\(\int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{3/2}} \, dx\) [308]

Optimal result

Integrand size = 28, antiderivative size = 391 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {b^2 x \sqrt {d+c^2 d x^2}}{4 c^4 d^2}-\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{4 c^5 d \sqrt {d+c^2 d x^2}}-\frac {b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c^3 d \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{c^5 d \sqrt {d+c^2 d x^2}}+\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 c^4 d^2}-\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{2 b c^5 d \sqrt {d+c^2 d x^2}}-\frac {2 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}}-\frac {b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{c^5 d \sqrt {d+c^2 d x^2}} \] Output:

1/4*b^2*x*(c^2*d*x^2+d)^(1/2)/c^4/d^2-1/4*b^2*(c^2*x^2+1)^(1/2)*arcsinh(c* 
x)/c^5/d/(c^2*d*x^2+d)^(1/2)-1/2*b*x^2*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x) 
)/c^3/d/(c^2*d*x^2+d)^(1/2)-x^3*(a+b*arcsinh(c*x))^2/c^2/d/(c^2*d*x^2+d)^( 
1/2)+(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))^2/c^5/d/(c^2*d*x^2+d)^(1/2)+3/2* 
x*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/c^4/d^2-1/2*(c^2*x^2+1)^(1/2)*( 
a+b*arcsinh(c*x))^3/b/c^5/d/(c^2*d*x^2+d)^(1/2)-2*b*(c^2*x^2+1)^(1/2)*(a+b 
*arcsinh(c*x))*ln(1+(c*x+(c^2*x^2+1)^(1/2))^2)/c^5/d/(c^2*d*x^2+d)^(1/2)-b 
^2*(c^2*x^2+1)^(1/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/c^5/d/(c^2*d*x^ 
2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.56 (sec) , antiderivative size = 288, normalized size of antiderivative = 0.74 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {4 a^2 c \sqrt {d} x \left (3+c^2 x^2\right )-12 a^2 \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+b^2 \sqrt {d} \left (8 c x \text {arcsinh}(c x)^2+8 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )+\sqrt {1+c^2 x^2} \left (-4 \text {arcsinh}(c x)^3-2 \text {arcsinh}(c x) \left (\cosh (2 \text {arcsinh}(c x))+8 \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )\right )+2 \text {arcsinh}(c x)^2 (-4+\sinh (2 \text {arcsinh}(c x)))+\sinh (2 \text {arcsinh}(c x))\right )\right )+2 a b \sqrt {d} \left (8 c x \text {arcsinh}(c x)-\sqrt {1+c^2 x^2} \left (6 \text {arcsinh}(c x)^2+\cosh (2 \text {arcsinh}(c x))+4 \log \left (1+c^2 x^2\right )-2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))\right )\right )}{8 c^5 d^{3/2} \sqrt {d+c^2 d x^2}} \] Input:

Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2),x]
 

Output:

(4*a^2*c*Sqrt[d]*x*(3 + c^2*x^2) - 12*a^2*Sqrt[d + c^2*d*x^2]*Log[c*d*x + 
Sqrt[d]*Sqrt[d + c^2*d*x^2]] + b^2*Sqrt[d]*(8*c*x*ArcSinh[c*x]^2 + 8*Sqrt[ 
1 + c^2*x^2]*PolyLog[2, -E^(-2*ArcSinh[c*x])] + Sqrt[1 + c^2*x^2]*(-4*ArcS 
inh[c*x]^3 - 2*ArcSinh[c*x]*(Cosh[2*ArcSinh[c*x]] + 8*Log[1 + E^(-2*ArcSin 
h[c*x])]) + 2*ArcSinh[c*x]^2*(-4 + Sinh[2*ArcSinh[c*x]]) + Sinh[2*ArcSinh[ 
c*x]])) + 2*a*b*Sqrt[d]*(8*c*x*ArcSinh[c*x] - Sqrt[1 + c^2*x^2]*(6*ArcSinh 
[c*x]^2 + Cosh[2*ArcSinh[c*x]] + 4*Log[1 + c^2*x^2] - 2*ArcSinh[c*x]*Sinh[ 
2*ArcSinh[c*x]])))/(8*c^5*d^(3/2)*Sqrt[d + c^2*d*x^2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 3.35 (sec) , antiderivative size = 383, normalized size of antiderivative = 0.98, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {6225, 6227, 262, 222, 6191, 262, 222, 6198, 6212, 3042, 26, 4201, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 6227

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6191

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

\(\Big \downarrow \) 262

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6198

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {x^3 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {3 \left (\frac {x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{2 c^2 d}-\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^3}{6 b c^3 \sqrt {c^2 d x^2+d}}-\frac {b \sqrt {c^2 x^2+1} \left (\frac {1}{2} x^2 (a+b \text {arcsinh}(c x))-\frac {1}{2} b c \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )\right )}{c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {i \left (2 i \left (\frac {1}{2} \log \left (e^{2 \text {arcsinh}(c x)}+1\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )\right )-\frac {i (a+b \text {arcsinh}(c x))^2}{2 b}\right )}{c^4}+\frac {x^2 (a+b \text {arcsinh}(c x))}{2 c^2}-\frac {b \left (\frac {x \sqrt {c^2 x^2+1}}{2 c^2}-\frac {\text {arcsinh}(c x)}{2 c^3}\right )}{2 c}\right )}{c d \sqrt {c^2 d x^2+d}}\)

Input:

Int[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(3/2),x]
 

Output:

-((x^3*(a + b*ArcSinh[c*x])^2)/(c^2*d*Sqrt[d + c^2*d*x^2])) + (3*((x*Sqrt[ 
d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(2*c^2*d) - (Sqrt[1 + c^2*x^2]*(a + 
 b*ArcSinh[c*x])^3)/(6*b*c^3*Sqrt[d + c^2*d*x^2]) - (b*Sqrt[1 + c^2*x^2]*( 
(x^2*(a + b*ArcSinh[c*x]))/2 - (b*c*((x*Sqrt[1 + c^2*x^2])/(2*c^2) - ArcSi 
nh[c*x]/(2*c^3)))/2))/(c*Sqrt[d + c^2*d*x^2])))/(c^2*d) + (2*b*Sqrt[1 + c^ 
2*x^2]*((x^2*(a + b*ArcSinh[c*x]))/(2*c^2) - (b*((x*Sqrt[1 + c^2*x^2])/(2* 
c^2) - ArcSinh[c*x]/(2*c^3)))/(2*c) + (I*(((-1/2*I)*(a + b*ArcSinh[c*x])^2 
)/b + (2*I)*(((a + b*ArcSinh[c*x])*Log[1 + E^(2*ArcSinh[c*x])])/2 + (b*Pol 
yLog[2, -E^(2*ArcSinh[c*x])])/4)))/c^4))/(c*d*Sqrt[d + c^2*d*x^2])
 

Defintions of rubi rules used

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 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 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4201

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x 
_Symbol] :> Simp[(-I)*((c + d*x)^(m + 1)/(d*(m + 1))), x] + Simp[2*I   Int[ 
(c + d*x)^m*(E^(2*((-I)*e + f*fz*x))/(1 + E^(2*((-I)*e + f*fz*x)))), x], x] 
 /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 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 6212

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

rule 6225

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/(2*e*(p + 1))), x] + (-Simp[f^2*((m - 1)/(2*e*(p + 1))) 
   Int[(f*x)^(m - 2)*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] - S 
imp[b*f*(n/(2*c*(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]) /; Fre 
eQ[{a, b, c, d, e, f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && IG 
tQ[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 [A] (verified)

Time = 1.46 (sec) , antiderivative size = 579, normalized size of antiderivative = 1.48

method	result	size
default	\(\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{4} d \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-2 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}+2 \operatorname {arcsinh}\left (x c \right )^{3} x^{2} c^{2}-\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}-4 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+8 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-6 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} x c +3 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+4 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+2 \operatorname {arcsinh}\left (x c \right )^{3}-\sqrt {c^{2} x^{2}+1}\, x c -4 \operatorname {arcsinh}\left (x c \right )^{2}+8 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (x c \right )+4 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right )}{4 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{2} c^{5}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 c^{4} x^{4}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-8 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-12 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +3 c^{2} x^{2}+6 \operatorname {arcsinh}\left (x c \right )^{2}+8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-8 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{4 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{2} c^{5}}\)	\(579\)
parts	\(\frac {a^{2} x^{3}}{2 c^{2} d \sqrt {c^{2} d \,x^{2}+d}}+\frac {3 a^{2} x}{2 c^{4} d \sqrt {c^{2} d \,x^{2}+d}}-\frac {3 a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{4} d \sqrt {c^{2} d}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-2 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}+2 \operatorname {arcsinh}\left (x c \right )^{3} x^{2} c^{2}-\sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}-4 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+8 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-6 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} x c +3 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+4 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+2 \operatorname {arcsinh}\left (x c \right )^{3}-\sqrt {c^{2} x^{2}+1}\, x c -4 \operatorname {arcsinh}\left (x c \right )^{2}+8 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\operatorname {arcsinh}\left (x c \right )+4 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )\right )}{4 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{2} c^{5}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-4 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+2 c^{4} x^{4}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-8 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-12 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +3 c^{2} x^{2}+6 \operatorname {arcsinh}\left (x c \right )^{2}+8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )-8 \,\operatorname {arcsinh}\left (x c \right )+1\right )}{4 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{2} c^{5}}\)	\(579\)

Input:

int(x^4*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/2*a^2*x^3/c^2/d/(c^2*d*x^2+d)^(1/2)+3/2*a^2/c^4*x/d/(c^2*d*x^2+d)^(1/2)- 
3/2*a^2/c^4/d*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)- 
1/4*b^2/(c^2*x^2+1)^(3/2)*(d*(c^2*x^2+1))^(1/2)*(-2*arcsinh(x*c)^2*(c^2*x^ 
2+1)^(1/2)*x^3*c^3+2*arcsinh(x*c)*c^4*x^4+2*arcsinh(x*c)^3*x^2*c^2-(c^2*x^ 
2+1)^(1/2)*c^3*x^3-4*arcsinh(x*c)^2*x^2*c^2+8*arcsinh(x*c)*ln(1+(x*c+(c^2* 
x^2+1)^(1/2))^2)*x^2*c^2-6*(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*x*c+3*arcsinh( 
x*c)*c^2*x^2+4*polylog(2,-(x*c+(c^2*x^2+1)^(1/2))^2)*x^2*c^2+2*arcsinh(x*c 
)^3-(c^2*x^2+1)^(1/2)*x*c-4*arcsinh(x*c)^2+8*arcsinh(x*c)*ln(1+(x*c+(c^2*x 
^2+1)^(1/2))^2)+arcsinh(x*c)+4*polylog(2,-(x*c+(c^2*x^2+1)^(1/2))^2))/d^2/ 
c^5-1/4*a*b/(c^2*x^2+1)^(3/2)*(d*(c^2*x^2+1))^(1/2)*(-4*arcsinh(x*c)*(c^2* 
x^2+1)^(1/2)*x^3*c^3+2*c^4*x^4+6*arcsinh(x*c)^2*x^2*c^2+8*ln(1+(x*c+(c^2*x 
^2+1)^(1/2))^2)*x^2*c^2-8*arcsinh(x*c)*c^2*x^2-12*arcsinh(x*c)*(c^2*x^2+1) 
^(1/2)*x*c+3*c^2*x^2+6*arcsinh(x*c)^2+8*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)-8* 
arcsinh(x*c)+1)/d^2/c^5
 

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(3/2),x, algorithm="frica 
s")
 

Output:

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

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

integrate(x**4*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(3/2),x)
 

Output:

Integral(x**4*(a + b*asinh(c*x))**2/(d*(c**2*x**2 + 1))**(3/2), x)
 

Maxima [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(3/2),x, algorithm="maxim 
a")
 

Output:

1/2*a^2*(x^3/(sqrt(c^2*d*x^2 + d)*c^2*d) + 3*x/(sqrt(c^2*d*x^2 + d)*c^4*d) 
 - 3*arcsinh(c*x)/(c^5*d^(3/2))) + integrate(b^2*x^4*log(c*x + sqrt(c^2*x^ 
2 + 1))^2/(c^2*d*x^2 + d)^(3/2) + 2*a*b*x^4*log(c*x + sqrt(c^2*x^2 + 1))/( 
c^2*d*x^2 + d)^(3/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(3/2),x, algorithm="giac" 
)
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{3/2}} \,d x \] Input:

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(3/2),x)
 

Output:

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(3/2), x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{3/2}} \, dx=\frac {4 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}+12 \sqrt {c^{2} x^{2}+1}\, a^{2} c x +16 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{7} x^{2}+16 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{5}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{7} x^{2}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{5}-12 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2} c^{2} x^{2}-12 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2}+9 a^{2} c^{2} x^{2}+9 a^{2}}{8 \sqrt {d}\, c^{5} d \left (c^{2} x^{2}+1\right )} \] Input:

int(x^4*(a+b*asinh(c*x))^2/(c^2*d*x^2+d)^(3/2),x)
                                                                                    
                                                                                    
 

Output:

(4*sqrt(c**2*x**2 + 1)*a**2*c**3*x**3 + 12*sqrt(c**2*x**2 + 1)*a**2*c*x + 
16*int((asinh(c*x)*x**4)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 
 1)),x)*a*b*c**7*x**2 + 16*int((asinh(c*x)*x**4)/(sqrt(c**2*x**2 + 1)*c**2 
*x**2 + sqrt(c**2*x**2 + 1)),x)*a*b*c**5 + 8*int((asinh(c*x)**2*x**4)/(sqr 
t(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c**7*x**2 + 8*in 
t((asinh(c*x)**2*x**4)/(sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1 
)),x)*b**2*c**5 - 12*log(sqrt(c**2*x**2 + 1) + c*x)*a**2*c**2*x**2 - 12*lo 
g(sqrt(c**2*x**2 + 1) + c*x)*a**2 + 9*a**2*c**2*x**2 + 9*a**2)/(8*sqrt(d)* 
c**5*d*(c**2*x**2 + 1))