\(\int (d+c^2 d x^2)^{5/2} \sqrt {a+b \text {arcsinh}(c x)} \, dx\) [102]

Optimal result

Integrand size = 27, antiderivative size = 632 \[ \int \left (d+c^2 d x^2\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)} \, dx=\frac {5}{16} d^2 x \sqrt {d+c^2 d x^2} \sqrt {a+b \text {arcsinh}(c x)}+\frac {5}{24} d x \left (d+c^2 d x^2\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{6} x \left (d+c^2 d x^2\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^{3/2}}{24 b c \sqrt {1+c^2 x^2}}+\frac {3 \sqrt {b} d^2 e^{\frac {4 a}{b}} \sqrt {\pi } \sqrt {d+c^2 d x^2} \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{512 c \sqrt {1+c^2 x^2}}+\frac {15 \sqrt {b} d^2 e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \sqrt {d+c^2 d x^2} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{256 c \sqrt {1+c^2 x^2}}+\frac {\sqrt {b} d^2 e^{\frac {6 a}{b}} \sqrt {\frac {\pi }{6}} \sqrt {d+c^2 d x^2} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{768 c \sqrt {1+c^2 x^2}}-\frac {3 \sqrt {b} d^2 e^{-\frac {4 a}{b}} \sqrt {\pi } \sqrt {d+c^2 d x^2} \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{512 c \sqrt {1+c^2 x^2}}-\frac {15 \sqrt {b} d^2 e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \sqrt {d+c^2 d x^2} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{256 c \sqrt {1+c^2 x^2}}-\frac {\sqrt {b} d^2 e^{-\frac {6 a}{b}} \sqrt {\frac {\pi }{6}} \sqrt {d+c^2 d x^2} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{768 c \sqrt {1+c^2 x^2}} \] Output:

5/16*d^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^(1/2)+5/24*d*x*(c^2*d*x^ 
2+d)^(3/2)*(a+b*arcsinh(c*x))^(1/2)+1/6*x*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh 
(c*x))^(1/2)+5/24*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^(3/2)/b/c/(c^ 
2*x^2+1)^(1/2)+3/512*b^(1/2)*d^2*exp(4*a/b)*Pi^(1/2)*(c^2*d*x^2+d)^(1/2)*e 
rf(2*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/c/(c^2*x^2+1)^(1/2)+15/512*b^(1/2)* 
d^2*exp(2*a/b)*2^(1/2)*Pi^(1/2)*(c^2*d*x^2+d)^(1/2)*erf(2^(1/2)*(a+b*arcsi 
nh(c*x))^(1/2)/b^(1/2))/c/(c^2*x^2+1)^(1/2)+1/4608*b^(1/2)*d^2*exp(6*a/b)* 
6^(1/2)*Pi^(1/2)*(c^2*d*x^2+d)^(1/2)*erf(6^(1/2)*(a+b*arcsinh(c*x))^(1/2)/ 
b^(1/2))/c/(c^2*x^2+1)^(1/2)-3/512*b^(1/2)*d^2*Pi^(1/2)*(c^2*d*x^2+d)^(1/2 
)*erfi(2*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/c/exp(4*a/b)/(c^2*x^2+1)^(1/2)- 
15/512*b^(1/2)*d^2*2^(1/2)*Pi^(1/2)*(c^2*d*x^2+d)^(1/2)*erfi(2^(1/2)*(a+b* 
arcsinh(c*x))^(1/2)/b^(1/2))/c/exp(2*a/b)/(c^2*x^2+1)^(1/2)-1/4608*b^(1/2) 
*d^2*6^(1/2)*Pi^(1/2)*(c^2*d*x^2+d)^(1/2)*erfi(6^(1/2)*(a+b*arcsinh(c*x))^ 
(1/2)/b^(1/2))/c/exp(6*a/b)/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 5.02 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.06 \[ \int \left (d+c^2 d x^2\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)} \, dx=-\frac {b d^2 e^{-\frac {6 a}{b}} \sqrt {d+c^2 d x^2} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}} \left (-\sqrt {6} b^2 \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}} \Gamma \left (\frac {3}{2},-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )+9 b e^{\frac {2 a}{b}} \left (4 a \sqrt {\frac {a}{b}+\text {arcsinh}(c x)}+4 b \text {arcsinh}(c x) \sqrt {\frac {a}{b}+\text {arcsinh}(c x)}+b \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+9 \sqrt {2} b e^{\frac {4 a}{b}} \left (16 a \sqrt {\frac {a}{b}+\text {arcsinh}(c x)}+16 b \text {arcsinh}(c x) \sqrt {\frac {a}{b}+\text {arcsinh}(c x)}+b \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+9 \sqrt {2} b e^{\frac {8 a}{b}} \left (-16 a \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}-16 b \text {arcsinh}(c x) \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}+b \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )+9 b e^{\frac {10 a}{b}} \left (-4 a \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}-4 b \text {arcsinh}(c x) \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}+b \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )+e^{\frac {6 a}{b}} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}} \left (480 (a+b \text {arcsinh}(c x))^2-\sqrt {6} b^2 e^{\frac {6 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \Gamma \left (\frac {3}{2},\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{2304 c \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^{5/2}} \] Input:

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

Output:

-1/2304*(b*d^2*Sqrt[d + c^2*d*x^2]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]*(-( 
Sqrt[6]*b^2*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcSinh[c*x])^2/ 
b^2)]*Gamma[3/2, (-6*(a + b*ArcSinh[c*x]))/b]) + 9*b*E^((2*a)/b)*(4*a*Sqrt 
[a/b + ArcSinh[c*x]] + 4*b*ArcSinh[c*x]*Sqrt[a/b + ArcSinh[c*x]] + b*Sqrt[ 
-((a + b*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, 
 (-4*(a + b*ArcSinh[c*x]))/b] + 9*Sqrt[2]*b*E^((4*a)/b)*(16*a*Sqrt[a/b + A 
rcSinh[c*x]] + 16*b*ArcSinh[c*x]*Sqrt[a/b + ArcSinh[c*x]] + b*Sqrt[-((a + 
b*ArcSinh[c*x])/b)]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, (-2*(a 
 + b*ArcSinh[c*x]))/b] + 9*Sqrt[2]*b*E^((8*a)/b)*(-16*a*Sqrt[-((a + b*ArcS 
inh[c*x])/b)] - 16*b*ArcSinh[c*x]*Sqrt[-((a + b*ArcSinh[c*x])/b)] + b*Sqrt 
[a/b + ArcSinh[c*x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, (2*(a 
 + b*ArcSinh[c*x]))/b] + 9*b*E^((10*a)/b)*(-4*a*Sqrt[-((a + b*ArcSinh[c*x] 
)/b)] - 4*b*ArcSinh[c*x]*Sqrt[-((a + b*ArcSinh[c*x])/b)] + b*Sqrt[a/b + Ar 
cSinh[c*x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, (4*(a + b*ArcS 
inh[c*x]))/b] + E^((6*a)/b)*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)]*(480*(a + 
b*ArcSinh[c*x])^2 - Sqrt[6]*b^2*E^((6*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*Gamma 
[3/2, (6*(a + b*ArcSinh[c*x]))/b])))/(c*E^((6*a)/b)*Sqrt[1 + c^2*x^2]*(a + 
 b*ArcSinh[c*x])^(5/2))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 5.62 (sec) , antiderivative size = 838, normalized size of antiderivative = 1.33, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {6201, 6201, 6200, 6195, 25, 5971, 27, 3042, 26, 3789, 2611, 2633, 2634, 6198, 6234, 25, 5971, 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.

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

\(\Big \downarrow \) 6201

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \int \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}dx+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 6201

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \int \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}dx+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 6200

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {b c \sqrt {c^2 d x^2+d} \int \frac {x}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{4 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 6195

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}-\frac {\sqrt {c^2 d x^2+d} \int -\frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{4 c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {\cosh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{4 c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 5971

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{2 \sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{4 c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{8 c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int -\frac {i \sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{8 c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}-\frac {i \sqrt {c^2 d x^2+d} \int \frac {\sin \left (\frac {2 i a}{b}-\frac {2 i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{8 c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 3789

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {i \sqrt {c^2 d x^2+d} \left (\frac {1}{2} i \int \frac {e^{-2 \text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i \int \frac {e^{2 \text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))\right )}{8 c \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 2611

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {i \sqrt {c^2 d x^2+d} \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-i \int e^{\frac {2 (a+b \text {arcsinh}(c x))}{b}-\frac {2 a}{b}}d\sqrt {a+b \text {arcsinh}(c x)}\right )}{8 c \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 2633

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (\frac {3}{4} d \left (-\frac {i \sqrt {c^2 d x^2+d} \left (i \int e^{\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{8 c \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 2634

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {c^2 x^2+1}}dx}{2 \sqrt {c^2 x^2+1}}-\frac {i \sqrt {c^2 d x^2+d} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{8 c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 6198

\(\displaystyle -\frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )^2}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{12 \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {b c d \sqrt {c^2 d x^2+d} \int \frac {x \left (c^2 x^2+1\right )}{\sqrt {a+b \text {arcsinh}(c x)}}dx}{8 \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {i \sqrt {c^2 d x^2+d} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{8 c \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^{3/2}}{3 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 6234

\(\displaystyle -\frac {d^2 \sqrt {c^2 d x^2+d} \int -\frac {\cosh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{12 c \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (-\frac {d \sqrt {c^2 d x^2+d} \int -\frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{8 c \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {i \sqrt {c^2 d x^2+d} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{8 c \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^{3/2}}{3 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \int \frac {\cosh ^5\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{12 c \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (\frac {d \sqrt {c^2 d x^2+d} \int \frac {\cosh ^3\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{8 c \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {i \sqrt {c^2 d x^2+d} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{8 c \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^{3/2}}{3 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \int \left (\frac {\sinh \left (\frac {6 a}{b}-\frac {6 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {5 \sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{32 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{12 c \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (\frac {d \sqrt {c^2 d x^2+d} \int \left (\frac {\sinh \left (\frac {4 a}{b}-\frac {4 (a+b \text {arcsinh}(c x))}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\sinh \left (\frac {2 a}{b}-\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{8 c \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (-\frac {i \sqrt {c^2 d x^2+d} \left (\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{\frac {2 a}{b}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\frac {\pi }{2}} \sqrt {b} e^{-\frac {2 a}{b}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{8 c \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^{3/2}}{3 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{4} x \left (c^2 d x^2+d\right )^{3/2} \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{6} x \left (c^2 d x^2+d\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{6} x \sqrt {a+b \text {arcsinh}(c x)} \left (c^2 d x^2+d\right )^{5/2}-\frac {d^2 \left (-\frac {1}{32} \sqrt {b} e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {5}{64} \sqrt {b} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{64} \sqrt {b} e^{\frac {6 a}{b}} \sqrt {\frac {\pi }{6}} \text {erf}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {b} e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {5}{64} \sqrt {b} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{64} \sqrt {b} e^{-\frac {6 a}{b}} \sqrt {\frac {\pi }{6}} \text {erfi}\left (\frac {\sqrt {6} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right ) \sqrt {c^2 d x^2+d}}{12 c \sqrt {c^2 x^2+1}}+\frac {5}{6} d \left (\frac {1}{4} x \sqrt {a+b \text {arcsinh}(c x)} \left (c^2 d x^2+d\right )^{3/2}-\frac {d \left (-\frac {1}{32} \sqrt {b} e^{\frac {4 a}{b}} \sqrt {\pi } \text {erf}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{8} \sqrt {b} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {b} e^{-\frac {4 a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {2 \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{8} \sqrt {b} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right ) \sqrt {c^2 d x^2+d}}{8 c \sqrt {c^2 x^2+1}}+\frac {3}{4} d \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^{3/2}}{3 b c \sqrt {c^2 x^2+1}}+\frac {1}{2} x \sqrt {c^2 d x^2+d} \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \sqrt {c^2 d x^2+d} \left (\frac {1}{2} i \sqrt {b} e^{\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {b} e^{-\frac {2 a}{b}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{8 c \sqrt {c^2 x^2+1}}\right )\right )\)

Input:

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

Output:

(x*(d + c^2*d*x^2)^(5/2)*Sqrt[a + b*ArcSinh[c*x]])/6 + (5*d*((x*(d + c^2*d 
*x^2)^(3/2)*Sqrt[a + b*ArcSinh[c*x]])/4 - (d*Sqrt[d + c^2*d*x^2]*(-1/32*(S 
qrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]]) - ( 
Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt 
[b]])/8 + (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(3 
2*E^((4*a)/b)) + (Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x] 
])/Sqrt[b]])/(8*E^((2*a)/b))))/(8*c*Sqrt[1 + c^2*x^2]) + (3*d*((x*Sqrt[d + 
 c^2*d*x^2]*Sqrt[a + b*ArcSinh[c*x]])/2 + (Sqrt[d + c^2*d*x^2]*(a + b*ArcS 
inh[c*x])^(3/2))/(3*b*c*Sqrt[1 + c^2*x^2]) - ((I/8)*Sqrt[d + c^2*d*x^2]*(( 
I/2)*Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]]) 
/Sqrt[b]] - ((I/2)*Sqrt[b]*Sqrt[Pi/2]*Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x 
]])/Sqrt[b]])/E^((2*a)/b)))/(c*Sqrt[1 + c^2*x^2])))/4))/6 - (d^2*Sqrt[d + 
c^2*d*x^2]*(-1/32*(Sqrt[b]*E^((4*a)/b)*Sqrt[Pi]*Erf[(2*Sqrt[a + b*ArcSinh[ 
c*x]])/Sqrt[b]]) - (5*Sqrt[b]*E^((2*a)/b)*Sqrt[Pi/2]*Erf[(Sqrt[2]*Sqrt[a + 
 b*ArcSinh[c*x]])/Sqrt[b]])/64 - (Sqrt[b]*E^((6*a)/b)*Sqrt[Pi/6]*Erf[(Sqrt 
[6]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/64 + (Sqrt[b]*Sqrt[Pi]*Erfi[(2*Sqr 
t[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*E^((4*a)/b)) + (5*Sqrt[b]*Sqrt[Pi/2]* 
Erfi[(Sqrt[2]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*E^((2*a)/b)) + (Sqrt 
[b]*Sqrt[Pi/6]*Erfi[(Sqrt[6]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(64*E^((6 
*a)/b))))/(12*c*Sqrt[1 + c^2*x^2])
 

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 5971

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]
 

rule 6195

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]
 

rule 6198

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[(1/(b*c*(n + 1)))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e*x^2]]*( 
a + b*ArcSinh[c*x])^(n + 1), x] /; FreeQ[{a, b, c, d, e, n}, x] && EqQ[e, c 
^2*d] && NeQ[n, -1]
 

rule 6200

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*Sqrt[(d_) + (e_.)*(x_)^2], x_ 
Symbol] :> Simp[x*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/2), x] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[b*c*(n/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2* 
x^2]]   Int[x*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e 
}, x] && EqQ[e, c^2*d] && GtQ[n, 0]
 

rule 6201

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), 
x_Symbol] :> Simp[x*(d + e*x^2)^p*((a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + 
(Simp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x 
], x] - Simp[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p]   Int[x* 
(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, 
b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 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]
 

Maple [F]

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

Input:

int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^(1/2),x)
 

Output:

int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^(1/2),x)
 

Fricas [F(-2)]

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

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^(1/2),x, algorithm="frica 
s")
 

Output:

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

Sympy [F(-1)]

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

integrate((c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^(1/2),x, algorithm="maxim 
a")
 

Output:

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

Giac [F(-2)]

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

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^(1/2),x, algorithm="giac" 
)
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

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

int((a + b*asinh(c*x))^(1/2)*(d + c^2*d*x^2)^(5/2),x)
 

Output:

int((a + b*asinh(c*x))^(1/2)*(d + c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \left (d+c^2 d x^2\right )^{5/2} \sqrt {a+b \text {arcsinh}(c x)} \, dx=\sqrt {d}\, d^{2} \left (\left (\int \sqrt {c^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{4}d x \right ) c^{4}+2 \left (\int \sqrt {c^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (c x \right ) b +a}\, x^{2}d x \right ) c^{2}+\int \sqrt {c^{2} x^{2}+1}\, \sqrt {\mathit {asinh} \left (c x \right ) b +a}d x \right ) \] Input:

int((c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x))^(1/2),x)
                                                                                    
                                                                                    
 

Output:

sqrt(d)*d**2*(int(sqrt(c**2*x**2 + 1)*sqrt(asinh(c*x)*b + a)*x**4,x)*c**4 
+ 2*int(sqrt(c**2*x**2 + 1)*sqrt(asinh(c*x)*b + a)*x**2,x)*c**2 + int(sqrt 
(c**2*x**2 + 1)*sqrt(asinh(c*x)*b + a),x))