Integrand size = 18, antiderivative size = 368 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=2 b^2 d^3 x-\frac {4 b^2 d e^2 x}{3 c^2}+\frac {3}{4} b^2 d^2 e x^2-\frac {3 b^2 e^3 x^2}{32 c^2}+\frac {2}{9} b^2 d e^2 x^3+\frac {1}{32} b^2 e^3 x^4-\frac {2 b d^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {4 b d e^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c^3}-\frac {3 b d^2 e x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{2 c}+\frac {3 b e^3 x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{16 c^3}-\frac {2 b d e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{3 c}-\frac {b e^3 x^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c}-\frac {d^4 (a+b \text {arcsinh}(c x))^2}{4 e}+\frac {3 d^2 e (a+b \text {arcsinh}(c x))^2}{4 c^2}-\frac {3 e^3 (a+b \text {arcsinh}(c x))^2}{32 c^4}+\frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^2}{4 e} \]
2*b^2*d^3*x-4/3*b^2*d*e^2*x/c^2+3/4*b^2*d^2*e*x^2-3/32*b^2*e^3*x^2/c^2+2/9 *b^2*d*e^2*x^3+1/32*b^2*e^3*x^4-1/4*d^4*(a+b*arcsinh(c*x))^2/e+3/4*d^2*e*( a+b*arcsinh(c*x))^2/c^2-3/32*e^3*(a+b*arcsinh(c*x))^2/c^4+1/4*(e*x+d)^4*(a +b*arcsinh(c*x))^2/e-2*b*d^3*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+4/3*b* d*e^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3-3/2*b*d^2*e*x*(a+b*arcsinh( c*x))*(c^2*x^2+1)^(1/2)/c+3/16*b*e^3*x*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2 )/c^3-2/3*b*d*e^2*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c-1/8*b*e^3*x^3 *(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c
Time = 0.36 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.96 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {c \left (72 a^2 c^3 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-6 a b \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )+b^2 c x \left (-3 e^2 (128 d+9 e x)+c^2 \left (576 d^3+216 d^2 e x+64 d e^2 x^2+9 e^3 x^3\right )\right )\right )-6 b \left (-3 a \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right )+b c \sqrt {1+c^2 x^2} \left (-e^2 (64 d+9 e x)+c^2 \left (96 d^3+72 d^2 e x+32 d e^2 x^2+6 e^3 x^3\right )\right )\right ) \text {arcsinh}(c x)+9 b^2 \left (24 c^2 d^2 e-3 e^3+8 c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )\right ) \text {arcsinh}(c x)^2}{288 c^4} \]
(c*(72*a^2*c^3*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) - 6*a*b*Sqrt[ 1 + c^2*x^2]*(-(e^2*(64*d + 9*e*x)) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2* x^2 + 6*e^3*x^3)) + b^2*c*x*(-3*e^2*(128*d + 9*e*x) + c^2*(576*d^3 + 216*d ^2*e*x + 64*d*e^2*x^2 + 9*e^3*x^3))) - 6*b*(-3*a*(24*c^2*d^2*e - 3*e^3 + 8 *c^4*x*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3)) + b*c*Sqrt[1 + c^2*x^2 ]*(-(e^2*(64*d + 9*e*x)) + c^2*(96*d^3 + 72*d^2*e*x + 32*d*e^2*x^2 + 6*e^3 *x^3)))*ArcSinh[c*x] + 9*b^2*(24*c^2*d^2*e - 3*e^3 + 8*c^4*x*(4*d^3 + 6*d^ 2*e*x + 4*d*e^2*x^2 + e^3*x^3))*ArcSinh[c*x]^2)/(288*c^4)
Time = 1.05 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.05, 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.
| \(\displaystyle \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6243 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^2}{4 e}-\frac {b c \int \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{2 e}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^2}{4 e}-\frac {b c \int \left (\frac {(a+b \text {arcsinh}(c x)) d^4}{\sqrt {c^2 x^2+1}}+\frac {4 e x (a+b \text {arcsinh}(c x)) d^3}{\sqrt {c^2 x^2+1}}+\frac {6 e^2 x^2 (a+b \text {arcsinh}(c x)) d^2}{\sqrt {c^2 x^2+1}}+\frac {4 e^3 x^3 (a+b \text {arcsinh}(c x)) d}{\sqrt {c^2 x^2+1}}+\frac {e^4 x^4 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}\right )dx}{2 e}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {(d+e x)^4 (a+b \text {arcsinh}(c x))^2}{4 e}-\frac {b c \left (\frac {3 e^4 (a+b \text {arcsinh}(c x))^2}{16 b c^5}-\frac {3 d^2 e^2 (a+b \text {arcsinh}(c x))^2}{2 b c^3}+\frac {4 d^3 e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}+\frac {3 d^2 e^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c^2}+\frac {4 d e^3 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^2}+\frac {e^4 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{4 c^2}-\frac {8 d e^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{3 c^4}-\frac {3 e^4 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{8 c^4}+\frac {d^4 (a+b \text {arcsinh}(c x))^2}{2 b c}+\frac {8 b d e^3 x}{3 c^3}+\frac {3 b e^4 x^2}{16 c^3}-\frac {4 b d^3 e x}{c}-\frac {3 b d^2 e^2 x^2}{2 c}-\frac {4 b d e^3 x^3}{9 c}-\frac {b e^4 x^4}{16 c}\right )}{2 e}\) |
((d + e*x)^4*(a + b*ArcSinh[c*x])^2)/(4*e) - (b*c*((-4*b*d^3*e*x)/c + (8*b *d*e^3*x)/(3*c^3) - (3*b*d^2*e^2*x^2)/(2*c) + (3*b*e^4*x^2)/(16*c^3) - (4* b*d*e^3*x^3)/(9*c) - (b*e^4*x^4)/(16*c) + (4*d^3*e*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c^2 - (8*d*e^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3 *c^4) + (3*d^2*e^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c^2 - (3*e^4* x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c^4) + (4*d*e^3*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2) + (e^4*x^3*Sqrt[1 + c^2*x^2]*(a + b *ArcSinh[c*x]))/(4*c^2) + (d^4*(a + b*ArcSinh[c*x])^2)/(2*b*c) - (3*d^2*e^ 2*(a + b*ArcSinh[c*x])^2)/(2*b*c^3) + (3*e^4*(a + b*ArcSinh[c*x])^2)/(16*b *c^5)))/(2*e)
3.1.12.3.1 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 = 0.44 (sec) , antiderivative size = 526, normalized size of antiderivative = 1.43
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {e^{3} \left (8 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-3 \operatorname {arcsinh}\left (c x \right )^{2}-3 c^{2} x^{2}-3\right )}{32}+\frac {d c \,e^{2} \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{9}+\frac {3 d^{2} c^{2} e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}+d^{3} c^{3} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c^{3}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arcsinh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arcsinh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arcsinh}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (c x \right )+e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{8}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+4 d c \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{4 e}\right )}{c^{3}}}{c}\) | \(526\) |
| default | \(\frac {\frac {a^{2} \left (e c x +c d \right )^{4}}{4 c^{3} e}+\frac {b^{2} \left (\frac {e^{3} \left (8 \operatorname {arcsinh}\left (c x \right )^{2} x^{4} c^{4}-4 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{3} c^{3}+c^{4} x^{4}+6 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}-3 \operatorname {arcsinh}\left (c x \right )^{2}-3 c^{2} x^{2}-3\right )}{32}+\frac {d c \,e^{2} \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{9}+\frac {3 d^{2} c^{2} e \left (2 \operatorname {arcsinh}\left (c x \right )^{2} x^{2} c^{2}-2 \,\operatorname {arcsinh}\left (c x \right ) c x \sqrt {c^{2} x^{2}+1}+\operatorname {arcsinh}\left (c x \right )^{2}+c^{2} x^{2}+1\right )}{4}+d^{3} c^{3} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )\right )}{c^{3}}+\frac {2 a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{4} d^{4}}{4 e}+\operatorname {arcsinh}\left (c x \right ) c^{4} d^{3} x +\frac {3 e \,\operatorname {arcsinh}\left (c x \right ) c^{4} d^{2} x^{2}}{2}+e^{2} \operatorname {arcsinh}\left (c x \right ) c^{4} d \,x^{3}+\frac {e^{3} \operatorname {arcsinh}\left (c x \right ) c^{4} x^{4}}{4}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (c x \right )+e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{8}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+4 d c \,e^{3} \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{4 e}\right )}{c^{3}}}{c}\) | \(526\) |
| parts | \(\frac {a^{2} \left (e x +d \right )^{4}}{4 e}+\frac {b^{2} \left (72 \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{4} e^{3}+288 \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{3} d \,e^{2}+432 \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x^{2} d^{2} e +288 \operatorname {arcsinh}\left (c x \right )^{2} c^{4} x \,d^{3}-36 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3} e^{3}-192 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x^{2} d \,e^{2}-432 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} x \,d^{2} e -576 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{3} d^{3}+216 \operatorname {arcsinh}\left (c x \right )^{2} c^{2} d^{2} e +9 c^{4} x^{4} e^{3}+64 c^{4} d \,e^{2} x^{3}+216 c^{4} x^{2} d^{2} e +576 x \,c^{4} d^{3}+54 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x \,e^{3}+384 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c d \,e^{2}-27 \operatorname {arcsinh}\left (c x \right )^{2} e^{3}-27 c^{2} x^{2} e^{3}-384 c^{2} x d \,e^{2}+216 c^{2} d^{2} e -27 e^{3}\right )}{288 c^{4}}+\frac {2 a b \left (\frac {c \,e^{3} \operatorname {arcsinh}\left (c x \right ) x^{4}}{4}+c \,e^{2} \operatorname {arcsinh}\left (c x \right ) x^{3} d +\frac {3 c \,\operatorname {arcsinh}\left (c x \right ) d^{2} e \,x^{2}}{2}+\operatorname {arcsinh}\left (c x \right ) c x \,d^{3}+\frac {c \,\operatorname {arcsinh}\left (c x \right ) d^{4}}{4 e}-\frac {c^{4} d^{4} \operatorname {arcsinh}\left (c x \right )+e^{4} \left (\frac {c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{4}-\frac {3 c x \sqrt {c^{2} x^{2}+1}}{8}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{8}\right )+4 d^{3} c^{3} e \sqrt {c^{2} x^{2}+1}+6 d^{2} c^{2} e^{2} \left (\frac {c x \sqrt {c^{2} x^{2}+1}}{2}-\frac {\operatorname {arcsinh}\left (c x \right )}{2}\right )+4 d c \,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 )}{4 c^{3} e}\right )}{c}\) | \(573\) |
1/c*(1/4*a^2/c^3*(c*e*x+c*d)^4/e+b^2/c^3*(1/32*e^3*(8*arcsinh(c*x)^2*x^4*c ^4-4*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^3*c^3+c^4*x^4+6*arcsinh(c*x)*c*x*(c^ 2*x^2+1)^(1/2)-3*arcsinh(c*x)^2-3*c^2*x^2-3)+1/9*d*c*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)+3/4*d^2*c^2*e*(2*arcsinh(c*x)^2*x^2*c^2-2*arcs inh(c*x)*c*x*(c^2*x^2+1)^(1/2)+arcsinh(c*x)^2+c^2*x^2+1)+d^3*c^3*(arcsinh( c*x)^2*x*c-2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x))+2*a*b/c^3*(1/4/e*arcsi nh(c*x)*c^4*d^4+arcsinh(c*x)*c^4*d^3*x+3/2*e*arcsinh(c*x)*c^4*d^2*x^2+e^2* arcsinh(c*x)*c^4*d*x^3+1/4*e^3*arcsinh(c*x)*c^4*x^4-1/4/e*(c^4*d^4*arcsinh (c*x)+e^4*(1/4*c^3*x^3*(c^2*x^2+1)^(1/2)-3/8*c*x*(c^2*x^2+1)^(1/2)+3/8*arc sinh(c*x))+4*d^3*c^3*e*(c^2*x^2+1)^(1/2)+6*d^2*c^2*e^2*(1/2*c*x*(c^2*x^2+1 )^(1/2)-1/2*arcsinh(c*x))+4*d*c*e^3*(1/3*c^2*x^2*(c^2*x^2+1)^(1/2)-2/3*(c^ 2*x^2+1)^(1/2)))))
Time = 0.27 (sec) , antiderivative size = 475, normalized size of antiderivative = 1.29 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {9 \, {\left (8 \, a^{2} + b^{2}\right )} c^{4} e^{3} x^{4} + 32 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{4} d e^{2} x^{3} + 27 \, {\left (8 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{2} e - b^{2} c^{2} e^{3}\right )} x^{2} + 9 \, {\left (8 \, b^{2} c^{4} e^{3} x^{4} + 32 \, b^{2} c^{4} d e^{2} x^{3} + 48 \, b^{2} c^{4} d^{2} e x^{2} + 32 \, b^{2} c^{4} d^{3} x + 24 \, b^{2} c^{2} d^{2} e - 3 \, b^{2} e^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 96 \, {\left (3 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{4} d^{3} - 4 \, b^{2} c^{2} d e^{2}\right )} x + 6 \, {\left (24 \, a b c^{4} e^{3} x^{4} + 96 \, a b c^{4} d e^{2} x^{3} + 144 \, a b c^{4} d^{2} e x^{2} + 96 \, a b c^{4} d^{3} x + 72 \, a b c^{2} d^{2} e - 9 \, a b e^{3} - {\left (6 \, b^{2} c^{3} e^{3} x^{3} + 32 \, b^{2} c^{3} d e^{2} x^{2} + 96 \, b^{2} c^{3} d^{3} - 64 \, b^{2} c d e^{2} + 9 \, {\left (8 \, b^{2} c^{3} d^{2} e - b^{2} c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (6 \, a b c^{3} e^{3} x^{3} + 32 \, a b c^{3} d e^{2} x^{2} + 96 \, a b c^{3} d^{3} - 64 \, a b c d e^{2} + 9 \, {\left (8 \, a b c^{3} d^{2} e - a b c e^{3}\right )} x\right )} \sqrt {c^{2} x^{2} + 1}}{288 \, c^{4}} \]
1/288*(9*(8*a^2 + b^2)*c^4*e^3*x^4 + 32*(9*a^2 + 2*b^2)*c^4*d*e^2*x^3 + 27 *(8*(2*a^2 + b^2)*c^4*d^2*e - b^2*c^2*e^3)*x^2 + 9*(8*b^2*c^4*e^3*x^4 + 32 *b^2*c^4*d*e^2*x^3 + 48*b^2*c^4*d^2*e*x^2 + 32*b^2*c^4*d^3*x + 24*b^2*c^2* d^2*e - 3*b^2*e^3)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 96*(3*(a^2 + 2*b^2)*c^ 4*d^3 - 4*b^2*c^2*d*e^2)*x + 6*(24*a*b*c^4*e^3*x^4 + 96*a*b*c^4*d*e^2*x^3 + 144*a*b*c^4*d^2*e*x^2 + 96*a*b*c^4*d^3*x + 72*a*b*c^2*d^2*e - 9*a*b*e^3 - (6*b^2*c^3*e^3*x^3 + 32*b^2*c^3*d*e^2*x^2 + 96*b^2*c^3*d^3 - 64*b^2*c*d* e^2 + 9*(8*b^2*c^3*d^2*e - b^2*c*e^3)*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt (c^2*x^2 + 1)) - 6*(6*a*b*c^3*e^3*x^3 + 32*a*b*c^3*d*e^2*x^2 + 96*a*b*c^3* d^3 - 64*a*b*c*d*e^2 + 9*(8*a*b*c^3*d^2*e - a*b*c*e^3)*x)*sqrt(c^2*x^2 + 1 ))/c^4
Leaf count of result is larger than twice the leaf count of optimal. 743 vs. \(2 (364) = 728\).
Time = 0.45 (sec) , antiderivative size = 743, normalized size of antiderivative = 2.02 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} a^{2} d^{3} x + \frac {3 a^{2} d^{2} e x^{2}}{2} + a^{2} d e^{2} x^{3} + \frac {a^{2} e^{3} x^{4}}{4} + 2 a b d^{3} x \operatorname {asinh}{\left (c x \right )} + 3 a b d^{2} e x^{2} \operatorname {asinh}{\left (c x \right )} + 2 a b d e^{2} x^{3} \operatorname {asinh}{\left (c x \right )} + \frac {a b e^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {2 a b d^{3} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {3 a b d^{2} e x \sqrt {c^{2} x^{2} + 1}}{2 c} - \frac {2 a b d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{3 c} - \frac {a b e^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{8 c} + \frac {3 a b d^{2} e \operatorname {asinh}{\left (c x \right )}}{2 c^{2}} + \frac {4 a b d e^{2} \sqrt {c^{2} x^{2} + 1}}{3 c^{3}} + \frac {3 a b e^{3} x \sqrt {c^{2} x^{2} + 1}}{16 c^{3}} - \frac {3 a b e^{3} \operatorname {asinh}{\left (c x \right )}}{16 c^{4}} + b^{2} d^{3} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{3} x + \frac {3 b^{2} d^{2} e x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {3 b^{2} d^{2} e x^{2}}{4} + b^{2} d e^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )} + \frac {2 b^{2} d e^{2} x^{3}}{9} + \frac {b^{2} e^{3} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {b^{2} e^{3} x^{4}}{32} - \frac {2 b^{2} d^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {3 b^{2} d^{2} e x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2 c} - \frac {2 b^{2} d e^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c} - \frac {b^{2} e^{3} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{8 c} + \frac {3 b^{2} d^{2} e \operatorname {asinh}^{2}{\left (c x \right )}}{4 c^{2}} - \frac {4 b^{2} d e^{2} x}{3 c^{2}} - \frac {3 b^{2} e^{3} x^{2}}{32 c^{2}} + \frac {4 b^{2} d e^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{3 c^{3}} + \frac {3 b^{2} e^{3} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{16 c^{3}} - \frac {3 b^{2} e^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{32 c^{4}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{3} x + \frac {3 d^{2} e x^{2}}{2} + d e^{2} x^{3} + \frac {e^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d**3*x + 3*a**2*d**2*e*x**2/2 + a**2*d*e**2*x**3 + a**2*e* *3*x**4/4 + 2*a*b*d**3*x*asinh(c*x) + 3*a*b*d**2*e*x**2*asinh(c*x) + 2*a*b *d*e**2*x**3*asinh(c*x) + a*b*e**3*x**4*asinh(c*x)/2 - 2*a*b*d**3*sqrt(c** 2*x**2 + 1)/c - 3*a*b*d**2*e*x*sqrt(c**2*x**2 + 1)/(2*c) - 2*a*b*d*e**2*x* *2*sqrt(c**2*x**2 + 1)/(3*c) - a*b*e**3*x**3*sqrt(c**2*x**2 + 1)/(8*c) + 3 *a*b*d**2*e*asinh(c*x)/(2*c**2) + 4*a*b*d*e**2*sqrt(c**2*x**2 + 1)/(3*c**3 ) + 3*a*b*e**3*x*sqrt(c**2*x**2 + 1)/(16*c**3) - 3*a*b*e**3*asinh(c*x)/(16 *c**4) + b**2*d**3*x*asinh(c*x)**2 + 2*b**2*d**3*x + 3*b**2*d**2*e*x**2*as inh(c*x)**2/2 + 3*b**2*d**2*e*x**2/4 + b**2*d*e**2*x**3*asinh(c*x)**2 + 2* b**2*d*e**2*x**3/9 + b**2*e**3*x**4*asinh(c*x)**2/4 + b**2*e**3*x**4/32 - 2*b**2*d**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/c - 3*b**2*d**2*e*x*sqrt(c**2*x **2 + 1)*asinh(c*x)/(2*c) - 2*b**2*d*e**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c *x)/(3*c) - b**2*e**3*x**3*sqrt(c**2*x**2 + 1)*asinh(c*x)/(8*c) + 3*b**2*d **2*e*asinh(c*x)**2/(4*c**2) - 4*b**2*d*e**2*x/(3*c**2) - 3*b**2*e**3*x**2 /(32*c**2) + 4*b**2*d*e**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(3*c**3) + 3*b** 2*e**3*x*sqrt(c**2*x**2 + 1)*asinh(c*x)/(16*c**3) - 3*b**2*e**3*asinh(c*x) **2/(32*c**4), Ne(c, 0)), (a**2*(d**3*x + 3*d**2*e*x**2/2 + d*e**2*x**3 + e**3*x**4/4), True))
Time = 0.21 (sec) , antiderivative size = 590, normalized size of antiderivative = 1.60 \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{4} \, b^{2} e^{3} x^{4} \operatorname {arsinh}\left (c x\right )^{2} + b^{2} d e^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{4} \, a^{2} e^{3} x^{4} + \frac {3}{2} \, b^{2} d^{2} e x^{2} \operatorname {arsinh}\left (c x\right )^{2} + a^{2} d e^{2} x^{3} + b^{2} d^{3} x \operatorname {arsinh}\left (c x\right )^{2} + \frac {3}{2} \, a^{2} d^{2} e x^{2} + \frac {3}{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^{2} e + \frac {3}{4} \, {\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^{2} e + \frac {2}{3} \, {\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 d e^{2} - \frac {2}{9} \, {\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} d e^{2} + \frac {1}{16} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} a b e^{3} + \frac {1}{32} \, {\left ({\left (\frac {x^{4}}{c^{2}} - \frac {3 \, x^{2}}{c^{4}} + \frac {3 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \, {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c \operatorname {arsinh}\left (c x\right )\right )} b^{2} e^{3} + 2 \, b^{2} d^{3} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{3} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{3}}{c} \]
1/4*b^2*e^3*x^4*arcsinh(c*x)^2 + b^2*d*e^2*x^3*arcsinh(c*x)^2 + 1/4*a^2*e^ 3*x^4 + 3/2*b^2*d^2*e*x^2*arcsinh(c*x)^2 + a^2*d*e^2*x^3 + b^2*d^3*x*arcsi nh(c*x)^2 + 3/2*a^2*d^2*e*x^2 + 3/2*(2*x^2*arcsinh(c*x) - c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^3))*a*b*d^2*e + 3/4*(c^2*(x^2/c^2 - log(c*x + sqrt(c^2*x^2 + 1))^2/c^4) - 2*c*(sqrt(c^2*x^2 + 1)*x/c^2 - arcsinh(c*x)/c^ 3)*arcsinh(c*x))*b^2*d^2*e + 2/3*(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*d*e^2 - 2/9*(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*d*e^2 + 1/16*(8*x^4*arcsinh(c*x) - (2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*s qrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c)*a*b*e^3 + 1/32*((x^4/c^2 - 3*x^2/c^4 + 3*log(c*x + sqrt(c^2*x^2 + 1))^2/c^6)*c^2 - 2*(2*sqrt(c^2*x^2 + 1)*x^3/c^2 - 3*sqrt(c^2*x^2 + 1)*x/c^4 + 3*arcsinh(c*x)/c^5)*c*arcsinh( c*x))*b^2*e^3 + 2*b^2*d^3*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x)/c) + a^2*d^3 *x + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*d^3/c
Exception generated. \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (d+e x)^3 (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \]