Integrand size = 23, antiderivative size = 136 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=-\frac {4}{9} b^2 e^2 x+\frac {2 b^2 e^2 (c+d x)^3}{27 d}+\frac {4 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{9 d}-\frac {2 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{3 d} \]
-4/9*b^2*e^2*x+2/27*b^2*e^2*(d*x+c)^3/d+1/3*e^2*(d*x+c)^3*(a+b*arcsinh(d*x +c))^2/d+4/9*b*e^2*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d-2/9*b*e^2*(d *x+c)^2*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d
Time = 0.21 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.08 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {e^2 \left (-12 b^2 (c+d x)+\left (9 a^2+2 b^2\right ) (c+d x)^3+6 a b \left (2-(c+d x)^2\right ) \sqrt {1+(c+d x)^2}+6 b \left (3 a (c+d x)^3+2 b \sqrt {1+(c+d x)^2}-b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+9 b^2 (c+d x)^3 \text {arcsinh}(c+d x)^2\right )}{27 d} \]
(e^2*(-12*b^2*(c + d*x) + (9*a^2 + 2*b^2)*(c + d*x)^3 + 6*a*b*(2 - (c + d* x)^2)*Sqrt[1 + (c + d*x)^2] + 6*b*(3*a*(c + d*x)^3 + 2*b*Sqrt[1 + (c + d*x )^2] - b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 9*b^2*(c + d*x)^3*ArcSinh[c + d*x]^2))/(27*d)
Time = 0.53 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.88, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {6274, 27, 6191, 6227, 15, 6213, 24}
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.
| \(\displaystyle \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{3} b \int (c+d x)^2d(c+d x)+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))-\frac {1}{9} b (c+d x)^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b \int 1d(c+d x)\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))-\frac {1}{9} b (c+d x)^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b (c+d x)\right )-\frac {1}{9} b (c+d x)^3\right )\right )}{d}\) |
(e^2*(((c + d*x)^3*(a + b*ArcSinh[c + d*x])^2)/3 - (2*b*(-1/9*(b*(c + d*x) ^3) + ((c + d*x)^2*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/3 - (2* (-(b*(c + d*x)) + Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])))/3))/3)) /d
3.2.29.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((e_.) + (f_.)*(x_))^( m_.), x_Symbol] :> Simp[1/d Subst[Int[((d*e - c*f)/d + f*(x/d))^m*(a + b* ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x]
Time = 0.41 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.20
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{9}-\frac {4 d x}{9}-\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(163\) |
| default | \(\frac {\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3}+e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{9}-\frac {4 d x}{9}-\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )+2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(163\) |
| parts | \(\frac {a^{2} e^{2} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b^{2} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {4 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {2 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{9}-\frac {4 d x}{9}-\frac {4 c}{9}+\frac {2 \left (d x +c \right )^{3}}{27}\right )}{d}+\frac {2 e^{2} a b \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{3}-\frac {\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{9}+\frac {2 \sqrt {1+\left (d x +c \right )^{2}}}{9}\right )}{d}\) | \(168\) |
1/d*(1/3*a^2*e^2*(d*x+c)^3+e^2*b^2*(1/3*(d*x+c)^3*arcsinh(d*x+c)^2+4/9*arc sinh(d*x+c)*(1+(d*x+c)^2)^(1/2)-2/9*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)*(d* x+c)^2-4/9*d*x-4/9*c+2/27*(d*x+c)^3)+2*e^2*a*b*(1/3*(d*x+c)^3*arcsinh(d*x+ c)-1/9*(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+2/9*(1+(d*x+c)^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (122) = 244\).
Time = 0.27 (sec) , antiderivative size = 358, normalized size of antiderivative = 2.63 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c d^{2} e^{2} x^{2} + 3 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{2} - 4 \, b^{2}\right )} d e^{2} x + 9 \, {\left (b^{2} d^{3} e^{2} x^{3} + 3 \, b^{2} c d^{2} e^{2} x^{2} + 3 \, b^{2} c^{2} d e^{2} x + b^{2} c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 6 \, {\left (3 \, a b d^{3} e^{2} x^{3} + 9 \, a b c d^{2} e^{2} x^{2} + 9 \, a b c^{2} d e^{2} x + 3 \, a b c^{3} e^{2} - {\left (b^{2} d^{2} e^{2} x^{2} + 2 \, b^{2} c d e^{2} x + {\left (b^{2} c^{2} - 2 \, b^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 6 \, {\left (a b d^{2} e^{2} x^{2} + 2 \, a b c d e^{2} x + {\left (a b c^{2} - 2 \, a b\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{27 \, d} \]
1/27*((9*a^2 + 2*b^2)*d^3*e^2*x^3 + 3*(9*a^2 + 2*b^2)*c*d^2*e^2*x^2 + 3*(( 9*a^2 + 2*b^2)*c^2 - 4*b^2)*d*e^2*x + 9*(b^2*d^3*e^2*x^3 + 3*b^2*c*d^2*e^2 *x^2 + 3*b^2*c^2*d*e^2*x + b^2*c^3*e^2)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d *x + c^2 + 1))^2 + 6*(3*a*b*d^3*e^2*x^3 + 9*a*b*c*d^2*e^2*x^2 + 9*a*b*c^2* d*e^2*x + 3*a*b*c^3*e^2 - (b^2*d^2*e^2*x^2 + 2*b^2*c*d*e^2*x + (b^2*c^2 - 2*b^2)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)) - 6*(a*b*d^2*e^2*x^2 + 2*a*b*c*d*e^2*x + (a*b*c^2 - 2*a*b)*e^2)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/d
Leaf count of result is larger than twice the leaf count of optimal. 610 vs. \(2 (126) = 252\).
Time = 0.32 (sec) , antiderivative size = 610, normalized size of antiderivative = 4.49 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\begin {cases} a^{2} c^{2} e^{2} x + a^{2} c d e^{2} x^{2} + \frac {a^{2} d^{2} e^{2} x^{3}}{3} + \frac {2 a b c^{3} e^{2} \operatorname {asinh}{\left (c + d x \right )}}{3 d} + 2 a b c^{2} e^{2} x \operatorname {asinh}{\left (c + d x \right )} - \frac {2 a b c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9 d} + 2 a b c d e^{2} x^{2} \operatorname {asinh}{\left (c + d x \right )} - \frac {4 a b c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9} + \frac {2 a b d^{2} e^{2} x^{3} \operatorname {asinh}{\left (c + d x \right )}}{3} - \frac {2 a b d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9} + \frac {4 a b e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{9 d} + \frac {b^{2} c^{3} e^{2} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3 d} + b^{2} c^{2} e^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c^{2} e^{2} x}{9} - \frac {2 b^{2} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9 d} + b^{2} c d e^{2} x^{2} \operatorname {asinh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c d e^{2} x^{2}}{9} - \frac {4 b^{2} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9} + \frac {b^{2} d^{2} e^{2} x^{3} \operatorname {asinh}^{2}{\left (c + d x \right )}}{3} + \frac {2 b^{2} d^{2} e^{2} x^{3}}{27} - \frac {2 b^{2} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9} - \frac {4 b^{2} e^{2} x}{9} + \frac {4 b^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{9 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \]
Piecewise((a**2*c**2*e**2*x + a**2*c*d*e**2*x**2 + a**2*d**2*e**2*x**3/3 + 2*a*b*c**3*e**2*asinh(c + d*x)/(3*d) + 2*a*b*c**2*e**2*x*asinh(c + d*x) - 2*a*b*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(9*d) + 2*a*b*c*d*e* *2*x**2*asinh(c + d*x) - 4*a*b*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/9 + 2*a*b*d**2*e**2*x**3*asinh(c + d*x)/3 - 2*a*b*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/9 + 4*a*b*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(9*d) + b**2*c**3*e**2*asinh(c + d*x)**2/(3*d) + b**2*c**2*e**2*x*as inh(c + d*x)**2 + 2*b**2*c**2*e**2*x/9 - 2*b**2*c**2*e**2*sqrt(c**2 + 2*c* d*x + d**2*x**2 + 1)*asinh(c + d*x)/(9*d) + b**2*c*d*e**2*x**2*asinh(c + d *x)**2 + 2*b**2*c*d*e**2*x**2/9 - 4*b**2*c*e**2*x*sqrt(c**2 + 2*c*d*x + d* *2*x**2 + 1)*asinh(c + d*x)/9 + b**2*d**2*e**2*x**3*asinh(c + d*x)**2/3 + 2*b**2*d**2*e**2*x**3/27 - 2*b**2*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x **2 + 1)*asinh(c + d*x)/9 - 4*b**2*e**2*x/9 + 4*b**2*e**2*sqrt(c**2 + 2*c* d*x + d**2*x**2 + 1)*asinh(c + d*x)/(9*d), Ne(d, 0)), (c**2*e**2*x*(a + b* asinh(c))**2, True))
\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
1/3*a^2*d^2*e^2*x^3 + a^2*c*d*e^2*x^2 + (2*x^2*arcsinh(d*x + c) - d*(3*c^2 *arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 + sqrt(d^ 2*x^2 + 2*c*d*x + c^2 + 1)*x/d^2 - (c^2 + 1)*arcsinh(2*(d^2*x + c*d)/sqrt( -4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^3 - 3*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c /d^3))*a*b*c*d*e^2 + 1/9*(6*x^3*arcsinh(d*x + c) - d*(2*sqrt(d^2*x^2 + 2*c *d*x + c^2 + 1)*x^2/d^2 - 15*c^3*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 - 5*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x/d^3 + 9*( c^2 + 1)*c*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^4 + 15*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^2/d^4 - 4*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)/d^4))*a*b*d^2*e^2 + a^2*c^2*e^2*x + 2*((d*x + c)*arc sinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a*b*c^2*e^2/d + 1/3*(b^2*d^2*e^2*x^ 3 + 3*b^2*c*d*e^2*x^2 + 3*b^2*c^2*e^2*x)*log(d*x + c + sqrt(d^2*x^2 + 2*c* d*x + c^2 + 1))^2 - integrate(2/3*(b^2*d^5*e^2*x^5 + 5*b^2*c*d^4*e^2*x^4 + (10*c^2*d^3*e^2 + d^3*e^2)*b^2*x^3 + 3*(3*c^3*d^2*e^2 + c*d^2*e^2)*b^2*x^ 2 + 3*(c^4*d*e^2 + c^2*d*e^2)*b^2*x + (b^2*d^4*e^2*x^4 + 4*b^2*c*d^3*e^2*x ^3 + 6*b^2*c^2*d^2*e^2*x^2 + 3*b^2*c^3*d*e^2*x)*sqrt(d^2*x^2 + 2*c*d*x + c ^2 + 1))*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))/(d^3*x^3 + 3*c*d ^2*x^2 + c^3 + (3*c^2*d + d)*x + (d^2*x^2 + 2*c*d*x + c^2 + 1)^(3/2) + c), x)
\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \]
Timed out. \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \]