Integrand size = 16, antiderivative size = 365 \[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}+\frac {2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2} \]
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Time = 0.65 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 22, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {5859, 5829, 5773, 5818, 5774, 3388, 2211, 2236, 2235, 5779, 5780, 5556, 12, 3389, 5783} \[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=-\frac {2 \sqrt {\pi } c e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 \sqrt {2 \pi } e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 \sqrt {\pi } c e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}+\frac {2 \sqrt {2 \pi } e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 \sqrt {(c+d x)^2+1} (c+d x)}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 c \sqrt {(c+d x)^2+1}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3388
Rule 3389
Rule 5556
Rule 5773
Rule 5774
Rule 5779
Rule 5780
Rule 5783
Rule 5818
Rule 5829
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {-\frac {c}{d}+\frac {x}{d}}{(a+b \text {arcsinh}(x))^{5/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {c}{d (a+b \text {arcsinh}(x))^{5/2}}+\frac {x}{d (a+b \text {arcsinh}(x))^{5/2}}\right ) \, dx,x,c+d x\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {x}{(a+b \text {arcsinh}(x))^{5/2}} \, dx,x,c+d x\right )}{d^2}-\frac {c \text {Subst}\left (\int \frac {1}{(a+b \text {arcsinh}(x))^{5/2}} \, dx,x,c+d x\right )}{d^2} \\ & = \frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{3 b d^2}+\frac {4 \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{3 b d^2}-\frac {(2 c) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{3 b d^2} \\ & = \frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {16 \text {Subst}\left (\int \frac {x}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{3 b^2 d^2}-\frac {(4 c) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{3 b^2 d^2} \\ & = \frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 \text {Subst}\left (\int \frac {\cosh \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+d x)\right )}{3 b^3 d^2}-\frac {(4 c) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^3 d^2} \\ & = \frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{2 \sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^3 d^2}-\frac {(2 c) \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+d x)\right )}{3 b^3 d^2}-\frac {(2 c) \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+d x)\right )}{3 b^3 d^2} \\ & = \frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {8 \text {Subst}\left (\int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^3 d^2}-\frac {(4 c) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{3 b^3 d^2}-\frac {(4 c) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{3 b^3 d^2} \\ & = \frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {4 \text {Subst}\left (\int \frac {e^{-i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^3 d^2}+\frac {4 \text {Subst}\left (\int \frac {e^{i \left (\frac {2 i a}{b}-\frac {2 i x}{b}\right )}}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{3 b^3 d^2} \\ & = \frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {8 \text {Subst}\left (\int e^{\frac {2 a}{b}-\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{3 b^3 d^2}+\frac {8 \text {Subst}\left (\int e^{-\frac {2 a}{b}+\frac {2 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{3 b^3 d^2} \\ & = \frac {2 c \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 (c+d x) \sqrt {1+(c+d x)^2}}{3 b d^2 (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {4 c (c+d x)}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {8 (c+d x)^2}{3 b^2 d^2 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {2 c e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 e^{\frac {2 a}{b}} \sqrt {2 \pi } \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}-\frac {2 c e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2}+\frac {2 e^{-\frac {2 a}{b}} \sqrt {2 \pi } \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{3 b^{5/2} d^2} \\ \end{align*}
Time = 3.61 (sec) , antiderivative size = 555, normalized size of antiderivative = 1.52 \[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=-\frac {-4 a \sqrt {b} c (c+d x)-2 b^{3/2} c \sqrt {1+(c+d x)^2}-4 b^{3/2} c (c+d x) \text {arcsinh}(c+d x)+4 a \sqrt {b} \cosh (2 \text {arcsinh}(c+d x))+4 b^{3/2} \text {arcsinh}(c+d x) \cosh (2 \text {arcsinh}(c+d x))+2 c \sqrt {\pi } (a+b \text {arcsinh}(c+d x))^{3/2} \cosh \left (\frac {a}{b}\right ) \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+2 \sqrt {2 \pi } (a+b \text {arcsinh}(c+d x))^{3/2} \cosh \left (\frac {2 a}{b}\right ) \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+2 c \sqrt {\pi } (a+b \text {arcsinh}(c+d x))^{3/2} \cosh \left (\frac {a}{b}\right ) \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-2 \sqrt {2 \pi } (a+b \text {arcsinh}(c+d x))^{3/2} \cosh \left (\frac {2 a}{b}\right ) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+2 c \sqrt {\pi } (a+b \text {arcsinh}(c+d x))^{3/2} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )-2 c \sqrt {\pi } (a+b \text {arcsinh}(c+d x))^{3/2} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {a}{b}\right )+2 \sqrt {2 \pi } (a+b \text {arcsinh}(c+d x))^{3/2} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )+2 \sqrt {2 \pi } (a+b \text {arcsinh}(c+d x))^{3/2} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right ) \sinh \left (\frac {2 a}{b}\right )+b^{3/2} \sinh (2 \text {arcsinh}(c+d x))}{3 b^{5/2} d^2 (a+b \text {arcsinh}(c+d x))^{3/2}} \]
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\[\int \frac {x}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int \frac {x}{\left (a + b \operatorname {asinh}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int { \frac {x}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int { \frac {x}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x}{(a+b \text {arcsinh}(c+d x))^{5/2}} \, dx=\int \frac {x}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2}} \,d x \]
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