\(\int (d+c^2 d x^2)^2 \sqrt {a+b \text {arcsinh}(c x)} \, dx\) [80]

Optimal result

Integrand size = 25, antiderivative size = 506 \[ \int \left (d+c^2 d x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)} \, dx=\frac {8}{15} d^2 x \sqrt {a+b \text {arcsinh}(c x)}+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) \sqrt {a+b \text {arcsinh}(c x)}+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)}+\frac {5 \sqrt {b} d^2 e^{a/b} \sqrt {\pi } \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c}+\frac {\sqrt {b} d^2 e^{\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{60 c}+\frac {\sqrt {b} d^2 e^{\frac {3 a}{b}} \sqrt {3 \pi } \text {erf}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{320 c}+\frac {\sqrt {b} d^2 e^{\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{320 c}-\frac {5 \sqrt {b} d^2 e^{-\frac {a}{b}} \sqrt {\pi } \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{32 c}-\frac {\sqrt {b} d^2 e^{-\frac {3 a}{b}} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{60 c}-\frac {\sqrt {b} d^2 e^{-\frac {3 a}{b}} \sqrt {3 \pi } \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{320 c}-\frac {\sqrt {b} d^2 e^{-\frac {5 a}{b}} \sqrt {\frac {\pi }{5}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )}{320 c} \] Output:

8/15*d^2*x*(a+b*arcsinh(c*x))^(1/2)+4/15*d^2*x*(c^2*x^2+1)*(a+b*arcsinh(c* 
x))^(1/2)+1/5*d^2*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^(1/2)+5/32*b^(1/2)*d^ 
2*exp(a/b)*Pi^(1/2)*erf((a+b*arcsinh(c*x))^(1/2)/b^(1/2))/c+5/576*b^(1/2)* 
d^2*exp(3*a/b)*3^(1/2)*Pi^(1/2)*erf(3^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/ 
2))/c+1/1600*b^(1/2)*d^2*exp(5*a/b)*5^(1/2)*Pi^(1/2)*erf(5^(1/2)*(a+b*arcs 
inh(c*x))^(1/2)/b^(1/2))/c-5/32*b^(1/2)*d^2*Pi^(1/2)*erfi((a+b*arcsinh(c*x 
))^(1/2)/b^(1/2))/c/exp(a/b)-5/576*b^(1/2)*d^2*3^(1/2)*Pi^(1/2)*erfi(3^(1/ 
2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/c/exp(3*a/b)-1/1600*b^(1/2)*d^2*5^(1/ 
2)*Pi^(1/2)*erfi(5^(1/2)*(a+b*arcsinh(c*x))^(1/2)/b^(1/2))/c/exp(5*a/b)
 

Mathematica [A] (verified)

Time = 2.31 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.83 \[ \int \left (d+c^2 d x^2\right )^2 \sqrt {a+b \text {arcsinh}(c x)} \, dx=-\frac {b d^2 e^{-\frac {5 a}{b}} \left (-450 e^{\frac {6 a}{b}} \left (3 a \sqrt {\frac {a}{b}+\text {arcsinh}(c x)}+3 b \text {arcsinh}(c x) \sqrt {\frac {a}{b}+\text {arcsinh}(c x)}+8 b \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},\frac {a}{b}+\text {arcsinh}(c x)\right )-9 \sqrt {5} b \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {3}{2},-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )-125 \sqrt {3} b e^{\frac {2 a}{b}} \left (-\frac {a+b \text {arcsinh}(c x)}{b}\right )^{3/2} \Gamma \left (\frac {3}{2},-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )-450 e^{\frac {4 a}{b}} \left (3 a \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}+3 b \text {arcsinh}(c x) \sqrt {-\frac {a+b \text {arcsinh}(c x)}{b}}-8 b \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} \sqrt {-\frac {(a+b \text {arcsinh}(c x))^2}{b^2}}\right ) \Gamma \left (\frac {3}{2},-\frac {a+b \text {arcsinh}(c x)}{b}\right )+e^{\frac {8 a}{b}} \sqrt {\frac {a}{b}+\text {arcsinh}(c x)} (a+b \text {arcsinh}(c x)) \left (125 \sqrt {3} \Gamma \left (\frac {3}{2},\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )+9 \sqrt {5} e^{\frac {2 a}{b}} \Gamma \left (\frac {3}{2},\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )\right )\right )}{7200 c (a+b \text {arcsinh}(c x))^{3/2}} \] Input:

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

Output:

-1/7200*(b*d^2*(-450*E^((6*a)/b)*(3*a*Sqrt[a/b + ArcSinh[c*x]] + 3*b*ArcSi 
nh[c*x]*Sqrt[a/b + ArcSinh[c*x]] + 8*b*Sqrt[-((a + b*ArcSinh[c*x])/b)]*Sqr 
t[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, a/b + ArcSinh[c*x]] - 9*Sqrt[ 
5]*b*(-((a + b*ArcSinh[c*x])/b))^(3/2)*Gamma[3/2, (-5*(a + b*ArcSinh[c*x]) 
)/b] - 125*Sqrt[3]*b*E^((2*a)/b)*(-((a + b*ArcSinh[c*x])/b))^(3/2)*Gamma[3 
/2, (-3*(a + b*ArcSinh[c*x]))/b] - 450*E^((4*a)/b)*(3*a*Sqrt[-((a + b*ArcS 
inh[c*x])/b)] + 3*b*ArcSinh[c*x]*Sqrt[-((a + b*ArcSinh[c*x])/b)] - 8*b*Sqr 
t[a/b + ArcSinh[c*x]]*Sqrt[-((a + b*ArcSinh[c*x])^2/b^2)])*Gamma[3/2, -((a 
 + b*ArcSinh[c*x])/b)] + E^((8*a)/b)*Sqrt[a/b + ArcSinh[c*x]]*(a + b*ArcSi 
nh[c*x])*(125*Sqrt[3]*Gamma[3/2, (3*(a + b*ArcSinh[c*x]))/b] + 9*Sqrt[5]*E 
^((2*a)/b)*Gamma[3/2, (5*(a + b*ArcSinh[c*x]))/b])))/(c*E^((5*a)/b)*(a + b 
*ArcSinh[c*x])^(3/2))
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.82 (sec) , antiderivative size = 657, normalized size of antiderivative = 1.30, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {6201, 27, 6201, 6187, 6234, 25, 3042, 26, 3789, 2611, 2633, 2634, 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.

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

\(\Big \downarrow \) 6201

\(\displaystyle -\frac {1}{10} b c d^2 \int \frac {x \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx+\frac {4}{5} d \int d \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}dx+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{10} b c d^2 \int \frac {x \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx+\frac {4}{5} d^2 \int \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}dx+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 6201

\(\displaystyle -\frac {1}{10} b c d^2 \int \frac {x \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx+\frac {4}{5} d^2 \left (-\frac {1}{6} b c \int \frac {x \sqrt {c^2 x^2+1}}{\sqrt {a+b \text {arcsinh}(c x)}}dx+\frac {2}{3} \int \sqrt {a+b \text {arcsinh}(c x)}dx+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 6187

\(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} b c \int \frac {x}{\sqrt {c^2 x^2+1} \sqrt {a+b \text {arcsinh}(c x)}}dx\right )-\frac {1}{6} b c \int \frac {x \sqrt {c^2 x^2+1}}{\sqrt {a+b \text {arcsinh}(c x)}}dx+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}\right )-\frac {1}{10} b c d^2 \int \frac {x \left (c^2 x^2+1\right )^{3/2}}{\sqrt {a+b \text {arcsinh}(c x)}}dx+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 6234

\(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {\int -\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}\right )-\frac {\int -\frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c}+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}\right )-\frac {d^2 \int -\frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{10 c}+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (\frac {\int \frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}+x \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c}+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{10 c}+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x \sqrt {a+b \text {arcsinh}(c x)}+\frac {\int -\frac {i \sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}\right )+\frac {\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c}+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{10 c}+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \int \frac {\sin \left (\frac {i a}{b}-\frac {i (a+b \text {arcsinh}(c x))}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{2 c}\right )+\frac {\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c}+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{10 c}+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 3789

\(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (\frac {1}{2} i \int \frac {e^{-\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))-\frac {1}{2} i \int \frac {e^{\text {arcsinh}(c x)}}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))\right )}{2 c}\right )+\frac {\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c}+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{10 c}+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 2611

\(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-i \int e^{\frac {a+b \text {arcsinh}(c x)}{b}-\frac {a}{b}}d\sqrt {a+b \text {arcsinh}(c x)}\right )}{2 c}\right )+\frac {\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c}+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{10 c}+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 2633

\(\displaystyle \frac {4}{5} d^2 \left (\frac {2}{3} \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (i \int e^{\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}}d\sqrt {a+b \text {arcsinh}(c x)}-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 c}\right )+\frac {\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c}+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}\right )+\frac {d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{10 c}+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 2634

\(\displaystyle \frac {4}{5} d^2 \left (\frac {\int \frac {\cosh ^2\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{6 c}+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}+\frac {2}{3} \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 c}\right )\right )+\frac {d^2 \int \frac {\cosh ^4\left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{\sqrt {a+b \text {arcsinh}(c x)}}d(a+b \text {arcsinh}(c x))}{10 c}+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 5971

\(\displaystyle \frac {4}{5} d^2 \left (\frac {\int \left (\frac {\sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{4 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{6 c}+\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}+\frac {2}{3} \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 c}\right )\right )+\frac {d^2 \int \left (\frac {\sinh \left (\frac {5 a}{b}-\frac {5 (a+b \text {arcsinh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {3 \sinh \left (\frac {3 a}{b}-\frac {3 (a+b \text {arcsinh}(c x))}{b}\right )}{16 \sqrt {a+b \text {arcsinh}(c x)}}+\frac {\sinh \left (\frac {a}{b}-\frac {a+b \text {arcsinh}(c x)}{b}\right )}{8 \sqrt {a+b \text {arcsinh}(c x)}}\right )d(a+b \text {arcsinh}(c x))}{10 c}+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4}{5} d^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) \sqrt {a+b \text {arcsinh}(c x)}+\frac {2}{3} \left (x \sqrt {a+b \text {arcsinh}(c x)}-\frac {i \left (\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{2} i \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{2 c}\right )-\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}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 \sqrt {a+b \text {arcsinh}(c x)}-\frac {d^2 \left (-\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{a/b} \text {erf}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )-\frac {1}{32} \sqrt {3 \pi } \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}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{\frac {5 a}{b}} \text {erf}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{16} \sqrt {\pi } \sqrt {b} e^{-\frac {a}{b}} \text {erfi}\left (\frac {\sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {3 \pi } \sqrt {b} e^{-\frac {3 a}{b}} \text {erfi}\left (\frac {\sqrt {3} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )+\frac {1}{32} \sqrt {\frac {\pi }{5}} \sqrt {b} e^{-\frac {5 a}{b}} \text {erfi}\left (\frac {\sqrt {5} \sqrt {a+b \text {arcsinh}(c x)}}{\sqrt {b}}\right )\right )}{10 c}\)

Input:

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

Output:

(d^2*x*(1 + c^2*x^2)^2*Sqrt[a + b*ArcSinh[c*x]])/5 + (4*d^2*((x*(1 + c^2*x 
^2)*Sqrt[a + b*ArcSinh[c*x]])/3 + (2*(x*Sqrt[a + b*ArcSinh[c*x]] - ((I/2)* 
((I/2)*Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]] - (( 
I/2)*Sqrt[b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/E^(a/b)))/c) 
)/3 - (-1/8*(Sqrt[b]*E^(a/b)*Sqrt[Pi]*Erf[Sqrt[a + b*ArcSinh[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)))/5 - (d^2*(-1/16*(Sqrt[b]*E^(a/b)*Sqrt[P 
i]*Erf[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]]) - (Sqrt[b]*E^((3*a)/b)*Sqrt[3*Pi 
]*Erf[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/32 - (Sqrt[b]*E^((5*a)/ 
b)*Sqrt[Pi/5]*Erf[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/32 + (Sqrt[ 
b]*Sqrt[Pi]*Erfi[Sqrt[a + b*ArcSinh[c*x]]/Sqrt[b]])/(16*E^(a/b)) + (Sqrt[b 
]*Sqrt[3*Pi]*Erfi[(Sqrt[3]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b]])/(32*E^((3*a 
)/b)) + (Sqrt[b]*Sqrt[Pi/5]*Erfi[(Sqrt[5]*Sqrt[a + b*ArcSinh[c*x]])/Sqrt[b 
]])/(32*E^((5*a)/b))))/(10*c)
 

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 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 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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

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

Output:

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

Reduce [F]

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

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

Output:

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