3.3.66 \(\int x^3 (d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx\) [266]

3.3.66.1 Optimal result

Integrand size = 28, antiderivative size = 482 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {304 b^2 d \sqrt {d+c^2 d x^2}}{3675 c^4}+\frac {4 a b d x \sqrt {d+c^2 d x^2}}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {152 b^2 d \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{11025 c^4}-\frac {38 b^2 d \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}}{6125 c^4}+\frac {2 b^2 d \left (1+c^2 x^2\right )^3 \sqrt {d+c^2 d x^2}}{343 c^4}+\frac {4 b^2 d x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{35 c^3 \sqrt {1+c^2 x^2}}-\frac {2 b d x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{105 c \sqrt {1+c^2 x^2}}-\frac {16 b c d x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{175 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d x^7 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{49 \sqrt {1+c^2 x^2}}-\frac {2 d \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{35 c^4}+\frac {d x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{35 c^2}+\frac {3}{35} d x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {1}{7} x^4 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \]

output

1/7*x^4*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2-304/3675*b^2*d*(c^2*d*x^2 
+d)^(1/2)/c^4-152/11025*b^2*d*(c^2*x^2+1)*(c^2*d*x^2+d)^(1/2)/c^4-38/6125* 
b^2*d*(c^2*x^2+1)^2*(c^2*d*x^2+d)^(1/2)/c^4+2/343*b^2*d*(c^2*x^2+1)^3*(c^2 
*d*x^2+d)^(1/2)/c^4-2/35*d*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^4+1/ 
35*d*x^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^2+3/35*d*x^4*(a+b*arcs 
inh(c*x))^2*(c^2*d*x^2+d)^(1/2)+4/35*a*b*d*x*(c^2*d*x^2+d)^(1/2)/c^3/(c^2* 
x^2+1)^(1/2)+4/35*b^2*d*x*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/c^3/(c^2*x^2+1) 
^(1/2)-2/105*b*d*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^ 
(1/2)-16/175*b*c*d*x^5*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^ 
(1/2)-2/49*b*c^3*d*x^7*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^ 
(1/2)
 

3.3.66.2 Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.52 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d \sqrt {d+c^2 d x^2} \left (11025 a^2 \left (1+c^2 x^2\right )^3 \left (-2+5 c^2 x^2\right )-210 a b c x \sqrt {1+c^2 x^2} \left (-210+35 c^2 x^2+168 c^4 x^4+75 c^6 x^6\right )+2 b^2 \left (-18692-20371 c^2 x^2+499 c^4 x^4+3303 c^6 x^6+1125 c^8 x^8\right )-210 b \left (-105 a \left (1+c^2 x^2\right )^3 \left (-2+5 c^2 x^2\right )+b c x \sqrt {1+c^2 x^2} \left (-210+35 c^2 x^2+168 c^4 x^4+75 c^6 x^6\right )\right ) \text {arcsinh}(c x)+11025 b^2 \left (1+c^2 x^2\right )^3 \left (-2+5 c^2 x^2\right ) \text {arcsinh}(c x)^2\right )}{385875 c^4 \left (1+c^2 x^2\right )} \]

input

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

output

(d*Sqrt[d + c^2*d*x^2]*(11025*a^2*(1 + c^2*x^2)^3*(-2 + 5*c^2*x^2) - 210*a 
*b*c*x*Sqrt[1 + c^2*x^2]*(-210 + 35*c^2*x^2 + 168*c^4*x^4 + 75*c^6*x^6) + 
2*b^2*(-18692 - 20371*c^2*x^2 + 499*c^4*x^4 + 3303*c^6*x^6 + 1125*c^8*x^8) 
 - 210*b*(-105*a*(1 + c^2*x^2)^3*(-2 + 5*c^2*x^2) + b*c*x*Sqrt[1 + c^2*x^2 
]*(-210 + 35*c^2*x^2 + 168*c^4*x^4 + 75*c^6*x^6))*ArcSinh[c*x] + 11025*b^2 
*(1 + c^2*x^2)^3*(-2 + 5*c^2*x^2)*ArcSinh[c*x]^2))/(385875*c^4*(1 + c^2*x^ 
2))
 

3.3.66.3 Rubi [A] (verified)

Time = 2.66 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.12, number of steps used = 19, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.643, Rules used = {6223, 6218, 27, 354, 86, 2009, 6221, 6191, 243, 53, 2009, 6227, 6191, 243, 53, 2009, 6213, 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.

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

\(\Big \downarrow \) 6223

\(\displaystyle -\frac {2 b c d \sqrt {c^2 d x^2+d} \int x^4 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{7 \sqrt {c^2 x^2+1}}+\frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 6218

\(\displaystyle \frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x^5 \left (5 c^2 x^2+7\right )}{35 \sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{7 \sqrt {c^2 x^2+1}}+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{35} b c \int \frac {x^5 \left (5 c^2 x^2+7\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{7 \sqrt {c^2 x^2+1}}+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 354

\(\displaystyle \frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{70} b c \int \frac {x^4 \left (5 c^2 x^2+7\right )}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{7 \sqrt {c^2 x^2+1}}+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 86

\(\displaystyle \frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{70} b c \int \left (\frac {5 \left (c^2 x^2+1\right )^{5/2}}{c^4}-\frac {8 \left (c^2 x^2+1\right )^{3/2}}{c^4}+\frac {\sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))\right )}{7 \sqrt {c^2 x^2+1}}+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{7} d \int x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6221

\(\displaystyle \frac {3}{7} d \left (-\frac {2 b c \sqrt {c^2 d x^2+d} \int x^4 (a+b \text {arcsinh}(c x))dx}{5 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {3}{7} d \left (-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{5} b c \int \frac {x^5}{\sqrt {c^2 x^2+1}}dx\right )}{5 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx^2\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \int \left (\frac {\left (c^2 x^2+1\right )^{3/2}}{c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {1}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \int x^2 (a+b \text {arcsinh}(c x))dx}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {c^2 x^2+1}}dx\right )}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 243

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx^2\right )}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 53

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \int \left (\frac {\sqrt {c^2 x^2+1}}{c^2}-\frac {1}{c^2 \sqrt {c^2 x^2+1}}\right )dx^2\right )}{3 c}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {3}{7} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \int (a+b \text {arcsinh}(c x))dx}{c}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}\right )+\frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{7} x^4 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{7} c^2 x^7 (a+b \text {arcsinh}(c x))+\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{70} b c \left (\frac {10 \left (c^2 x^2+1\right )^{7/2}}{7 c^6}-\frac {16 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}+\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {4 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{7 \sqrt {c^2 x^2+1}}+\frac {3}{7} d \left (\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 \sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \left (\frac {x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{3 c^2}-\frac {2 \left (\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{c^2}-\frac {2 b \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c}\right )}{3 c^2}-\frac {2 b \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c}\right )}{5 \sqrt {c^2 x^2+1}}\right )\)

input

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

output

(x^4*(d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/7 - (2*b*c*d*Sqrt[d + c 
^2*d*x^2]*(-1/70*(b*c*((4*Sqrt[1 + c^2*x^2])/c^6 + (2*(1 + c^2*x^2)^(3/2)) 
/(3*c^6) - (16*(1 + c^2*x^2)^(5/2))/(5*c^6) + (10*(1 + c^2*x^2)^(7/2))/(7* 
c^6))) + (x^5*(a + b*ArcSinh[c*x]))/5 + (c^2*x^7*(a + b*ArcSinh[c*x]))/7)) 
/(7*Sqrt[1 + c^2*x^2]) + (3*d*((x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x 
])^2)/5 - (2*b*c*Sqrt[d + c^2*d*x^2]*(-1/10*(b*c*((2*Sqrt[1 + c^2*x^2])/c^ 
6 - (4*(1 + c^2*x^2)^(3/2))/(3*c^6) + (2*(1 + c^2*x^2)^(5/2))/(5*c^6))) + 
(x^5*(a + b*ArcSinh[c*x]))/5))/(5*Sqrt[1 + c^2*x^2]) + (Sqrt[d + c^2*d*x^2 
]*((x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(3*c^2) - (2*b*(-1/6*(b* 
c*((-2*Sqrt[1 + c^2*x^2])/c^4 + (2*(1 + c^2*x^2)^(3/2))/(3*c^4))) + (x^3*( 
a + b*ArcSinh[c*x]))/3))/(3*c) - (2*((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*ArcSinh[c*x]))/c))/( 
3*c^2)))/(5*Sqrt[1 + c^2*x^2])))/7
 

3.3.66.3.1 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 86

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ 
.), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 
] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p 
+ 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
 

rule 243

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2   Subst[In 
t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I 
ntegerQ[(m - 1)/2]
 

rule 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(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] && IntegerQ 
[(m - 1)/2]
 

rule 2009

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

rule 6191

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] 
 :> Simp[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* 
(n/(d*(m + 1)))   Int[(d*x)^(m + 1)*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + 
c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
 

rule 6213

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

rule 6218

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ 
)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(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, f, m}, x] && EqQ[e, c^2*d] 
&& IGtQ[p, 0]
 

rule 6221

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + 
 (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc 
Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt 
[1 + c^2*x^2]]   Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] 
, x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   I 
nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d 
, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
 

rule 6223

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

rule 6227

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

3.3.66.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1765\) vs. \(2(420)=840\).

Time = 0.39 (sec) , antiderivative size = 1766, normalized size of antiderivative = 3.66

method	result	size
default	\(\text {Expression too large to display}\)	\(1766\)
parts	\(\text {Expression too large to display}\)	\(1766\)

input

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

output

a^2*(1/7*x^2*(c^2*d*x^2+d)^(5/2)/c^2/d-2/35/d/c^4*(c^2*d*x^2+d)^(5/2))+b^2 
*(1/43904*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8+64*c^7*x^7*(c^2*x^2+1)^(1/2)+1 
44*c^6*x^6+112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4+56*c^3*x^3*(c^2*x^2+1 
)^(1/2)+25*c^2*x^2+7*c*x*(c^2*x^2+1)^(1/2)+1)*(49*arcsinh(c*x)^2-14*arcsin 
h(c*x)+2)*d/c^4/(c^2*x^2+1)+1/16000*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6+16*c 
^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4+20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^ 
2+5*c*x*(c^2*x^2+1)^(1/2)+1)*(25*arcsinh(c*x)^2-10*arcsinh(c*x)+2)*d/c^4/( 
c^2*x^2+1)-1/1152*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^( 
1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(9*arcsinh(c*x)^2-6*arcsinh(c*x) 
+2)*d/c^4/(c^2*x^2+1)-3/128*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1) 
^(1/2)+1)*(arcsinh(c*x)^2-2*arcsinh(c*x)+2)*d/c^4/(c^2*x^2+1)-3/128*(d*(c^ 
2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)^2+2*arcsin 
h(c*x)+2)*d/c^4/(c^2*x^2+1)-1/1152*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3* 
x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*(9*arcsinh(c*x) 
^2+6*arcsinh(c*x)+2)*d/c^4/(c^2*x^2+1)+1/16000*(d*(c^2*x^2+1))^(1/2)*(16*c 
^6*x^6-16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4-20*c^3*x^3*(c^2*x^2+1)^(1/2 
)+13*c^2*x^2-5*c*x*(c^2*x^2+1)^(1/2)+1)*(25*arcsinh(c*x)^2+10*arcsinh(c*x) 
+2)*d/c^4/(c^2*x^2+1)+1/43904*(d*(c^2*x^2+1))^(1/2)*(64*c^8*x^8-64*c^7*x^7 
*(c^2*x^2+1)^(1/2)+144*c^6*x^6-112*c^5*x^5*(c^2*x^2+1)^(1/2)+104*c^4*x^4-5 
6*c^3*x^3*(c^2*x^2+1)^(1/2)+25*c^2*x^2-7*c*x*(c^2*x^2+1)^(1/2)+1)*(49*a...
 

3.3.66.5 Fricas [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 402, normalized size of antiderivative = 0.83 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {11025 \, {\left (5 \, b^{2} c^{8} d x^{8} + 13 \, b^{2} c^{6} d x^{6} + 9 \, b^{2} c^{4} d x^{4} - b^{2} c^{2} d x^{2} - 2 \, b^{2} d\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 210 \, {\left (525 \, a b c^{8} d x^{8} + 1365 \, a b c^{6} d x^{6} + 945 \, a b c^{4} d x^{4} - 105 \, a b c^{2} d x^{2} - 210 \, a b d - {\left (75 \, b^{2} c^{7} d x^{7} + 168 \, b^{2} c^{5} d x^{5} + 35 \, b^{2} c^{3} d x^{3} - 210 \, b^{2} c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (1125 \, {\left (49 \, a^{2} + 2 \, b^{2}\right )} c^{8} d x^{8} + 9 \, {\left (15925 \, a^{2} + 734 \, b^{2}\right )} c^{6} d x^{6} + {\left (99225 \, a^{2} + 998 \, b^{2}\right )} c^{4} d x^{4} - {\left (11025 \, a^{2} + 40742 \, b^{2}\right )} c^{2} d x^{2} - 2 \, {\left (11025 \, a^{2} + 18692 \, b^{2}\right )} d - 210 \, {\left (75 \, a b c^{7} d x^{7} + 168 \, a b c^{5} d x^{5} + 35 \, a b c^{3} d x^{3} - 210 \, a b c d x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{385875 \, {\left (c^{6} x^{2} + c^{4}\right )}} \]

input

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

output

1/385875*(11025*(5*b^2*c^8*d*x^8 + 13*b^2*c^6*d*x^6 + 9*b^2*c^4*d*x^4 - b^ 
2*c^2*d*x^2 - 2*b^2*d)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))^2 
+ 210*(525*a*b*c^8*d*x^8 + 1365*a*b*c^6*d*x^6 + 945*a*b*c^4*d*x^4 - 105*a* 
b*c^2*d*x^2 - 210*a*b*d - (75*b^2*c^7*d*x^7 + 168*b^2*c^5*d*x^5 + 35*b^2*c 
^3*d*x^3 - 210*b^2*c*d*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d)*log(c*x + 
 sqrt(c^2*x^2 + 1)) + (1125*(49*a^2 + 2*b^2)*c^8*d*x^8 + 9*(15925*a^2 + 73 
4*b^2)*c^6*d*x^6 + (99225*a^2 + 998*b^2)*c^4*d*x^4 - (11025*a^2 + 40742*b^ 
2)*c^2*d*x^2 - 2*(11025*a^2 + 18692*b^2)*d - 210*(75*a*b*c^7*d*x^7 + 168*a 
*b*c^5*d*x^5 + 35*a*b*c^3*d*x^3 - 210*a*b*c*d*x)*sqrt(c^2*x^2 + 1))*sqrt(c 
^2*d*x^2 + d))/(c^6*x^2 + c^4)
 

3.3.66.6 Sympy [F]

\[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^{3} \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]

input

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

output

Integral(x**3*(d*(c**2*x**2 + 1))**(3/2)*(a + b*asinh(c*x))**2, x)
 

3.3.66.7 Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 346, normalized size of antiderivative = 0.72 \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{35} \, {\left (\frac {5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} b^{2} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{35} \, {\left (\frac {5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a b \operatorname {arsinh}\left (c x\right ) + \frac {1}{35} \, {\left (\frac {5 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}}{c^{4} d}\right )} a^{2} + \frac {2}{385875} \, b^{2} {\left (\frac {1125 \, \sqrt {c^{2} x^{2} + 1} c^{4} d^{\frac {3}{2}} x^{6} + 2178 \, \sqrt {c^{2} x^{2} + 1} c^{2} d^{\frac {3}{2}} x^{4} - 1679 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {3}{2}} x^{2} - \frac {18692 \, \sqrt {c^{2} x^{2} + 1} d^{\frac {3}{2}}}{c^{2}}}{c^{2}} - \frac {105 \, {\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} + 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} - 210 \, d^{\frac {3}{2}} x\right )} \operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (75 \, c^{6} d^{\frac {3}{2}} x^{7} + 168 \, c^{4} d^{\frac {3}{2}} x^{5} + 35 \, c^{2} d^{\frac {3}{2}} x^{3} - 210 \, d^{\frac {3}{2}} x\right )} a b}{3675 \, c^{3}} \]

input

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

output

1/35*(5*(c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(5/2)/(c^4*d 
))*b^2*arcsinh(c*x)^2 + 2/35*(5*(c^2*d*x^2 + d)^(5/2)*x^2/(c^2*d) - 2*(c^2 
*d*x^2 + d)^(5/2)/(c^4*d))*a*b*arcsinh(c*x) + 1/35*(5*(c^2*d*x^2 + d)^(5/2 
)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(5/2)/(c^4*d))*a^2 + 2/385875*b^2*((1125 
*sqrt(c^2*x^2 + 1)*c^4*d^(3/2)*x^6 + 2178*sqrt(c^2*x^2 + 1)*c^2*d^(3/2)*x^ 
4 - 1679*sqrt(c^2*x^2 + 1)*d^(3/2)*x^2 - 18692*sqrt(c^2*x^2 + 1)*d^(3/2)/c 
^2)/c^2 - 105*(75*c^6*d^(3/2)*x^7 + 168*c^4*d^(3/2)*x^5 + 35*c^2*d^(3/2)*x 
^3 - 210*d^(3/2)*x)*arcsinh(c*x)/c^3) - 2/3675*(75*c^6*d^(3/2)*x^7 + 168*c 
^4*d^(3/2)*x^5 + 35*c^2*d^(3/2)*x^3 - 210*d^(3/2)*x)*a*b/c^3
 

3.3.66.8 Giac [F(-2)]

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

input

integrate(x^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^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
 

3.3.66.9 Mupad [F(-1)]

Timed out. \[ \int x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]

input

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

output

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