\(\int x^2 (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx\) [281]

Optimal result

Integrand size = 28, antiderivative size = 536 \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {359 b^2 d^2 x \sqrt {d+c^2 d x^2}}{36864 c^2}+\frac {1079 b^2 d^2 x^3 \sqrt {d+c^2 d x^2}}{55296}+\frac {209 b^2 c^2 d^2 x^5 \sqrt {d+c^2 d x^2}}{13824}+\frac {1}{256} b^2 c^4 d^2 x^7 \sqrt {d+c^2 d x^2}+\frac {359 b^2 d^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{36864 c^3 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{128 c \sqrt {1+c^2 x^2}}-\frac {59 b c d^2 x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{384 \sqrt {1+c^2 x^2}}-\frac {17 b c^3 d^2 x^6 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{144 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{32 \sqrt {1+c^2 x^2}}+\frac {5 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{128 c^2}+\frac {5}{64} d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{48} d x^3 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{8} x^3 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{384 b c^3 \sqrt {1+c^2 x^2}} \] Output:

-359/36864*b^2*d^2*x*(c^2*d*x^2+d)^(1/2)/c^2+1079/55296*b^2*d^2*x^3*(c^2*d 
*x^2+d)^(1/2)+209/13824*b^2*c^2*d^2*x^5*(c^2*d*x^2+d)^(1/2)+1/256*b^2*c^4* 
d^2*x^7*(c^2*d*x^2+d)^(1/2)+359/36864*b^2*d^2*(c^2*d*x^2+d)^(1/2)*arcsinh( 
c*x)/c^3/(c^2*x^2+1)^(1/2)-5/128*b*d^2*x^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsin 
h(c*x))/c/(c^2*x^2+1)^(1/2)-59/384*b*c*d^2*x^4*(c^2*d*x^2+d)^(1/2)*(a+b*ar 
csinh(c*x))/(c^2*x^2+1)^(1/2)-17/144*b*c^3*d^2*x^6*(c^2*d*x^2+d)^(1/2)*(a+ 
b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-1/32*b*c^5*d^2*x^8*(c^2*d*x^2+d)^(1/2)*( 
a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)+5/128*d^2*x*(c^2*d*x^2+d)^(1/2)*(a+b*a 
rcsinh(c*x))^2/c^2+5/64*d^2*x^3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2+5 
/48*d*x^3*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2+1/8*x^3*(c^2*d*x^2+d)^( 
5/2)*(a+b*arcsinh(c*x))^2-5/384*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x)) 
^3/b/c^3/(c^2*x^2+1)^(1/2)
 

Mathematica [A] (verified)

Time = 2.74 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.15 \[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (34560 a^2 c x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+271872 a^2 c^3 x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+313344 a^2 c^5 x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+110592 a^2 c^7 x^7 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}-11520 b^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^3+13824 a b \sqrt {d+c^2 d x^2} \cosh (2 \text {arcsinh}(c x))-3456 a b \sqrt {d+c^2 d x^2} \cosh (4 \text {arcsinh}(c x))-1536 a b \sqrt {d+c^2 d x^2} \cosh (6 \text {arcsinh}(c x))-216 a b \sqrt {d+c^2 d x^2} \cosh (8 \text {arcsinh}(c x))-34560 a^2 \sqrt {d} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-6912 b^2 \sqrt {d+c^2 d x^2} \sinh (2 \text {arcsinh}(c x))+864 b^2 \sqrt {d+c^2 d x^2} \sinh (4 \text {arcsinh}(c x))+256 b^2 \sqrt {d+c^2 d x^2} \sinh (6 \text {arcsinh}(c x))+27 b^2 \sqrt {d+c^2 d x^2} \sinh (8 \text {arcsinh}(c x))+24 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) (576 b \cosh (2 \text {arcsinh}(c x))-144 b \cosh (4 \text {arcsinh}(c x))-64 b \cosh (6 \text {arcsinh}(c x))-9 b \cosh (8 \text {arcsinh}(c x))-1152 a \sinh (2 \text {arcsinh}(c x))+576 a \sinh (4 \text {arcsinh}(c x))+384 a \sinh (6 \text {arcsinh}(c x))+72 a \sinh (8 \text {arcsinh}(c x)))+288 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2 (-120 a-48 b \sinh (2 \text {arcsinh}(c x))+24 b \sinh (4 \text {arcsinh}(c x))+16 b \sinh (6 \text {arcsinh}(c x))+3 b \sinh (8 \text {arcsinh}(c x)))\right )}{884736 c^3 \sqrt {1+c^2 x^2}} \] Input:

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

Output:

(d^2*(34560*a^2*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 271872*a^2*c^3 
*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 313344*a^2*c^5*x^5*Sqrt[1 + c 
^2*x^2]*Sqrt[d + c^2*d*x^2] + 110592*a^2*c^7*x^7*Sqrt[1 + c^2*x^2]*Sqrt[d 
+ c^2*d*x^2] - 11520*b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 + 13824*a*b*Sq 
rt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 3456*a*b*Sqrt[d + c^2*d*x^2]*Cosh 
[4*ArcSinh[c*x]] - 1536*a*b*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] - 216 
*a*b*Sqrt[d + c^2*d*x^2]*Cosh[8*ArcSinh[c*x]] - 34560*a^2*Sqrt[d]*Sqrt[1 + 
 c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 6912*b^2*Sqrt[d + c^2 
*d*x^2]*Sinh[2*ArcSinh[c*x]] + 864*b^2*Sqrt[d + c^2*d*x^2]*Sinh[4*ArcSinh[ 
c*x]] + 256*b^2*Sqrt[d + c^2*d*x^2]*Sinh[6*ArcSinh[c*x]] + 27*b^2*Sqrt[d + 
 c^2*d*x^2]*Sinh[8*ArcSinh[c*x]] + 24*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*( 
576*b*Cosh[2*ArcSinh[c*x]] - 144*b*Cosh[4*ArcSinh[c*x]] - 64*b*Cosh[6*ArcS 
inh[c*x]] - 9*b*Cosh[8*ArcSinh[c*x]] - 1152*a*Sinh[2*ArcSinh[c*x]] + 576*a 
*Sinh[4*ArcSinh[c*x]] + 384*a*Sinh[6*ArcSinh[c*x]] + 72*a*Sinh[8*ArcSinh[c 
*x]]) + 288*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(-120*a - 48*b*Sinh[2*Arc 
Sinh[c*x]] + 24*b*Sinh[4*ArcSinh[c*x]] + 16*b*Sinh[6*ArcSinh[c*x]] + 3*b*S 
inh[8*ArcSinh[c*x]])))/(884736*c^3*Sqrt[1 + c^2*x^2])
 

Rubi [A] (verified)

Time = 3.94 (sec) , antiderivative size = 743, normalized size of antiderivative = 1.39, number of steps used = 26, number of rules used = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.929, Rules used = {6223, 6218, 27, 1590, 27, 363, 262, 262, 222, 6223, 6218, 27, 363, 262, 262, 222, 6221, 6191, 262, 262, 222, 6227, 6191, 262, 222, 6198}

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.

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

\(\Big \downarrow \) 6223

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

\(\Big \downarrow \) 6218

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x^4 \left (3 c^4 x^4+8 c^2 x^2+6\right )}{24 \sqrt {c^2 x^2+1}}dx+\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{24} b c \int \frac {x^4 \left (3 c^4 x^4+8 c^2 x^2+6\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 1590

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{24} b c \left (\frac {\int \frac {c^2 x^4 \left (43 c^2 x^2+48\right )}{\sqrt {c^2 x^2+1}}dx}{8 c^2}+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )+\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{24} b c \left (\frac {1}{8} \int \frac {x^4 \left (43 c^2 x^2+48\right )}{\sqrt {c^2 x^2+1}}dx+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )+\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )+\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )+\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )+\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {5}{8} d \int x^2 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6223

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \int x^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6218

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x^4 \left (2 c^2 x^2+3\right )}{12 \sqrt {c^2 x^2+1}}dx+\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{12} b c \int \frac {x^4 \left (2 c^2 x^2+3\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{12} b c \left (\frac {4}{3} \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )+\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )+\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx-\frac {b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )+\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \int x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2dx+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6221

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \int x^3 (a+b \text {arcsinh}(c x))dx}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6227

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}-\frac {b \int x (a+b \text {arcsinh}(c x))dx}{c}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx\right )}{c}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\int \frac {1}{\sqrt {c^2 x^2+1}}dx}{2 c^2}\right )\right )}{c}-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {5}{8} d \left (\frac {1}{2} d \left (\frac {\sqrt {c^2 d x^2+d} \left (-\frac {\int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 x^2+1}}dx}{2 c^2}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{c}\right )}{4 \sqrt {c^2 x^2+1}}+\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )+\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {1}{8} x^3 \left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2+\frac {5}{8} d \left (\frac {1}{6} x^3 \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{2} d \left (\frac {1}{4} x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2+\frac {\sqrt {c^2 d x^2+d} \left (-\frac {(a+b \text {arcsinh}(c x))^3}{6 b c^3}+\frac {x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 c^2}-\frac {b \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{c}\right )}{4 \sqrt {c^2 x^2+1}}-\frac {b c \sqrt {c^2 d x^2+d} \left (\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{4} b c \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )\right )}{2 \sqrt {c^2 x^2+1}}\right )-\frac {b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{6} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{12} b c \left (\frac {4}{3} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {1}{3} x^5 \sqrt {c^2 x^2+1}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )-\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{8} c^4 x^8 (a+b \text {arcsinh}(c x))+\frac {1}{3} c^2 x^6 (a+b \text {arcsinh}(c x))+\frac {1}{4} x^4 (a+b \text {arcsinh}(c x))-\frac {1}{24} b c \left (\frac {1}{8} \left (\frac {73}{6} \left (\frac {x^3 \sqrt {c^2 x^2+1}}{4 c^2}-\frac {3 \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{4 c^2}\right )+\frac {43}{6} x^5 \sqrt {c^2 x^2+1}\right )+\frac {3}{8} c^2 x^7 \sqrt {c^2 x^2+1}\right )\right )}{4 \sqrt {c^2 x^2+1}}\)

Input:

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

Output:

(x^3*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/8 - (b*c*d^2*Sqrt[d + c 
^2*d*x^2]*((x^4*(a + b*ArcSinh[c*x]))/4 + (c^2*x^6*(a + b*ArcSinh[c*x]))/3 
 + (c^4*x^8*(a + b*ArcSinh[c*x]))/8 - (b*c*((3*c^2*x^7*Sqrt[1 + c^2*x^2])/ 
8 + ((43*x^5*Sqrt[1 + c^2*x^2])/6 + (73*((x^3*Sqrt[1 + c^2*x^2])/(4*c^2) - 
 (3*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)))/(4*c^2)))/6)/8 
))/24))/(4*Sqrt[1 + c^2*x^2]) + (5*d*((x^3*(d + c^2*d*x^2)^(3/2)*(a + b*Ar 
cSinh[c*x])^2)/6 - (b*c*d*Sqrt[d + c^2*d*x^2]*((x^4*(a + b*ArcSinh[c*x]))/ 
4 + (c^2*x^6*(a + b*ArcSinh[c*x]))/6 - (b*c*((x^5*Sqrt[1 + c^2*x^2])/3 + ( 
4*((x^3*Sqrt[1 + c^2*x^2])/(4*c^2) - (3*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - A 
rcSinh[c*x]/(2*c^3)))/(4*c^2)))/3))/12))/(3*Sqrt[1 + c^2*x^2]) + (d*((x^3* 
Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/4 - (b*c*Sqrt[d + c^2*d*x^2]*( 
(x^4*(a + b*ArcSinh[c*x]))/4 - (b*c*((x^3*Sqrt[1 + c^2*x^2])/(4*c^2) - (3* 
((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c^3)))/(4*c^2)))/4))/(2*S 
qrt[1 + c^2*x^2]) + (Sqrt[d + c^2*d*x^2]*((x*Sqrt[1 + c^2*x^2]*(a + b*ArcS 
inh[c*x])^2)/(2*c^2) - (a + b*ArcSinh[c*x])^3/(6*b*c^3) - (b*((x^2*(a + b* 
ArcSinh[c*x]))/2 - (b*c*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSinh[c*x]/(2*c 
^3)))/2))/c))/(4*Sqrt[1 + c^2*x^2])))/2))/8
 

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 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt 
[a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
 

rule 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 1590

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( 
c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[c^p*(f*x)^(m + 4*p - 1)*((d + e*x^2)^ 
(q + 1)/(e*f^(4*p - 1)*(m + 4*p + 2*q + 1))), x] + Simp[1/(e*(m + 4*p + 2*q 
 + 1))   Int[(f*x)^m*(d + e*x^2)^q*ExpandToSum[e*(m + 4*p + 2*q + 1)*((a + 
b*x^2 + c*x^4)^p - c^p*x^(4*p)) - d*c^p*(m + 4*p - 1)*x^(4*p - 2), x], x], 
x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[p, 
0] &&  !IntegerQ[q] && NeQ[m + 4*p + 2*q + 1, 0]
 

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 6198

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

rule 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2279\) vs. \(2(468)=936\).

Time = 0.90 (sec) , antiderivative size = 2280, normalized size of antiderivative = 4.25

method	result	size
default	\(\text {Expression too large to display}\)	\(2280\)
parts	\(\text {Expression too large to display}\)	\(2280\)

Input:

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

Output:

1/8*a^2*x*(c^2*d*x^2+d)^(7/2)/c^2/d-1/48*a^2/c^2*x*(c^2*d*x^2+d)^(5/2)-5/1 
92*a^2/c^2*d*x*(c^2*d*x^2+d)^(3/2)-5/128*a^2/c^2*d^2*x*(c^2*d*x^2+d)^(1/2) 
-5/128*a^2/c^2*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^( 
1/2)+b^2*(-5/384*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^3*arcsinh(x*c)^ 
3*d^2+1/65536*(d*(c^2*x^2+1))^(1/2)*(128*c^9*x^9+128*(c^2*x^2+1)^(1/2)*x^8 
*c^8+320*x^7*c^7+256*x^6*c^6*(c^2*x^2+1)^(1/2)+272*x^5*c^5+160*x^4*c^4*(c^ 
2*x^2+1)^(1/2)+88*x^3*c^3+32*x^2*c^2*(c^2*x^2+1)^(1/2)+8*x*c+(c^2*x^2+1)^( 
1/2))*(32*arcsinh(x*c)^2-8*arcsinh(x*c)+1)*d^2/c^3/(c^2*x^2+1)+1/6912*(d*( 
c^2*x^2+1))^(1/2)*(32*x^7*c^7+32*x^6*c^6*(c^2*x^2+1)^(1/2)+64*x^5*c^5+48*x 
^4*c^4*(c^2*x^2+1)^(1/2)+38*x^3*c^3+18*x^2*c^2*(c^2*x^2+1)^(1/2)+6*x*c+(c^ 
2*x^2+1)^(1/2))*(18*arcsinh(x*c)^2-6*arcsinh(x*c)+1)*d^2/c^3/(c^2*x^2+1)+1 
/2048*(d*(c^2*x^2+1))^(1/2)*(8*x^5*c^5+8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3* 
c^3+8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2+1)^(1/2))*(8*arcsinh(x*c)^2 
-4*arcsinh(x*c)+1)*d^2/c^3/(c^2*x^2+1)-1/256*(d*(c^2*x^2+1))^(1/2)*(2*x^3* 
c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2+1)^(1/2))*(2*arcsinh(x*c)^2 
-2*arcsinh(x*c)+1)*d^2/c^3/(c^2*x^2+1)-1/256*(d*(c^2*x^2+1))^(1/2)*(2*x^3* 
c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c-(c^2*x^2+1)^(1/2))*(2*arcsinh(x*c)^2 
+2*arcsinh(x*c)+1)*d^2/c^3/(c^2*x^2+1)+1/2048*(d*(c^2*x^2+1))^(1/2)*(8*x^5 
*c^5-8*x^4*c^4*(c^2*x^2+1)^(1/2)+12*x^3*c^3-8*x^2*c^2*(c^2*x^2+1)^(1/2)+4* 
x*c-(c^2*x^2+1)^(1/2))*(8*arcsinh(x*c)^2+4*arcsinh(x*c)+1)*d^2/c^3/(c^2...
 

Fricas [F]

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

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

Output:

integral((a^2*c^4*d^2*x^6 + 2*a^2*c^2*d^2*x^4 + a^2*d^2*x^2 + (b^2*c^4*d^2 
*x^6 + 2*b^2*c^2*d^2*x^4 + b^2*d^2*x^2)*arcsinh(c*x)^2 + 2*(a*b*c^4*d^2*x^ 
6 + 2*a*b*c^2*d^2*x^4 + a*b*d^2*x^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

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

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

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int x^2 \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d}\, d^{2} \left (48 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{7} x^{7}+136 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{5} x^{5}+118 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}+15 \sqrt {c^{2} x^{2}+1}\, a^{2} c x +768 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{6}d x \right ) a b \,c^{7}+1536 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{4}d x \right ) a b \,c^{5}+768 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) a b \,c^{3}+384 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{6}d x \right ) b^{2} c^{7}+768 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{4}d x \right ) b^{2} c^{5}+384 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x^{2}d x \right ) b^{2} c^{3}-15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2}\right )}{384 c^{3}} \] Input:

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

Output:

(sqrt(d)*d**2*(48*sqrt(c**2*x**2 + 1)*a**2*c**7*x**7 + 136*sqrt(c**2*x**2 
+ 1)*a**2*c**5*x**5 + 118*sqrt(c**2*x**2 + 1)*a**2*c**3*x**3 + 15*sqrt(c** 
2*x**2 + 1)*a**2*c*x + 768*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**6,x)*a*b* 
c**7 + 1536*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x**4,x)*a*b*c**5 + 768*int( 
sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2,x)*a*b*c**3 + 384*int(sqrt(c**2*x**2 + 
 1)*asinh(c*x)**2*x**6,x)*b**2*c**7 + 768*int(sqrt(c**2*x**2 + 1)*asinh(c* 
x)**2*x**4,x)*b**2*c**5 + 384*int(sqrt(c**2*x**2 + 1)*asinh(c*x)**2*x**2,x 
)*b**2*c**3 - 15*log(sqrt(c**2*x**2 + 1) + c*x)*a**2))/(384*c**3)