Integrand size = 16, antiderivative size = 183 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=-\frac {b c \sqrt {1+c^2 x^2}}{6 \left (c^2 d^2+e^2\right ) (d+e x)^2}-\frac {b c^3 d \sqrt {1+c^2 x^2}}{2 \left (c^2 d^2+e^2\right )^2 (d+e x)}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}-\frac {b c^3 \left (2 c^2 d^2-e^2\right ) \text {arctanh}\left (\frac {e-c^2 d x}{\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}}\right )}{6 e \left (c^2 d^2+e^2\right )^{5/2}} \]
1/3*(-a-b*arcsinh(c*x))/e/(e*x+d)^3-1/6*b*c^3*(2*c^2*d^2-e^2)*arctanh((-c^ 2*d*x+e)/(c^2*d^2+e^2)^(1/2)/(c^2*x^2+1)^(1/2))/e/(c^2*d^2+e^2)^(5/2)-1/6* b*c*(c^2*x^2+1)^(1/2)/(c^2*d^2+e^2)/(e*x+d)^2-1/2*b*c^3*d*(c^2*x^2+1)^(1/2 )/(c^2*d^2+e^2)^2/(e*x+d)
Time = 0.30 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=\frac {1}{6} \left (-\frac {2 a}{e (d+e x)^3}-\frac {b c \sqrt {1+c^2 x^2} \left (e^2+c^2 d (4 d+3 e x)\right )}{\left (c^2 d^2+e^2\right )^2 (d+e x)^2}-\frac {2 b \text {arcsinh}(c x)}{e (d+e x)^3}-\frac {b c^3 \left (-2 c^2 d^2+e^2\right ) \log (d+e x)}{e \left (c^2 d^2+e^2\right )^{5/2}}+\frac {b c^3 \left (-2 c^2 d^2+e^2\right ) \log \left (e-c^2 d x+\sqrt {c^2 d^2+e^2} \sqrt {1+c^2 x^2}\right )}{e \left (c^2 d^2+e^2\right )^{5/2}}\right ) \]
((-2*a)/(e*(d + e*x)^3) - (b*c*Sqrt[1 + c^2*x^2]*(e^2 + c^2*d*(4*d + 3*e*x )))/((c^2*d^2 + e^2)^2*(d + e*x)^2) - (2*b*ArcSinh[c*x])/(e*(d + e*x)^3) - (b*c^3*(-2*c^2*d^2 + e^2)*Log[d + e*x])/(e*(c^2*d^2 + e^2)^(5/2)) + (b*c^ 3*(-2*c^2*d^2 + e^2)*Log[e - c^2*d*x + Sqrt[c^2*d^2 + e^2]*Sqrt[1 + c^2*x^ 2]])/(e*(c^2*d^2 + e^2)^(5/2)))/6
Time = 0.35 (sec) , antiderivative size = 199, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {6243, 498, 25, 679, 488, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx\) |
| \(\Big \downarrow \) 6243 |
| \(\displaystyle \frac {b c \int \frac {1}{(d+e x)^3 \sqrt {c^2 x^2+1}}dx}{3 e}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 498 |
| \(\displaystyle \frac {b c \left (-\frac {c^2 \int -\frac {2 d-e x}{(d+e x)^2 \sqrt {c^2 x^2+1}}dx}{2 \left (c^2 d^2+e^2\right )}-\frac {e \sqrt {c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right ) (d+e x)^2}\right )}{3 e}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b c \left (\frac {c^2 \int \frac {2 d-e x}{(d+e x)^2 \sqrt {c^2 x^2+1}}dx}{2 \left (c^2 d^2+e^2\right )}-\frac {e \sqrt {c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right ) (d+e x)^2}\right )}{3 e}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {b c \left (\frac {c^2 \left (\frac {\left (2 c^2 d^2-e^2\right ) \int \frac {1}{(d+e x) \sqrt {c^2 x^2+1}}dx}{c^2 d^2+e^2}-\frac {3 d e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) (d+e x)}\right )}{2 \left (c^2 d^2+e^2\right )}-\frac {e \sqrt {c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right ) (d+e x)^2}\right )}{3 e}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {b c \left (\frac {c^2 \left (-\frac {\left (2 c^2 d^2-e^2\right ) \int \frac {1}{c^2 d^2+e^2-\frac {\left (e-c^2 d x\right )^2}{c^2 x^2+1}}d\frac {e-c^2 d x}{\sqrt {c^2 x^2+1}}}{c^2 d^2+e^2}-\frac {3 d e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) (d+e x)}\right )}{2 \left (c^2 d^2+e^2\right )}-\frac {e \sqrt {c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right ) (d+e x)^2}\right )}{3 e}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {b c \left (\frac {c^2 \left (-\frac {\left (2 c^2 d^2-e^2\right ) \text {arctanh}\left (\frac {e-c^2 d x}{\sqrt {c^2 x^2+1} \sqrt {c^2 d^2+e^2}}\right )}{\left (c^2 d^2+e^2\right )^{3/2}}-\frac {3 d e \sqrt {c^2 x^2+1}}{\left (c^2 d^2+e^2\right ) (d+e x)}\right )}{2 \left (c^2 d^2+e^2\right )}-\frac {e \sqrt {c^2 x^2+1}}{2 \left (c^2 d^2+e^2\right ) (d+e x)^2}\right )}{3 e}-\frac {a+b \text {arcsinh}(c x)}{3 e (d+e x)^3}\) |
-1/3*(a + b*ArcSinh[c*x])/(e*(d + e*x)^3) + (b*c*(-1/2*(e*Sqrt[1 + c^2*x^2 ])/((c^2*d^2 + e^2)*(d + e*x)^2) + (c^2*((-3*d*e*Sqrt[1 + c^2*x^2])/((c^2* d^2 + e^2)*(d + e*x)) - ((2*c^2*d^2 - e^2)*ArcTanh[(e - c^2*d*x)/(Sqrt[c^2 *d^2 + e^2]*Sqrt[1 + c^2*x^2])])/(c^2*d^2 + e^2)^(3/2)))/(2*(c^2*d^2 + e^2 ))))/(3*e)
3.1.11.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ d*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))), x] + S imp[b/((n + 1)*(b*c^2 + a*d^2)) Int[(c + d*x)^(n + 1)*(a + b*x^2)^p*(c*(n + 1) - d*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[n , -1] && ((LtQ[n, -1] && IntQuadraticQ[a, 0, b, c, d, n, p, x]) || (SumSimp lerQ[n, 1] && IntegerQ[p]) || ILtQ[Simplify[n + 2*p + 3], 0])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcSinh[c*x])^( n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n , 0] && NeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(509\) vs. \(2(168)=336\).
Time = 0.41 (sec) , antiderivative size = 510, normalized size of antiderivative = 2.79
| method | result | size |
| parts | \(-\frac {a}{3 \left (e x +d \right )^{3} e}-\frac {b \,c^{3} \operatorname {arcsinh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {b \,c^{3} \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}-\frac {b \,c^{4} d \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{5} d^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}+e^{2}\right )^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}+\frac {b \,c^{3} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}\) | \(510\) |
| derivativedivides | \(\frac {-\frac {a \,c^{4}}{3 \left (e c x +c d \right )^{3} e}-\frac {b \,c^{4} \operatorname {arcsinh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {b \,c^{4} \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}-\frac {b \,c^{5} d \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{6} d^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}+e^{2}\right )^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}+\frac {b \,c^{4} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\) | \(520\) |
| default | \(\frac {-\frac {a \,c^{4}}{3 \left (e c x +c d \right )^{3} e}-\frac {b \,c^{4} \operatorname {arcsinh}\left (c x \right )}{3 \left (e c x +c d \right )^{3} e}-\frac {b \,c^{4} \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \left (c x +\frac {d c}{e}\right )^{2}}-\frac {b \,c^{5} d \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{2 e \left (c^{2} d^{2}+e^{2}\right )^{2} \left (c x +\frac {d c}{e}\right )}-\frac {b \,c^{6} d^{2} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{2 e^{2} \left (c^{2} d^{2}+e^{2}\right )^{2} \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}+\frac {b \,c^{4} \ln \left (\frac {\frac {2 c^{2} d^{2}+2 e^{2}}{e^{2}}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+2 \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}\, \sqrt {\left (c x +\frac {d c}{e}\right )^{2}-\frac {2 d c \left (c x +\frac {d c}{e}\right )}{e}+\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}{c x +\frac {d c}{e}}\right )}{6 e^{2} \left (c^{2} d^{2}+e^{2}\right ) \sqrt {\frac {c^{2} d^{2}+e^{2}}{e^{2}}}}}{c}\) | \(520\) |
-1/3*a/(e*x+d)^3/e-1/3*b*c^3/(c*e*x+c*d)^3/e*arcsinh(c*x)-1/6*b*c^3/e^2/(c ^2*d^2+e^2)/(c*x+d*c/e)^2*((c*x+d*c/e)^2-2*d*c/e*(c*x+d*c/e)+(c^2*d^2+e^2) /e^2)^(1/2)-1/2*b*c^4/e*d/(c^2*d^2+e^2)^2/(c*x+d*c/e)*((c*x+d*c/e)^2-2*d*c /e*(c*x+d*c/e)+(c^2*d^2+e^2)/e^2)^(1/2)-1/2*b*c^5/e^2*d^2/(c^2*d^2+e^2)^2/ ((c^2*d^2+e^2)/e^2)^(1/2)*ln((2*(c^2*d^2+e^2)/e^2-2*d*c/e*(c*x+d*c/e)+2*(( c^2*d^2+e^2)/e^2)^(1/2)*((c*x+d*c/e)^2-2*d*c/e*(c*x+d*c/e)+(c^2*d^2+e^2)/e ^2)^(1/2))/(c*x+d*c/e))+1/6*b*c^3/e^2/(c^2*d^2+e^2)/((c^2*d^2+e^2)/e^2)^(1 /2)*ln((2*(c^2*d^2+e^2)/e^2-2*d*c/e*(c*x+d*c/e)+2*((c^2*d^2+e^2)/e^2)^(1/2 )*((c*x+d*c/e)^2-2*d*c/e*(c*x+d*c/e)+(c^2*d^2+e^2)/e^2)^(1/2))/(c*x+d*c/e) )
Leaf count of result is larger than twice the leaf count of optimal. 977 vs. \(2 (167) = 334\).
Time = 0.62 (sec) , antiderivative size = 977, normalized size of antiderivative = 5.34 \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=-\frac {{\left (2 \, a + 3 \, b\right )} c^{6} d^{9} + 3 \, {\left (2 \, a + b\right )} c^{4} d^{7} e^{2} + 6 \, a c^{2} d^{5} e^{4} + 2 \, a d^{3} e^{6} + 3 \, {\left (b c^{6} d^{6} e^{3} + b c^{4} d^{4} e^{5}\right )} x^{3} + 9 \, {\left (b c^{6} d^{7} e^{2} + b c^{4} d^{5} e^{4}\right )} x^{2} + {\left (2 \, b c^{5} d^{8} - b c^{3} d^{6} e^{2} + {\left (2 \, b c^{5} d^{5} e^{3} - b c^{3} d^{3} e^{5}\right )} x^{3} + 3 \, {\left (2 \, b c^{5} d^{6} e^{2} - b c^{3} d^{4} e^{4}\right )} x^{2} + 3 \, {\left (2 \, b c^{5} d^{7} e - b c^{3} d^{5} e^{3}\right )} x\right )} \sqrt {c^{2} d^{2} + e^{2}} \log \left (-\frac {c^{3} d^{2} x - c d e - \sqrt {c^{2} d^{2} + e^{2}} {\left (c^{2} d x - e\right )} + {\left (c^{2} d^{2} - \sqrt {c^{2} d^{2} + e^{2}} c d + e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{e x + d}\right ) + 9 \, {\left (b c^{6} d^{8} e + b c^{4} d^{6} e^{3}\right )} x - 2 \, {\left ({\left (b c^{6} d^{6} e^{3} + 3 \, b c^{4} d^{4} e^{5} + 3 \, b c^{2} d^{2} e^{7} + b e^{9}\right )} x^{3} + 3 \, {\left (b c^{6} d^{7} e^{2} + 3 \, b c^{4} d^{5} e^{4} + 3 \, b c^{2} d^{3} e^{6} + b d e^{8}\right )} x^{2} + 3 \, {\left (b c^{6} d^{8} e + 3 \, b c^{4} d^{6} e^{3} + 3 \, b c^{2} d^{4} e^{5} + b d^{2} e^{7}\right )} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 2 \, {\left (b c^{6} d^{9} + 3 \, b c^{4} d^{7} e^{2} + 3 \, b c^{2} d^{5} e^{4} + b d^{3} e^{6} + {\left (b c^{6} d^{6} e^{3} + 3 \, b c^{4} d^{4} e^{5} + 3 \, b c^{2} d^{2} e^{7} + b e^{9}\right )} x^{3} + 3 \, {\left (b c^{6} d^{7} e^{2} + 3 \, b c^{4} d^{5} e^{4} + 3 \, b c^{2} d^{3} e^{6} + b d e^{8}\right )} x^{2} + 3 \, {\left (b c^{6} d^{8} e + 3 \, b c^{4} d^{6} e^{3} + 3 \, b c^{2} d^{4} e^{5} + b d^{2} e^{7}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (4 \, b c^{5} d^{8} e + 5 \, b c^{3} d^{6} e^{3} + b c d^{4} e^{5} + 3 \, {\left (b c^{5} d^{6} e^{3} + b c^{3} d^{4} e^{5}\right )} x^{2} + {\left (7 \, b c^{5} d^{7} e^{2} + 8 \, b c^{3} d^{5} e^{4} + b c d^{3} e^{6}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{6 \, {\left (c^{6} d^{12} e + 3 \, c^{4} d^{10} e^{3} + 3 \, c^{2} d^{8} e^{5} + d^{6} e^{7} + {\left (c^{6} d^{9} e^{4} + 3 \, c^{4} d^{7} e^{6} + 3 \, c^{2} d^{5} e^{8} + d^{3} e^{10}\right )} x^{3} + 3 \, {\left (c^{6} d^{10} e^{3} + 3 \, c^{4} d^{8} e^{5} + 3 \, c^{2} d^{6} e^{7} + d^{4} e^{9}\right )} x^{2} + 3 \, {\left (c^{6} d^{11} e^{2} + 3 \, c^{4} d^{9} e^{4} + 3 \, c^{2} d^{7} e^{6} + d^{5} e^{8}\right )} x\right )}} \]
-1/6*((2*a + 3*b)*c^6*d^9 + 3*(2*a + b)*c^4*d^7*e^2 + 6*a*c^2*d^5*e^4 + 2* a*d^3*e^6 + 3*(b*c^6*d^6*e^3 + b*c^4*d^4*e^5)*x^3 + 9*(b*c^6*d^7*e^2 + b*c ^4*d^5*e^4)*x^2 + (2*b*c^5*d^8 - b*c^3*d^6*e^2 + (2*b*c^5*d^5*e^3 - b*c^3* d^3*e^5)*x^3 + 3*(2*b*c^5*d^6*e^2 - b*c^3*d^4*e^4)*x^2 + 3*(2*b*c^5*d^7*e - b*c^3*d^5*e^3)*x)*sqrt(c^2*d^2 + e^2)*log(-(c^3*d^2*x - c*d*e - sqrt(c^2 *d^2 + e^2)*(c^2*d*x - e) + (c^2*d^2 - sqrt(c^2*d^2 + e^2)*c*d + e^2)*sqrt (c^2*x^2 + 1))/(e*x + d)) + 9*(b*c^6*d^8*e + b*c^4*d^6*e^3)*x - 2*((b*c^6* d^6*e^3 + 3*b*c^4*d^4*e^5 + 3*b*c^2*d^2*e^7 + b*e^9)*x^3 + 3*(b*c^6*d^7*e^ 2 + 3*b*c^4*d^5*e^4 + 3*b*c^2*d^3*e^6 + b*d*e^8)*x^2 + 3*(b*c^6*d^8*e + 3* b*c^4*d^6*e^3 + 3*b*c^2*d^4*e^5 + b*d^2*e^7)*x)*log(c*x + sqrt(c^2*x^2 + 1 )) - 2*(b*c^6*d^9 + 3*b*c^4*d^7*e^2 + 3*b*c^2*d^5*e^4 + b*d^3*e^6 + (b*c^6 *d^6*e^3 + 3*b*c^4*d^4*e^5 + 3*b*c^2*d^2*e^7 + b*e^9)*x^3 + 3*(b*c^6*d^7*e ^2 + 3*b*c^4*d^5*e^4 + 3*b*c^2*d^3*e^6 + b*d*e^8)*x^2 + 3*(b*c^6*d^8*e + 3 *b*c^4*d^6*e^3 + 3*b*c^2*d^4*e^5 + b*d^2*e^7)*x)*log(-c*x + sqrt(c^2*x^2 + 1)) + (4*b*c^5*d^8*e + 5*b*c^3*d^6*e^3 + b*c*d^4*e^5 + 3*(b*c^5*d^6*e^3 + b*c^3*d^4*e^5)*x^2 + (7*b*c^5*d^7*e^2 + 8*b*c^3*d^5*e^4 + b*c*d^3*e^6)*x) *sqrt(c^2*x^2 + 1))/(c^6*d^12*e + 3*c^4*d^10*e^3 + 3*c^2*d^8*e^5 + d^6*e^7 + (c^6*d^9*e^4 + 3*c^4*d^7*e^6 + 3*c^2*d^5*e^8 + d^3*e^10)*x^3 + 3*(c^6*d ^10*e^3 + 3*c^4*d^8*e^5 + 3*c^2*d^6*e^7 + d^4*e^9)*x^2 + 3*(c^6*d^11*e^2 + 3*c^4*d^9*e^4 + 3*c^2*d^7*e^6 + d^5*e^8)*x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=\int \frac {a + b \operatorname {asinh}{\left (c x \right )}}{\left (d + e x\right )^{4}}\, dx \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x + d\right )}^{4}} \,d x } \]
1/6*(6*c*integrate(1/3/(c^3*e^4*x^6 + 3*c^3*d*e^3*x^5 + 3*c*d^2*e^2*x^2 + c*d^3*e*x + (3*c^3*d^2*e^2 + c*e^4)*x^4 + (c^3*d^3*e + 3*c*d*e^3)*x^3 + (c ^2*e^4*x^5 + 3*c^2*d*e^3*x^4 + 3*d^2*e^2*x + d^3*e + (3*c^2*d^2*e^2 + e^4) *x^3 + (c^2*d^3*e + 3*d*e^3)*x^2)*sqrt(c^2*x^2 + 1)), x) - 2*(c^6*d^3 - 3* c^4*d*e^2)*log(e*x + d)/(c^6*d^6*e + 3*c^4*d^4*e^3 + 3*c^2*d^2*e^5 + e^7) + (3*c^6*d^6 + 2*c^4*d^4*e^2 - c^2*d^2*e^4 + 2*(c^6*d^4*e^2 - c^2*e^6)*x^2 + (5*c^6*d^5*e + 2*c^4*d^3*e^3 - 3*c^2*d*e^5)*x + (c^6*d^6 - 3*c^4*d^4*e^ 2 + (c^6*d^3*e^3 - 3*c^4*d*e^5)*x^3 + 3*(c^6*d^4*e^2 - 3*c^4*d^2*e^4)*x^2 + 3*(c^6*d^5*e - 3*c^4*d^3*e^3)*x)*log(c^2*x^2 + 1) - 2*(c^6*d^6 + 3*c^4*d ^4*e^2 + 3*c^2*d^2*e^4 + e^6)*log(c*x + sqrt(c^2*x^2 + 1)))/(c^6*d^9*e + 3 *c^4*d^7*e^3 + 3*c^2*d^5*e^5 + d^3*e^7 + (c^6*d^6*e^4 + 3*c^4*d^4*e^6 + 3* c^2*d^2*e^8 + e^10)*x^3 + 3*(c^6*d^7*e^3 + 3*c^4*d^5*e^5 + 3*c^2*d^3*e^7 + d*e^9)*x^2 + 3*(c^6*d^8*e^2 + 3*c^4*d^6*e^4 + 3*c^2*d^4*e^6 + d^2*e^8)*x) - I*(3*c^6*d^2 - c^4*e^2)*(log(I*c*x + 1) - log(-I*c*x + 1))/((c^6*d^6 + 3*c^4*d^4*e^2 + 3*c^2*d^2*e^4 + e^6)*c))*b - 1/3*a/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e)
\[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (e x + d\right )}^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{(d+e x)^4} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{{\left (d+e\,x\right )}^4} \,d x \]