\(\int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{(d+c^2 d x^2)^{5/2}} \, dx\) [319]

Optimal result

Integrand size = 28, antiderivative size = 307 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=-\frac {b^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {b x (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {10 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}-\frac {5 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {5 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c^4 d^2 \sqrt {d+c^2 d x^2}} \] Output:

-1/3*b^2/c^4/d^2/(c^2*d*x^2+d)^(1/2)-1/3*b*x*(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^2*(a+b*arcsinh(c*x))^2/c^2/d/(c^ 
2*d*x^2+d)^(3/2)-2/3*(a+b*arcsinh(c*x))^2/c^4/d^2/(c^2*d*x^2+d)^(1/2)+10/3 
*b*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x^2+1)^(1/2))/c^4/ 
d^2/(c^2*d*x^2+d)^(1/2)-5/3*I*b^2*(c^2*x^2+1)^(1/2)*polylog(2,-I*(c*x+(c^2 
*x^2+1)^(1/2)))/c^4/d^2/(c^2*d*x^2+d)^(1/2)+5/3*I*b^2*(c^2*x^2+1)^(1/2)*po 
lylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c^4/d^2/(c^2*d*x^2+d)^(1/2)
 

Mathematica [A] (verified)

Time = 1.03 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.98 \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {-a^2 \left (2+3 c^2 x^2\right )+a b \left (-2 \left (2+3 c^2 x^2\right ) \text {arcsinh}(c x)+\sqrt {1+c^2 x^2} \left (-c x+10 \left (1+c^2 x^2\right ) \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )\right )-b^2 \left (1+c^2 x^2+c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+2 \text {arcsinh}(c x)^2+3 c^2 x^2 \text {arcsinh}(c x)^2+5 i \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1-i e^{-\text {arcsinh}(c x)}\right )-5 i \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \log \left (1+i e^{-\text {arcsinh}(c x)}\right )+5 i \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-5 i \left (1+c^2 x^2\right )^{3/2} \operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )\right )}{3 c^4 d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}} \] Input:

Integrate[(x^3*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]
 

Output:

(-(a^2*(2 + 3*c^2*x^2)) + a*b*(-2*(2 + 3*c^2*x^2)*ArcSinh[c*x] + Sqrt[1 + 
c^2*x^2]*(-(c*x) + 10*(1 + c^2*x^2)*ArcTan[Tanh[ArcSinh[c*x]/2]])) - b^2*( 
1 + c^2*x^2 + c*x*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 2*ArcSinh[c*x]^2 + 3*c^ 
2*x^2*ArcSinh[c*x]^2 + (5*I)*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 - I/E^ 
ArcSinh[c*x]] - (5*I)*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*Log[1 + I/E^ArcSinh 
[c*x]] + (5*I)*(1 + c^2*x^2)^(3/2)*PolyLog[2, (-I)/E^ArcSinh[c*x]] - (5*I) 
*(1 + c^2*x^2)^(3/2)*PolyLog[2, I/E^ArcSinh[c*x]]))/(3*c^4*d^2*(1 + c^2*x^ 
2)*Sqrt[d + c^2*d*x^2])
 

Rubi [A] (verified)

Time = 2.10 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6225, 6213, 6204, 3042, 4668, 2715, 2838, 6225, 241, 6204, 3042, 4668, 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.

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 6213

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

\(\Big \downarrow \) 6225

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

\(\Big \downarrow \) 241

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

\(\Big \downarrow \) 6204

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4668

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[(x^3*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]
 

Output:

-1/3*(x^2*(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*b/(c^3*Sqrt[1 + c^2*x^2]) - (x*(a + b*ArcSinh[c*x]))/ 
(2*c^2*(1 + c^2*x^2)) + (2*(a + b*ArcSinh[c*x])*ArcTan[E^ArcSinh[c*x]] - I 
*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] + I*b*PolyLog[2, I*E^ArcSinh[c*x]])/(2* 
c^3)))/(3*c*d^2*Sqrt[d + c^2*d*x^2]) + (2*(-((a + b*ArcSinh[c*x])^2/(c^2*d 
*Sqrt[d + c^2*d*x^2])) + (2*b*Sqrt[1 + c^2*x^2]*(2*(a + b*ArcSinh[c*x])*Ar 
cTan[E^ArcSinh[c*x]] - I*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] + I*b*PolyLog[2 
, I*E^ArcSinh[c*x]]))/(c^2*d*Sqrt[d + c^2*d*x^2])))/(3*c^2*d)
 

Defintions of rubi rules used

rule 241

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

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 4668

Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ 
))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( 
I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I))   Int[(c + d*x)^(m - 1)*Log[ 
1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I))   Int[(c 
+ d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c 
, d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
 

rule 6204

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

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)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 703 vs. \(2 (294 ) = 588\).

Time = 1.18 (sec) , antiderivative size = 704, normalized size of antiderivative = 2.29

method	result	size
default	\(a^{2} \left (-\frac {x^{2}}{c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} x^{2}}{\left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x}{3 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3} c^{3}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}-\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}+\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}-\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}+\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x^{2}}{\left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{3 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3} c^{3}}-\frac {4 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}+\frac {5 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}-\frac {5 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}\)	\(704\)
parts	\(a^{2} \left (-\frac {x^{2}}{c^{2} d \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}-\frac {2}{3 d \,c^{4} \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2} x^{2}}{\left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x}{3 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3} c^{3}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, x^{2}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {2 b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )^{2}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}-\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}+\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) \ln \left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}-\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1+i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}+\frac {5 i b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {dilog}\left (1-i \left (x c +\sqrt {c^{2} x^{2}+1}\right )\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}-\frac {2 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right ) x^{2}}{\left (c^{2} x^{2}+1\right )^{2} d^{3} c^{2}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, x}{3 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}} d^{3} c^{3}}-\frac {4 a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (x c \right )}{3 \left (c^{2} x^{2}+1\right )^{2} d^{3} c^{4}}+\frac {5 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}+i\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}-\frac {5 i a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \ln \left (x c +\sqrt {c^{2} x^{2}+1}-i\right )}{3 \sqrt {c^{2} x^{2}+1}\, d^{3} c^{4}}\)	\(704\)

Input:

int(x^3*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

a^2*(-x^2/c^2/d/(c^2*d*x^2+d)^(3/2)-2/3/d/c^4/(c^2*d*x^2+d)^(3/2))-b^2*(d* 
(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^2*arcsinh(x*c)^2*x^2-1/3*b^2*(d*(c^ 
2*x^2+1))^(1/2)/(c^2*x^2+1)^(3/2)/d^3/c^3*arcsinh(x*c)*x-1/3*b^2*(d*(c^2*x 
^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^2*x^2-2/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2* 
x^2+1)^2/d^3/c^4*arcsinh(x*c)^2-1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^ 
2/d^3/c^4-5/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3/c^4*arcsin 
h(x*c)*ln(1+I*(x*c+(c^2*x^2+1)^(1/2)))+5/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^ 
2*x^2+1)^(1/2)/d^3/c^4*arcsinh(x*c)*ln(1-I*(x*c+(c^2*x^2+1)^(1/2)))-5/3*I* 
b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3/c^4*dilog(1+I*(x*c+(c^2*x^ 
2+1)^(1/2)))+5/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3/c^4*dil 
og(1-I*(x*c+(c^2*x^2+1)^(1/2)))-2*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/ 
d^3/c^2*arcsinh(x*c)*x^2-1/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(3/2)/d 
^3/c^3*x-4/3*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^4*arcsinh(x*c)+ 
5/3*I*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3/c^4*ln(x*c+(c^2*x^2+ 
1)^(1/2)+I)-5/3*I*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3/c^4*ln(x 
*c+(c^2*x^2+1)^(1/2)-I)
 

Fricas [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{3}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="frica 
s")
 

Output:

integral((b^2*x^3*arcsinh(c*x)^2 + 2*a*b*x^3*arcsinh(c*x) + a^2*x^3)*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]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{3} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(x**3*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(5/2),x)
 

Output:

Integral(x**3*(a + b*asinh(c*x))**2/(d*(c**2*x**2 + 1))**(5/2), x)
 

Maxima [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{3}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="maxim 
a")
 

Output:

-1/3*a*b*c*(x/(c^6*d^(5/2)*x^2 + c^4*d^(5/2)) - 5*arctan(c*x)/(c^5*d^(5/2) 
)) - 2/3*a*b*(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))*arcsinh(c*x) - 1/3*a^2*(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)) + b^2*integrate(x^3*log(c*x + sqrt(c^2*x^2 
 + 1))^2/(c^2*d*x^2 + d)^(5/2), x)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^3*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),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. \[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \] Input:

int((x^3*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(5/2),x)
 

Output:

int((x^3*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {-3 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-2 \sqrt {c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{8} x^{4}+12 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{6} x^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{4}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{8} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{6} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{3}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{4}}{3 \sqrt {d}\, c^{4} d^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:

int(x^3*(a+b*asinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x)
 

Output:

( - 3*sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - 2*sqrt(c**2*x**2 + 1)*a**2 + 6* 
int((asinh(c*x)*x**3)/(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**8*x**4 + 12*int((asinh(c*x)* 
x**3)/(sqrt(c**2*x**2 + 1)*c**4*x**4 + 2*sqrt(c**2*x**2 + 1)*c**2*x**2 + s 
qrt(c**2*x**2 + 1)),x)*a*b*c**6*x**2 + 6*int((asinh(c*x)*x**3)/(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**4 + 3*int((asinh(c*x)**2*x**3)/(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**8*x** 
4 + 6*int((asinh(c*x)**2*x**3)/(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**6*x**2 + 3*int((as 
inh(c*x)**2*x**3)/(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**4)/(3*sqrt(d)*c**4*d**2*(c**4*x 
**4 + 2*c**2*x**2 + 1))