\(\int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{5/2}} \, dx\) [318]

Optimal result

Integrand size = 28, antiderivative size = 398 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2 x}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x^2 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^3 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {x (a+b \text {arcsinh}(c x))^2}{c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{3 b c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \log \left (1+e^{2 \text {arcsinh}(c x)}\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}+\frac {4 b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{3 c^5 d^2 \sqrt {d+c^2 d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.11 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.90 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {-a^2 c \sqrt {d} x \left (3+4 c^2 x^2\right )+a b \sqrt {d} \left (\sqrt {1+c^2 x^2}+2 c x \text {arcsinh}(c x)-8 c x \left (1+c^2 x^2\right ) \text {arcsinh}(c x)+\left (1+c^2 x^2\right )^{3/2} \left (3 \text {arcsinh}(c x)^2+4 \log \left (1+c^2 x^2\right )\right )\right )+3 a^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-b^2 \sqrt {d} \left (c x+c^3 x^3-\sqrt {1+c^2 x^2} \text {arcsinh}(c x)+3 c x \text {arcsinh}(c x)^2+4 c^3 x^3 \text {arcsinh}(c x)^2-4 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^2-\left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)^3-8 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1+e^{-2 \text {arcsinh}(c x)}\right )+4 \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-e^{-2 \text {arcsinh}(c x)}\right )\right )}{3 c^5 d^{5/2} \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}} \] Input:

Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]
 

Output:

(-(a^2*c*Sqrt[d]*x*(3 + 4*c^2*x^2)) + a*b*Sqrt[d]*(Sqrt[1 + c^2*x^2] + 2*c 
*x*ArcSinh[c*x] - 8*c*x*(1 + c^2*x^2)*ArcSinh[c*x] + (1 + c^2*x^2)^(3/2)*( 
3*ArcSinh[c*x]^2 + 4*Log[1 + c^2*x^2])) + 3*a^2*(1 + c^2*x^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]*(c*x + c^3* 
x^3 - Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 3*c*x*ArcSinh[c*x]^2 + 4*c^3*x^3*Ar 
cSinh[c*x]^2 - 4*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]^2 - (1 + c^2*x^2)^(3/2)* 
ArcSinh[c*x]^3 - 8*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 + E^(-2*ArcSinh[ 
c*x])] + 4*(1 + c^2*x^2)^(3/2)*PolyLog[2, -E^(-2*ArcSinh[c*x])]))/(3*c^5*d 
^(5/2)*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])
 

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 2.26 (sec) , antiderivative size = 408, normalized size of antiderivative = 1.03, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6225, 6225, 252, 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.

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 252

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

\(\Big \downarrow \) 222

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

\(\Big \downarrow \) 6198

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

\(\Big \downarrow \) 6212

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 26

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

\(\Big \downarrow \) 4201

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(x^4*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]
 

Output:

-1/3*(x^3*(a + b*ArcSinh[c*x])^2)/(c^2*d*(d + c^2*d*x^2)^(3/2)) + (2*b*Sqr 
t[1 + c^2*x^2]*(-1/2*(x^2*(a + b*ArcSinh[c*x]))/(c^2*(1 + c^2*x^2)) + (b*( 
-(x/(c^2*Sqrt[1 + c^2*x^2])) + ArcSinh[c*x]/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*ArcSin 
h[c*x])])/2 + (b*PolyLog[2, -E^(2*ArcSinh[c*x])])/4)))/c^4))/(3*c*d^2*Sqrt 
[d + c^2*d*x^2]) + (-((x*(a + b*ArcSinh[c*x])^2)/(c^2*d*Sqrt[d + c^2*d*x^2 
])) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(3*b*c^3*d*Sqrt[d + c^2*d 
*x^2]) - ((2*I)*b*Sqrt[1 + c^2*x^2]*(((-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*PolyLog[ 
2, -E^(2*ArcSinh[c*x])])/4)))/(c^3*d*Sqrt[d + c^2*d*x^2]))/(c^2*d)
 

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 252

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x 
)^(m - 1)*((a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] - Simp[c^2*((m - 1)/(2*b* 
(p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c 
}, x] && LtQ[p, -1] && GtQ[m, 1] &&  !ILtQ[(m + 2*p + 3)/2, 0] && IntBinomi 
alQ[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 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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(765\) vs. \(2(370)=740\).

Time = 1.34 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.92

method	result	size
default	\(-\frac {a^{2} x^{3}}{3 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a^{2} x}{c^{4} d^{2} \sqrt {c^{2} d \,x^{2}+d}}+\frac {a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{4} d^{2} \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (\operatorname {arcsinh}\left (x c \right )^{3} x^{4} c^{4}-4 \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}+8 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-4 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+4 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+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}-8 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+16 \,\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}-3 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} x c +\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+8 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+2 c^{2} x^{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 )+1\right )}{3 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) d^{3} c^{5}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (3 \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}+8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+16 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-16 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+3 \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 )}{3 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) d^{3} c^{5}}\)	\(766\)
parts	\(-\frac {a^{2} x^{3}}{3 c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {a^{2} x}{c^{4} d^{2} \sqrt {c^{2} d \,x^{2}+d}}+\frac {a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{c^{4} d^{2} \sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (\operatorname {arcsinh}\left (x c \right )^{3} x^{4} c^{4}-4 \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}+8 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-4 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+4 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}+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}-8 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+16 \,\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}-3 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} x c +\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+8 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}+2 c^{2} x^{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 )+1\right )}{3 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) d^{3} c^{5}}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \sqrt {c^{2} x^{2}+1}\, \left (3 \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}+8 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{4} c^{4}-8 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-8 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+6 \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+16 \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right ) x^{2} c^{2}-16 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x c +c^{2} x^{2}+3 \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 )}{3 \left (c^{6} x^{6}+3 c^{4} x^{4}+3 c^{2} x^{2}+1\right ) d^{3} c^{5}}\)	\(766\)

Input:

int(x^4*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/3*a^2*x^3/c^2/d/(c^2*d*x^2+d)^(3/2)-a^2/c^4/d^2*x/(c^2*d*x^2+d)^(1/2)+a 
^2/c^4/d^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/3 
*b^2*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/2)/(c^6*x^6+3*c^4*x^4+3*c^2*x^2+ 
1)/d^3/c^5*(arcsinh(x*c)^3*x^4*c^4-4*arcsinh(x*c)^2*x^4*c^4+8*arcsinh(x*c) 
*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^4*c^4-4*arcsinh(x*c)^2*(c^2*x^2+1)^(1/2 
)*x^3*c^3+4*polylog(2,-(x*c+(c^2*x^2+1)^(1/2))^2)*x^4*c^4+c^4*x^4+2*arcsin 
h(x*c)^3*x^2*c^2-(c^2*x^2+1)^(1/2)*c^3*x^3-8*arcsinh(x*c)^2*x^2*c^2+16*arc 
sinh(x*c)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^2*c^2-3*(c^2*x^2+1)^(1/2)*arcs 
inh(x*c)^2*x*c+arcsinh(x*c)*c^2*x^2+8*polylog(2,-(x*c+(c^2*x^2+1)^(1/2))^2 
)*x^2*c^2+2*c^2*x^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)+1)+1/3*a*b*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+1)^(1/ 
2)/(c^6*x^6+3*c^4*x^4+3*c^2*x^2+1)/d^3/c^5*(3*arcsinh(x*c)^2*x^4*c^4+8*ln( 
1+(x*c+(c^2*x^2+1)^(1/2))^2)*x^4*c^4-8*arcsinh(x*c)*c^4*x^4-8*arcsinh(x*c) 
*(c^2*x^2+1)^(1/2)*x^3*c^3+6*arcsinh(x*c)^2*x^2*c^2+16*ln(1+(x*c+(c^2*x^2+ 
1)^(1/2))^2)*x^2*c^2-16*arcsinh(x*c)*c^2*x^2-6*arcsinh(x*c)*(c^2*x^2+1)^(1 
/2)*x*c+c^2*x^2+3*arcsinh(x*c)^2+8*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)-8*arcsi 
nh(x*c)+1)
 

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),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^6*d^3*x^6 + 3*c^4*d^3*x^4 + 3*c^2*d^3*x^2 + d^3), x)
 

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**4*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral(x**4*(a + b*asinh(c*x))**2/(d*(c**2*x**2 + 1))**(5/2), x)
 

Maxima [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="maxim 
a")
 

Output:

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

Giac [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="giac" 
)
 

Output:

integrate((b*arcsinh(c*x) + a)^2*x^4/(c^2*d*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \] Input:

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(5/2),x)
 

Output:

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {-4 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{3} x^{3}-3 \sqrt {c^{2} x^{2}+1}\, a^{2} c x +6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{9} x^{4}+12 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{7} x^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{5}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{9} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{7} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{4}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{5}+3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2} c^{4} x^{4}+6 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2} c^{2} x^{2}+3 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x \right ) a^{2}}{3 \sqrt {d}\, c^{5} d^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

int(x^4*(a+b*asinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x)
 

Output:

( - 4*sqrt(c**2*x**2 + 1)*a**2*c**3*x**3 - 3*sqrt(c**2*x**2 + 1)*a**2*c*x 
+ 6*int((asinh(c*x)*x**4)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x** 
2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*a*b*c**9*x**4 + 12*int((asinh(c 
*x)*x**4)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 
 + sqrt(c**2*x**2 + 1)),x)*a*b*c**7*x**2 + 6*int((asinh(c*x)*x**4)/(sqrt(c 
**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 
 + 1)),x)*a*b*c**5 + 3*int((asinh(c*x)**2*x**4)/(sqrt(c**2*x**2 + 1)*c**4* 
x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c**9 
*x**4 + 6*int((asinh(c*x)**2*x**4)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt 
(c**2*x**2 + 1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c**7*x**2 + 3*int 
((asinh(c*x)**2*x**4)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 
1)*c**2*x**2 + sqrt(c**2*x**2 + 1)),x)*b**2*c**5 + 3*log(sqrt(c**2*x**2 + 
1) + c*x)*a**2*c**4*x**4 + 6*log(sqrt(c**2*x**2 + 1) + c*x)*a**2*c**2*x**2 
 + 3*log(sqrt(c**2*x**2 + 1) + c*x)*a**2)/(3*sqrt(d)*c**5*d**2*(c**4*x**4 
+ 2*c**2*x**2 + 1))