Integrand size = 28, antiderivative size = 418 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {b^2}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \sqrt {d+c^2 d x^2}}{c^6 d^3}-\frac {b x^3 (a+b \text {arcsinh}(c x))}{3 c^3 d^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}}-\frac {5 b x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^5 d^2 \sqrt {d+c^2 d x^2}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (d+c^2 d x^2\right )^{3/2}}-\frac {4 x^2 (a+b \text {arcsinh}(c x))^2}{3 c^4 d^2 \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{3 c^6 d^3}-\frac {22 b \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}+\frac {11 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}}-\frac {11 i b^2 \sqrt {1+c^2 x^2} \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{3 c^6 d^2 \sqrt {d+c^2 d x^2}} \] Output:
1/3*b^2/c^6/d^2/(c^2*d*x^2+d)^(1/2)+2*b^2*(c^2*d*x^2+d)^(1/2)/c^6/d^3-1/3* b*x^3*(a+b*arcsinh(c*x))/c^3/d^2/(c^2*x^2+1)^(1/2)/(c^2*d*x^2+d)^(1/2)-5/3 *b*x*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))/c^5/d^2/(c^2*d*x^2+d)^(1/2)-1/3* x^4*(a+b*arcsinh(c*x))^2/c^2/d/(c^2*d*x^2+d)^(3/2)-4/3*x^2*(a+b*arcsinh(c* x))^2/c^4/d^2/(c^2*d*x^2+d)^(1/2)+8/3*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x ))^2/c^6/d^3-22/3*b*(c^2*x^2+1)^(1/2)*(a+b*arcsinh(c*x))*arctan(c*x+(c^2*x ^2+1)^(1/2))/c^6/d^2/(c^2*d*x^2+d)^(1/2)+11/3*I*b^2*(c^2*x^2+1)^(1/2)*poly log(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/c^6/d^2/(c^2*d*x^2+d)^(1/2)-11/3*I*b^2*( c^2*x^2+1)^(1/2)*polylog(2,I*(c*x+(c^2*x^2+1)^(1/2)))/c^6/d^2/(c^2*d*x^2+d )^(1/2)
Time = 1.45 (sec) , antiderivative size = 333, normalized size of antiderivative = 0.80 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {\sqrt {d+c^2 d x^2} \left (a^2 \left (8+12 c^2 x^2+3 c^4 x^4\right )+a b \left (2 \left (8+12 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)-\sqrt {1+c^2 x^2} \left (c x \left (5+6 c^2 x^2\right )+22 \left (1+c^2 x^2\right ) \arctan \left (\tanh \left (\frac {1}{2} \text {arcsinh}(c x)\right )\right )\right )\right )+b^2 \left (c x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)-6 c x \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)-\text {arcsinh}(c x)^2+3 \left (1+c^2 x^2\right )^2 \left (2+\text {arcsinh}(c x)^2\right )+\left (1+c^2 x^2\right ) \left (1+6 \text {arcsinh}(c x)^2\right )+11 i \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x) \left (\log \left (1-i e^{-\text {arcsinh}(c x)}\right )-\log \left (1+i e^{-\text {arcsinh}(c x)}\right )\right )+11 i \left (1+c^2 x^2\right )^{3/2} \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arcsinh}(c x)}\right )\right )\right )\right )}{3 c^6 d^3 \left (1+c^2 x^2\right )^2} \] Input:
Integrate[(x^5*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]
Output:
(Sqrt[d + c^2*d*x^2]*(a^2*(8 + 12*c^2*x^2 + 3*c^4*x^4) + a*b*(2*(8 + 12*c^ 2*x^2 + 3*c^4*x^4)*ArcSinh[c*x] - Sqrt[1 + c^2*x^2]*(c*x*(5 + 6*c^2*x^2) + 22*(1 + c^2*x^2)*ArcTan[Tanh[ArcSinh[c*x]/2]])) + b^2*(c*x*Sqrt[1 + c^2*x ^2]*ArcSinh[c*x] - 6*c*x*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x] - ArcSinh[c*x]^2 + 3*(1 + c^2*x^2)^2*(2 + ArcSinh[c*x]^2) + (1 + c^2*x^2)*(1 + 6*ArcSinh[c *x]^2) + (11*I)*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]*(Log[1 - I/E^ArcSinh[c*x] ] - Log[1 + I/E^ArcSinh[c*x]]) + (11*I)*(1 + c^2*x^2)^(3/2)*(PolyLog[2, (- I)/E^ArcSinh[c*x]] - PolyLog[2, I/E^ArcSinh[c*x]]))))/(3*c^6*d^3*(1 + c^2* x^2)^2)
Time = 3.18 (sec) , antiderivative size = 523, normalized size of antiderivative = 1.25, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6225, 6225, 243, 53, 2009, 6213, 2009, 6227, 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.
| \(\displaystyle \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 6225 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \int \frac {x^4 (a+b \text {arcsinh}(c x))}{\left (c^2 x^2+1\right )^2}dx}{3 c d^2 \sqrt {c^2 d x^2+d}}+\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\left (c^2 d x^2+d\right )^{3/2}}dx}{3 c^2 d}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6225 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{2 c^2}+\frac {b \int \frac {x^3}{\left (c^2 x^2+1\right )^{3/2}}dx}{2 c}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c d \sqrt {c^2 d x^2+d}}+\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{2 c^2}+\frac {b \int \frac {x^2}{\left (c^2 x^2+1\right )^{3/2}}dx^2}{4 c}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c d \sqrt {c^2 d x^2+d}}+\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{2 c^2}+\frac {b \int \left (\frac {1}{c^2 \sqrt {c^2 x^2+1}}-\frac {1}{c^2 \left (c^2 x^2+1\right )^{3/2}}\right )dx^2}{4 c}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}+\frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c d \sqrt {c^2 d x^2+d}}+\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c d \sqrt {c^2 d x^2+d}}+\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{c^2 d}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{2 c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )}{4 c}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {4 \left (\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x))dx}{c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c d \sqrt {c^2 d x^2+d}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}\right )}{3 c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{2 c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )}{4 c}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{c d \sqrt {c^2 d x^2+d}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))}{c^2 x^2+1}dx}{2 c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )}{4 c}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{c^2}-\frac {b \int \frac {x}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{c^2}-\frac {b \int \frac {x}{\sqrt {c^2 x^2+1}}dx}{c}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}\right )}{2 c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )}{4 c}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle \frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{c^2}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx}{c^2}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{2 c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )}{4 c}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 6204 |
| \(\displaystyle \frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \left (-\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{2 c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )}{4 c}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {\int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \left (-\frac {\int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{2 c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )}{4 c}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle \frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \left (-\frac {-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{2 c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )}{4 c}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle \frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \left (-\frac {-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{2 c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )}{4 c}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle \frac {4 \left (\frac {2 b \sqrt {c^2 x^2+1} \left (-\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{c d \sqrt {c^2 d x^2+d}}-\frac {x^2 (a+b \text {arcsinh}(c x))^2}{c^2 d \sqrt {c^2 d x^2+d}}+\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{c^2 d}\right )}{3 c^2 d}+\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {3 \left (-\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{c^3}+\frac {x (a+b \text {arcsinh}(c x))}{c^2}-\frac {b \sqrt {c^2 x^2+1}}{c^3}\right )}{2 c^2}-\frac {x^3 (a+b \text {arcsinh}(c x))}{2 c^2 \left (c^2 x^2+1\right )}+\frac {b \left (\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {2}{c^4 \sqrt {c^2 x^2+1}}\right )}{4 c}\right )}{3 c d^2 \sqrt {c^2 d x^2+d}}-\frac {x^4 (a+b \text {arcsinh}(c x))^2}{3 c^2 d \left (c^2 d x^2+d\right )^{3/2}}\) |
Input:
Int[(x^5*(a + b*ArcSinh[c*x])^2)/(d + c^2*d*x^2)^(5/2),x]
Output:
-1/3*(x^4*(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]*((b*(2/(c^4*Sqrt[1 + c^2*x^2]) + (2*Sqrt[1 + c^2*x^2])/c^4) )/(4*c) - (x^3*(a + b*ArcSinh[c*x]))/(2*c^2*(1 + c^2*x^2)) + (3*(-((b*Sqrt [1 + c^2*x^2])/c^3) + (x*(a + b*ArcSinh[c*x]))/c^2 - (2*(a + b*ArcSinh[c*x ])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] + I*b*Poly Log[2, I*E^ArcSinh[c*x]])/c^3))/(2*c^2)))/(3*c*d^2*Sqrt[d + c^2*d*x^2]) + (4*(-((x^2*(a + b*ArcSinh[c*x])^2)/(c^2*d*Sqrt[d + c^2*d*x^2])) + (2*((Sqr t[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(c^2*d) - (2*b*Sqrt[1 + c^2*x^2]* (a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c*x]))/(c*Sqrt[d + c^2*d*x^2] )))/(c^2*d) + (2*b*Sqrt[1 + c^2*x^2]*(-((b*Sqrt[1 + c^2*x^2])/c^3) + (x*(a + b*ArcSinh[c*x]))/c^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]])/c ^3))/(c*d*Sqrt[d + c^2*d*x^2])))/(3*c^2*d)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
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]
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]
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]
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]
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]
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]
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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1041 vs. \(2 (393 ) = 786\).
Time = 1.33 (sec) , antiderivative size = 1042, normalized size of antiderivative = 2.49
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1042\) |
| parts | \(\text {Expression too large to display}\) | \(1042\) |
Input:
int(x^5*(a+b*arcsinh(x*c))^2/(c^2*d*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
Output:
a^2*(x^4/c^2/d/(c^2*d*x^2+d)^(3/2)-4/c^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)))+2*b^2*(d*(c^2*x^2+1))^(1/2)/d^3/c^6/(c^2*x^ 2+1)+b^2*(d*(c^2*x^2+1))^(1/2)/d^3/c^4/(c^2*x^2+1)*arcsinh(x*c)^2*x^2+11/3 *I*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3/c^6*ln(x*c+(c^2*x^2+1)^ (1/2)-I)+2*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^4*arcsinh(x*c)^2* x^2-11/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/d^3/c^6*dilog(1-I*( x*c+(c^2*x^2+1)^(1/2)))+11/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2) /d^3/c^6*arcsinh(x*c)*ln(1+I*(x*c+(c^2*x^2+1)^(1/2)))-2*b^2*(d*(c^2*x^2+1) )^(1/2)/d^3/c^5/(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x+1/3*b^2*(d*(c^2*x^2+1))^( 1/2)/(c^2*x^2+1)^(3/2)/d^3/c^5*arcsinh(x*c)*x+2*b^2*(d*(c^2*x^2+1))^(1/2)/ d^3/c^4/(c^2*x^2+1)*x^2+b^2*(d*(c^2*x^2+1))^(1/2)/d^3/c^6/(c^2*x^2+1)*arcs inh(x*c)^2+1/3*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^6+1/3*b^2*(d* (c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^4*x^2+5/3*b^2*(d*(c^2*x^2+1))^(1/2) /(c^2*x^2+1)^2/d^3/c^6*arcsinh(x*c)^2+11/3*I*b^2*(d*(c^2*x^2+1))^(1/2)/(c^ 2*x^2+1)^(1/2)/d^3/c^6*dilog(1+I*(x*c+(c^2*x^2+1)^(1/2)))+2*a*b*(d*(c^2*x^ 2+1))^(1/2)/d^3/c^4/(c^2*x^2+1)*arcsinh(x*c)*x^2-2*a*b*(d*(c^2*x^2+1))^(1/ 2)/d^3/c^5/(c^2*x^2+1)^(1/2)*x+2*a*b*(d*(c^2*x^2+1))^(1/2)/d^3/c^6/(c^2*x^ 2+1)*arcsinh(x*c)+4*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^4*arcsin h(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^5*x+10/3* a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^2/d^3/c^6*arcsinh(x*c)-11/3*I*b^2...
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^5*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="frica s")
Output:
integral((b^2*x^5*arcsinh(c*x)^2 + 2*a*b*x^5*arcsinh(c*x) + a^2*x^5)*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)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \] Input:
integrate(x**5*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(5/2),x)
Output:
Integral(x**5*(a + b*asinh(c*x))**2/(d*(c**2*x**2 + 1))**(5/2), x)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{5}}{{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate(x^5*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(5/2),x, algorithm="maxim a")
Output:
1/3*a^2*(3*x^4/((c^2*d*x^2 + d)^(3/2)*c^2*d) + 12*x^2/((c^2*d*x^2 + d)^(3/ 2)*c^4*d) + 8/((c^2*d*x^2 + d)^(3/2)*c^6*d)) + 1/3*(3*b^2*c^4*sqrt(d)*x^4 + 12*b^2*c^2*sqrt(d)*x^2 + 8*b^2*sqrt(d))*sqrt(c^2*x^2 + 1)*log(c*x + sqrt (c^2*x^2 + 1))^2/(c^10*d^3*x^4 + 2*c^8*d^3*x^2 + c^6*d^3) + integrate(-2/3 *((12*b^2*c^3*x^3 - 3*(a*b*c^5 - b^2*c^5)*x^5 + 8*b^2*c*x)*(c^2*x^2 + 1) + (15*b^2*c^4*x^4 - 3*(a*b*c^6 - b^2*c^6)*x^6 + 20*b^2*c^2*x^2 + 8*b^2)*sqr t(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))/(c^12*d^(5/2)*x^7 + 3*c^10*d^ (5/2)*x^5 + 3*c^8*d^(5/2)*x^3 + c^6*d^(5/2)*x + (c^11*d^(5/2)*x^6 + 3*c^9* d^(5/2)*x^4 + 3*c^7*d^(5/2)*x^2 + c^5*d^(5/2))*sqrt(c^2*x^2 + 1)), x)
Exception generated. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x^5*(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
Timed out. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{{\left (d\,c^2\,x^2+d\right )}^{5/2}} \,d x \] Input:
int((x^5*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(5/2),x)
Output:
int((x^5*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(5/2), x)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\left (d+c^2 d x^2\right )^{5/2}} \, dx=\frac {3 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+12 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}+8 \sqrt {c^{2} x^{2}+1}\, a^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{10} x^{4}+12 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{8} x^{2}+6 \left (\int \frac {\mathit {asinh} \left (c x \right ) x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) a b \,c^{6}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{10} x^{4}+6 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{8} x^{2}+3 \left (\int \frac {\mathit {asinh} \left (c x \right )^{2} x^{5}}{\sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+2 \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}}d x \right ) b^{2} c^{6}}{3 \sqrt {d}\, c^{6} d^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:
int(x^5*(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**4*x**4 + 12*sqrt(c**2*x**2 + 1)*a**2*c**2*x **2 + 8*sqrt(c**2*x**2 + 1)*a**2 + 6*int((asinh(c*x)*x**5)/(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**10*x**4 + 12*int((asinh(c*x)*x**5)/(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**2 + 6*int((asinh(c*x)*x**5)/(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**6 + 3*int((asinh(c*x)** 2*x**5)/(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**10*x**4 + 6*int((asinh(c*x)**2*x**5)/(sqr t(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**2 + 3*int((asinh(c*x)**2*x**5)/(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)/(3*sqrt(d)*c**6*d**2*(c**4*x**4 + 2*c**2*x**2 + 1))