Integrand size = 25, antiderivative size = 410 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=-\frac {2 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {16 e^2 \sqrt {1+(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {24 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {3 e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d} \]
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Time = 0.77 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.00, number of steps used = 24, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {5859, 12, 5779, 5818, 5778, 3389, 2211, 2236, 2235, 5773, 5819} \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\frac {\sqrt {\pi } e^2 e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {3 \sqrt {3 \pi } e^2 e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {\sqrt {\pi } e^2 e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 \sqrt {3 \pi } e^2 e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {24 e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {16 e^2 \sqrt {(c+d x)^2+1}}{15 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {2 e^2 \sqrt {(c+d x)^2+1} (c+d x)^2}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}} \]
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Rule 12
Rule 2211
Rule 2235
Rule 2236
Rule 3389
Rule 5773
Rule 5778
Rule 5779
Rule 5818
Rule 5819
Rule 5859
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^2 x^2}{(a+b \text {arcsinh}(x))^{7/2}} \, dx,x,c+d x\right )}{d} \\ & = \frac {e^2 \text {Subst}\left (\int \frac {x^2}{(a+b \text {arcsinh}(x))^{7/2}} \, dx,x,c+d x\right )}{d} \\ & = -\frac {2 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}+\frac {\left (4 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^{5/2}} \, dx,x,c+d x\right )}{5 b d}+\frac {\left (6 e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {1+x^2} (a+b \text {arcsinh}(x))^{5/2}} \, dx,x,c+d x\right )}{5 b d} \\ & = -\frac {2 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}+\frac {\left (8 e^2\right ) \text {Subst}\left (\int \frac {1}{(a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{15 b^2 d}+\frac {\left (12 e^2\right ) \text {Subst}\left (\int \frac {x^2}{(a+b \text {arcsinh}(x))^{3/2}} \, dx,x,c+d x\right )}{5 b^2 d} \\ & = -\frac {2 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {16 e^2 \sqrt {1+(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {24 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {\left (24 e^2\right ) \text {Subst}\left (\int \left (-\frac {3 \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+d x)\right )}{5 b^4 d}+\frac {\left (16 e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2} \sqrt {a+b \text {arcsinh}(x)}} \, dx,x,c+d x\right )}{15 b^3 d} \\ & = -\frac {2 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {16 e^2 \sqrt {1+(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {24 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\left (16 e^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{15 b^4 d}+\frac {\left (6 e^2\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}-\frac {x}{b}\right )}{\sqrt {x}} \, dx,x,a+b \text {arcsinh}(c+d x)\right )}{5 b^4 d}-\frac {\left (18 e^2\right ) \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+d x)\right )}{5 b^4 d} \\ & = -\frac {2 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {16 e^2 \sqrt {1+(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {24 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\left (8 e^2\right ) \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 )}{15 b^4 d}+\frac {\left (8 e^2\right ) \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 )}{15 b^4 d}+\frac {\left (3 e^2\right ) \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 )}{5 b^4 d}-\frac {\left (3 e^2\right ) \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 )}{5 b^4 d}-\frac {\left (9 e^2\right ) \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+d x)\right )}{5 b^4 d}+\frac {\left (9 e^2\right ) \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+d x)\right )}{5 b^4 d} \\ & = -\frac {2 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {16 e^2 \sqrt {1+(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {24 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\left (16 e^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{15 b^4 d}+\frac {\left (16 e^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{15 b^4 d}+\frac {\left (6 e^2\right ) \text {Subst}\left (\int e^{\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{5 b^4 d}-\frac {\left (6 e^2\right ) \text {Subst}\left (\int e^{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{5 b^4 d}-\frac {\left (18 e^2\right ) \text {Subst}\left (\int e^{\frac {3 a}{b}-\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{5 b^4 d}+\frac {\left (18 e^2\right ) \text {Subst}\left (\int e^{-\frac {3 a}{b}+\frac {3 x^2}{b}} \, dx,x,\sqrt {a+b \text {arcsinh}(c+d x)}\right )}{5 b^4 d} \\ & = -\frac {2 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b d (a+b \text {arcsinh}(c+d x))^{5/2}}-\frac {8 e^2 (c+d x)}{15 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {4 e^2 (c+d x)^3}{5 b^2 d (a+b \text {arcsinh}(c+d x))^{3/2}}-\frac {16 e^2 \sqrt {1+(c+d x)^2}}{15 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {24 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2}}{5 b^3 d \sqrt {a+b \text {arcsinh}(c+d x)}}+\frac {e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}-\frac {3 e^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d}-\frac {e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{15 b^{7/2} d}+\frac {3 e^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{5 b^{7/2} d} \\ \end{align*}
Time = 1.20 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.16 \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\frac {e^2 \left (3 b^2 e^{\text {arcsinh}(c+d x)}+e^{-\text {arcsinh}(c+d x)} \left (4 a^2-2 a b+3 b^2+2 (4 a-b) b \text {arcsinh}(c+d x)+4 b^2 \text {arcsinh}(c+d x)^2-4 e^{\frac {a}{b}+\text {arcsinh}(c+d x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x))^2 \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )\right )-3 \left (b^2 e^{3 \text {arcsinh}(c+d x)}+2 e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c+d x)) \left (e^{3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} (6 a+b+6 b \text {arcsinh}(c+d x))+6 \sqrt {3} b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )+2 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c+d x)) \left (e^{\frac {a}{b}+\text {arcsinh}(c+d x)} (2 a+b+2 b \text {arcsinh}(c+d x))+2 b \left (-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )\right )-3 e^{-3 \text {arcsinh}(c+d x)} \left (b^2+2 (a+b \text {arcsinh}(c+d x)) \left (6 a-b+6 b \text {arcsinh}(c+d x)-6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arcsinh}(c+d x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} (a+b \text {arcsinh}(c+d x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )\right )\right )}{60 b^3 d (a+b \text {arcsinh}(c+d x))^{5/2}} \]
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\[\int \frac {\left (d e x +c e \right )^{2}}{\left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {7}{2}}}d x\]
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Exception generated. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=e^{2} \left (\int \frac {c^{2}}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {d^{2} x^{2}}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx + \int \frac {2 c d x}{a^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} + 3 a^{2} b \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )} + 3 a b^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )} + b^{3} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{3}{\left (c + d x \right )}}\, dx\right ) \]
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\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int { \frac {{\left (d e x + c e\right )}^{2}}{{\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(c e+d e x)^2}{(a+b \text {arcsinh}(c+d x))^{7/2}} \, dx=\int \frac {{\left (c\,e+d\,e\,x\right )}^2}{{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{7/2}} \,d x \]
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