Integrand size = 26, antiderivative size = 335 \[ \int \frac {x^2 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=-\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}-\frac {d e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {d e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}+\frac {d e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {d e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3} \]
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Time = 0.88 (sec) , antiderivative size = 335, normalized size of antiderivative = 1.00, number of steps used = 32, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5814, 5819, 5556, 3389, 2211, 2236, 2235} \[ \int \frac {x^2 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {\sqrt {\pi } d e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}-\frac {\sqrt {3 \pi } d e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {\sqrt {5 \pi } d e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {\sqrt {\pi } d e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}+\frac {\sqrt {3 \pi } d e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}+\frac {\sqrt {5 \pi } d e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {2 d x^2 \left (c^2 x^2+1\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}} \]
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Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5556
Rule 5814
Rule 5819
Rubi steps \begin{align*} \text {integral}& = -\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {(4 d) \int \frac {x \sqrt {1+c^2 x^2}}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b c}+\frac {(10 c d) \int \frac {x^3 \sqrt {1+c^2 x^2}}{\sqrt {a+b \text {arcsinh}(c x)}} \, dx}{b} \\ & = -\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {(4 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^3}-\frac {(10 d) \text {Subst}\left (\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {x}{b}\right ) \sinh ^3\left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^3} \\ & = -\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {(4 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^3}-\frac {(10 d) \text {Subst}\left (\int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{16 \sqrt {x}}-\frac {\sinh \left (\frac {3 a}{b}-\frac {3 x}{b}\right )}{16 \sqrt {x}}-\frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{8 \sqrt {x}}\right ) \, dx,x,a+b \text {arcsinh}(c x)\right )}{b^2 c^3} \\ & = -\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {(5 d) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}-\frac {5 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{8 b^2 c^3}+\frac {(5 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 )}{8 b^2 c^3}-\frac {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 )}{b^2 c^3}-\frac {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 )}{b^2 c^3}+\frac {(5 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 )}{4 b^2 c^3} \\ & = -\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {(5 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 )}{16 b^2 c^3}-\frac {(5 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 )}{16 b^2 c^3}-\frac {(5 d) \text {Subst}\left (\int \frac {e^{-i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3}+\frac {(5 d) \text {Subst}\left (\int \frac {e^{i \left (\frac {5 i a}{b}-\frac {5 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c x)\right )}{16 b^2 c^3}-\frac {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 )}{2 b^2 c^3}+\frac {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 )}{2 b^2 c^3}-\frac {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 )}{2 b^2 c^3}+\frac {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 )}{2 b^2 c^3}+\frac {(5 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 )}{8 b^2 c^3}-\frac {(5 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 )}{8 b^2 c^3} \\ & = -\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {(5 d) \text {Subst}\left (\int e^{\frac {5 a}{b}-\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{8 b^2 c^3}+\frac {(5 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 )}{8 b^2 c^3}-\frac {(5 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 )}{8 b^2 c^3}+\frac {(5 d) \text {Subst}\left (\int e^{-\frac {5 a}{b}+\frac {5 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{8 b^2 c^3}-\frac {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 )}{b^2 c^3}-\frac {d \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^3}+\frac {d \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c^3}+\frac {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 )}{b^2 c^3}+\frac {(5 d) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 b^2 c^3}-\frac {(5 d) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c x)}\right )}{4 b^2 c^3} \\ & = -\frac {2 d x^2 \left (1+c^2 x^2\right )^{3/2}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {d e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}-\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 )}{16 b^{3/2} c^3}-\frac {d e^{\frac {5 a}{b}} \sqrt {5 \pi } \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3}-\frac {d e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{8 b^{3/2} c^3}+\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 )}{16 b^{3/2} c^3}+\frac {d e^{-\frac {5 a}{b}} \sqrt {5 \pi } \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{16 b^{3/2} c^3} \\ \end{align*}
Time = 0.68 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.30 \[ \int \frac {x^2 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\frac {d e^{-5 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \left (-e^{\frac {5 a}{b}}-e^{\frac {5 a}{b}+2 \text {arcsinh}(c x)}+2 e^{\frac {5 a}{b}+4 \text {arcsinh}(c x)}+2 e^{\frac {5 a}{b}+6 \text {arcsinh}(c x)}-e^{\frac {5 a}{b}+8 \text {arcsinh}(c x)}-e^{\frac {5 a}{b}+10 \text {arcsinh}(c x)}-2 e^{\frac {6 a}{b}+5 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {5} e^{5 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )+\sqrt {3} e^{\frac {2 a}{b}+5 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-2 e^{\frac {4 a}{b}+5 \text {arcsinh}(c x)} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+\sqrt {3} e^{\frac {8 a}{b}+5 \text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+\sqrt {5} e^{5 \left (\frac {2 a}{b}+\text {arcsinh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {1}{2},\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{16 b c^3 \sqrt {a+b \text {arcsinh}(c x)}} \]
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\[\int \frac {x^{2} \left (c^{2} d \,x^{2}+d \right )}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {x^2 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x^2 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=d \left (\int \frac {x^{2}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx + \int \frac {c^{2} x^{4}}{a \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asinh}{\left (c x \right )}} \operatorname {asinh}{\left (c x \right )}}\, dx\right ) \]
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\[ \int \frac {x^2 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )} x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2 \left (d+c^2 d x^2\right )}{(a+b \text {arcsinh}(c x))^{3/2}} \, dx=\int \frac {x^2\,\left (d\,c^2\,x^2+d\right )}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2}} \,d x \]
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