Integrand size = 23, antiderivative size = 281 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^4 \, dx=-\frac {160}{27} b^4 e^2 x+\frac {8 b^4 e^2 (c+d x)^3}{81 d}+\frac {160 b^3 e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{27 d}-\frac {8 b^3 e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{27 d}-\frac {8 b^2 e^2 (c+d x) (a+b \text {arcsinh}(c+d x))^2}{3 d}+\frac {4 b^2 e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2}{9 d}+\frac {8 b e^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{9 d}-\frac {4 b e^2 (c+d x)^2 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))^3}{9 d}+\frac {e^2 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4}{3 d} \]
-160/27*b^4*e^2*x+8/81*b^4*e^2*(d*x+c)^3/d-8/3*b^2*e^2*(d*x+c)*(a+b*arcsin h(d*x+c))^2/d+4/9*b^2*e^2*(d*x+c)^3*(a+b*arcsinh(d*x+c))^2/d+1/3*e^2*(d*x+ c)^3*(a+b*arcsinh(d*x+c))^4/d+160/27*b^3*e^2*(a+b*arcsinh(d*x+c))*(1+(d*x+ c)^2)^(1/2)/d-8/27*b^3*e^2*(d*x+c)^2*(a+b*arcsinh(d*x+c))*(1+(d*x+c)^2)^(1 /2)/d+8/9*b*e^2*(a+b*arcsinh(d*x+c))^3*(1+(d*x+c)^2)^(1/2)/d-4/9*b*e^2*(d* x+c)^2*(a+b*arcsinh(d*x+c))^3*(1+(d*x+c)^2)^(1/2)/d
Time = 0.45 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.47 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {e^2 \left (-24 b^2 \left (9 a^2+20 b^2\right ) (c+d x)+\left (27 a^4+36 a^2 b^2+8 b^4\right ) (c+d x)^3+12 a b \sqrt {1+(c+d x)^2} \left (6 a^2+40 b^2-\left (3 a^2+2 b^2\right ) (c+d x)^2\right )+12 b \left (-36 a b^2 (c+d x)+9 a^3 (c+d x)^3+6 a b^2 (c+d x)^3+18 a^2 b \sqrt {1+(c+d x)^2}+40 b^3 \sqrt {1+(c+d x)^2}-9 a^2 b (c+d x)^2 \sqrt {1+(c+d x)^2}-2 b^3 (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+18 b^2 \left (-12 b^2 (c+d x)+9 a^2 (c+d x)^3+2 b^2 (c+d x)^3+12 a b \sqrt {1+(c+d x)^2}-6 a b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^2-36 b^3 \left (-3 a (c+d x)^3-2 b \sqrt {1+(c+d x)^2}+b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)^3+27 b^4 (c+d x)^3 \text {arcsinh}(c+d x)^4\right )}{81 d} \]
(e^2*(-24*b^2*(9*a^2 + 20*b^2)*(c + d*x) + (27*a^4 + 36*a^2*b^2 + 8*b^4)*( c + d*x)^3 + 12*a*b*Sqrt[1 + (c + d*x)^2]*(6*a^2 + 40*b^2 - (3*a^2 + 2*b^2 )*(c + d*x)^2) + 12*b*(-36*a*b^2*(c + d*x) + 9*a^3*(c + d*x)^3 + 6*a*b^2*( c + d*x)^3 + 18*a^2*b*Sqrt[1 + (c + d*x)^2] + 40*b^3*Sqrt[1 + (c + d*x)^2] - 9*a^2*b*(c + d*x)^2*Sqrt[1 + (c + d*x)^2] - 2*b^3*(c + d*x)^2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x] + 18*b^2*(-12*b^2*(c + d*x) + 9*a^2*(c + d* x)^3 + 2*b^2*(c + d*x)^3 + 12*a*b*Sqrt[1 + (c + d*x)^2] - 6*a*b*(c + d*x)^ 2*Sqrt[1 + (c + d*x)^2])*ArcSinh[c + d*x]^2 - 36*b^3*(-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]^3 + 27*b^4*(c + d*x)^3*ArcSinh[c + d*x]^4))/(81*d)
Time = 1.43 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.99, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.565, Rules used = {6274, 27, 6191, 6227, 6191, 6213, 6187, 6213, 24, 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.
| \(\displaystyle \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 6274 |
| \(\displaystyle \frac {\int e^2 (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {e^2 \int (c+d x)^2 (a+b \text {arcsinh}(c+d x))^4d(c+d x)}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \left (-b \int (c+d x)^2 (a+b \text {arcsinh}(c+d x))^2d(c+d x)-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6191 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \left (-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))^3}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \left (-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \int (a+b \text {arcsinh}(c+d x))^2d(c+d x)\right )+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6187 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^2-2 b \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )\right )-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^2-2 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b \int 1d(c+d x)\right )\right )\right )-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3\right )\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \left (-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \int \frac {(c+d x)^3 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^2-2 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b (c+d x)\right )\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6227 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \left (-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{3} b \int (c+d x)^2d(c+d x)+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))\right )\right )+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^2-2 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b (c+d x)\right )\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 15 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \left (-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (-\frac {2}{3} \int \frac {(c+d x) (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))-\frac {1}{9} b (c+d x)^3\right )\right )+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^2-2 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b (c+d x)\right )\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 6213 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \left (-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b \int 1d(c+d x)\right )+\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))-\frac {1}{9} b (c+d x)^3\right )\right )+\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^2-2 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b (c+d x)\right )\right )\right )\right )\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {e^2 \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^4-\frac {4}{3} b \left (\frac {1}{3} (c+d x)^2 \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-b \left (\frac {1}{3} (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2-\frac {2}{3} b \left (\frac {1}{3} \sqrt {(c+d x)^2+1} (c+d x)^2 (a+b \text {arcsinh}(c+d x))-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b (c+d x)\right )-\frac {1}{9} b (c+d x)^3\right )\right )-\frac {2}{3} \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))^3-3 b \left ((c+d x) (a+b \text {arcsinh}(c+d x))^2-2 b \left (\sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))-b (c+d x)\right )\right )\right )\right )\right )}{d}\) |
(e^2*(((c + d*x)^3*(a + b*ArcSinh[c + d*x])^4)/3 - (4*b*(((c + d*x)^2*Sqrt [1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])^3)/3 - b*(((c + d*x)^3*(a + b*A rcSinh[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) - (2*(Sqrt[1 + (c + d*x)^2] *(a + b*ArcSinh[c + d*x])^3 - 3*b*((c + d*x)*(a + b*ArcSinh[c + d*x])^2 - 2*b*(-(b*(c + d*x)) + Sqrt[1 + (c + d*x)^2]*(a + b*ArcSinh[c + d*x])))))/3 ))/3))/d
3.2.48.3.1 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_.), x_Symbol] :> Simp[x*(a + b*A rcSinh[c*x])^n, x] - Simp[b*c*n Int[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[ 1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]
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 = 0.39 (sec) , antiderivative size = 473, normalized size of antiderivative = 1.68
| method | result | size |
| derivativedivides | \(\frac {\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3}+e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{4}}{3}+\frac {8 \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {4 \left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {8 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {160 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{27}-\frac {160 d x}{27}-\frac {160 c}{27}+\frac {4 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{9}-\frac {8 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )+4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\frac {2 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{3}+\frac {40 \sqrt {1+\left (d x +c \right )^{2}}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{27}\right )+6 e^{2} a^{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 )+4 e^{2} b \,a^{3} \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}\) | \(473\) |
| default | \(\frac {\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3}+e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{4}}{3}+\frac {8 \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {4 \left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {8 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {160 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{27}-\frac {160 d x}{27}-\frac {160 c}{27}+\frac {4 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{9}-\frac {8 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )+4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\frac {2 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{3}+\frac {40 \sqrt {1+\left (d x +c \right )^{2}}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{27}\right )+6 e^{2} a^{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 )+4 e^{2} b \,a^{3} \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}\) | \(473\) |
| parts | \(\frac {e^{2} a^{4} \left (d x +c \right )^{3}}{3 d}+\frac {e^{2} b^{4} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{4}}{3}+\frac {8 \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {4 \left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{9}-\frac {8 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )^{2}}{3}+\frac {160 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{27}-\frac {160 d x}{27}-\frac {160 c}{27}+\frac {4 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{2}}{9}-\frac {8 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )^{2}}{27}+\frac {8 \left (d x +c \right )^{3}}{81}\right )}{d}+\frac {4 e^{2} a \,b^{3} \left (\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )^{3}}{3}+\frac {2 \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {\left (d x +c \right )^{2} \operatorname {arcsinh}\left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{3}-\frac {4 \left (d x +c \right ) \operatorname {arcsinh}\left (d x +c \right )}{3}+\frac {40 \sqrt {1+\left (d x +c \right )^{2}}}{27}+\frac {2 \left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right )}{9}-\frac {2 \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}}{27}\right )}{d}+\frac {6 e^{2} a^{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 {4 e^{2} b \,a^{3} \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}\) | \(484\) |
1/d*(1/3*e^2*a^4*(d*x+c)^3+e^2*b^4*(1/3*(d*x+c)^3*arcsinh(d*x+c)^4+8/9*arc sinh(d*x+c)^3*(1+(d*x+c)^2)^(1/2)-4/9*(d*x+c)^2*arcsinh(d*x+c)^3*(1+(d*x+c )^2)^(1/2)-8/3*(d*x+c)*arcsinh(d*x+c)^2+160/27*arcsinh(d*x+c)*(1+(d*x+c)^2 )^(1/2)-160/27*d*x-160/27*c+4/9*(d*x+c)^3*arcsinh(d*x+c)^2-8/27*arcsinh(d* x+c)*(1+(d*x+c)^2)^(1/2)*(d*x+c)^2+8/81*(d*x+c)^3)+4*e^2*a*b^3*(1/3*(d*x+c )^3*arcsinh(d*x+c)^3+2/3*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)-1/3*(d*x+c)^ 2*arcsinh(d*x+c)^2*(1+(d*x+c)^2)^(1/2)-4/3*(d*x+c)*arcsinh(d*x+c)+40/27*(1 +(d*x+c)^2)^(1/2)+2/9*(d*x+c)^3*arcsinh(d*x+c)-2/27*(d*x+c)^2*(1+(d*x+c)^2 )^(1/2))+6*e^2*a^2*b^2*(1/3*(d*x+c)^3*arcsinh(d*x+c)^2+4/9*arcsinh(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)+4*e^2*b*a^3*(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. 900 vs. \(2 (255) = 510\).
Time = 0.29 (sec) , antiderivative size = 900, normalized size of antiderivative = 3.20 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^4 \, dx=\frac {{\left (27 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (27 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (72 \, a^{2} b^{2} + 160 \, b^{4} - {\left (27 \, a^{4} + 36 \, a^{2} b^{2} + 8 \, b^{4}\right )} c^{2}\right )} d e^{2} x + 27 \, {\left (b^{4} d^{3} e^{2} x^{3} + 3 \, b^{4} c d^{2} e^{2} x^{2} + 3 \, b^{4} c^{2} d e^{2} x + b^{4} c^{3} e^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + 36 \, {\left (3 \, a b^{3} d^{3} e^{2} x^{3} + 9 \, a b^{3} c d^{2} e^{2} x^{2} + 9 \, a b^{3} c^{2} d e^{2} x + 3 \, a b^{3} c^{3} e^{2} - {\left (b^{4} d^{2} e^{2} x^{2} + 2 \, b^{4} c d e^{2} x + {\left (b^{4} c^{2} - 2 \, b^{4}\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 )^{3} + 18 \, {\left ({\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} d^{3} e^{2} x^{3} + 3 \, {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c d^{2} e^{2} x^{2} - 3 \, {\left (4 \, b^{4} - {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c^{2}\right )} d e^{2} x - {\left (12 \, b^{4} c - {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c^{3}\right )} e^{2} - 6 \, {\left (a b^{3} d^{2} e^{2} x^{2} + 2 \, a b^{3} c d e^{2} x + {\left (a b^{3} c^{2} - 2 \, a b^{3}\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 )^{2} + 12 \, {\left (3 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} d^{3} e^{2} x^{3} + 9 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c d^{2} e^{2} x^{2} - 9 \, {\left (4 \, a b^{3} - {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c^{2}\right )} d e^{2} x - 3 \, {\left (12 \, a b^{3} c - {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c^{3}\right )} e^{2} - {\left ({\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c d e^{2} x - {\left (18 \, a^{2} b^{2} + 40 \, b^{4} - {\left (9 \, a^{2} b^{2} + 2 \, b^{4}\right )} c^{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 ) - 12 \, {\left ({\left (3 \, a^{3} b + 2 \, a b^{3}\right )} d^{2} e^{2} x^{2} + 2 \, {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c d e^{2} x - {\left (6 \, a^{3} b + 40 \, a b^{3} - {\left (3 \, a^{3} b + 2 \, a b^{3}\right )} c^{2}\right )} e^{2}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{81 \, d} \]
1/81*((27*a^4 + 36*a^2*b^2 + 8*b^4)*d^3*e^2*x^3 + 3*(27*a^4 + 36*a^2*b^2 + 8*b^4)*c*d^2*e^2*x^2 - 3*(72*a^2*b^2 + 160*b^4 - (27*a^4 + 36*a^2*b^2 + 8 *b^4)*c^2)*d*e^2*x + 27*(b^4*d^3*e^2*x^3 + 3*b^4*c*d^2*e^2*x^2 + 3*b^4*c^2 *d*e^2*x + b^4*c^3*e^2)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^4 + 36*(3*a*b^3*d^3*e^2*x^3 + 9*a*b^3*c*d^2*e^2*x^2 + 9*a*b^3*c^2*d*e^2*x + 3*a*b^3*c^3*e^2 - (b^4*d^2*e^2*x^2 + 2*b^4*c*d*e^2*x + (b^4*c^2 - 2*b^4)* 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))^3 + 18*((9*a^2*b^2 + 2*b^4)*d^3*e^2*x^3 + 3*(9*a^2*b^2 + 2* b^4)*c*d^2*e^2*x^2 - 3*(4*b^4 - (9*a^2*b^2 + 2*b^4)*c^2)*d*e^2*x - (12*b^4 *c - (9*a^2*b^2 + 2*b^4)*c^3)*e^2 - 6*(a*b^3*d^2*e^2*x^2 + 2*a*b^3*c*d*e^2 *x + (a*b^3*c^2 - 2*a*b^3)*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))^2 + 12*(3*(3*a^3*b + 2*a*b^3)*d^ 3*e^2*x^3 + 9*(3*a^3*b + 2*a*b^3)*c*d^2*e^2*x^2 - 9*(4*a*b^3 - (3*a^3*b + 2*a*b^3)*c^2)*d*e^2*x - 3*(12*a*b^3*c - (3*a^3*b + 2*a*b^3)*c^3)*e^2 - ((9 *a^2*b^2 + 2*b^4)*d^2*e^2*x^2 + 2*(9*a^2*b^2 + 2*b^4)*c*d*e^2*x - (18*a^2* b^2 + 40*b^4 - (9*a^2*b^2 + 2*b^4)*c^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)) - 12*((3*a^3*b + 2* a*b^3)*d^2*e^2*x^2 + 2*(3*a^3*b + 2*a*b^3)*c*d*e^2*x - (6*a^3*b + 40*a*b^3 - (3*a^3*b + 2*a*b^3)*c^2)*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. 1889 vs. \(2 (264) = 528\).
Time = 0.70 (sec) , antiderivative size = 1889, normalized size of antiderivative = 6.72 \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^4 \, dx=\text {Too large to display} \]
Piecewise((a**4*c**2*e**2*x + a**4*c*d*e**2*x**2 + a**4*d**2*e**2*x**3/3 + 4*a**3*b*c**3*e**2*asinh(c + d*x)/(3*d) + 4*a**3*b*c**2*e**2*x*asinh(c + d*x) - 4*a**3*b*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(9*d) + 4*a **3*b*c*d*e**2*x**2*asinh(c + d*x) - 8*a**3*b*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/9 + 4*a**3*b*d**2*e**2*x**3*asinh(c + d*x)/3 - 4*a**3*b* d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/9 + 8*a**3*b*e**2*sqrt(c* *2 + 2*c*d*x + d**2*x**2 + 1)/(9*d) + 2*a**2*b**2*c**3*e**2*asinh(c + d*x) **2/d + 6*a**2*b**2*c**2*e**2*x*asinh(c + d*x)**2 + 4*a**2*b**2*c**2*e**2* x/3 - 4*a**2*b**2*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(3*d) + 6*a**2*b**2*c*d*e**2*x**2*asinh(c + d*x)**2 + 4*a**2*b**2*c* d*e**2*x**2/3 - 8*a**2*b**2*c*e**2*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)* asinh(c + d*x)/3 + 2*a**2*b**2*d**2*e**2*x**3*asinh(c + d*x)**2 + 4*a**2*b **2*d**2*e**2*x**3/9 - 4*a**2*b**2*d*e**2*x**2*sqrt(c**2 + 2*c*d*x + d**2* x**2 + 1)*asinh(c + d*x)/3 - 8*a**2*b**2*e**2*x/3 + 8*a**2*b**2*e**2*sqrt( c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(3*d) + 4*a*b**3*c**3*e**2* asinh(c + d*x)**3/(3*d) + 8*a*b**3*c**3*e**2*asinh(c + d*x)/(9*d) + 4*a*b* *3*c**2*e**2*x*asinh(c + d*x)**3 + 8*a*b**3*c**2*e**2*x*asinh(c + d*x)/3 - 4*a*b**3*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)**2 /(3*d) - 8*a*b**3*c**2*e**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(27*d) + 4*a*b**3*c*d*e**2*x**2*asinh(c + d*x)**3 + 8*a*b**3*c*d*e**2*x**2*asinh...
\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
1/3*a^4*d^2*e^2*x^3 + a^4*c*d*e^2*x^2 + 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)/sqr t(-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*d*e^2 + 2/9*(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^3*b*d^2*e^2 + a^4*c^2*e^2*x + 4*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a^3*b*c^2*e^2/d + 1/3*(b^4*d^ 2*e^2*x^3 + 3*b^4*c*d*e^2*x^2 + 3*b^4*c^2*e^2*x)*log(d*x + c + sqrt(d^2*x^ 2 + 2*c*d*x + c^2 + 1))^4 + integrate(2/3*(2*((3*a*b^3*d^5*e^2 - b^4*d^5*e ^2)*x^5 + 3*(c^5*e^2 + c^3*e^2)*a*b^3 + 5*(3*a*b^3*c*d^4*e^2 - b^4*c*d^4*e ^2)*x^4 + (3*(10*c^2*d^3*e^2 + d^3*e^2)*a*b^3 - (10*c^2*d^3*e^2 + d^3*e^2) *b^4)*x^3 + 3*((10*c^3*d^2*e^2 + 3*c*d^2*e^2)*a*b^3 - (3*c^3*d^2*e^2 + c*d ^2*e^2)*b^4)*x^2 + 3*((5*c^4*d*e^2 + 3*c^2*d*e^2)*a*b^3 - (c^4*d*e^2 + c^2 *d*e^2)*b^4)*x + (3*(c^4*e^2 + c^2*e^2)*a*b^3 + (3*a*b^3*d^4*e^2 - b^4*d^4 *e^2)*x^4 + 4*(3*a*b^3*c*d^3*e^2 - b^4*c*d^3*e^2)*x^3 - 3*(2*b^4*c^2*d^2*e ^2 - (6*c^2*d^2*e^2 + d^2*e^2)*a*b^3)*x^2 - 3*(b^4*c^3*d*e^2 - 2*(2*c^3...
\[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^4 \, dx=\int { {\left (d e x + c e\right )}^{2} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{4} \,d x } \]
Timed out. \[ \int (c e+d e x)^2 (a+b \text {arcsinh}(c+d x))^4 \, dx=\int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^4 \,d x \]