Integrand size = 18, antiderivative size = 239 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=2 b^2 d^2 x-\frac {4 b^2 e^2 x}{9 c^2}+\frac {1}{2} b^2 d e x^2+\frac {2}{27} b^2 e^2 x^3-\frac {2 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b e^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3}-\frac {b d e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}-\frac {2 b e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}-\frac {d^3 (a+b \text {arcsinh}(c x))^2}{3 e}+\frac {d e (a+b \text {arcsinh}(c x))^2}{2 c^2}+\frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e} \]
2*b^2*d^2*x-4/9*b^2*e^2*x/c^2+1/2*b^2*d*e*x^2+2/27*b^2*e^2*x^3-1/3*d^3*(a+ b*arcsinh(c*x))^2/e+1/2*d*e*(a+b*arcsinh(c*x))^2/c^2+1/3*(e*x+d)^3*(a+b*ar csinh(c*x))^2/e-2*b*d^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+4/9*b*e^2*( a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-b*d*e*x*(a+b*arcsinh(c*x))*(c^2*x^ 2+1)^(1/2)/c-2/9*b*e^2*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c
Time = 0.23 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.04 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {18 a^2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )-6 a b \sqrt {1+c^2 x^2} \left (-4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+b^2 c x \left (-24 e^2+c^2 \left (108 d^2+27 d e x+4 e^2 x^2\right )\right )-6 b \left (-3 a \left (3 c d e+2 c^3 x \left (3 d^2+3 d e x+e^2 x^2\right )\right )+b \sqrt {1+c^2 x^2} \left (-4 e^2+c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )\right ) \text {arcsinh}(c x)+9 b^2 c \left (6 c^2 d^2 x+2 c^2 e^2 x^3+3 d \left (e+2 c^2 e x^2\right )\right ) \text {arcsinh}(c x)^2}{54 c^3} \]
(18*a^2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2) - 6*a*b*Sqrt[1 + c^2*x^2]*(-4*e^ 2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x^2)) + b^2*c*x*(-24*e^2 + c^2*(108*d^2 + 27*d*e*x + 4*e^2*x^2)) - 6*b*(-3*a*(3*c*d*e + 2*c^3*x*(3*d^2 + 3*d*e*x + e^2*x^2)) + b*Sqrt[1 + c^2*x^2]*(-4*e^2 + c^2*(18*d^2 + 9*d*e*x + 2*e^2*x ^2)))*ArcSinh[c*x] + 9*b^2*c*(6*c^2*d^2*x + 2*c^2*e^2*x^3 + 3*d*(e + 2*c^2 *e*x^2))*ArcSinh[c*x]^2)/(54*c^3)
Time = 0.78 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.10, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {6243, 6253, 2009}
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.
| \(\displaystyle \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6243 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e}-\frac {2 b c \int \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{3 e}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e}-\frac {2 b c \int \left (\frac {(a+b \text {arcsinh}(c x)) d^3}{\sqrt {c^2 x^2+1}}+\frac {3 e x (a+b \text {arcsinh}(c x)) d^2}{\sqrt {c^2 x^2+1}}+\frac {3 e^2 x^2 (a+b \text {arcsinh}(c x)) d}{\sqrt {c^2 x^2+1}}+\frac {e^3 x^3 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}\right )dx}{3 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^3 (a+b \text {arcsinh}(c x))^2}{3 e}-\frac {2 b c \left (-\frac {3 d e^2 (a+b \text {arcsinh}(c x))^2}{4 b c^3}+\frac {3 d^2 e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}+\frac {3 d e^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{2 c^2}+\frac {e^3 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}-\frac {2 e^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^4}+\frac {d^3 (a+b \text {arcsinh}(c x))^2}{2 b c}+\frac {2 b e^3 x}{3 c^3}-\frac {3 b d^2 e x}{c}-\frac {3 b d e^2 x^2}{4 c}-\frac {b e^3 x^3}{9 c}\right )}{3 e}\) |
((d + e*x)^3*(a + b*ArcSinh[c*x])^2)/(3*e) - (2*b*c*((-3*b*d^2*e*x)/c + (2 *b*e^3*x)/(3*c^3) - (3*b*d*e^2*x^2)/(4*c) - (b*e^3*x^3)/(9*c) + (3*d^2*e*S qrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c^2 - (2*e^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^4) + (3*d*e^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c* x]))/(2*c^2) + (e^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2) + (d^3*(a + b*ArcSinh[c*x])^2)/(2*b*c) - (3*d*e^2*(a + b*ArcSinh[c*x])^2)/(4 *b*c^3)))/(3*e)
3.1.13.3.1 Defintions of rubi rules used
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]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n , 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
Time = 0.46 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.52
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (\frac {e^{2} \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {d c e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{2}+d^{2} c^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsinh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arcsinh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 e}\right )}{c^{2}}}{c}\) | \(364\) |
| default | \(\frac {\frac {a^{2} \left (e c x +c d \right )^{3}}{3 c^{2} e}+\frac {b^{2} \left (\frac {e^{2} \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {d c e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{2}+d^{2} c^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c^{2}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{3} d^{3}}{3 e}+\operatorname {arcsinh}\left (c x \right ) c^{3} d^{2} x +e \,\operatorname {arcsinh}\left (c x \right ) c^{3} d \,x^{2}+\frac {e^{2} \operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 e}\right )}{c^{2}}}{c}\) | \(364\) |
| parts | \(\frac {a^{2} \left (e x +d \right )^{3}}{3 e}+\frac {b^{2} \left (18 \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{3} e^{2}+54 \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x^{2} d e +54 \operatorname {arcsinh}\left (c x \right )^{2} c^{3} x \,d^{2}-12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2} e^{2}-54 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x d e -108 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} d^{2}+27 \operatorname {arcsinh}\left (c x \right )^{2} c d e +4 c^{3} x^{3} e^{2}+27 c^{3} x^{2} d e +108 c^{3} x \,d^{2}+24 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, e^{2}-24 c x \,e^{2}+27 c d e \right )}{54 c^{3}}+\frac {2 a b \left (\frac {c \,e^{2} \operatorname {arcsinh}\left (c x \right ) x^{3}}{3}+c \,\operatorname {arcsinh}\left (c x \right ) d e \,x^{2}+\operatorname {arcsinh}\left (c x \right ) c x \,d^{2}+\frac {c \,\operatorname {arcsinh}\left (c x \right ) d^{3}}{3 e}-\frac {c^{3} d^{3} \operatorname {arcsinh}\left (c x \right )+e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )+3 d^{2} c^{2} e \sqrt {c^{2} x^{2}+1}+3 d c \,e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )}{3 c^{2} e}\right )}{c}\) | \(380\) |
1/c*(1/3*a^2/c^2*(c*e*x+c*d)^3/e+b^2/c^2*(1/27*e^2*(9*arcsinh(c*x)^2*x^3*c ^3-6*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^3+12*arcsinh(c*x)*(c^2 *x^2+1)^(1/2)-12*c*x)+1/2*d*c*e*(2*arcsinh(c*x)^2*x^2*c^2-2*arcsinh(c*x)*c *x*(c^2*x^2+1)^(1/2)+arcsinh(c*x)^2+c^2*x^2+1)+d^2*c^2*(arcsinh(c*x)^2*x*c -2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x))+2*a*b/c^2*(1/3/e*arcsinh(c*x)*c^ 3*d^3+arcsinh(c*x)*c^3*d^2*x+e*arcsinh(c*x)*c^3*d*x^2+1/3*e^2*arcsinh(c*x) *c^3*x^3-1/3/e*(c^3*d^3*arcsinh(c*x)+e^3*(1/3*c^2*x^2*(c^2*x^2+1)^(1/2)-2/ 3*(c^2*x^2+1)^(1/2))+3*d^2*c^2*e*(c^2*x^2+1)^(1/2)+3*d*c*e^2*(1/2*c*x*(c^2 *x^2+1)^(1/2)-1/2*arcsinh(c*x)))))
Time = 0.26 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.33 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {2 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} e^{2} x^{3} + 27 \, {\left (2 \, a^{2} + b^{2}\right )} c^{3} d e x^{2} + 9 \, {\left (2 \, b^{2} c^{3} e^{2} x^{3} + 6 \, b^{2} c^{3} d e x^{2} + 6 \, b^{2} c^{3} d^{2} x + 3 \, b^{2} c d e\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (9 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{3} d^{2} - 4 \, b^{2} c e^{2}\right )} x + 6 \, {\left (6 \, a b c^{3} e^{2} x^{3} + 18 \, a b c^{3} d e x^{2} + 18 \, a b c^{3} d^{2} x + 9 \, a b c d e - {\left (2 \, b^{2} c^{2} e^{2} x^{2} + 9 \, b^{2} c^{2} d e x + 18 \, b^{2} c^{2} d^{2} - 4 \, b^{2} e^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (2 \, a b c^{2} e^{2} x^{2} + 9 \, a b c^{2} d e x + 18 \, a b c^{2} d^{2} - 4 \, a b e^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{54 \, c^{3}} \]
1/54*(2*(9*a^2 + 2*b^2)*c^3*e^2*x^3 + 27*(2*a^2 + b^2)*c^3*d*e*x^2 + 9*(2* b^2*c^3*e^2*x^3 + 6*b^2*c^3*d*e*x^2 + 6*b^2*c^3*d^2*x + 3*b^2*c*d*e)*log(c *x + sqrt(c^2*x^2 + 1))^2 + 6*(9*(a^2 + 2*b^2)*c^3*d^2 - 4*b^2*c*e^2)*x + 6*(6*a*b*c^3*e^2*x^3 + 18*a*b*c^3*d*e*x^2 + 18*a*b*c^3*d^2*x + 9*a*b*c*d*e - (2*b^2*c^2*e^2*x^2 + 9*b^2*c^2*d*e*x + 18*b^2*c^2*d^2 - 4*b^2*e^2)*sqrt (c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 6*(2*a*b*c^2*e^2*x^2 + 9*a*b *c^2*d*e*x + 18*a*b*c^2*d^2 - 4*a*b*e^2)*sqrt(c^2*x^2 + 1))/c^3
Time = 0.33 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.90 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} a^{2} d^{2} x + a^{2} d e x^{2} + \frac {a^{2} e^{2} x^{3}}{3} + 2 a b d^{2} x \operatorname {asinh}{\left (c x \right )} + 2 a b d e x^{2} \operatorname {asinh}{\left (c x \right )} + \frac {2 a b e^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {2 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {a b d e x \sqrt {c^{2} x^{2} + 1}}{c} - \frac {2 a b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {a b d e \operatorname {asinh}{\left (c x \right )}}{c^{2}} + \frac {4 a b e^{2} \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + b^{2} d^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{2} x + b^{2} d e x^{2} \operatorname {asinh}^{2}{\left (c x \right )} + \frac {b^{2} d e x^{2}}{2} + \frac {b^{2} e^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} e^{2} x^{3}}{27} - \frac {2 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {b^{2} d e x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {2 b^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} + \frac {b^{2} d e \operatorname {asinh}^{2}{\left (c x \right )}}{2 c^{2}} - \frac {4 b^{2} e^{2} x}{9 c^{2}} + \frac {4 b^{2} e^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c^{3}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{2} x + d e x^{2} + \frac {e^{2} x^{3}}{3}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d**2*x + a**2*d*e*x**2 + a**2*e**2*x**3/3 + 2*a*b*d**2*x*a sinh(c*x) + 2*a*b*d*e*x**2*asinh(c*x) + 2*a*b*e**2*x**3*asinh(c*x)/3 - 2*a *b*d**2*sqrt(c**2*x**2 + 1)/c - a*b*d*e*x*sqrt(c**2*x**2 + 1)/c - 2*a*b*e* *2*x**2*sqrt(c**2*x**2 + 1)/(9*c) + a*b*d*e*asinh(c*x)/c**2 + 4*a*b*e**2*s qrt(c**2*x**2 + 1)/(9*c**3) + b**2*d**2*x*asinh(c*x)**2 + 2*b**2*d**2*x + b**2*d*e*x**2*asinh(c*x)**2 + b**2*d*e*x**2/2 + b**2*e**2*x**3*asinh(c*x)* *2/3 + 2*b**2*e**2*x**3/27 - 2*b**2*d**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/c - b**2*d*e*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/c - 2*b**2*e**2*x**2*sqrt(c**2 *x**2 + 1)*asinh(c*x)/(9*c) + b**2*d*e*asinh(c*x)**2/(2*c**2) - 4*b**2*e** 2*x/(9*c**2) + 4*b**2*e**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(9*c**3), Ne(c, 0)), (a**2*(d**2*x + d*e*x**2 + e**2*x**3/3), True))
Time = 0.19 (sec) , antiderivative size = 378, normalized size of antiderivative = 1.58 \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{3} \, b^{2} e^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + b^{2} d e x^{2} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} e^{2} x^{3} + b^{2} d^{2} x \operatorname {arsinh}\left (c x\right )^{2} + a^{2} d e x^{2} + {\left (2 \, x^{2} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )}\right )} a b d e + \frac {1}{2} \, {\left (c^{2} {\left (\frac {x^{2}}{c^{2}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{4}}\right )} - 2 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x}{c^{2}} - \frac {\operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} \operatorname {arsinh}\left (c x\right )\right )} b^{2} d e + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b e^{2} - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} e^{2} + 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \]
1/3*b^2*e^2*x^3*arcsinh(c*x)^2 + b^2*d*e*x^2*arcsinh(c*x)^2 + 1/3*a^2*e^2* x^3 + b^2*d^2*x*arcsinh(c*x)^2 + a^2*d*e*x^2 + (2*x^2*arcsinh(c*x) - c*(sq rt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3))*a*b*d*e + 1/2*(c^2*(x^2/c^2 - l og(c*x + sqrt(c^2*x^2 + 1))^2/c^4) - 2*c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsin h(c*x)/c^3)*arcsinh(c*x))*b^2*d*e + 2/9*(3*x^3*arcsinh(c*x) - c*(sqrt(c^2* x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*e^2 - 2/27*(3*c*(sqrt(c^2 *x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4)*arcsinh(c*x) - (c^2*x^3 - 6*x )/c^2)*b^2*e^2 + 2*b^2*d^2*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d^ 2*x + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*d^2/c
Exception generated. \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int (d+e x)^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2 \,d x \]