Integrand size = 23, antiderivative size = 349 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=-\frac {45 b^4 e^3 (c+d x)^2}{128 d}+\frac {3 b^4 e^3 (c+d x)^4}{128 d}+\frac {45 b^3 e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{64 d}-\frac {3 b^3 e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{32 d}-\frac {45 b^2 e^3 (a+b \text {arcsinh}(c+d x))^2}{128 d}-\frac {9 b^2 e^3 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b^2 e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2}{16 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{8 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{4 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^4}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4}{4 d} \]
-45/128*b^4*e^3*(d*x+c)^2/d+3/128*b^4*e^3*(d*x+c)^4/d-45/128*b^2*e^3*(a+b* arcsinh(d*x+c))^2/d-9/16*b^2*e^3*(d*x+c)^2*(a+b*arcsinh(d*x+c))^2/d+3/16*b ^2*e^3*(d*x+c)^4*(a+b*arcsinh(d*x+c))^2/d-3/32*e^3*(a+b*arcsinh(d*x+c))^4/ d+1/4*e^3*(d*x+c)^4*(a+b*arcsinh(d*x+c))^4/d+45/64*b^3*e^3*(d*x+c)*(a+b*ar csinh(d*x+c))*(1+(d*x+c)^2)^(1/2)/d-3/32*b^3*e^3*(d*x+c)^3*(a+b*arcsinh(d* x+c))*(1+(d*x+c)^2)^(1/2)/d+3/8*b*e^3*(d*x+c)*(a+b*arcsinh(d*x+c))^3*(1+(d *x+c)^2)^(1/2)/d-1/4*b*e^3*(d*x+c)^3*(a+b*arcsinh(d*x+c))^3*(1+(d*x+c)^2)^ (1/2)/d
Time = 0.54 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.36 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {e^3 \left (-9 b^2 \left (8 a^2+5 b^2\right ) (c+d x)^2+\left (32 a^4+24 a^2 b^2+3 b^4\right ) (c+d x)^4+2 a b (c+d x) \sqrt {1+(c+d x)^2} \left (24 a^2+45 b^2-2 \left (8 a^2+3 b^2\right ) (c+d x)^2\right )-6 a b \left (8 a^2+15 b^2\right ) \text {arcsinh}(c+d x)+2 b (c+d x) \left (-72 a b^2 (c+d x)+64 a^3 (c+d x)^3+24 a b^2 (c+d x)^3+72 a^2 b \sqrt {1+(c+d x)^2}+45 b^3 \sqrt {1+(c+d x)^2}-48 a^2 b (c+d x)^2 \sqrt {1+(c+d x)^2}-6 b^3 (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+3 b^2 \left (-24 a^2-15 b^2-24 b^2 (c+d x)^2+64 a^2 (c+d x)^4+8 b^2 (c+d x)^4+48 a b (c+d x) \sqrt {1+(c+d x)^2}-32 a b (c+d x)^3 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2+16 b^3 \left (-3 a+8 a (c+d x)^4+3 b (c+d x) \sqrt {1+(c+d x)^2}-2 b (c+d x)^3 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^3+4 b^4 \left (-3+8 (c+d x)^4\right ) \text {arcsinh}(c+d x)^4\right )}{128 d} \]
(e^3*(-9*b^2*(8*a^2 + 5*b^2)*(c + d*x)^2 + (32*a^4 + 24*a^2*b^2 + 3*b^4)*( c + d*x)^4 + 2*a*b*(c + d*x)*Sqrt[1 + (c + d*x)^2]*(24*a^2 + 45*b^2 - 2*(8 *a^2 + 3*b^2)*(c + d*x)^2) - 6*a*b*(8*a^2 + 15*b^2)*ArcSinh[c + d*x] + 2*b *(c + d*x)*(-72*a*b^2*(c + d*x) + 64*a^3*(c + d*x)^3 + 24*a*b^2*(c + d*x)^ 3 + 72*a^2*b*Sqrt[1 + (c + d*x)^2] + 45*b^3*Sqrt[1 + (c + d*x)^2] - 48*a^2 *b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] - 6*b^3*(c + d*x)^2*Sqrt[1 + (c + d*x )^2])*ArcSinh[c + d*x] + 3*b^2*(-24*a^2 - 15*b^2 - 24*b^2*(c + d*x)^2 + 64 *a^2*(c + d*x)^4 + 8*b^2*(c + d*x)^4 + 48*a*b*(c + d*x)*Sqrt[1 + (c + d*x) ^2] - 32*a*b*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 + 16*b^ 3*(-3*a + 8*a*(c + d*x)^4 + 3*b*(c + d*x)*Sqrt[1 + (c + d*x)^2] - 2*b*(c + d*x)^3*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^3 + 4*b^4*(-3 + 8*(c + d*x )^4)*ArcSinh[c + d*x]^4))/(128*d)
Time = 1.76 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {6274, 27, 6191, 6227, 6191, 6227, 15, 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.
| \(\displaystyle \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int e^3 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^3 \int (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4-b \int \frac {(c+d x)^4 (a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4-b \left (-\frac {3}{4} b \int (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2d(c+d x)-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \int \frac {(c+d x)^4 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{4} b \int (c+d x)^3d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))\right )\right )-\frac {3}{4} \left (-\frac {3}{2} b \int (c+d x) (a+b \text {arcsinh}(c+d x))^2d(c+d x)-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )+\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )-\frac {3}{4} \left (-\frac {3}{2} b \int (c+d x) (a+b \text {arcsinh}(c+d x))^2d(c+d x)-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )+\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )-\frac {3}{4} \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2-b \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {1}{2} \int \frac {(a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )+\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )-\frac {3}{4} \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2-b \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{8 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )+\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (-\frac {3}{4} \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{2} b \int (c+d x)d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))\right )+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )-\frac {3}{4} \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2-b \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{2} b \int (c+d x)d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{8 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )+\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4-b \left (-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (-\frac {3}{4} \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b (c+d x)^2\right )+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )-\frac {3}{4} \left (-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2-b \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b (c+d x)^2\right )\right )-\frac {(a+b \text {arcsinh}(c+d x))^4}{8 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )+\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6198 |
| \(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^4-b \left (\frac {1}{4} (c+d x)^3 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-\frac {3}{4} b \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {3}{4} \left (\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) (a+b \text {arcsinh}(c+d x))-\frac {(a+b \text {arcsinh}(c+d x))^2}{4 b}-\frac {1}{4} b (c+d x)^2\right )-\frac {1}{16} b (c+d x)^4\right )\right )-\frac {3}{4} \left (-\frac {(a+b \text {arcsinh}(c+d x))^4}{8 b}+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-\frac {3}{2} b \left (\frac {1}{2} (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2-b \left (\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) (a+b \text {arcsinh}(c+d x))-\frac {(a+b \text {arcsinh}(c+d x))^2}{4 b}-\frac {1}{4} b (c+d x)^2\right )\right )\right )\right )\right )}{d}\) |
(e^3*(((c + d*x)^4*(a + b*ArcSinh[c + d*x])^4)/4 - b*(((c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/4 - (3*b*(((c + d*x)^4*(a + b*A rcSinh[c + d*x])^2)/4 - (b*(-1/16*(b*(c + d*x)^4) + ((c + d*x)^3*Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x]))/4 - (3*(-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)))/4))/2))/4 - (3*(((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))/4)))/d
3.2.47.3.1 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]
Time = 0.44 (sec) , antiderivative size = 573, normalized size of antiderivative = 1.64
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{3} a^{4} \left (d x +c \right )^{4}}{4}+e^{3} b^{4} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{4}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {3 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{16}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}+\frac {45 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{64}+\frac {27 \operatorname {arcsinh}\left (d x +c \right )^{2}}{128}+\frac {3 \left (d x +c \right )^{4}}{128}-\frac {45 \left (d x +c \right )^{2}}{128}-\frac {45}{128}-\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{16}\right )+4 e^{3} a \,b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{32}-\frac {3 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{128}+\frac {45 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{256}+\frac {27 \,\operatorname {arcsinh}\left (d x +c \right )}{256}-\frac {9 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{32}\right )+6 e^{3} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )+4 e^{3} b \,a^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(573\) |
| default | \(\frac {\frac {e^{3} a^{4} \left (d x +c \right )^{4}}{4}+e^{3} b^{4} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{4}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {3 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{16}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}+\frac {45 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{64}+\frac {27 \operatorname {arcsinh}\left (d x +c \right )^{2}}{128}+\frac {3 \left (d x +c \right )^{4}}{128}-\frac {45 \left (d x +c \right )^{2}}{128}-\frac {45}{128}-\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{16}\right )+4 e^{3} a \,b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{32}-\frac {3 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{128}+\frac {45 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{256}+\frac {27 \,\operatorname {arcsinh}\left (d x +c \right )}{256}-\frac {9 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{32}\right )+6 e^{3} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )+4 e^{3} b \,a^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(573\) |
| parts | \(\frac {e^{3} a^{4} \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b^{4} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{4}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{4}+\frac {3 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{8}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{4}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{16}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}+\frac {45 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{64}+\frac {27 \operatorname {arcsinh}\left (d x +c \right )^{2}}{128}+\frac {3 \left (d x +c \right )^{4}}{128}-\frac {45 \left (d x +c \right )^{2}}{128}-\frac {45}{128}-\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \left (1+\left (d x +c \right )^{2}\right )}{16}\right )}{d}+\frac {4 e^{3} a \,b^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{3}}{4}-\frac {3 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {9 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{32}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{3}}{32}+\frac {3 \left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{32}-\frac {3 \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{128}+\frac {45 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{256}+\frac {27 \,\operatorname {arcsinh}\left (d x +c \right )}{256}-\frac {9 \left (1+\left (d x +c \right )^{2}\right ) \operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}+\frac {6 e^{3} a^{2} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )}{d}+\frac {4 e^{3} b \,a^{3} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(584\) |
1/d*(1/4*e^3*a^4*(d*x+c)^4+e^3*b^4*(1/4*(d*x+c)^4*arcsinh(d*x+c)^4-1/4*(d* x+c)^3*arcsinh(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+3/8*(d*x+c)*arcsinh(d*x+c)^3*( 1+(d*x+c)^2)^(1/2)-3/32*arcsinh(d*x+c)^4+3/16*(d*x+c)^4*arcsinh(d*x+c)^2-3 /32*(d*x+c)^3*arcsinh(d*x+c)*(1+(d*x+c)^2)^(1/2)+45/64*(1+(d*x+c)^2)^(1/2) *(d*x+c)*arcsinh(d*x+c)+27/128*arcsinh(d*x+c)^2+3/128*(d*x+c)^4-45/128*(d* x+c)^2-45/128-9/16*arcsinh(d*x+c)^2*(1+(d*x+c)^2))+4*e^3*a*b^3*(1/4*(d*x+c )^4*arcsinh(d*x+c)^3-3/16*(d*x+c)^3*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)+9 /32*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)*(d*x+c)-3/32*arcsinh(d*x+c)^3+3/3 2*(d*x+c)^4*arcsinh(d*x+c)-3/128*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+45/256*(d*x +c)*(1+(d*x+c)^2)^(1/2)+27/256*arcsinh(d*x+c)-9/32*(1+(d*x+c)^2)*arcsinh(d *x+c))+6*e^3*a^2*b^2*(1/4*(d*x+c)^4*arcsinh(d*x+c)^2-1/8*(d*x+c)^3*arcsinh (d*x+c)*(1+(d*x+c)^2)^(1/2)+3/16*(1+(d*x+c)^2)^(1/2)*(d*x+c)*arcsinh(d*x+c )-3/32*arcsinh(d*x+c)^2+1/32*(d*x+c)^4-3/32*(d*x+c)^2-3/32)+4*e^3*b*a^3*(1 /4*(d*x+c)^4*arcsinh(d*x+c)-1/16*(d*x+c)^3*(1+(d*x+c)^2)^(1/2)+3/32*(d*x+c )*(1+(d*x+c)^2)^(1/2)-3/32*arcsinh(d*x+c)))
Leaf count of result is larger than twice the leaf count of optimal. 1241 vs. \(2 (319) = 638\).
Time = 0.29 (sec) , antiderivative size = 1241, normalized size of antiderivative = 3.56 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Too large to display} \]
1/128*((32*a^4 + 24*a^2*b^2 + 3*b^4)*d^4*e^3*x^4 + 4*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c*d^3*e^3*x^3 - 3*(24*a^2*b^2 + 15*b^4 - 2*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c^2)*d^2*e^3*x^2 + 2*(2*(32*a^4 + 24*a^2*b^2 + 3*b^4)*c^3 - 9*(8*a ^2*b^2 + 5*b^4)*c)*d*e^3*x + 4*(8*b^4*d^4*e^3*x^4 + 32*b^4*c*d^3*e^3*x^3 + 48*b^4*c^2*d^2*e^3*x^2 + 32*b^4*c^3*d*e^3*x + (8*b^4*c^4 - 3*b^4)*e^3)*lo g(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + 16*(8*a*b^3*d^4*e^3*x^4 + 32*a*b^3*c*d^3*e^3*x^3 + 48*a*b^3*c^2*d^2*e^3*x^2 + 32*a*b^3*c^3*d*e^3* x + (8*a*b^3*c^4 - 3*a*b^3)*e^3 - (2*b^4*d^3*e^3*x^3 + 6*b^4*c*d^2*e^3*x^2 + 3*(2*b^4*c^2 - b^4)*d*e^3*x + (2*b^4*c^3 - 3*b^4*c)*e^3)*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*(8*(8*a^2*b^2 + b^4)*d^4*e^3*x^4 + 32*(8*a^2*b^2 + b^4)*c*d^3*e^3*x^3 - 24*(b^4 - 2*(8*a^2*b^2 + b^4)*c^2)*d^2*e^3*x^2 - 16*(3*b^4*c - 2*(8*a^2*b^ 2 + b^4)*c^3)*d*e^3*x - (24*b^4*c^2 - 8*(8*a^2*b^2 + b^4)*c^4 + 24*a^2*b^2 + 15*b^4)*e^3 - 16*(2*a*b^3*d^3*e^3*x^3 + 6*a*b^3*c*d^2*e^3*x^2 + 3*(2*a* b^3*c^2 - a*b^3)*d*e^3*x + (2*a*b^3*c^3 - 3*a*b^3*c)*e^3)*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* (8*(8*a^3*b + 3*a*b^3)*d^4*e^3*x^4 + 32*(8*a^3*b + 3*a*b^3)*c*d^3*e^3*x^3 - 24*(3*a*b^3 - 2*(8*a^3*b + 3*a*b^3)*c^2)*d^2*e^3*x^2 - 16*(9*a*b^3*c - 2 *(8*a^3*b + 3*a*b^3)*c^3)*d*e^3*x - (72*a*b^3*c^2 - 8*(8*a^3*b + 3*a*b^3)* c^4 + 24*a^3*b + 45*a*b^3)*e^3 - 3*(2*(8*a^2*b^2 + b^4)*d^3*e^3*x^3 + 6...
Leaf count of result is larger than twice the leaf count of optimal. 2876 vs. \(2 (325) = 650\).
Time = 1.12 (sec) , antiderivative size = 2876, normalized size of antiderivative = 8.24 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Too large to display} \]
Piecewise((a**4*c**3*e**3*x + 3*a**4*c**2*d*e**3*x**2/2 + a**4*c*d**2*e**3 *x**3 + a**4*d**3*e**3*x**4/4 + a**3*b*c**4*e**3*asinh(c + d*x)/d + 4*a**3 *b*c**3*e**3*x*asinh(c + d*x) - a**3*b*c**3*e**3*sqrt(c**2 + 2*c*d*x + d** 2*x**2 + 1)/(4*d) + 6*a**3*b*c**2*d*e**3*x**2*asinh(c + d*x) - 3*a**3*b*c* *2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/4 + 4*a**3*b*c*d**2*e**3*x* *3*asinh(c + d*x) - 3*a**3*b*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/4 + 3*a**3*b*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(8*d) + a** 3*b*d**3*e**3*x**4*asinh(c + d*x) - a**3*b*d**2*e**3*x**3*sqrt(c**2 + 2*c* d*x + d**2*x**2 + 1)/4 + 3*a**3*b*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/8 - 3*a**3*b*e**3*asinh(c + d*x)/(8*d) + 3*a**2*b**2*c**4*e**3*asinh(c + d*x)**2/(2*d) + 6*a**2*b**2*c**3*e**3*x*asinh(c + d*x)**2 + 3*a**2*b**2 *c**3*e**3*x/4 - 3*a**2*b**2*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1 )*asinh(c + d*x)/(4*d) + 9*a**2*b**2*c**2*d*e**3*x**2*asinh(c + d*x)**2 + 9*a**2*b**2*c**2*d*e**3*x**2/8 - 9*a**2*b**2*c**2*e**3*x*sqrt(c**2 + 2*c*d *x + d**2*x**2 + 1)*asinh(c + d*x)/4 + 6*a**2*b**2*c*d**2*e**3*x**3*asinh( c + d*x)**2 + 3*a**2*b**2*c*d**2*e**3*x**3/4 - 9*a**2*b**2*c*d*e**3*x**2*s qrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/4 - 9*a**2*b**2*c*e**3* x/8 + 9*a**2*b**2*c*e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d* x)/(8*d) + 3*a**2*b**2*d**3*e**3*x**4*asinh(c + d*x)**2/2 + 3*a**2*b**2*d* *3*e**3*x**4/16 - 3*a**2*b**2*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2...
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
1/4*a^4*d^3*e^3*x^4 + a^4*c*d^2*e^3*x^3 + 3/2*a^4*c^2*d*e^3*x^2 + 3*(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)*a rcsinh(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*c^2*d*e^3 + 2/3*(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^3*b*c*d^2*e^ 3 + 1/24*(24*x^4*arcsinh(d*x + c) - (6*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*x ^3/d^2 - 14*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c*x^2/d^3 + 105*c^4*arcsinh( 2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 + 35*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^2*x/d^4 - 90*(c^2 + 1)*c^2*arcsinh(2*(d^2*x + c*d)/ sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 - 105*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*c^3/d^5 - 9*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*x/d^4 + 9*(c ^2 + 1)^2*arcsinh(2*(d^2*x + c*d)/sqrt(-4*c^2*d^2 + 4*(c^2 + 1)*d^2))/d^5 + 55*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1)*(c^2 + 1)*c/d^5)*d)*a^3*b*d^3*e^3 + a^4*c^3*e^3*x + 4*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a^ 3*b*c^3*e^3/d + 1/4*(b^4*d^3*e^3*x^4 + 4*b^4*c*d^2*e^3*x^3 + 6*b^4*c^2*...
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
Timed out. \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^4 \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]