\(\int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx\) [132]

Optimal result

Integrand size = 25, antiderivative size = 394 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=-\frac {5 b^2 e^2 (c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}}{6 d}+\frac {5 b^2 e^2 (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}}{36 d}+\frac {5 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{9 d}-\frac {5 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^{3/2}}{18 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}}{3 d}-\frac {15 b^{5/2} e^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d}+\frac {5 b^{5/2} e^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{576 d}+\frac {15 b^{5/2} e^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{64 d}-\frac {5 b^{5/2} e^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )}{576 d} \] Output:

-5/6*b^2*e^2*(d*x+c)*(a+b*arcsinh(d*x+c))^(1/2)/d+5/36*b^2*e^2*(d*x+c)^3*( 
a+b*arcsinh(d*x+c))^(1/2)/d+5/9*b*e^2*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x 
+c))^(3/2)/d-5/18*b*e^2*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c)) 
^(3/2)/d+1/3*e^2*(d*x+c)^3*(a+b*arcsinh(d*x+c))^(5/2)/d-15/64*b^(5/2)*e^2* 
exp(a/b)*Pi^(1/2)*erf((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2))/d+5/1728*b^(5/2) 
*e^2*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arcsinh(d*x+c))^(1/2)/b^ 
(1/2))/d+15/64*b^(5/2)*e^2*Pi^(1/2)*erfi((a+b*arcsinh(d*x+c))^(1/2)/b^(1/2 
))/d/exp(a/b)-5/1728*b^(5/2)*e^2*3^(1/2)*Pi^(1/2)*erfi(3^(1/2)*(a+b*arcsin 
h(d*x+c))^(1/2)/b^(1/2))/d/exp(3*a/b)
 

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.56 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=-\frac {b^3 e^2 e^{-\frac {3 a}{b}} \left (-81 e^{\frac {4 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {7}{2},\frac {a}{b}+\text {arcsinh}(c+d x)\right )+\sqrt {3} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {7}{2},-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )-81 e^{\frac {2 a}{b}} \sqrt {-\frac {a+b \text {arcsinh}(c+d x)}{b}} \Gamma \left (\frac {7}{2},-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )+\sqrt {3} e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c+d x)} \Gamma \left (\frac {7}{2},\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )\right )}{648 d \sqrt {a+b \text {arcsinh}(c+d x)}} \] Input:

Integrate[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^(5/2),x]
 

Output:

-1/648*(b^3*e^2*(-81*E^((4*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, a 
/b + ArcSinh[c + d*x]] + Sqrt[3]*Sqrt[-((a + b*ArcSinh[c + d*x])/b)]*Gamma 
[7/2, (-3*(a + b*ArcSinh[c + d*x]))/b] - 81*E^((2*a)/b)*Sqrt[-((a + b*ArcS 
inh[c + d*x])/b)]*Gamma[7/2, -((a + b*ArcSinh[c + d*x])/b)] + Sqrt[3]*E^(( 
6*a)/b)*Sqrt[a/b + ArcSinh[c + d*x]]*Gamma[7/2, (3*(a + b*ArcSinh[c + d*x] 
))/b]))/(d*E^((3*a)/b)*Sqrt[a + b*ArcSinh[c + d*x]])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.77 (sec) , antiderivative size = 467, normalized size of antiderivative = 1.19, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {6274, 27, 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.

\(\displaystyle \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx\)

\(\Big \downarrow \) 6274

\(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{5/2}d(c+d x)}{d}\)

\(\Big \downarrow \) 6192

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \int (c+d x)^2 \sqrt {a+b \text {arcsinh}(c+d x)}d(c+d x)-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 6192

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{6} b \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^{3/2}}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{6} b \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \int \sqrt {a+b \text {arcsinh}(c+d x)}d(c+d x)\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 6187

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} b \int \frac {c+d x}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{6} b \int \frac {(c+d x)^3}{\sqrt {(c+d x)^2+1} \sqrt {a+b \text {arcsinh}(c+d x)}}d(c+d x)\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{6} \int -\frac {\sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} \int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{6} \int \frac {\sinh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left (\frac {1}{2} \int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))+(c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{2} \int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} \int \frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \int \frac {e^{\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))-\frac {1}{2} i \int \frac {e^{-\frac {a-c-d x}{b}}}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-i \int e^{\frac {a+b \text {arcsinh}(c+d x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}\right )\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}}d\sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c+d x))}{b}\right )^3}{\sqrt {a+b \text {arcsinh}(c+d x)}}d(a+b \text {arcsinh}(c+d x))\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 3793

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \int \left (\frac {3 i \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c+d x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}-\frac {i \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c+d x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c+d x)}}\right )d(a+b \text {arcsinh}(c+d x))\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^{5/2}-\frac {5}{6} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^{3/2}-\frac {3}{2} b \left ((c+d x) \sqrt {a+b \text {arcsinh}(c+d x)}-\frac {1}{2} i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )\right )-\frac {1}{2} b \left (\frac {1}{3} (c+d x)^3 \sqrt {a+b \text {arcsinh}(c+d x)}+\frac {1}{6} i \left (\frac {3}{8} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {1}{8} i \sqrt {\frac {\pi }{3}} \sqrt {b} e^{\frac {3 a}{b}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )-\frac {3}{8} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )+\frac {1}{8} i \sqrt {\frac {\pi }{3}} \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c+d x)}}{\sqrt {b}}\right )\right )\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^{3/2}\right )\right )}{d}\)

Input:

Int[(c*e + d*e*x)^2*(a + b*ArcSinh[c + d*x])^(5/2),x]
 

Output:

(e^2*(((c + d*x)^3*(a + b*ArcSinh[c + d*x])^(5/2))/3 - (5*b*(((c + d*x)^2* 
Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2))/3 - (2*(Sqrt[1 + (c 
+ d*x)^2]*(a + b*ArcSinh[c + d*x])^(3/2) - (3*b*((c + d*x)*Sqrt[a + b*ArcS 
inh[c + d*x]] - (I/2)*((I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSi 
nh[c + d*x]]/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c 
+ d*x]]/Sqrt[b]])/E^(a/b))))/2))/3 - (b*(((c + d*x)^3*Sqrt[a + b*ArcSinh[c 
 + d*x]])/3 + (I/6)*(((3*I)/8)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*Arc 
Sinh[c + d*x]]/Sqrt[b]] - (I/8)*Sqrt[b]*E^((3*a)/b)*Sqrt[Pi/3]*Erf[(Sqrt[3 
]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]] - (((3*I)/8)*Sqrt[b]*Sqrt[Pi]*Erf 
i[Sqrt[a + b*ArcSinh[c + d*x]]/Sqrt[b]])/E^(a/b) + ((I/8)*Sqrt[b]*Sqrt[Pi/ 
3]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c + d*x]])/Sqrt[b]])/E^((3*a)/b))))/2) 
)/6))/d
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2611

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]
 

rule 2633

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]
 

rule 2634

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]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3789

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]
 

rule 3793

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]))
 

rule 6187

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]
 

rule 6192

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]
 

rule 6213

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]
 

rule 6227

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]
 

rule 6234

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]
 

rule 6274

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]
 

Maple [F]

\[\int \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{\frac {5}{2}}d x\]

Input:

int((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c))^(5/2),x)
 

Output:

int((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c))^(5/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="fricas")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=e^{2} \left (\int a^{2} c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int a^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b^{2} c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b c^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 2 a^{2} c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}}\, dx + \int b^{2} d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}\, dx + \int 2 a b d^{2} x^{2} \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx + \int 2 b^{2} c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}^{2}{\left (c + d x \right )}\, dx + \int 4 a b c d x \sqrt {a + b \operatorname {asinh}{\left (c + d x \right )}} \operatorname {asinh}{\left (c + d x \right )}\, dx\right ) \] Input:

integrate((d*e*x+c*e)**2*(a+b*asinh(d*x+c))**(5/2),x)
 

Output:

e**2*(Integral(a**2*c**2*sqrt(a + b*asinh(c + d*x)), x) + Integral(a**2*d* 
*2*x**2*sqrt(a + b*asinh(c + d*x)), x) + Integral(b**2*c**2*sqrt(a + b*asi 
nh(c + d*x))*asinh(c + d*x)**2, x) + Integral(2*a*b*c**2*sqrt(a + b*asinh( 
c + d*x))*asinh(c + d*x), x) + Integral(2*a**2*c*d*x*sqrt(a + b*asinh(c + 
d*x)), x) + Integral(b**2*d**2*x**2*sqrt(a + b*asinh(c + d*x))*asinh(c + d 
*x)**2, x) + Integral(2*a*b*d**2*x**2*sqrt(a + b*asinh(c + d*x))*asinh(c + 
 d*x), x) + Integral(2*b**2*c*d*x*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x 
)**2, x) + Integral(4*a*b*c*d*x*sqrt(a + b*asinh(c + d*x))*asinh(c + d*x), 
 x))
 

Maxima [F]

\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="maxima")
 

Output:

integrate((d*e*x + c*e)^2*(b*arcsinh(d*x + c) + a)^(5/2), x)
 

Giac [F]

\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((d*e*x+c*e)^2*(a+b*arcsinh(d*x+c))^(5/2),x, algorithm="giac")
 

Output:

integrate((d*e*x + c*e)^2*(b*arcsinh(d*x + c) + a)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^{5/2} \,d x \] Input:

int((c*e + d*e*x)^2*(a + b*asinh(c + d*x))^(5/2),x)
 

Output:

int((c*e + d*e*x)^2*(a + b*asinh(c + d*x))^(5/2), x)
 

Reduce [F]

\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^{5/2} \, dx=e^{2} \left (\left (\int \sqrt {a +b \mathit {asinh} \left (d x +c \right )}d x \right ) a^{2} c^{2}+2 \left (\int \sqrt {a +b \mathit {asinh} \left (d x +c \right )}\, \mathit {asinh} \left (d x +c \right ) x^{2}d x \right ) a b \,d^{2}+4 \left (\int \sqrt {a +b \mathit {asinh} \left (d x +c \right )}\, \mathit {asinh} \left (d x +c \right ) x d x \right ) a b c d +2 \left (\int \sqrt {a +b \mathit {asinh} \left (d x +c \right )}\, \mathit {asinh} \left (d x +c \right )d x \right ) a b \,c^{2}+\left (\int \sqrt {a +b \mathit {asinh} \left (d x +c \right )}\, \mathit {asinh} \left (d x +c \right )^{2} x^{2}d x \right ) b^{2} d^{2}+2 \left (\int \sqrt {a +b \mathit {asinh} \left (d x +c \right )}\, \mathit {asinh} \left (d x +c \right )^{2} x d x \right ) b^{2} c d +\left (\int \sqrt {a +b \mathit {asinh} \left (d x +c \right )}\, \mathit {asinh} \left (d x +c \right )^{2}d x \right ) b^{2} c^{2}+\left (\int \sqrt {a +b \mathit {asinh} \left (d x +c \right )}\, x^{2}d x \right ) a^{2} d^{2}+2 \left (\int \sqrt {a +b \mathit {asinh} \left (d x +c \right )}\, x d x \right ) a^{2} c d \right ) \] Input:

int((d*e*x+c*e)^2*(a+b*asinh(d*x+c))^(5/2),x)
 

Output:

e**2*(int(sqrt(asinh(c + d*x)*b + a),x)*a**2*c**2 + 2*int(sqrt(asinh(c + d 
*x)*b + a)*asinh(c + d*x)*x**2,x)*a*b*d**2 + 4*int(sqrt(asinh(c + d*x)*b + 
 a)*asinh(c + d*x)*x,x)*a*b*c*d + 2*int(sqrt(asinh(c + d*x)*b + a)*asinh(c 
 + d*x),x)*a*b*c**2 + int(sqrt(asinh(c + d*x)*b + a)*asinh(c + d*x)**2*x** 
2,x)*b**2*d**2 + 2*int(sqrt(asinh(c + d*x)*b + a)*asinh(c + d*x)**2*x,x)*b 
**2*c*d + int(sqrt(asinh(c + d*x)*b + a)*asinh(c + d*x)**2,x)*b**2*c**2 + 
int(sqrt(asinh(c + d*x)*b + a)*x**2,x)*a**2*d**2 + 2*int(sqrt(asinh(c + d* 
x)*b + a)*x,x)*a**2*c*d)