Integrand size = 28, antiderivative size = 383 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {16 a b x \sqrt {1+c^2 x^2}}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {298 b^2 \left (1+c^2 x^2\right )}{225 c^6 \sqrt {d+c^2 d x^2}}-\frac {76 b^2 \left (1+c^2 x^2\right )^2}{675 c^6 \sqrt {d+c^2 d x^2}}+\frac {2 b^2 \left (1+c^2 x^2\right )^3}{125 c^6 \sqrt {d+c^2 d x^2}}-\frac {16 b^2 x \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{15 c^5 \sqrt {d+c^2 d x^2}}+\frac {8 b x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{45 c^3 \sqrt {d+c^2 d x^2}}-\frac {2 b x^5 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{25 c \sqrt {d+c^2 d x^2}}+\frac {8 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^6 d}-\frac {4 x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4 d}+\frac {x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{5 c^2 d} \]
298/225*b^2*(c^2*x^2+1)/c^6/(c^2*d*x^2+d)^(1/2)-76/675*b^2*(c^2*x^2+1)^2/c ^6/(c^2*d*x^2+d)^(1/2)+2/125*b^2*(c^2*x^2+1)^3/c^6/(c^2*d*x^2+d)^(1/2)-16/ 15*a*b*x*(c^2*x^2+1)^(1/2)/c^5/(c^2*d*x^2+d)^(1/2)-16/15*b^2*x*arcsinh(c*x )*(c^2*x^2+1)^(1/2)/c^5/(c^2*d*x^2+d)^(1/2)+8/45*b*x^3*(a+b*arcsinh(c*x))* (c^2*x^2+1)^(1/2)/c^3/(c^2*d*x^2+d)^(1/2)-2/25*b*x^5*(a+b*arcsinh(c*x))*(c ^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)+8/15*(a+b*arcsinh(c*x))^2*(c^2*d*x^2 +d)^(1/2)/c^6/d-4/15*x^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^4/d+1/ 5*x^4*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^2/d
Time = 0.31 (sec) , antiderivative size = 230, normalized size of antiderivative = 0.60 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {-30 a b c x \sqrt {1+c^2 x^2} \left (120-20 c^2 x^2+9 c^4 x^4\right )+225 a^2 \left (8+4 c^2 x^2-c^4 x^4+3 c^6 x^6\right )+2 b^2 \left (2072+1936 c^2 x^2-109 c^4 x^4+27 c^6 x^6\right )+30 b \left (b c x \sqrt {1+c^2 x^2} \left (-120+20 c^2 x^2-9 c^4 x^4\right )+15 a \left (8+4 c^2 x^2-c^4 x^4+3 c^6 x^6\right )\right ) \text {arcsinh}(c x)+225 b^2 \left (8+4 c^2 x^2-c^4 x^4+3 c^6 x^6\right ) \text {arcsinh}(c x)^2}{3375 c^6 \sqrt {d+c^2 d x^2}} \]
(-30*a*b*c*x*Sqrt[1 + c^2*x^2]*(120 - 20*c^2*x^2 + 9*c^4*x^4) + 225*a^2*(8 + 4*c^2*x^2 - c^4*x^4 + 3*c^6*x^6) + 2*b^2*(2072 + 1936*c^2*x^2 - 109*c^4 *x^4 + 27*c^6*x^6) + 30*b*(b*c*x*Sqrt[1 + c^2*x^2]*(-120 + 20*c^2*x^2 - 9* c^4*x^4) + 15*a*(8 + 4*c^2*x^2 - c^4*x^4 + 3*c^6*x^6))*ArcSinh[c*x] + 225* b^2*(8 + 4*c^2*x^2 - c^4*x^4 + 3*c^6*x^6)*ArcSinh[c*x]^2)/(3375*c^6*Sqrt[d + c^2*d*x^2])
Time = 1.64 (sec) , antiderivative size = 397, normalized size of antiderivative = 1.04, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6227, 6191, 243, 53, 2009, 6227, 6191, 243, 53, 2009, 6213, 2009}
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.
| \(\displaystyle \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}} \, dx\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \int x^4 (a+b \text {arcsinh}(c x))dx}{5 c \sqrt {c^2 d x^2+d}}-\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle -\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{5} b c \int \frac {x^5}{\sqrt {c^2 x^2+1}}dx\right )}{5 c \sqrt {c^2 d x^2+d}}-\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{5 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \int \frac {x^4}{\sqrt {c^2 x^2+1}}dx^2\right )}{5 c \sqrt {c^2 d x^2+d}}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle -\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{5 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \int \left (\frac {\left (c^2 x^2+1\right )^{3/2}}{c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}+\frac {1}{c^4 \sqrt {c^2 x^2+1}}\right )dx^2\right )}{5 c \sqrt {c^2 d x^2+d}}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle -\frac {4 \left (-\frac {2 b \sqrt {c^2 x^2+1} \int x^2 (a+b \text {arcsinh}(c x))dx}{3 c \sqrt {c^2 d x^2+d}}-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{3} b c \int \frac {x^3}{\sqrt {c^2 x^2+1}}dx\right )}{3 c \sqrt {c^2 d x^2+d}}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \int \frac {x^2}{\sqrt {c^2 x^2+1}}dx^2\right )}{3 c \sqrt {c^2 d x^2+d}}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \int \left (\frac {\sqrt {c^2 x^2+1}}{c^2}-\frac {1}{c^2 \sqrt {c^2 x^2+1}}\right )dx^2\right )}{3 c \sqrt {c^2 d x^2+d}}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {c^2 d x^2+d}}dx}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle -\frac {4 \left (-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \int (a+b \text {arcsinh}(c x))dx}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}+\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{5 c^2}+\frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 c \sqrt {c^2 d x^2+d}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{5 c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{5} x^5 (a+b \text {arcsinh}(c x))-\frac {1}{10} b c \left (\frac {2 \left (c^2 x^2+1\right )^{5/2}}{5 c^6}-\frac {4 \left (c^2 x^2+1\right )^{3/2}}{3 c^6}+\frac {2 \sqrt {c^2 x^2+1}}{c^6}\right )\right )}{5 c \sqrt {c^2 d x^2+d}}-\frac {4 \left (\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{3 c^2 d}-\frac {2 \left (\frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{c^2 d}-\frac {2 b \sqrt {c^2 x^2+1} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{c \sqrt {c^2 d x^2+d}}\right )}{3 c^2}-\frac {2 b \sqrt {c^2 x^2+1} \left (\frac {1}{3} x^3 (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^4}-\frac {2 \sqrt {c^2 x^2+1}}{c^4}\right )\right )}{3 c \sqrt {c^2 d x^2+d}}\right )}{5 c^2}\) |
(x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(5*c^2*d) - (2*b*Sqrt[1 + c^2*x^2]*(-1/10*(b*c*((2*Sqrt[1 + c^2*x^2])/c^6 - (4*(1 + c^2*x^2)^(3/2)) /(3*c^6) + (2*(1 + c^2*x^2)^(5/2))/(5*c^6))) + (x^5*(a + b*ArcSinh[c*x]))/ 5))/(5*c*Sqrt[d + c^2*d*x^2]) - (4*((x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSin h[c*x])^2)/(3*c^2*d) - (2*b*Sqrt[1 + c^2*x^2]*(-1/6*(b*c*((-2*Sqrt[1 + c^2 *x^2])/c^4 + (2*(1 + c^2*x^2)^(3/2))/(3*c^4))) + (x^3*(a + b*ArcSinh[c*x]) )/3))/(3*c*Sqrt[d + c^2*d*x^2]) - (2*((Sqrt[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])))/(3*c^2)))/(5*c^2)
3.3.90.3.1 Defintions of rubi rules used
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_)^(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[((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]
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/(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]
Leaf count of result is larger than twice the leaf count of optimal. \(1226\) vs. \(2(335)=670\).
Time = 0.33 (sec) , antiderivative size = 1227, normalized size of antiderivative = 3.20
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1227\) |
| parts | \(\text {Expression too large to display}\) | \(1227\) |
a^2*(1/5*x^4/c^2/d*(c^2*d*x^2+d)^(1/2)-4/5/c^2*(1/3*x^2/c^2/d*(c^2*d*x^2+d )^(1/2)-2/3/d/c^4*(c^2*d*x^2+d)^(1/2)))+b^2*(1/4000*(d*(c^2*x^2+1))^(1/2)* (16*c^6*x^6+16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4+20*c^3*x^3*(c^2*x^2+1) ^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^2+1)^(1/2)+1)*(25*arcsinh(c*x)^2-10*arcsinh (c*x)+2)/c^6/d/(c^2*x^2+1)-5/864*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^ 3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(9*arcsinh(c*x)^2 -6*arcsinh(c*x)+2)/c^6/d/(c^2*x^2+1)+5/16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c *x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)^2-2*arcsinh(c*x)+2)/c^6/d/(c^2*x^2+1 )+5/16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c* x)^2+2*arcsinh(c*x)+2)/c^6/d/(c^2*x^2+1)-5/864*(d*(c^2*x^2+1))^(1/2)*(4*c^ 4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*(9* arcsinh(c*x)^2+6*arcsinh(c*x)+2)/c^6/d/(c^2*x^2+1)+1/4000*(d*(c^2*x^2+1))^ (1/2)*(16*c^6*x^6-16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4-20*c^3*x^3*(c^2* x^2+1)^(1/2)+13*c^2*x^2-5*c*x*(c^2*x^2+1)^(1/2)+1)*(25*arcsinh(c*x)^2+10*a rcsinh(c*x)+2)/c^6/d/(c^2*x^2+1))+2*a*b*(1/800*(d*(c^2*x^2+1))^(1/2)*(16*c ^6*x^6+16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4+20*c^3*x^3*(c^2*x^2+1)^(1/2 )+13*c^2*x^2+5*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+5*arcsinh(c*x))/c^6/d/(c^2*x^2 +1)-5/288*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c ^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+3*arcsinh(c*x))/c^6/d/(c^2*x^2+1)+5/ 16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(-1+arcsinh(...
Time = 0.28 (sec) , antiderivative size = 319, normalized size of antiderivative = 0.83 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {225 \, {\left (3 \, b^{2} c^{6} x^{6} - b^{2} c^{4} x^{4} + 4 \, b^{2} c^{2} x^{2} + 8 \, b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (45 \, a b c^{6} x^{6} - 15 \, a b c^{4} x^{4} + 60 \, a b c^{2} x^{2} + 120 \, a b - {\left (9 \, b^{2} c^{5} x^{5} - 20 \, b^{2} c^{3} x^{3} + 120 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} x^{6} - {\left (225 \, a^{2} + 218 \, b^{2}\right )} c^{4} x^{4} + 4 \, {\left (225 \, a^{2} + 968 \, b^{2}\right )} c^{2} x^{2} + 1800 \, a^{2} + 4144 \, b^{2} - 30 \, {\left (9 \, a b c^{5} x^{5} - 20 \, a b c^{3} x^{3} + 120 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{3375 \, {\left (c^{8} d x^{2} + c^{6} d\right )}} \]
1/3375*(225*(3*b^2*c^6*x^6 - b^2*c^4*x^4 + 4*b^2*c^2*x^2 + 8*b^2)*sqrt(c^2 *d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 30*(45*a*b*c^6*x^6 - 15*a*b*c ^4*x^4 + 60*a*b*c^2*x^2 + 120*a*b - (9*b^2*c^5*x^5 - 20*b^2*c^3*x^3 + 120* b^2*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1 )) + (27*(25*a^2 + 2*b^2)*c^6*x^6 - (225*a^2 + 218*b^2)*c^4*x^4 + 4*(225*a ^2 + 968*b^2)*c^2*x^2 + 1800*a^2 + 4144*b^2 - 30*(9*a*b*c^5*x^5 - 20*a*b*c ^3*x^3 + 120*a*b*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^8*d*x^2 + c^6*d)
\[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^{5} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]
Time = 0.21 (sec) , antiderivative size = 353, normalized size of antiderivative = 0.92 \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {1}{15} \, {\left (\frac {3 \, \sqrt {c^{2} d x^{2} + d} x^{4}}{c^{2} d} - \frac {4 \, \sqrt {c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {c^{2} d x^{2} + d}}{c^{6} d}\right )} b^{2} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{15} \, {\left (\frac {3 \, \sqrt {c^{2} d x^{2} + d} x^{4}}{c^{2} d} - \frac {4 \, \sqrt {c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {c^{2} d x^{2} + d}}{c^{6} d}\right )} a b \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, {\left (\frac {3 \, \sqrt {c^{2} d x^{2} + d} x^{4}}{c^{2} d} - \frac {4 \, \sqrt {c^{2} d x^{2} + d} x^{2}}{c^{4} d} + \frac {8 \, \sqrt {c^{2} d x^{2} + d}}{c^{6} d}\right )} a^{2} + \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {c^{2} x^{2} + 1} c^{2} x^{4} - 136 \, \sqrt {c^{2} x^{2} + 1} x^{2} + \frac {2072 \, \sqrt {c^{2} x^{2} + 1}}{c^{2}}}{c^{4} \sqrt {d}} - \frac {15 \, {\left (9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x\right )} \operatorname {arsinh}\left (c x\right )}{c^{5} \sqrt {d}}\right )} - \frac {2 \, {\left (9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x\right )} a b}{225 \, c^{5} \sqrt {d}} \]
1/15*(3*sqrt(c^2*d*x^2 + d)*x^4/(c^2*d) - 4*sqrt(c^2*d*x^2 + d)*x^2/(c^4*d ) + 8*sqrt(c^2*d*x^2 + d)/(c^6*d))*b^2*arcsinh(c*x)^2 + 2/15*(3*sqrt(c^2*d *x^2 + d)*x^4/(c^2*d) - 4*sqrt(c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(c^2*d*x ^2 + d)/(c^6*d))*a*b*arcsinh(c*x) + 1/15*(3*sqrt(c^2*d*x^2 + d)*x^4/(c^2*d ) - 4*sqrt(c^2*d*x^2 + d)*x^2/(c^4*d) + 8*sqrt(c^2*d*x^2 + d)/(c^6*d))*a^2 + 2/3375*b^2*((27*sqrt(c^2*x^2 + 1)*c^2*x^4 - 136*sqrt(c^2*x^2 + 1)*x^2 + 2072*sqrt(c^2*x^2 + 1)/c^2)/(c^4*sqrt(d)) - 15*(9*c^4*x^5 - 20*c^2*x^3 + 120*x)*arcsinh(c*x)/(c^5*sqrt(d))) - 2/225*(9*c^4*x^5 - 20*c^2*x^3 + 120*x )*a*b/(c^5*sqrt(d))
Exception generated. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\text {Exception raised: TypeError} \]
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. \[ \int \frac {x^5 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^5\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \]