3.3.77 \(\int (d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx\) [277]

3.3.77.1 Optimal result

Integrand size = 25, antiderivative size = 420 \[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {245 b^2 d^2 x \sqrt {d+c^2 d x^2}}{1152}+\frac {65 b^2 d^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{1728}+\frac {1}{108} b^2 d^2 x \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}-\frac {115 b^2 d^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{1152 c \sqrt {1+c^2 x^2}}-\frac {5 b c d^2 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 \sqrt {1+c^2 x^2}}-\frac {5 b d^2 \left (1+c^2 x^2\right )^{3/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{48 c}-\frac {b d^2 \left (1+c^2 x^2\right )^{5/2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{18 c}+\frac {5}{16} d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{24} d x \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2+\frac {1}{6} x \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2+\frac {5 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^3}{48 b c \sqrt {1+c^2 x^2}} \]

output

5/24*d*x*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2+1/6*x*(c^2*d*x^2+d)^(5/2 
)*(a+b*arcsinh(c*x))^2+245/1152*b^2*d^2*x*(c^2*d*x^2+d)^(1/2)+65/1728*b^2* 
d^2*x*(c^2*x^2+1)*(c^2*d*x^2+d)^(1/2)+1/108*b^2*d^2*x*(c^2*x^2+1)^2*(c^2*d 
*x^2+d)^(1/2)-5/48*b*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d 
)^(1/2)/c-1/18*b*d^2*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1 
/2)/c+5/16*d^2*x*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)-115/1152*b^2*d^2 
*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-5/16*b*c*d^2*x^2*(a+ 
b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+5/48*d^2*(a+b*arcsin 
h(c*x))^3*(c^2*d*x^2+d)^(1/2)/b/c/(c^2*x^2+1)^(1/2)
 

3.3.77.2 Mathematica [A] (verified)

Time = 2.30 (sec) , antiderivative size = 499, normalized size of antiderivative = 1.19 \[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (9504 a^2 c x \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+7488 a^2 c^3 x^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+2304 a^2 c^5 x^5 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}+1440 b^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^3-3240 a b \sqrt {d+c^2 d x^2} \cosh (2 \text {arcsinh}(c x))-324 a b \sqrt {d+c^2 d x^2} \cosh (4 \text {arcsinh}(c x))-24 a b \sqrt {d+c^2 d x^2} \cosh (6 \text {arcsinh}(c x))+4320 a^2 \sqrt {d} \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+1620 b^2 \sqrt {d+c^2 d x^2} \sinh (2 \text {arcsinh}(c x))+81 b^2 \sqrt {d+c^2 d x^2} \sinh (4 \text {arcsinh}(c x))+4 b^2 \sqrt {d+c^2 d x^2} \sinh (6 \text {arcsinh}(c x))-12 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) (270 b \cosh (2 \text {arcsinh}(c x))+27 b \cosh (4 \text {arcsinh}(c x))+2 b \cosh (6 \text {arcsinh}(c x))-540 a \sinh (2 \text {arcsinh}(c x))-108 a \sinh (4 \text {arcsinh}(c x))-12 a \sinh (6 \text {arcsinh}(c x)))+72 b \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)^2 (60 a+45 b \sinh (2 \text {arcsinh}(c x))+9 b \sinh (4 \text {arcsinh}(c x))+b \sinh (6 \text {arcsinh}(c x)))\right )}{13824 c \sqrt {1+c^2 x^2}} \]

input

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

output

(d^2*(9504*a^2*c*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 7488*a^2*c^3*x^ 
3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 2304*a^2*c^5*x^5*Sqrt[1 + c^2*x^ 
2]*Sqrt[d + c^2*d*x^2] + 1440*b^2*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^3 - 324 
0*a*b*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c*x]] - 324*a*b*Sqrt[d + c^2*d*x^ 
2]*Cosh[4*ArcSinh[c*x]] - 24*a*b*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]] 
+ 4320*a^2*Sqrt[d]*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^ 
2]] + 1620*b^2*Sqrt[d + c^2*d*x^2]*Sinh[2*ArcSinh[c*x]] + 81*b^2*Sqrt[d + 
c^2*d*x^2]*Sinh[4*ArcSinh[c*x]] + 4*b^2*Sqrt[d + c^2*d*x^2]*Sinh[6*ArcSinh 
[c*x]] - 12*b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]*(270*b*Cosh[2*ArcSinh[c*x]] 
 + 27*b*Cosh[4*ArcSinh[c*x]] + 2*b*Cosh[6*ArcSinh[c*x]] - 540*a*Sinh[2*Arc 
Sinh[c*x]] - 108*a*Sinh[4*ArcSinh[c*x]] - 12*a*Sinh[6*ArcSinh[c*x]]) + 72* 
b*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2*(60*a + 45*b*Sinh[2*ArcSinh[c*x]] + 9 
*b*Sinh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]])))/(13824*c*Sqrt[1 + c^2* 
x^2])
 

3.3.77.3 Rubi [A] (verified)

Time = 1.69 (sec) , antiderivative size = 502, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {6201, 6201, 6200, 6191, 262, 222, 6198, 6213, 211, 211, 211, 222}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 6201

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

\(\Big \downarrow \) 6200

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6198

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 211

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

\(\Big \downarrow \) 222

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

input

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

output

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

3.3.77.3.1 Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

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 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 6200

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

rule 6201

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

rule 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]
 

3.3.77.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1567\) vs. \(2(366)=732\).

Time = 0.30 (sec) , antiderivative size = 1568, normalized size of antiderivative = 3.73

method	result	size
default	\(\text {Expression too large to display}\)	\(1568\)
parts	\(\text {Expression too large to display}\)	\(1568\)

input

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

output

1/6*x*(c^2*d*x^2+d)^(5/2)*a^2+5/24*a^2*d*x*(c^2*d*x^2+d)^(3/2)+5/16*a^2*d^ 
2*x*(c^2*d*x^2+d)^(1/2)+5/16*a^2*d^3*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d 
)^(1/2))/(c^2*d)^(1/2)+b^2*(5/48*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c 
*arcsinh(c*x)^3*d^2+1/6912*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7+32*c^6*x^6*(c 
^2*x^2+1)^(1/2)+64*c^5*x^5+48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3+18*c^2* 
x^2*(c^2*x^2+1)^(1/2)+6*c*x+(c^2*x^2+1)^(1/2))*(18*arcsinh(c*x)^2-6*arcsin 
h(c*x)+1)*d^2/c/(c^2*x^2+1)+3/1024*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5+8*c^4* 
x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3+8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x+(c^2*x^ 
2+1)^(1/2))*(8*arcsinh(c*x)^2-4*arcsinh(c*x)+1)*d^2/c/(c^2*x^2+1)+15/256*( 
d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2 
+1)^(1/2))*(2*arcsinh(c*x)^2-2*arcsinh(c*x)+1)*d^2/c/(c^2*x^2+1)+15/256*(d 
*(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x-(c^2*x^2+ 
1)^(1/2))*(2*arcsinh(c*x)^2+2*arcsinh(c*x)+1)*d^2/c/(c^2*x^2+1)+3/1024*(d* 
(c^2*x^2+1))^(1/2)*(8*c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3-8*c^2 
*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*x^2+1)^(1/2))*(8*arcsinh(c*x)^2+4*arcsin 
h(c*x)+1)*d^2/c/(c^2*x^2+1)+1/6912*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7-32*c^ 
6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5-48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3 
-18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x-(c^2*x^2+1)^(1/2))*(18*arcsinh(c*x)^2+ 
6*arcsinh(c*x)+1)*d^2/c/(c^2*x^2+1))+2*a*b*(5/32*(d*(c^2*x^2+1))^(1/2)/(c^ 
2*x^2+1)^(1/2)/c*arcsinh(c*x)^2*d^2+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^...
 

3.3.77.5 Fricas [F]

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

input

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

output

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

3.3.77.6 Sympy [F]

\[ \int \left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2 \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}\, dx \]

input

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

output

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

3.3.77.7 Maxima [F(-2)]

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

input

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

output

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

3.3.77.8 Giac [F(-2)]

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

input

integrate((c^2*d*x^2+d)^(5/2)*(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
 

3.3.77.9 Mupad [F(-1)]

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

input

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

output

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