (\int \frac {c^{2} x^{2}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {1}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \]
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\[ \int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {c^{2} d x^{2} + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {c^{2} d x^{2} + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {d+c^2 d x^2}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {d\,c^2\,x^2+d}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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