\(\int (d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x))^3 \, dx\) [13]

Optimal result

Integrand size = 23, antiderivative size = 322 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^3 \, dx=-\frac {4144 b^3 d^2 \sqrt {1+c^2 x^2}}{1125 c}-\frac {272 b^3 d^2 \left (1+c^2 x^2\right )^{3/2}}{3375 c}-\frac {6 b^3 d^2 \left (1+c^2 x^2\right )^{5/2}}{625 c}+\frac {298}{75} b^2 d^2 x (a+b \text {arcsinh}(c x))+\frac {76}{225} b^2 c^2 d^2 x^3 (a+b \text {arcsinh}(c x))+\frac {6}{125} b^2 c^4 d^2 x^5 (a+b \text {arcsinh}(c x))-\frac {8 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{5 c}-\frac {4 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{15 c}-\frac {3 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{25 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^3+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^3+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^3 \] Output:

-4144/1125*b^3*d^2*(c^2*x^2+1)^(1/2)/c-272/3375*b^3*d^2*(c^2*x^2+1)^(3/2)/ 
c-6/625*b^3*d^2*(c^2*x^2+1)^(5/2)/c+298/75*b^2*d^2*x*(a+b*arcsinh(c*x))+76 
/225*b^2*c^2*d^2*x^3*(a+b*arcsinh(c*x))+6/125*b^2*c^4*d^2*x^5*(a+b*arcsinh 
(c*x))-8/5*b*d^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/c-4/15*b*d^2*(c^2* 
x^2+1)^(3/2)*(a+b*arcsinh(c*x))^2/c-3/25*b*d^2*(c^2*x^2+1)^(5/2)*(a+b*arcs 
inh(c*x))^2/c+8/15*d^2*x*(a+b*arcsinh(c*x))^3+4/15*d^2*x*(c^2*x^2+1)*(a+b* 
arcsinh(c*x))^3+1/5*d^2*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^3
 

Mathematica [A] (verified)

Time = 1.26 (sec) , antiderivative size = 327, normalized size of antiderivative = 1.02 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^3 \, dx=\frac {d^2 \left (1125 a^3 c x \left (15+10 c^2 x^2+3 c^4 x^4\right )-225 a^2 b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )+30 a b^2 c x \left (2235+190 c^2 x^2+27 c^4 x^4\right )-2 b^3 \sqrt {1+c^2 x^2} \left (31841+842 c^2 x^2+81 c^4 x^4\right )+15 b \left (225 a^2 c x \left (15+10 c^2 x^2+3 c^4 x^4\right )-30 a b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )+2 b^2 c x \left (2235+190 c^2 x^2+27 c^4 x^4\right )\right ) \text {arcsinh}(c x)-225 b^2 \left (-15 a c x \left (15+10 c^2 x^2+3 c^4 x^4\right )+b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )\right ) \text {arcsinh}(c x)^2+1125 b^3 c x \left (15+10 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)^3\right )}{16875 c} \] Input:

Integrate[(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^3,x]
 

Output:

(d^2*(1125*a^3*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) - 225*a^2*b*Sqrt[1 + c^2* 
x^2]*(149 + 38*c^2*x^2 + 9*c^4*x^4) + 30*a*b^2*c*x*(2235 + 190*c^2*x^2 + 2 
7*c^4*x^4) - 2*b^3*Sqrt[1 + c^2*x^2]*(31841 + 842*c^2*x^2 + 81*c^4*x^4) + 
15*b*(225*a^2*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) - 30*a*b*Sqrt[1 + c^2*x^2] 
*(149 + 38*c^2*x^2 + 9*c^4*x^4) + 2*b^2*c*x*(2235 + 190*c^2*x^2 + 27*c^4*x 
^4))*ArcSinh[c*x] - 225*b^2*(-15*a*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) + b*S 
qrt[1 + c^2*x^2]*(149 + 38*c^2*x^2 + 9*c^4*x^4))*ArcSinh[c*x]^2 + 1125*b^3 
*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]^3))/(16875*c)
 

Rubi [A] (verified)

Time = 1.83 (sec) , antiderivative size = 424, normalized size of antiderivative = 1.32, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6201, 27, 6201, 6187, 6213, 2009, 6199, 27, 353, 53, 1576, 1140, 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.

\(\displaystyle \int \left (c^2 d x^2+d\right )^2 (a+b \text {arcsinh}(c x))^3 \, dx\)

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 6187

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 2009

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

\(\Big \downarrow \) 6199

\(\displaystyle \frac {4}{5} d^2 \left (-b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (-b c \int \frac {x \left (c^2 x^2+3\right )}{3 \sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^3-3 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2}-\frac {2 b \left (-b c \int \frac {x \left (3 c^4 x^4+10 c^2 x^2+15\right )}{15 \sqrt {c^2 x^2+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^3\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4}{5} d^2 \left (-b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (-\frac {1}{3} b c \int \frac {x \left (c^2 x^2+3\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^3-3 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4+10 c^2 x^2+15\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^3\)

\(\Big \downarrow \) 353

\(\displaystyle \frac {4}{5} d^2 \left (-b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (-\frac {1}{6} b c \int \frac {c^2 x^2+3}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^3-3 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4+10 c^2 x^2+15\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^3\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {4}{5} d^2 \left (-b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (-\frac {1}{6} b c \int \left (\sqrt {c^2 x^2+1}+\frac {2}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^3-3 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {1}{15} b c \int \frac {x \left (3 c^4 x^4+10 c^2 x^2+15\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^3\)

\(\Big \downarrow \) 1576

\(\displaystyle \frac {4}{5} d^2 \left (-b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (-\frac {1}{6} b c \int \left (\sqrt {c^2 x^2+1}+\frac {2}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^3-3 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {1}{30} b c \int \frac {3 c^4 x^4+10 c^2 x^2+15}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^3\)

\(\Big \downarrow \) 1140

\(\displaystyle \frac {4}{5} d^2 \left (-b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (-\frac {1}{6} b c \int \left (\sqrt {c^2 x^2+1}+\frac {2}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 c}\right )+\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^3-3 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c}\right )\right )\right )-\frac {3}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2}-\frac {2 b \left (-\frac {1}{30} b c \int \left (3 \left (c^2 x^2+1\right )^{3/2}+4 \sqrt {c^2 x^2+1}+\frac {8}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{5 c}\right )+\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^3\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^3+\frac {4}{5} d^2 \left (\frac {1}{3} x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^3+\frac {2}{3} \left (x (a+b \text {arcsinh}(c x))^3-3 b c \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c}\right )\right )-b c \left (\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 c}\right )\right )-\frac {3}{5} b c d^2 \left (\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{5 c^2}-\frac {2 b \left (\frac {1}{5} c^4 x^5 (a+b \text {arcsinh}(c x))+\frac {2}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{30} b c \left (\frac {6 \left (c^2 x^2+1\right )^{5/2}}{5 c^2}+\frac {8 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {16 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{5 c}\right )\)

Input:

Int[(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^3,x]
 

Output:

(d^2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^3)/5 - (3*b*c*d^2*(((1 + c^2*x 
^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/(5*c^2) - (2*b*(-1/30*(b*c*((16*Sqrt[1 + 
 c^2*x^2])/c^2 + (8*(1 + c^2*x^2)^(3/2))/(3*c^2) + (6*(1 + c^2*x^2)^(5/2)) 
/(5*c^2))) + x*(a + b*ArcSinh[c*x]) + (2*c^2*x^3*(a + b*ArcSinh[c*x]))/3 + 
 (c^4*x^5*(a + b*ArcSinh[c*x]))/5))/(5*c)))/5 + (4*d^2*((x*(1 + c^2*x^2)*( 
a + b*ArcSinh[c*x])^3)/3 - b*c*(((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^ 
2)/(3*c^2) - (2*b*(-1/6*(b*c*((4*Sqrt[1 + c^2*x^2])/c^2 + (2*(1 + c^2*x^2) 
^(3/2))/(3*c^2))) + x*(a + b*ArcSinh[c*x]) + (c^2*x^3*(a + b*ArcSinh[c*x]) 
)/3))/(3*c)) + (2*(x*(a + b*ArcSinh[c*x])^3 - 3*b*c*((Sqrt[1 + c^2*x^2]*(a 
 + b*ArcSinh[c*x])^2)/c^2 - (2*b*(a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcS 
inh[c*x]))/c)))/3))/5
 

Defintions of rubi rules used

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 53

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, 
x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) 
|| LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
 

rule 353

Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] 
 :> Simp[1/2   Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ 
{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
 

rule 1140

Int[((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x 
_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*x + c*x^2)^p, x], x] /; 
FreeQ[{a, b, c, d, e, m}, x] && IGtQ[p, 0]
 

rule 1576

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^( 
p_.), x_Symbol] :> Simp[1/2   Subst[Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x] 
, x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]
 

rule 2009

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

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 6199

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb 
ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSinh[c*x])   u, 
 x] - Simp[b*c   Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; 
FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 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 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]
 

Maple [A] (verified)

Time = 2.32 (sec) , antiderivative size = 466, normalized size of antiderivative = 1.45

method	result	size
derivativedivides	\(\frac {a^{3} d^{2} \left (\frac {1}{5} x^{5} c^{5}+\frac {2}{3} x^{3} c^{3}+x c \right )+d^{2} b^{3} \left (\frac {8 \operatorname {arcsinh}\left (x c \right )^{3} x c}{15}+\frac {\operatorname {arcsinh}\left (x c \right )^{3} x c \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (x c \right )^{3} x c \left (c^{2} x^{2}+1\right )}{15}-\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}}{5}+\frac {4144 x c \,\operatorname {arcsinh}\left (x c \right )}{1125}-\frac {4144 \sqrt {c^{2} x^{2}+1}}{1125}-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {6 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )}{1125}-\frac {6 \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{625}-\frac {272 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3375}-\frac {4 \operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{15}\right )+3 d^{2} a \,b^{2} \left (\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 x c}{3375}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 x c \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+3 a^{2} b \,d^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {38 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{225}-\frac {x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{25}\right )}{c}\)	\(466\)
default	\(\frac {a^{3} d^{2} \left (\frac {1}{5} x^{5} c^{5}+\frac {2}{3} x^{3} c^{3}+x c \right )+d^{2} b^{3} \left (\frac {8 \operatorname {arcsinh}\left (x c \right )^{3} x c}{15}+\frac {\operatorname {arcsinh}\left (x c \right )^{3} x c \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (x c \right )^{3} x c \left (c^{2} x^{2}+1\right )}{15}-\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}}{5}+\frac {4144 x c \,\operatorname {arcsinh}\left (x c \right )}{1125}-\frac {4144 \sqrt {c^{2} x^{2}+1}}{1125}-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {6 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )}{1125}-\frac {6 \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{625}-\frac {272 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3375}-\frac {4 \operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{15}\right )+3 d^{2} a \,b^{2} \left (\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 x c}{3375}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 x c \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+3 a^{2} b \,d^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {38 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{225}-\frac {x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{25}\right )}{c}\)	\(466\)
parts	\(a^{3} d^{2} \left (\frac {1}{5} c^{4} x^{5}+\frac {2}{3} x^{3} c^{2}+x \right )+\frac {d^{2} b^{3} \left (\frac {8 \operatorname {arcsinh}\left (x c \right )^{3} x c}{15}+\frac {\operatorname {arcsinh}\left (x c \right )^{3} x c \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (x c \right )^{3} x c \left (c^{2} x^{2}+1\right )}{15}-\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}}{5}+\frac {4144 x c \,\operatorname {arcsinh}\left (x c \right )}{1125}-\frac {4144 \sqrt {c^{2} x^{2}+1}}{1125}-\frac {3 \operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {6 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 \,\operatorname {arcsinh}\left (x c \right ) x c \left (c^{2} x^{2}+1\right )}{1125}-\frac {6 \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{625}-\frac {272 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{3375}-\frac {4 \operatorname {arcsinh}\left (x c \right )^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{15}\right )}{c}+\frac {3 d^{2} a \,b^{2} \left (\frac {8 \operatorname {arcsinh}\left (x c \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (x c \right )^{2} x c \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 x c}{3375}-\frac {2 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 x c \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 x c \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (x c \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )}{c}+\frac {3 a^{2} b \,d^{2} \left (\frac {\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}}{3}+x c \,\operatorname {arcsinh}\left (x c \right )-\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {38 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}}{225}-\frac {x^{4} c^{4} \sqrt {c^{2} x^{2}+1}}{25}\right )}{c}\)	\(469\)
orering	\(\frac {x \left (29889 c^{6} x^{6}+179507 c^{4} x^{4}+2768347 c^{2} x^{2}-1732471\right ) \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{3}}{50625 \left (c^{2} x^{2}+1\right )^{3}}-\frac {\left (7857 c^{6} x^{6}+60788 c^{4} x^{4}+1445605 c^{2} x^{2}-316726\right ) \left (4 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{3} c^{2} d x +\frac {3 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} b c}{\sqrt {c^{2} x^{2}+1}}\right )}{50625 c^{2} \left (c^{2} x^{2}+1\right )^{2}}+\frac {2 x \left (189 c^{4} x^{4}+1738 c^{2} x^{2}+53349\right ) \left (8 c^{4} d^{2} x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{3}+\frac {24 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{3} d x b}{\sqrt {c^{2} x^{2}+1}}+4 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{3} c^{2} d +\frac {6 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b^{2} c^{2}}{c^{2} x^{2}+1}-\frac {3 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} b \,c^{3} x}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{16875 c^{2} \left (c^{2} x^{2}+1\right )}-\frac {\left (81 c^{4} x^{4}+842 c^{2} x^{2}+31841\right ) \left (24 c^{4} d^{2} x \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{3}+\frac {72 c^{5} d^{2} x^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} b}{\sqrt {c^{2} x^{2}+1}}+\frac {72 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) c^{4} d x \,b^{2}}{c^{2} x^{2}+1}+\frac {36 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{3} d b}{\sqrt {c^{2} x^{2}+1}}-\frac {36 \left (c^{2} d \,x^{2}+d \right ) \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} c^{5} d \,x^{2} b}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {6 \left (c^{2} d \,x^{2}+d \right )^{2} b^{3} c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {18 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right ) b^{2} c^{4} x}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {9 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} b \,c^{5} x^{2}}{\left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}-\frac {3 \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )^{2} b \,c^{3}}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}\right )}{50625 c^{4}}\)	\(718\)

Input:

int((c^2*d*x^2+d)^2*(a+b*arcsinh(x*c))^3,x,method=_RETURNVERBOSE)
 

Output:

1/c*(a^3*d^2*(1/5*x^5*c^5+2/3*x^3*c^3+x*c)+d^2*b^3*(8/15*arcsinh(x*c)^3*x* 
c+1/5*arcsinh(x*c)^3*x*c*(c^2*x^2+1)^2+4/15*arcsinh(x*c)^3*x*c*(c^2*x^2+1) 
-8/5*arcsinh(x*c)^2*(c^2*x^2+1)^(1/2)+4144/1125*x*c*arcsinh(x*c)-4144/1125 
*(c^2*x^2+1)^(1/2)-3/25*arcsinh(x*c)^2*(c^2*x^2+1)^(5/2)+6/125*arcsinh(x*c 
)*x*c*(c^2*x^2+1)^2+272/1125*arcsinh(x*c)*x*c*(c^2*x^2+1)-6/625*(c^2*x^2+1 
)^(5/2)-272/3375*(c^2*x^2+1)^(3/2)-4/15*arcsinh(x*c)^2*(c^2*x^2+1)^(3/2))+ 
3*d^2*a*b^2*(8/15*arcsinh(x*c)^2*x*c+1/5*arcsinh(x*c)^2*x*c*(c^2*x^2+1)^2+ 
4/15*arcsinh(x*c)^2*x*c*(c^2*x^2+1)-16/15*arcsinh(x*c)*(c^2*x^2+1)^(1/2)+4 
144/3375*x*c-2/25*arcsinh(x*c)*(c^2*x^2+1)^(5/2)+2/125*x*c*(c^2*x^2+1)^2+2 
72/3375*x*c*(c^2*x^2+1)-8/45*arcsinh(x*c)*(c^2*x^2+1)^(3/2))+3*a^2*b*d^2*( 
1/5*arcsinh(x*c)*x^5*c^5+2/3*arcsinh(x*c)*x^3*c^3+x*c*arcsinh(x*c)-149/225 
*(c^2*x^2+1)^(1/2)-38/225*x^2*c^2*(c^2*x^2+1)^(1/2)-1/25*x^4*c^4*(c^2*x^2+ 
1)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.41 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^3 \, dx=\frac {135 \, {\left (25 \, a^{3} + 6 \, a b^{2}\right )} c^{5} d^{2} x^{5} + 150 \, {\left (75 \, a^{3} + 38 \, a b^{2}\right )} c^{3} d^{2} x^{3} + 225 \, {\left (75 \, a^{3} + 298 \, a b^{2}\right )} c d^{2} x + 1125 \, {\left (3 \, b^{3} c^{5} d^{2} x^{5} + 10 \, b^{3} c^{3} d^{2} x^{3} + 15 \, b^{3} c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{3} + 225 \, {\left (45 \, a b^{2} c^{5} d^{2} x^{5} + 150 \, a b^{2} c^{3} d^{2} x^{3} + 225 \, a b^{2} c d^{2} x - {\left (9 \, b^{3} c^{4} d^{2} x^{4} + 38 \, b^{3} c^{2} d^{2} x^{2} + 149 \, b^{3} d^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 15 \, {\left (27 \, {\left (25 \, a^{2} b + 2 \, b^{3}\right )} c^{5} d^{2} x^{5} + 10 \, {\left (225 \, a^{2} b + 38 \, b^{3}\right )} c^{3} d^{2} x^{3} + 15 \, {\left (225 \, a^{2} b + 298 \, b^{3}\right )} c d^{2} x - 30 \, {\left (9 \, a b^{2} c^{4} d^{2} x^{4} + 38 \, a b^{2} c^{2} d^{2} x^{2} + 149 \, a b^{2} d^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (81 \, {\left (25 \, a^{2} b + 2 \, b^{3}\right )} c^{4} d^{2} x^{4} + 2 \, {\left (4275 \, a^{2} b + 842 \, b^{3}\right )} c^{2} d^{2} x^{2} + {\left (33525 \, a^{2} b + 63682 \, b^{3}\right )} d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{16875 \, c} \] Input:

integrate((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^3,x, algorithm="fricas")
 

Output:

1/16875*(135*(25*a^3 + 6*a*b^2)*c^5*d^2*x^5 + 150*(75*a^3 + 38*a*b^2)*c^3* 
d^2*x^3 + 225*(75*a^3 + 298*a*b^2)*c*d^2*x + 1125*(3*b^3*c^5*d^2*x^5 + 10* 
b^3*c^3*d^2*x^3 + 15*b^3*c*d^2*x)*log(c*x + sqrt(c^2*x^2 + 1))^3 + 225*(45 
*a*b^2*c^5*d^2*x^5 + 150*a*b^2*c^3*d^2*x^3 + 225*a*b^2*c*d^2*x - (9*b^3*c^ 
4*d^2*x^4 + 38*b^3*c^2*d^2*x^2 + 149*b^3*d^2)*sqrt(c^2*x^2 + 1))*log(c*x + 
 sqrt(c^2*x^2 + 1))^2 + 15*(27*(25*a^2*b + 2*b^3)*c^5*d^2*x^5 + 10*(225*a^ 
2*b + 38*b^3)*c^3*d^2*x^3 + 15*(225*a^2*b + 298*b^3)*c*d^2*x - 30*(9*a*b^2 
*c^4*d^2*x^4 + 38*a*b^2*c^2*d^2*x^2 + 149*a*b^2*d^2)*sqrt(c^2*x^2 + 1))*lo 
g(c*x + sqrt(c^2*x^2 + 1)) - (81*(25*a^2*b + 2*b^3)*c^4*d^2*x^4 + 2*(4275* 
a^2*b + 842*b^3)*c^2*d^2*x^2 + (33525*a^2*b + 63682*b^3)*d^2)*sqrt(c^2*x^2 
 + 1))/c
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 717 vs. \(2 (311) = 622\).

Time = 0.69 (sec) , antiderivative size = 717, normalized size of antiderivative = 2.23 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^3 \, dx =\text {Too large to display} \] Input:

integrate((c**2*d*x**2+d)**2*(a+b*asinh(c*x))**3,x)
                                                                                    
                                                                                    
 

Output:

Piecewise((a**3*c**4*d**2*x**5/5 + 2*a**3*c**2*d**2*x**3/3 + a**3*d**2*x + 
 3*a**2*b*c**4*d**2*x**5*asinh(c*x)/5 - 3*a**2*b*c**3*d**2*x**4*sqrt(c**2* 
x**2 + 1)/25 + 2*a**2*b*c**2*d**2*x**3*asinh(c*x) - 38*a**2*b*c*d**2*x**2* 
sqrt(c**2*x**2 + 1)/75 + 3*a**2*b*d**2*x*asinh(c*x) - 149*a**2*b*d**2*sqrt 
(c**2*x**2 + 1)/(75*c) + 3*a*b**2*c**4*d**2*x**5*asinh(c*x)**2/5 + 6*a*b** 
2*c**4*d**2*x**5/125 - 6*a*b**2*c**3*d**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c 
*x)/25 + 2*a*b**2*c**2*d**2*x**3*asinh(c*x)**2 + 76*a*b**2*c**2*d**2*x**3/ 
225 - 76*a*b**2*c*d**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/75 + 3*a*b**2*d 
**2*x*asinh(c*x)**2 + 298*a*b**2*d**2*x/75 - 298*a*b**2*d**2*sqrt(c**2*x** 
2 + 1)*asinh(c*x)/(75*c) + b**3*c**4*d**2*x**5*asinh(c*x)**3/5 + 6*b**3*c* 
*4*d**2*x**5*asinh(c*x)/125 - 3*b**3*c**3*d**2*x**4*sqrt(c**2*x**2 + 1)*as 
inh(c*x)**2/25 - 6*b**3*c**3*d**2*x**4*sqrt(c**2*x**2 + 1)/625 + 2*b**3*c* 
*2*d**2*x**3*asinh(c*x)**3/3 + 76*b**3*c**2*d**2*x**3*asinh(c*x)/225 - 38* 
b**3*c*d**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)**2/75 - 1684*b**3*c*d**2*x 
**2*sqrt(c**2*x**2 + 1)/16875 + b**3*d**2*x*asinh(c*x)**3 + 298*b**3*d**2* 
x*asinh(c*x)/75 - 149*b**3*d**2*sqrt(c**2*x**2 + 1)*asinh(c*x)**2/(75*c) - 
 63682*b**3*d**2*sqrt(c**2*x**2 + 1)/(16875*c), Ne(c, 0)), (a**3*d**2*x, T 
rue))
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 861 vs. \(2 (286) = 572\).

Time = 0.07 (sec) , antiderivative size = 861, normalized size of antiderivative = 2.67 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^3 \, dx =\text {Too large to display} \] Input:

integrate((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^3,x, algorithm="maxima")
 

Output:

1/5*b^3*c^4*d^2*x^5*arcsinh(c*x)^3 + 3/5*a*b^2*c^4*d^2*x^5*arcsinh(c*x)^2 
+ 1/5*a^3*c^4*d^2*x^5 + 2/3*b^3*c^2*d^2*x^3*arcsinh(c*x)^3 + 2*a*b^2*c^2*d 
^2*x^3*arcsinh(c*x)^2 + 1/25*(15*x^5*arcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x 
^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a^2*b*c 
^4*d^2 - 2/375*(15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/ 
c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 + 
120*x)/c^4)*a*b^2*c^4*d^2 - 1/5625*(225*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*s 
qrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcsinh(c*x)^2 + 2*c 
*((27*sqrt(c^2*x^2 + 1)*c^2*x^4 - 136*sqrt(c^2*x^2 + 1)*x^2 + 2072*sqrt(c^ 
2*x^2 + 1)/c^2)/c^4 - 15*(9*c^4*x^5 - 20*c^2*x^3 + 120*x)*arcsinh(c*x)/c^5 
))*b^3*c^4*d^2 + 2/3*a^3*c^2*d^2*x^3 + b^3*d^2*x*arcsinh(c*x)^3 + 2/3*(3*x 
^3*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4)) 
*a^2*b*c^2*d^2 - 4/9*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1) 
/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x)/c^2)*a*b^2*c^2*d^2 - 2/27*(9*c*(sqrt( 
c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4)*arcsinh(c*x)^2 + 2*c*((sqr 
t(c^2*x^2 + 1)*x^2 - 20*sqrt(c^2*x^2 + 1)/c^2)/c^2 - 3*(c^2*x^3 - 6*x)*arc 
sinh(c*x)/c^3))*b^3*c^2*d^2 + 3*a*b^2*d^2*x*arcsinh(c*x)^2 - 3*(sqrt(c^2*x 
^2 + 1)*arcsinh(c*x)^2/c - 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))/c)*b^3 
*d^2 + 6*a*b^2*d^2*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^3*d^2*x + 3* 
(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a^2*b*d^2/c
 

Giac [F(-2)]

Exception generated. \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^3 \, dx=\text {Exception raised: TypeError} \] Input:

integrate((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^3,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. \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^3 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^3\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \] Input:

int((a + b*asinh(c*x))^3*(d + c^2*d*x^2)^2,x)
 

Output:

int((a + b*asinh(c*x))^3*(d + c^2*d*x^2)^2, x)
 

Reduce [F]

\[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^3 \, dx=\frac {d^{2} \left (75 \mathit {asinh} \left (c x \right )^{3} b^{3} c x -225 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} b^{3}+225 \mathit {asinh} \left (c x \right )^{2} a \,b^{2} c x -450 \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) a \,b^{2}+45 \mathit {asinh} \left (c x \right ) a^{2} b \,c^{5} x^{5}+150 \mathit {asinh} \left (c x \right ) a^{2} b \,c^{3} x^{3}+225 \mathit {asinh} \left (c x \right ) a^{2} b c x +450 \mathit {asinh} \left (c x \right ) b^{3} c x -9 \sqrt {c^{2} x^{2}+1}\, a^{2} b \,c^{4} x^{4}-38 \sqrt {c^{2} x^{2}+1}\, a^{2} b \,c^{2} x^{2}-149 \sqrt {c^{2} x^{2}+1}\, a^{2} b -450 \sqrt {c^{2} x^{2}+1}\, b^{3}+75 \left (\int \mathit {asinh} \left (c x \right )^{3} x^{4}d x \right ) b^{3} c^{5}+150 \left (\int \mathit {asinh} \left (c x \right )^{3} x^{2}d x \right ) b^{3} c^{3}+225 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{4}d x \right ) a \,b^{2} c^{5}+450 \left (\int \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) a \,b^{2} c^{3}+15 a^{3} c^{5} x^{5}+50 a^{3} c^{3} x^{3}+75 a^{3} c x +450 a \,b^{2} c x \right )}{75 c} \] Input:

int((c^2*d*x^2+d)^2*(a+b*asinh(c*x))^3,x)
 

Output:

(d**2*(75*asinh(c*x)**3*b**3*c*x - 225*sqrt(c**2*x**2 + 1)*asinh(c*x)**2*b 
**3 + 225*asinh(c*x)**2*a*b**2*c*x - 450*sqrt(c**2*x**2 + 1)*asinh(c*x)*a* 
b**2 + 45*asinh(c*x)*a**2*b*c**5*x**5 + 150*asinh(c*x)*a**2*b*c**3*x**3 + 
225*asinh(c*x)*a**2*b*c*x + 450*asinh(c*x)*b**3*c*x - 9*sqrt(c**2*x**2 + 1 
)*a**2*b*c**4*x**4 - 38*sqrt(c**2*x**2 + 1)*a**2*b*c**2*x**2 - 149*sqrt(c* 
*2*x**2 + 1)*a**2*b - 450*sqrt(c**2*x**2 + 1)*b**3 + 75*int(asinh(c*x)**3* 
x**4,x)*b**3*c**5 + 150*int(asinh(c*x)**3*x**2,x)*b**3*c**3 + 225*int(asin 
h(c*x)**2*x**4,x)*a*b**2*c**5 + 450*int(asinh(c*x)**2*x**2,x)*a*b**2*c**3 
+ 15*a**3*c**5*x**5 + 50*a**3*c**3*x**3 + 75*a**3*c*x + 450*a*b**2*c*x))/( 
75*c)