Integrand size = 16, antiderivative size = 346 \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{7/2}} \, dx=-\frac {2 x^2 \sqrt {1+c^2 x^2}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}-\frac {8 x}{15 b^2 c^2 (a+b \text {arcsinh}(c x))^{3/2}}-\frac {4 x^3}{5 b^2 (a+b \text {arcsinh}(c x))^{3/2}}-\frac {16 \sqrt {1+c^2 x^2}}{15 b^3 c^3 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {24 x^2 \sqrt {1+c^2 x^2}}{5 b^3 c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}-\frac {3 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3}-\frac {e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{15 b^{7/2} c^3}+\frac {3 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{5 b^{7/2} c^3} \]
-8/15*x/b^2/c^2/(a+b*arcsinh(c*x))^(3/2)-4/5*x^3/b^2/(a+b*arcsinh(c*x))^(3 /2)+1/15*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/c ^3-1/15*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/b^(7/2)/c^3/exp(a/ b)-3/5*exp(3*a/b)*erf(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi ^(1/2)/b^(7/2)/c^3+3/5*erfi(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*3^(1 /2)*Pi^(1/2)/b^(7/2)/c^3/exp(3*a/b)-2/5*x^2*(c^2*x^2+1)^(1/2)/b/c/(a+b*arc sinh(c*x))^(5/2)-16/15*(c^2*x^2+1)^(1/2)/b^3/c^3/(a+b*arcsinh(c*x))^(1/2)- 24/5*x^2*(c^2*x^2+1)^(1/2)/b^3/c/(a+b*arcsinh(c*x))^(1/2)
Time = 1.18 (sec) , antiderivative size = 417, normalized size of antiderivative = 1.21 \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{7/2}} \, dx=\frac {3 b^2 e^{\text {arcsinh}(c x)}+e^{-\text {arcsinh}(c x)} \left (4 a^2-2 a b+3 b^2+2 (4 a-b) b \text {arcsinh}(c x)+4 b^2 \text {arcsinh}(c x)^2-4 e^{\frac {a}{b}+\text {arcsinh}(c x)} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x))^2 \Gamma \left (\frac {1}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )\right )-3 \left (b^2 e^{3 \text {arcsinh}(c x)}+2 e^{-\frac {3 a}{b}} (a+b \text {arcsinh}(c x)) \left (e^{3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} (6 a+b+6 b \text {arcsinh}(c x))+6 \sqrt {3} b \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )+2 e^{-\frac {a}{b}} (a+b \text {arcsinh}(c x)) \left (e^{\frac {a}{b}+\text {arcsinh}(c x)} (2 a+b+2 b \text {arcsinh}(c x))+2 b \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {1}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )\right )-3 e^{-3 \text {arcsinh}(c x)} \left (b^2+2 (a+b \text {arcsinh}(c x)) \left (6 a-b+6 b \text {arcsinh}(c x)-6 \sqrt {3} e^{3 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x)) \Gamma \left (\frac {1}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{60 b^3 c^3 (a+b \text {arcsinh}(c x))^{5/2}} \]
(3*b^2*E^ArcSinh[c*x] + (4*a^2 - 2*a*b + 3*b^2 + 2*(4*a - b)*b*ArcSinh[c*x ] + 4*b^2*ArcSinh[c*x]^2 - 4*E^(a/b + ArcSinh[c*x])*Sqrt[a/b + ArcSinh[c*x ]]*(a + b*ArcSinh[c*x])^2*Gamma[1/2, a/b + ArcSinh[c*x]])/E^ArcSinh[c*x] - 3*(b^2*E^(3*ArcSinh[c*x]) + (2*(a + b*ArcSinh[c*x])*(E^(3*(a/b + ArcSinh[ c*x]))*(6*a + b + 6*b*ArcSinh[c*x]) + 6*Sqrt[3]*b*(-((a + b*ArcSinh[c*x])/ b))^(3/2)*Gamma[1/2, (-3*(a + b*ArcSinh[c*x]))/b]))/E^((3*a)/b)) + (2*(a + b*ArcSinh[c*x])*(E^(a/b + ArcSinh[c*x])*(2*a + b + 2*b*ArcSinh[c*x]) + 2* b*(-((a + b*ArcSinh[c*x])/b))^(3/2)*Gamma[1/2, -((a + b*ArcSinh[c*x])/b)]) )/E^(a/b) - (3*(b^2 + 2*(a + b*ArcSinh[c*x])*(6*a - b + 6*b*ArcSinh[c*x] - 6*Sqrt[3]*E^(3*(a/b + ArcSinh[c*x]))*Sqrt[a/b + ArcSinh[c*x]]*(a + b*ArcS inh[c*x])*Gamma[1/2, (3*(a + b*ArcSinh[c*x]))/b])))/E^(3*ArcSinh[c*x]))/(6 0*b^3*c^3*(a + b*ArcSinh[c*x])^(5/2))
Result contains complex when optimal does not.
Time = 2.18 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.38, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {6194, 6233, 6188, 6193, 2009, 6234, 25, 3042, 26, 3789, 2611, 2633, 2634}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 6194 |
| \(\displaystyle \frac {4 \int \frac {x}{\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{5/2}}dx}{5 b c}+\frac {6 c \int \frac {x^3}{\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{5/2}}dx}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6233 |
| \(\displaystyle \frac {6 c \left (\frac {2 \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{3/2}}dx}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}+\frac {4 \left (\frac {2 \int \frac {1}{(a+b \text {arcsinh}(c x))^{3/2}}dx}{3 b c}-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b c}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6188 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (\frac {2 c \int \frac {x}{\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b c}+\frac {6 c \left (\frac {2 \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{3/2}}dx}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6193 |
| \(\displaystyle \frac {6 c \left (\frac {2 \left (\frac {2 \int \left (\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}+\frac {4 \left (\frac {2 \left (\frac {2 c \int \frac {x}{\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b c}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (\frac {2 c \int \frac {x}{\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}dx}{b}-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (\frac {2 \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c}-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (-\frac {2 \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c}-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{3 b c}-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}-\frac {2 \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c}\right )}{3 b c}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{b^2 c}\right )}{3 b c}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 i \left (\frac {1}{2} i \int \frac {e^{-\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i \int \frac {e^{\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))\right )}{b^2 c}\right )}{3 b c}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-i \int e^{\frac {a+b \text {arcsinh}(c x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c x)}\right )}{b^2 c}\right )}{3 b c}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c}\right )}{3 b c}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {4 \left (-\frac {2 x}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}+\frac {2 \left (-\frac {2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}+\frac {2 i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c}\right )}{3 b c}\right )}{5 b c}+\frac {6 c \left (\frac {2 \left (\frac {2 \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{b^2 c^3}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{b c \sqrt {a+b \text {arcsinh}(c x)}}\right )}{b c}-\frac {2 x^3}{3 b c (a+b \text {arcsinh}(c x))^{3/2}}\right )}{5 b}-\frac {2 x^2 \sqrt {c^2 x^2+1}}{5 b c (a+b \text {arcsinh}(c x))^{5/2}}\) |
(-2*x^2*Sqrt[1 + c^2*x^2])/(5*b*c*(a + b*ArcSinh[c*x])^(5/2)) + (4*((-2*x) /(3*b*c*(a + b*ArcSinh[c*x])^(3/2)) + (2*((-2*Sqrt[1 + c^2*x^2])/(b*c*Sqrt [a + b*ArcSinh[c*x]]) + ((2*I)*((I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSi nh[c*x]]/Sqrt[b]])/E^(a/b)))/(b^2*c)))/(3*b*c)))/(5*b*c) + (6*c*((-2*x^3)/ (3*b*c*(a + b*ArcSinh[c*x])^(3/2)) + (2*((-2*x^2*Sqrt[1 + c^2*x^2])/(b*c*S qrt[a + b*ArcSinh[c*x]]) + (2*((Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*Ar cSinh[c*x]]/Sqrt[b]])/8 - (Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi]*Erf[(Sqrt[3]*Sqr t[a + b*ArcSinh[c*x]])/Sqrt[b]])/8 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*Arc Sinh[c*x]]/Sqrt[b]])/(8*E^(a/b)) + (Sqrt[b]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[ a + b*ArcSinh[c*x]])/Sqrt[b]])/(8*E^((3*a)/b))))/(b^2*c^3)))/(b*c)))/(5*b)
3.2.54.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[2/d Subst[Int[F^(g*(e - c*(f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d *x]], x] /; FreeQ[{F, c, d, e, f, g}, x] && !TrueQ[$UseGamma]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{ F, a, b, c, d}, x] && PosQ[b]
Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt [Pi]*(Erf[(c + d*x)*Rt[(-b)*Log[F], 2]]/(2*d*Rt[(-b)*Log[F], 2])), x] /; Fr eeQ[{F, a, b, c, d}, x] && NegQ[b]
Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I /2 Int[(c + d*x)^m/E^(I*(e + f*x)), x], x] - Simp[I/2 Int[(c + d*x)^m*E ^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e, f, m}, x]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[Sqrt[1 + c^ 2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Simp[c/(b*(n + 1) ) Int[x*((a + b*ArcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ [{a, b, c}, x] && LtQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] - Si mp[1/(b^2*c^(m + 1)*(n + 1)) Subst[Int[ExpandTrigReduce[x^(n + 1), Sinh[- a/b + x/b]^(m - 1)*(m + (m + 1)*Sinh[-a/b + x/b]^2), x], x], x, a + b*ArcSi nh[c*x]], x] /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, - 1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^m*Sqrt[1 + c^2*x^2]*((a + b*ArcSinh[c*x])^(n + 1)/(b*c*(n + 1))), x] + (- Simp[c*((m + 1)/(b*(n + 1))) Int[x^(m + 1)*((a + b*ArcSinh[c*x])^(n + 1)/ Sqrt[1 + c^2*x^2]), x], x] - Simp[m/(b*c*(n + 1)) Int[x^(m - 1)*((a + b*A rcSinh[c*x])^(n + 1)/Sqrt[1 + c^2*x^2]), x], x]) /; FreeQ[{a, b, c}, x] && IGtQ[m, 0] && LtQ[n, -2]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((f_.)*(x_))^(m_.))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[((f*x)^m/(b*c*(n + 1)))*Simp[Sqrt[1 + c ^2*x^2]/Sqrt[d + e*x^2]]*(a + b*ArcSinh[c*x])^(n + 1), x] - Simp[f*(m/(b*c* (n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]] Int[(f*x)^(m - 1)*(a + b*ArcSinh[c*x])^(n + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e , c^2*d] && LtQ[n, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_) ^2)^(p_.), x_Symbol] :> Simp[(1/(b*c^(m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2* x^2)^p] Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c^2*d] && IGtQ[2*p + 2, 0] && IGtQ[m, 0]
\[\int \frac {x^{2}}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {7}{2}}}d x\]
Exception generated. \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{7/2}} \, dx=\int \frac {x^{2}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{7/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{7/2}} \, dx=\int { \frac {x^{2}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {x^2}{(a+b \text {arcsinh}(c x))^{7/2}} \, dx=\int \frac {x^2}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{7/2}} \,d x \]