Integrand size = 16, antiderivative size = 327 \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=-\frac {5 b^2 x \sqrt {a+b \text {arcsinh}(c x)}}{6 c^2}+\frac {5}{36} b^2 x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{9 c^3}-\frac {5 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{3/2}}{18 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {15 b^{5/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^3}+\frac {5 b^{5/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{576 c^3}+\frac {15 b^{5/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{64 c^3}-\frac {5 b^{5/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{576 c^3} \]
1/3*x^3*(a+b*arcsinh(c*x))^(5/2)+5/1728*b^(5/2)*exp(3*a/b)*erf(3^(1/2)*(a+ b*arcsinh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^3-5/1728*b^(5/2)*erfi(3^ (1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/c^3/exp(3*a/b)-15 /64*b^(5/2)*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3+15 /64*b^(5/2)*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3/exp(a/b)+5 /9*b*(a+b*arcsinh(c*x))^(3/2)*(c^2*x^2+1)^(1/2)/c^3-5/18*b*x^2*(a+b*arcsin h(c*x))^(3/2)*(c^2*x^2+1)^(1/2)/c-5/6*b^2*x*(a+b*arcsinh(c*x))^(1/2)/c^2+5 /36*b^2*x^3*(a+b*arcsinh(c*x))^(1/2)
Time = 0.23 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.61 \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=-\frac {b^3 e^{-\frac {3 a}{b}} \left (-81 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {7}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {7}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-81 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {7}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {7}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{648 c^3 \sqrt {a+b \text {arcsinh}(c x)}} \]
-1/648*(b^3*(-81*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma[7/2, a/b + Arc Sinh[c*x]] + Sqrt[3]*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[7/2, (-3*(a + b *ArcSinh[c*x]))/b] - 81*E^((2*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Gamma[ 7/2, -((a + b*ArcSinh[c*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[a/b + ArcSinh[c *x]]*Gamma[7/2, (3*(a + b*ArcSinh[c*x]))/b]))/(c^3*E^((3*a)/b)*Sqrt[a + b* ArcSinh[c*x]])
Result contains complex when optimal does not.
Time = 2.46 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.36, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.938, Rules used = {6192, 6227, 6192, 6213, 6187, 6234, 25, 3042, 26, 3789, 2611, 2633, 2634, 3793, 2009}
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.
| \(\displaystyle \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \int \frac {x^3 (a+b \text {arcsinh}(c x))^{3/2}}{\sqrt {c^2 x^2+1}}dx\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^{3/2}}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {b \int x^2 \sqrt {a+b \text {arcsinh}(c x)}dx}{2 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^{3/2}}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{6} b c \int \frac {x^3}{\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}dx\right )}{2 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \int \sqrt {a+b \text {arcsinh}(c x)}dx}{2 c}\right )}{3 c^2}-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{6} b c \int \frac {x^3}{\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}dx\right )}{2 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} b c \int \frac {x}{\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}dx\right )}{2 c}\right )}{3 c^2}-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{6} b c \int \frac {x^3}{\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}dx\right )}{2 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 6234 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}-\frac {\int -\frac {\sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c^3}\right )}{2 c}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {\int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {\int \frac {\sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c^3}+\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}\right )}{2 c}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (\frac {\int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}+x \sqrt {a+b \text {arcsinh}(c x)}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {\int \frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c^3}\right )}{2 c}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arcsinh}(c x)}+\frac {\int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c^3}\right )}{2 c}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3789 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c^3}\right )}{2 c}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (\frac {1}{2} i \int \frac {e^{-\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i \int \frac {e^{\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))\right )}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c^3}\right )}{2 c}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-i \int e^{\frac {a+b \text {arcsinh}(c x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c^3}\right )}{2 c}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c^3}\right )}{2 c}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3793 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {i \int \left (\frac {3 i \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {i \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{6 c^3}\right )}{2 c}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{5/2}-\frac {5}{6} b c \left (-\frac {b \left (\frac {1}{3} x^3 \sqrt {a+b \text {arcsinh}(c x)}+\frac {i \left (\frac {3}{8} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} i \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {3}{8} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} i \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{6 c^3}\right )}{2 c}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{c^2}-\frac {3 b \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 c}\right )}{2 c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^{3/2}}{3 c^2}\right )\) |
(x^3*(a + b*ArcSinh[c*x])^(5/2))/3 - (5*b*c*((x^2*Sqrt[1 + c^2*x^2]*(a + b *ArcSinh[c*x])^(3/2))/(3*c^2) - (2*((Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x] )^(3/2))/c^2 - (3*b*(x*Sqrt[a + b*ArcSinh[c*x]] - ((I/2)*((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*ArcSinh[c*x]]/Sqrt[b]])/E^(a/b)))/c))/(2*c)))/(3*c^2) - (b*((x^3*Sqrt[a + b*ArcSinh[c*x]])/3 + ((I/6)*(((3*I)/8)*Sqrt[b]*E^(a/b )*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]] - (I/8)*Sqrt[b]*E^((3*a)/ b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]] - (((3*I)/8) *Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/E^(a/b) + ((I/8) *Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/E^(( 3*a)/b)))/c^3))/(2*c)))/6
3.2.42.3.1 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[((c_.) + (d_.)*(x_))^(m_)*sin[(e_.) + (f_.)*(x_)]^(n_), x_Symbol] :> In t[ExpandTrigReduce[(c + d*x)^m, Sin[e + f*x]^n, x], x] /; FreeQ[{c, d, e, f , m}, x] && IGtQ[n, 1] && ( !RationalQ[m] || (GeQ[m, -1] && LtQ[m, 1]))
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[ x^(m + 1)*((a + b*ArcSinh[c*x])^n/(m + 1)), x] - Simp[b*c*(n/(m + 1)) Int [x^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; Free Q[{a, b, c}, x] && IGtQ[m, 0] && GtQ[n, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] /; FreeQ[ {a, b, c, d, e, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && NeQ[p, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_), x_Symbol] :> Simp[f*(f*x)^(m - 1)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 2*p + 1))), x] + (-Simp[f^2*((m - 1)/(c^2*(m + 2*p + 1))) Int[(f*x)^(m - 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*f*(n/(c*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int [(f*x)^(m - 1)*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x] ) /; FreeQ[{a, b, c, d, e, f, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && IGtQ[ m, 1] && NeQ[m + 2*p + 1, 0]
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]
\[\int x^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {5}{2}}d x\]
Exception generated. \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=\int x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {5}{2}}\, dx \]
\[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {5}{2}} x^{2} \,d x } \]
Exception generated. \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int x^2 (a+b \text {arcsinh}(c x))^{5/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{5/2} \,d x \]