Integrand size = 21, antiderivative size = 195 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {3 b^4 e (c+d x)^2}{4 d}-\frac {3 b^3 e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{2 d}+\frac {3 b^2 e (a+b \text {arcsinh}(c+d x))^2}{4 d}+\frac {3 b^2 e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{2 d}-\frac {b e (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{d}+\frac {e (a+b \text {arcsinh}(c+d x))^4}{4 d}+\frac {e (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4}{2 d} \]
3/4*b^4*e*(d*x+c)^2/d+3/4*b^2*e*(a+b*arcsinh(d*x+c))^2/d+3/2*b^2*e*(d*x+c) ^2*(a+b*arcsinh(d*x+c))^2/d+1/4*e*(a+b*arcsinh(d*x+c))^4/d+1/2*e*(d*x+c)^2 *(a+b*arcsinh(d*x+c))^4/d-3/2*b^3*e*(d*x+c)*(a+b*arcsinh(d*x+c))*(1+(d*x+c )^2)^(1/2)/d-b*e*(d*x+c)*(a+b*arcsinh(d*x+c))^3*(1+(d*x+c)^2)^(1/2)/d
Time = 0.35 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.54 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {e \left (\left (2 a^4+6 a^2 b^2+3 b^4\right ) (c+d x)^2-2 a b \left (2 a^2+3 b^2\right ) (c+d x) \sqrt {1+(c+d x)^2}+2 a b \left (2 a^2+3 b^2\right ) \text {arcsinh}(c+d x)-2 b (c+d x) \left (-4 a^3 (c+d x)-6 a b^2 (c+d x)+6 a^2 b \sqrt {1+(c+d x)^2}+3 b^3 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+3 b^2 \left (2 a^2+b^2+4 a^2 (c+d x)^2+2 b^2 (c+d x)^2-4 a b (c+d x) \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2+4 b^3 \left (a+2 a (c+d x)^2-b (c+d x) \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^3+b^4 \left (1+2 (c+d x)^2\right ) \text {arcsinh}(c+d x)^4\right )}{4 d} \]
(e*((2*a^4 + 6*a^2*b^2 + 3*b^4)*(c + d*x)^2 - 2*a*b*(2*a^2 + 3*b^2)*(c + d *x)*Sqrt[1 + (c + d*x)^2] + 2*a*b*(2*a^2 + 3*b^2)*ArcSinh[c + d*x] - 2*b*( c + d*x)*(-4*a^3*(c + d*x) - 6*a*b^2*(c + d*x) + 6*a^2*b*Sqrt[1 + (c + d*x )^2] + 3*b^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 3*b^2*(2*a^2 + b^2 + 4*a^2*(c + d*x)^2 + 2*b^2*(c + d*x)^2 - 4*a*b*(c + d*x)*Sqrt[1 + (c + d* x)^2])*ArcSinh[c + d*x]^2 + 4*b^3*(a + 2*a*(c + d*x)^2 - b*(c + d*x)*Sqrt[ 1 + (c + d*x)^2])*ArcSinh[c + d*x]^3 + b^4*(1 + 2*(c + d*x)^2)*ArcSinh[c + d*x]^4))/(4*d)
Time = 0.93 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.93, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6274, 27, 6191, 6227, 6191, 6198, 6227, 15, 6198}
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.
| \(\displaystyle \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int e (c+d x) (a+b \text {arcsinh}(c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e \int (c+d x) (a+b \text {arcsinh}(c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4-2 b \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4-2 b \left (-\frac {3}{2} b \int (c+d x) (a+b \text {arcsinh}(c+d x))^2d(c+d x)-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4-2 b \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2-b \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4-2 b \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2-b \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{8 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4-2 b \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2-b \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{2} b \int (c+d x)d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{8 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4-2 b \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2-b \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b (c+d x)^2\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{8 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {e \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4-2 b \left (-\frac {(a+b \text {arcsinh}(c+d x))^4}{8 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2-b \left (\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) (a+b \text {arcsinh}(c+d x))-\frac {(a+b \text {arcsinh}(c+d x))^2}{4 b}-\frac {1}{4} b (c+d x)^2\right )\right )\right )\right )}{d}\) |
(e*(((c + d*x)^2*(a + b*ArcSinh[c + d*x])^4)/2 - 2*b*(((c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/2 - (a + b*ArcSinh[c + d*x])^4/(8 *b) - (3*b*(((c + d*x)^2*(a + b*ArcSinh[c + d*x])^2)/2 - b*(-1/4*(b*(c + d *x)^2) + ((c + d*x)*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/2 - (a + b*ArcSinh[c + d*x])^2/(4*b))))/2)))/d
3.2.49.3.1 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_.)/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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(370\) vs. \(2(179)=358\).
Time = 0.08 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.90
| method | result | size |
| derivativedivides | \(\frac {\frac {e \,a^{4} \left (d x +c \right )^{2}}{2}+e \,b^{4} \left (\frac {\left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )^{4}}{2}-\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{4}+\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}+\frac {3}{4}\right )+4 e a \,b^{3} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{8}\right )+6 e \,a^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+4 e b \,a^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\) | \(371\) |
| default | \(\frac {\frac {e \,a^{4} \left (d x +c \right )^{2}}{2}+e \,b^{4} \left (\frac {\left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )^{4}}{2}-\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{4}+\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}+\frac {3}{4}\right )+4 e a \,b^{3} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{8}\right )+6 e \,a^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )+4 e b \,a^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\) | \(371\) |
| parts | \(e \,a^{4} \left (\frac {1}{2} d \,x^{2}+c x \right )+\frac {e \,b^{4} \left (\frac {\left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )^{4}}{2}-\left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{4}}{4}+\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {3 \left (d x +c \right )^{2}}{4}+\frac {3}{4}\right )}{d}+\frac {4 e a \,b^{3} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{3} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{4}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{3}}{4}+\frac {3 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{8}\right )}{d}+\frac {6 e \,a^{2} b^{2} \left (\frac {\operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{2}-\frac {\sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\operatorname {arcsinh}\left (d x +c \right )^{2}}{4}+\frac {\left (d x +c \right )^{2}}{4}+\frac {1}{4}\right )}{d}+\frac {4 e b \,a^{3} \left (\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )}{2}-\frac {\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {\operatorname {arcsinh}\left (d x +c \right )}{4}\right )}{d}\) | \(381\) |
1/d*(1/2*e*a^4*(d*x+c)^2+e*b^4*(1/2*(1+(d*x+c)^2)*arcsinh(d*x+c)^4-(d*x+c) *arcsinh(d*x+c)^3*(1+(d*x+c)^2)^(1/2)-1/4*arcsinh(d*x+c)^4+3/2*arcsinh(d*x +c)^2*(1+(d*x+c)^2)-3/2*(1+(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d*x+c)-3/4*arc sinh(d*x+c)^2+3/4*(d*x+c)^2+3/4)+4*e*a*b^3*(1/2*arcsinh(d*x+c)^3*(1+(d*x+c )^2)-3/4*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)*(d*x+c)-1/4*arcsinh(d*x+c)^3 +3/4*(1+(d*x+c)^2)*arcsinh(d*x+c)-3/8*(d*x+c)*(1+(d*x+c)^2)^(1/2)-3/8*arcs inh(d*x+c))+6*e*a^2*b^2*(1/2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)-1/2*(1+(d*x+c) ^2)^(1/2)*(d*x+c)*arcsinh(d*x+c)-1/4*arcsinh(d*x+c)^2+1/4*(d*x+c)^2+1/4)+4 *e*b*a^3*(1/2*(d*x+c)^2*arcsinh(d*x+c)-1/4*(d*x+c)*(1+(d*x+c)^2)^(1/2)+1/4 *arcsinh(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 574 vs. \(2 (179) = 358\).
Time = 0.27 (sec) , antiderivative size = 574, normalized size of antiderivative = 2.94 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {{\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} d^{2} e x^{2} + 2 \, {\left (2 \, a^{4} + 6 \, a^{2} b^{2} + 3 \, b^{4}\right )} c d e x + {\left (2 \, b^{4} d^{2} e x^{2} + 4 \, b^{4} c d e x + {\left (2 \, b^{4} c^{2} + b^{4}\right )} e\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + 4 \, {\left (2 \, a b^{3} d^{2} e x^{2} + 4 \, a b^{3} c d e x + {\left (2 \, a b^{3} c^{2} + a b^{3}\right )} e - {\left (b^{4} d e x + b^{4} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + 3 \, {\left (2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c d e x + {\left (2 \, a^{2} b^{2} + b^{4} + 2 \, {\left (2 \, a^{2} b^{2} + b^{4}\right )} c^{2}\right )} e - 4 \, {\left (a b^{3} d e x + a b^{3} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 2 \, {\left (2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d^{2} e x^{2} + 4 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c d e x + {\left (2 \, a^{3} b + 3 \, a b^{3} + 2 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c^{2}\right )} e - 3 \, {\left ({\left (2 \, a^{2} b^{2} + b^{4}\right )} d e x + {\left (2 \, a^{2} b^{2} + b^{4}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \, {\left ({\left (2 \, a^{3} b + 3 \, a b^{3}\right )} d e x + {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} c e\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{4 \, d} \]
1/4*((2*a^4 + 6*a^2*b^2 + 3*b^4)*d^2*e*x^2 + 2*(2*a^4 + 6*a^2*b^2 + 3*b^4) *c*d*e*x + (2*b^4*d^2*e*x^2 + 4*b^4*c*d*e*x + (2*b^4*c^2 + b^4)*e)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + 4*(2*a*b^3*d^2*e*x^2 + 4*a*b ^3*c*d*e*x + (2*a*b^3*c^2 + a*b^3)*e - (b^4*d*e*x + b^4*c*e)*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))^3 + 3*(2*(2*a^2*b^2 + b^4)*d^2*e*x^2 + 4*(2*a^2*b^2 + b^4)*c*d*e*x + (2*a^2*b ^2 + b^4 + 2*(2*a^2*b^2 + b^4)*c^2)*e - 4*(a*b^3*d*e*x + a*b^3*c*e)*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))^2 + 2*(2*(2*a^3*b + 3*a*b^3)*d^2*e*x^2 + 4*(2*a^3*b + 3*a*b^3)*c*d*e*x + (2*a^3*b + 3*a*b^3 + 2*(2*a^3*b + 3*a*b^3)*c^2)*e - 3*((2*a^2*b^2 + b^4 )*d*e*x + (2*a^2*b^2 + b^4)*c*e)*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)) - 2*((2*a^3*b + 3*a*b^3)*d*e*x + (2*a^3*b + 3*a*b^3)*c*e)*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. 1027 vs. \(2 (178) = 356\).
Time = 0.45 (sec) , antiderivative size = 1027, normalized size of antiderivative = 5.27 \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Too large to display} \]
Piecewise((a**4*c*e*x + a**4*d*e*x**2/2 + 2*a**3*b*c**2*e*asinh(c + d*x)/d + 4*a**3*b*c*e*x*asinh(c + d*x) - a**3*b*c*e*sqrt(c**2 + 2*c*d*x + d**2*x **2 + 1)/d + 2*a**3*b*d*e*x**2*asinh(c + d*x) - a**3*b*e*x*sqrt(c**2 + 2*c *d*x + d**2*x**2 + 1) + a**3*b*e*asinh(c + d*x)/d + 3*a**2*b**2*c**2*e*asi nh(c + d*x)**2/d + 6*a**2*b**2*c*e*x*asinh(c + d*x)**2 + 3*a**2*b**2*c*e*x - 3*a**2*b**2*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/d + 3*a**2*b**2*d*e*x**2*asinh(c + d*x)**2 + 3*a**2*b**2*d*e*x**2/2 - 3*a**2* b**2*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x) + 3*a**2*b**2 *e*asinh(c + d*x)**2/(2*d) + 2*a*b**3*c**2*e*asinh(c + d*x)**3/d + 3*a*b** 3*c**2*e*asinh(c + d*x)/d + 4*a*b**3*c*e*x*asinh(c + d*x)**3 + 6*a*b**3*c* e*x*asinh(c + d*x) - 3*a*b**3*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asi nh(c + d*x)**2/d - 3*a*b**3*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(2*d) + 2*a*b**3*d*e*x**2*asinh(c + d*x)**3 + 3*a*b**3*d*e*x**2*asinh(c + d*x) - 3*a*b**3*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2 - 3* a*b**3*e*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/2 + a*b**3*e*asinh(c + d*x )**3/d + 3*a*b**3*e*asinh(c + d*x)/(2*d) + b**4*c**2*e*asinh(c + d*x)**4/( 2*d) + 3*b**4*c**2*e*asinh(c + d*x)**2/(2*d) + b**4*c*e*x*asinh(c + d*x)** 4 + 3*b**4*c*e*x*asinh(c + d*x)**2 + 3*b**4*c*e*x/2 - b**4*c*e*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**3/d - 3*b**4*c*e*sqrt(c**2 + 2*c *d*x + d**2*x**2 + 1)*asinh(c + d*x)/(2*d) + b**4*d*e*x**2*asinh(c + d*...
\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
1/2*a^4*d*e*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^3*b*d*e + a^4 *c*e*x + 4*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a^3*b*c*e/ d + 1/2*(b^4*d*e*x^2 + 2*b^4*c*e*x)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + integrate(2*((2*(c^4*e + c^2*e)*a*b^3 + (2*a*b^3*d^4*e - b^ 4*d^4*e)*x^4 + 4*(2*a*b^3*c*d^3*e - b^4*c*d^3*e)*x^3 + (2*(6*c^2*d^2*e + d ^2*e)*a*b^3 - (5*c^2*d^2*e + d^2*e)*b^4)*x^2 + 2*(2*(2*c^3*d*e + c*d*e)*a* b^3 - (c^3*d*e + c*d*e)*b^4)*x + (2*(c^3*e + c*e)*a*b^3 + (2*a*b^3*d^3*e - b^4*d^3*e)*x^3 + 3*(2*a*b^3*c*d^2*e - b^4*c*d^2*e)*x^2 - 2*(b^4*c^2*d*e - (3*c^2*d*e + d*e)*a*b^3)*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))^3 + 3*(a^2*b^2*d^4*e*x^4 + 4*a^2*b^ 2*c*d^3*e*x^3 + (6*c^2*d^2*e + d^2*e)*a^2*b^2*x^2 + 2*(2*c^3*d*e + c*d*e)* a^2*b^2*x + (c^4*e + c^2*e)*a^2*b^2 + (a^2*b^2*d^3*e*x^3 + 3*a^2*b^2*c*d^2 *e*x^2 + (3*c^2*d*e + d*e)*a^2*b^2*x + (c^3*e + c*e)*a^2*b^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))^2) /(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)
\[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
Timed out. \[ \int (c e+d e x) (a+b \text {arcsinh}(c+d x))^4 \, dx=\int \left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]