Integrand size = 25, antiderivative size = 328 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\frac {b e^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{3 d}-\frac {b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{3 d}-\frac {3 b^{3/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {b^{3/2} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{96 d}-\frac {3 b^{3/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{32 d}+\frac {b^{3/2} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{96 d} \]
1/3*e^2*(d*x+c)^3*(a+b*arcsinh(d*x+c))^(3/2)/d+1/288*b^(3/2)*e^2*exp(3*a/b )*erf(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^(1/2)/d+1/288 *b^(3/2)*e^2*erfi(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))*3^(1/2)*Pi^( 1/2)/d/exp(3*a/b)-3/32*b^(3/2)*e^2*exp(a/b)*erf((a+b*arcsinh(d*x+c))^(1/2) /b^(1/2))*Pi^(1/2)/d-3/32*b^(3/2)*e^2*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1 /2))*Pi^(1/2)/d/exp(a/b)+1/3*b*e^2*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c) )^(1/2)/d-1/6*b*e^2*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))^(1/ 2)/d
Time = 0.23 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.73 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=-\frac {b e^2 e^{-\frac {3 a}{b}} \sqrt {a+b \text {arcsinh}(c+d x)} \left (-27 e^{\frac {4 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {5}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\sqrt {3} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {5}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-27 e^{\frac {2 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {5}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {5}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{216 d \sqrt {-\frac {(a+b \text {arcsinh}(c+d x))^2}{b^2}}} \]
-1/216*(b*e^2*Sqrt[a + b*ArcSinh[c + d*x]]*(-27*E^((4*a)/b)*Sqrt[-((a + b* ArcSinh[c + d*x])/b)]*Gamma[5/2, a/b + ArcSinh[c + d*x]] + Sqrt[3]*Sqrt[a/ b + ArcSinh[c + d*x]]*Gamma[5/2, (-3*(a + b*ArcSinh[c + d*x]))/b] - 27*E^( (2*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[5/2, -((a + b*ArcSinh[c + d*x] )/b)] + Sqrt[3]*E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma[5/2, (3*(a + b*ArcSinh[c + d*x]))/b]))/(d*E^((3*a)/b)*Sqrt[-((a + b*ArcSinh[c + d*x])^2/b^2)])
Time = 1.92 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.20, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6274, 27, 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.
| \(\displaystyle \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6192 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \int \frac {(c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {1}{6} b \int \frac {(c+d x)^2}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)-\frac {2}{3} \int \frac {(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6195 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \int \frac {(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{6} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right ) \sinh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 5971 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \int \frac {(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{6} \int \left (\frac {\cosh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \int \frac {(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} b \int \frac {1}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 6189 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}+\frac {\pi }{2}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 3788 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{2} \left (\frac {1}{2} i \int \frac {i e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int -\frac {i e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \left (\frac {1}{2} \left (-\frac {1}{2} \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \left (\frac {1}{2} \left (-\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-\int e^{\frac {a+b \text {arcsinh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}\right )+\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \left (\frac {1}{2} \left (-\int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
| \(\Big \downarrow \) 2634 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {1}{2} b \left (-\frac {2}{3} \left (\frac {1}{2} \left (-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}\right )+\frac {1}{6} \left (\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )}{d}\) |
(e^2*(((c + d*x)^3*(a + b*ArcSinh[c + d*x])^(3/2))/3 - (b*(((c + d*x)^2*Sq rt[1 + (c + d*x)^2]*Sqrt[a + b*ArcSinh[c + d*x]])/3 - (2*(Sqrt[1 + (c + d* x)^2]*Sqrt[a + b*ArcSinh[c + d*x]] + (-1/2*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[S qrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]]) - (Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b* ArcSinh[c + d*x]]/Sqrt[b]])/(2*E^(a/b)))/2))/3 + ((Sqrt[b]*E^(a/b)*Sqrt[Pi ]*Erf[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/8 - (Sqrt[b]*E^((3*a)/b)*Sqrt [Pi/3]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/8 + (Sqrt[b]*S qrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/(8*E^(a/b)) - (Sqrt[b] *Sqrt[Pi/3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/(8*E^((3 *a)/b)))/6))/2))/d
3.2.88.3.1 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[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
\[\int \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {3}{2}}d x\]
Exception generated. \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=e^{2} \left (\int a c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int a d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 2 a c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 2 b c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx\right ) \]
e**2*(Integral(a*c**2*sqrt(a + b*asinh(c + d*x)), x) + Integral(a*d**2*x** 2*sqrt(a + b*asinh(c + d*x)), x) + Integral(b*c**2*sqrt(a + b*asinh(c + d* x))*asinh(c + d*x), x) + Integral(2*a*c*d*x*sqrt(a + b*asinh(c + d*x)), x) + Integral(b*d**2*x**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x), x) + In tegral(2*b*c*d*x*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x), x))
\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{3/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{3/2} \,d x \]