Integrand size = 16, antiderivative size = 282 \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\frac {b \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^3}-\frac {b x^2 \sqrt {1+c^2 x^2} \sqrt {a+b \text {arcsinh}(c x)}}{6 c}+\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {3 b^{3/2} e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3}-\frac {3 b^{3/2} e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c^3}+\frac {b^{3/2} e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{96 c^3} \]
1/3*x^3*(a+b*arcsinh(c*x))^(3/2)+1/288*b^(3/2)*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+1/288*b^(3/2)*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)-3/32 *b^(3/2)*exp(a/b)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3-3/32* b^(3/2)*erfi((a+b*arcsinh(c*x))^(1/2)/b^(1/2))*Pi^(1/2)/c^3/exp(a/b)+1/3*b *(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^(1/2)/c^3-1/6*b*x^2*(c^2*x^2+1)^(1/2 )*(a+b*arcsinh(c*x))^(1/2)/c
Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.76 \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=-\frac {b e^{-\frac {3 a}{b}} \sqrt {a+b \text {arcsinh}(c x)} \left (-27 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {5}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {5}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-27 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {5}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \Gamma \left (\frac {5}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )\right )}{216 c^3 \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}} \]
-1/216*(b*Sqrt[a + b*ArcSinh[c*x]]*(-27*E^((4*a)/b)*Sqrt[-((a + b*ArcSinh[ c*x])/b)]*Gamma[5/2, a/b + ArcSinh[c*x]] + Sqrt[3]*Sqrt[a/b + ArcSinh[c*x] ]*Gamma[5/2, (-3*(a + b*ArcSinh[c*x]))/b] - 27*E^((2*a)/b)*Sqrt[a/b + ArcS inh[c*x]]*Gamma[5/2, -((a + b*ArcSinh[c*x])/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt [-((a + b*ArcSinh[c*x])/b)]*Gamma[5/2, (3*(a + b*ArcSinh[c*x]))/b]))/(c^3* E^((3*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])
Time = 1.96 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.34, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.812, Rules used = {6192, 6227, 6195, 5971, 2009, 6213, 6189, 3042, 3788, 26, 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.
| \(\displaystyle \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \int \frac {x^3 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {2 \int \frac {x \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {b \int \frac {x^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{6 c}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c^4}-\frac {2 \int \frac {x \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {\int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{6 c^4}-\frac {2 \int \frac {x \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {2 \int \frac {x \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{c^2}-\frac {b \int \frac {1}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{2 c}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 6189 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{c^2}-\frac {\int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{c^2}-\frac {\int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{c^2}-\frac {\frac {1}{2} i \int -\frac {i e^{-\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i \int \frac {i e^{\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{c^2}-\frac {\frac {1}{2} \int \frac {e^{-\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))+\frac {1}{2} \int \frac {e^{\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{c^2}-\frac {\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}+\int e^{\frac {a+b \text {arcsinh}(c x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c x)}}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{c^2}-\frac {\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 c^2}\right )}{3 c^2}-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 c^4}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {1}{3} x^3 (a+b \text {arcsinh}(c x))^{3/2}-\frac {1}{2} b c \left (-\frac {-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{6 c^4}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{c^2}-\frac {\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{2 c^2}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}{3 c^2}\right )\) |
(x^3*(a + b*ArcSinh[c*x])^(3/2))/3 - (b*c*((x^2*Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x]])/(3*c^2) - (2*((Sqrt[1 + c^2*x^2]*Sqrt[a + b*ArcSinh[c*x] ])/c^2 - ((Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]) /2 + (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(2*E^(a/b)) )/(2*c^2)))/(3*c^2) - (-1/8*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSi nh[c*x]]/Sqrt[b]]) + (Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/8 - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[ c*x]]/Sqrt[b]])/(8*E^(a/b)) + (Sqrt[b]*Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b *ArcSinh[c*x]])/Sqrt[b]])/(8*E^((3*a)/b)))/(6*c^4)))/2
3.2.39.3.1 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_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol ] :> Simp[I/2 Int[(c + d*x)^m/(E^(I*k*Pi)*E^(I*(e + f*x))), x], x] - Simp [I/2 Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d, e , f, m}, x] && IntegerQ[2*k]
Int[Cosh[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.)*Sinh[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Int[ExpandTrigReduce[(c + d*x)^m, Sinh[a + b*x]^n*Cosh[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[1/(b*c) S ubst[Int[x^n*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x]
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_)^(m_.), x_Symbol] :> Simp[ 1/(b*c^(m + 1)) Subst[Int[x^n*Sinh[-a/b + x/b]^m*Cosh[-a/b + x/b], x], x, a + b*ArcSinh[c*x]], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[m, 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]
\[\int x^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{\frac {3}{2}}d x\]
Exception generated. \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{\frac {3}{2}}\, dx \]
\[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int { {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{\frac {3}{2}} x^{2} \,d x } \]
Exception generated. \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int x^2 (a+b \text {arcsinh}(c x))^{3/2} \, dx=\int x^2\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^{3/2} \,d x \]