Integrand size = 23, antiderivative size = 172 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^2 \, dx=-\frac {3 b^2 e^3 (c+d x)^2}{32 d}+\frac {b^2 e^3 (c+d x)^4}{32 d}+\frac {3 b e^3 (c+d x) \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{16 d}-\frac {b e^3 (c+d x)^3 \sqrt {1+(c+d x)^2} (a+b \text {arcsinh}(c+d x))}{8 d}-\frac {3 e^3 (a+b \text {arcsinh}(c+d x))^2}{32 d}+\frac {e^3 (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2}{4 d} \] Output:
-3/32*b^2*e^3*(d*x+c)^2/d+1/32*b^2*e^3*(d*x+c)^4/d+3/16*b*e^3*(d*x+c)*(1+( d*x+c)^2)^(1/2)*(a+b*arcsinh(d*x+c))/d-1/8*b*e^3*(d*x+c)^3*(1+(d*x+c)^2)^( 1/2)*(a+b*arcsinh(d*x+c))/d-3/32*e^3*(a+b*arcsinh(d*x+c))^2/d+1/4*e^3*(d*x +c)^4*(a+b*arcsinh(d*x+c))^2/d
Time = 0.23 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.99 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {e^3 \left (-3 b^2 (c+d x)^2+\left (8 a^2+b^2\right ) (c+d x)^4+2 a b (c+d x) \left (3-2 (c+d x)^2\right ) \sqrt {1+(c+d x)^2}-6 a b \text {arcsinh}(c+d x)+2 b (c+d x) \left (8 a (c+d x)^3+3 b \sqrt {1+(c+d x)^2}-2 b (c+d x)^2 \sqrt {1+(c+d x)^2}\right ) \text {arcsinh}(c+d x)+b^2 \left (-3+8 (c+d x)^4\right ) \text {arcsinh}(c+d x)^2\right )}{32 d} \] Input:
Integrate[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^2,x]
Output:
(e^3*(-3*b^2*(c + d*x)^2 + (8*a^2 + b^2)*(c + d*x)^4 + 2*a*b*(c + d*x)*(3 - 2*(c + d*x)^2)*Sqrt[1 + (c + d*x)^2] - 6*a*b*ArcSinh[c + d*x] + 2*b*(c + d*x)*(8*a*(c + d*x)^3 + 3*b*Sqrt[1 + (c + d*x)^2] - 2*b*(c + d*x)^2*Sqrt[ 1 + (c + d*x)^2])*ArcSinh[c + d*x] + b^2*(-3 + 8*(c + d*x)^4)*ArcSinh[c + d*x]^2))/(32*d)
Time = 0.64 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.88, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {6274, 27, 6191, 6227, 15, 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.
\(\displaystyle \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^2 \, dx\) |
\(\Big \downarrow \) 6274 |
\(\displaystyle \frac {\int e^3 (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2d(c+d x)}{d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {e^3 \int (c+d x)^3 (a+b \text {arcsinh}(c+d x))^2d(c+d x)}{d}\) |
\(\Big \downarrow \) 6191 |
\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \int \frac {(c+d x)^4 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)\right )}{d}\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{4} b \int (c+d x)^3d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))\right )\right )}{d}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (-\frac {3}{4} \int \frac {(c+d x)^2 (a+b \text {arcsinh}(c+d x))}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )}{d}\) |
\(\Big \downarrow \) 6227 |
\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (-\frac {3}{4} \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{\sqrt {(c+d x)^2+1}}d(c+d x)-\frac {1}{2} b \int (c+d x)d(c+d x)+\frac {1}{2} (c+d x) \sqrt {(c+d x)^2+1} (a+b \text {arcsinh}(c+d x))\right )+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )}{d}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (-\frac {3}{4} \left (-\frac {1}{2} \int \frac {a+b \text {arcsinh}(c+d x)}{\sqrt {(c+d x)^2+1}}d(c+d x)+\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) (a+b \text {arcsinh}(c+d x))-\frac {1}{4} b (c+d x)^2\right )+\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {1}{16} b (c+d x)^4\right )\right )}{d}\) |
\(\Big \downarrow \) 6198 |
\(\displaystyle \frac {e^3 \left (\frac {1}{4} (c+d x)^4 (a+b \text {arcsinh}(c+d x))^2-\frac {1}{2} b \left (\frac {1}{4} \sqrt {(c+d x)^2+1} (c+d x)^3 (a+b \text {arcsinh}(c+d x))-\frac {3}{4} \left (\frac {1}{2} \sqrt {(c+d x)^2+1} (c+d x) (a+b \text {arcsinh}(c+d x))-\frac {(a+b \text {arcsinh}(c+d x))^2}{4 b}-\frac {1}{4} b (c+d x)^2\right )-\frac {1}{16} b (c+d x)^4\right )\right )}{d}\) |
Input:
Int[(c*e + d*e*x)^3*(a + b*ArcSinh[c + d*x])^2,x]
Output:
(e^3*(((c + d*x)^4*(a + b*ArcSinh[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))/d
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 = 0.74 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.13
method | result | size |
derivativedivides | \(\frac {\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4}+e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )+2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(194\) |
default | \(\frac {\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4}+e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )+2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(194\) |
parts | \(\frac {e^{3} a^{2} \left (d x +c \right )^{4}}{4 d}+\frac {e^{3} b^{2} \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )^{2}}{4}-\frac {\left (d x +c \right )^{3} \operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{8}+\frac {3 \,\operatorname {arcsinh}\left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}\, \left (d x +c \right )}{16}-\frac {3 \operatorname {arcsinh}\left (d x +c \right )^{2}}{32}+\frac {\left (d x +c \right )^{4}}{32}-\frac {3 \left (d x +c \right )^{2}}{32}-\frac {3}{32}\right )}{d}+\frac {2 e^{3} a b \left (\frac {\left (d x +c \right )^{4} \operatorname {arcsinh}\left (d x +c \right )}{4}-\frac {\left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}}{16}+\frac {3 \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}}{32}-\frac {3 \,\operatorname {arcsinh}\left (d x +c \right )}{32}\right )}{d}\) | \(199\) |
orering | \(\frac {\left (37 d^{6} x^{6}+222 c \,d^{5} x^{5}+555 c^{2} d^{4} x^{4}+740 c^{3} d^{3} x^{3}+546 c^{4} d^{2} x^{2}-21 d^{4} x^{4}+204 c^{5} d x -84 c \,d^{3} x^{3}+28 c^{6}-99 c^{2} d^{2} x^{2}-30 c^{3} d x -6 c^{4}-60 d^{2} x^{2}-120 c d x -24 c^{2}\right ) \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2}}{64 d \left (d x +c \right )^{5}}-\frac {\left (9 d^{6} x^{6}+54 c \,d^{5} x^{5}+135 c^{2} d^{4} x^{4}+180 c^{3} d^{3} x^{3}+130 c^{4} d^{2} x^{2}-11 d^{4} x^{4}+44 c^{5} d x -44 c \,d^{3} x^{3}+4 c^{6}-51 c^{2} d^{2} x^{2}-14 c^{3} d x -2 c^{4}-24 d^{2} x^{2}-48 c d x -6 c^{2}\right ) \left (3 \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2} d e +\frac {2 \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right ) b d}{\sqrt {1+\left (d x +c \right )^{2}}}\right )}{64 d^{2} \left (d x +c \right )^{4}}+\frac {x \left (d^{3} x^{3}+4 c \,d^{2} x^{2}+6 c^{2} d x +4 c^{3}-3 d x -6 c \right ) \left (d^{2} x^{2}+2 c d x +c^{2}+1\right ) \left (6 \left (d e x +c e \right ) \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right )^{2} d^{2} e^{2}+\frac {12 \left (d e x +c e \right )^{2} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right ) d^{2} e b}{\sqrt {1+\left (d x +c \right )^{2}}}+\frac {2 \left (d e x +c e \right )^{3} b^{2} d^{2}}{1+\left (d x +c \right )^{2}}-\frac {2 \left (d e x +c e \right )^{3} \left (a +b \,\operatorname {arcsinh}\left (d x +c \right )\right ) b \,d^{2} \left (d x +c \right )}{\left (1+\left (d x +c \right )^{2}\right )^{\frac {3}{2}}}\right )}{64 d^{2} \left (d x +c \right )^{3}}\) | \(556\) |
Input:
int((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(1/4*e^3*a^2*(d*x+c)^4+e^3*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*arcsinh(d*x+c)*(1+(d*x+c)^2 )^(1/2)*(d*x+c)-3/32*arcsinh(d*x+c)^2+1/32*(d*x+c)^4-3/32*(d*x+c)^2-3/32)+ 2*e^3*a*b*(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. 486 vs. \(2 (156) = 312\).
Time = 0.11 (sec) , antiderivative size = 486, normalized size of antiderivative = 2.83 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {{\left (8 \, a^{2} + b^{2}\right )} d^{4} e^{3} x^{4} + 4 \, {\left (8 \, a^{2} + b^{2}\right )} c d^{3} e^{3} x^{3} + 3 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{2} - b^{2}\right )} d^{2} e^{3} x^{2} + 2 \, {\left (2 \, {\left (8 \, a^{2} + b^{2}\right )} c^{3} - 3 \, b^{2} c\right )} d e^{3} x + {\left (8 \, b^{2} d^{4} e^{3} x^{4} + 32 \, b^{2} c d^{3} e^{3} x^{3} + 48 \, b^{2} c^{2} d^{2} e^{3} x^{2} + 32 \, b^{2} c^{3} d e^{3} x + {\left (8 \, b^{2} c^{4} - 3 \, b^{2}\right )} e^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 2 \, {\left (8 \, a b d^{4} e^{3} x^{4} + 32 \, a b c d^{3} e^{3} x^{3} + 48 \, a b c^{2} d^{2} e^{3} x^{2} + 32 \, a b c^{3} d e^{3} x + {\left (8 \, a b c^{4} - 3 \, a b\right )} e^{3} - {\left (2 \, b^{2} d^{3} e^{3} x^{3} + 6 \, b^{2} c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, b^{2} c^{2} - b^{2}\right )} d e^{3} x + {\left (2 \, b^{2} c^{3} - 3 \, b^{2} c\right )} e^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 2 \, {\left (2 \, a b d^{3} e^{3} x^{3} + 6 \, a b c d^{2} e^{3} x^{2} + 3 \, {\left (2 \, a b c^{2} - a b\right )} d e^{3} x + {\left (2 \, a b c^{3} - 3 \, a b c\right )} e^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{32 \, d} \] Input:
integrate((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^2,x, algorithm="fricas")
Output:
1/32*((8*a^2 + b^2)*d^4*e^3*x^4 + 4*(8*a^2 + b^2)*c*d^3*e^3*x^3 + 3*(2*(8* a^2 + b^2)*c^2 - b^2)*d^2*e^3*x^2 + 2*(2*(8*a^2 + b^2)*c^3 - 3*b^2*c)*d*e^ 3*x + (8*b^2*d^4*e^3*x^4 + 32*b^2*c*d^3*e^3*x^3 + 48*b^2*c^2*d^2*e^3*x^2 + 32*b^2*c^3*d*e^3*x + (8*b^2*c^4 - 3*b^2)*e^3)*log(d*x + c + sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))^2 + 2*(8*a*b*d^4*e^3*x^4 + 32*a*b*c*d^3*e^3*x^3 + 48 *a*b*c^2*d^2*e^3*x^2 + 32*a*b*c^3*d*e^3*x + (8*a*b*c^4 - 3*a*b)*e^3 - (2*b ^2*d^3*e^3*x^3 + 6*b^2*c*d^2*e^3*x^2 + 3*(2*b^2*c^2 - b^2)*d*e^3*x + (2*b^ 2*c^3 - 3*b^2*c)*e^3)*sqrt(d^2*x^2 + 2*c*d*x + c^2 + 1))*log(d*x + c + sqr t(d^2*x^2 + 2*c*d*x + c^2 + 1)) - 2*(2*a*b*d^3*e^3*x^3 + 6*a*b*c*d^2*e^3*x ^2 + 3*(2*a*b*c^2 - a*b)*d*e^3*x + (2*a*b*c^3 - 3*a*b*c)*e^3)*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. 916 vs. \(2 (155) = 310\).
Time = 0.49 (sec) , antiderivative size = 916, normalized size of antiderivative = 5.33 \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^2 \, dx =\text {Too large to display} \] Input:
integrate((d*e*x+c*e)**3*(a+b*asinh(d*x+c))**2,x)
Output:
Piecewise((a**2*c**3*e**3*x + 3*a**2*c**2*d*e**3*x**2/2 + a**2*c*d**2*e**3 *x**3 + a**2*d**3*e**3*x**4/4 + a*b*c**4*e**3*asinh(c + d*x)/(2*d) + 2*a*b *c**3*e**3*x*asinh(c + d*x) - a*b*c**3*e**3*sqrt(c**2 + 2*c*d*x + d**2*x** 2 + 1)/(8*d) + 3*a*b*c**2*d*e**3*x**2*asinh(c + d*x) - 3*a*b*c**2*e**3*x*s qrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/8 + 2*a*b*c*d**2*e**3*x**3*asinh(c + d *x) - 3*a*b*c*d*e**3*x**2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/8 + 3*a*b*c *e**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/(16*d) + a*b*d**3*e**3*x**4*asi nh(c + d*x)/2 - a*b*d**2*e**3*x**3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/8 + 3*a*b*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)/16 - 3*a*b*e**3*asinh( c + d*x)/(16*d) + b**2*c**4*e**3*asinh(c + d*x)**2/(4*d) + b**2*c**3*e**3* x*asinh(c + d*x)**2 + b**2*c**3*e**3*x/8 - b**2*c**3*e**3*sqrt(c**2 + 2*c* d*x + d**2*x**2 + 1)*asinh(c + d*x)/(8*d) + 3*b**2*c**2*d*e**3*x**2*asinh( c + d*x)**2/2 + 3*b**2*c**2*d*e**3*x**2/16 - 3*b**2*c**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/8 + b**2*c*d**2*e**3*x**3*asinh( c + d*x)**2 + b**2*c*d**2*e**3*x**3/8 - 3*b**2*c*d*e**3*x**2*sqrt(c**2 + 2 *c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/8 - 3*b**2*c*e**3*x/16 + 3*b**2*c*e **3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/(16*d) + b**2*d**3 *e**3*x**4*asinh(c + d*x)**2/4 + b**2*d**3*e**3*x**4/32 - b**2*d**2*e**3*x **3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)/8 - 3*b**2*d*e**3* x**2/32 + 3*b**2*e**3*x*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + ...
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^2,x, algorithm="maxima")
Output:
1/4*a^2*d^3*e^3*x^4 + a^2*c*d^2*e^3*x^3 + 3/2*a^2*c^2*d*e^3*x^2 + 3/2*(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)/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*b*c^2*d*e^3 + 1/3*(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*b*c*d^2*e^3 + 1/48*(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)/sq rt(-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*b*d^3*e^3 + a^2 *c^3*e^3*x + 2*((d*x + c)*arcsinh(d*x + c) - sqrt((d*x + c)^2 + 1))*a*b*c^ 3*e^3/d + 1/4*(b^2*d^3*e^3*x^4 + 4*b^2*c*d^2*e^3*x^3 + 6*b^2*c^2*d*e^3*...
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int { {\left (d e x + c e\right )}^{3} {\left (b \operatorname {arsinh}\left (d x + c\right ) + a\right )}^{2} \,d x } \] Input:
integrate((d*e*x+c*e)^3*(a+b*arcsinh(d*x+c))^2,x, algorithm="giac")
Output:
integrate((d*e*x + c*e)^3*(b*arcsinh(d*x + c) + a)^2, x)
Timed out. \[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^2 \, dx=\int {\left (c\,e+d\,e\,x\right )}^3\,{\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \] Input:
int((c*e + d*e*x)^3*(a + b*asinh(c + d*x))^2,x)
Output:
int((c*e + d*e*x)^3*(a + b*asinh(c + d*x))^2, x)
\[ \int (c e+d e x)^3 (a+b \text {arcsinh}(c+d x))^2 \, dx=\frac {e^{3} \left (16 \left (\int \mathit {asinh} \left (d x +c \right )^{2} x^{3}d x \right ) b^{2} d^{4}-8 \mathit {asinh} \left (d x +c \right )^{2} b^{2} c^{4}+12 \mathit {asinh} \left (d x +c \right )^{2} b^{2} c^{2}-3 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}+c +d x \right ) a b +4 a^{2} d^{4} x^{4}+48 \left (\int \mathit {asinh} \left (d x +c \right )^{2} x^{2}d x \right ) b^{2} c \,d^{3}+16 \mathit {asinh} \left (d x +c \right )^{2} b^{2} c^{3} d x +24 \mathit {asinh} \left (d x +c \right )^{2} b^{2} c^{2} d^{2} x^{2}+8 \mathit {asinh} \left (d x +c \right ) a b \,d^{4} x^{4}-2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, a b \,d^{3} x^{3}+3 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, a b d x -52 b^{2} c^{4}+12 b^{2} c^{2}+40 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, \mathit {asinh} \left (d x +c \right ) b^{2} c^{3}+32 \mathit {asinh} \left (d x +c \right ) a b \,c^{4}-2 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, a b \,c^{3}+3 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, a b c -24 \,\mathrm {log}\left (\sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}+c +d x \right ) a b \,c^{4}+16 a^{2} c^{3} d x +24 a^{2} c^{2} d^{2} x^{2}+16 a^{2} c \,d^{3} x^{3}-40 b^{2} c^{3} d x +12 b^{2} c^{2} d^{2} x^{2}-24 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, \mathit {asinh} \left (d x +c \right ) b^{2} c^{2} d x +32 \mathit {asinh} \left (d x +c \right ) a b \,c^{3} d x +48 \mathit {asinh} \left (d x +c \right ) a b \,c^{2} d^{2} x^{2}+32 \mathit {asinh} \left (d x +c \right ) a b c \,d^{3} x^{3}-6 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, a b \,c^{2} d x -6 \sqrt {d^{2} x^{2}+2 c d x +c^{2}+1}\, a b c \,d^{2} x^{2}\right )}{16 d} \] Input:
int((d*e*x+c*e)^3*(a+b*asinh(d*x+c))^2,x)
Output:
(e**3*( - 8*asinh(c + d*x)**2*b**2*c**4 + 16*asinh(c + d*x)**2*b**2*c**3*d *x + 24*asinh(c + d*x)**2*b**2*c**2*d**2*x**2 + 12*asinh(c + d*x)**2*b**2* c**2 + 40*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)*b**2*c**3 - 24*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*asinh(c + d*x)*b**2*c**2*d*x + 32* asinh(c + d*x)*a*b*c**4 + 32*asinh(c + d*x)*a*b*c**3*d*x + 48*asinh(c + d* x)*a*b*c**2*d**2*x**2 + 32*asinh(c + d*x)*a*b*c*d**3*x**3 + 8*asinh(c + d* x)*a*b*d**4*x**4 - 2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b*c**3 - 6*sqr t(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b*c**2*d*x - 6*sqrt(c**2 + 2*c*d*x + d **2*x**2 + 1)*a*b*c*d**2*x**2 + 3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b *c - 2*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b*d**3*x**3 + 3*sqrt(c**2 + 2*c*d*x + d**2*x**2 + 1)*a*b*d*x + 16*int(asinh(c + d*x)**2*x**3,x)*b**2*d **4 + 48*int(asinh(c + d*x)**2*x**2,x)*b**2*c*d**3 - 24*log(sqrt(c**2 + 2* c*d*x + d**2*x**2 + 1) + c + d*x)*a*b*c**4 - 3*log(sqrt(c**2 + 2*c*d*x + d **2*x**2 + 1) + c + d*x)*a*b + 16*a**2*c**3*d*x + 24*a**2*c**2*d**2*x**2 + 16*a**2*c*d**3*x**3 + 4*a**2*d**4*x**4 - 52*b**2*c**4 - 40*b**2*c**3*d*x + 12*b**2*c**2*d**2*x**2 + 12*b**2*c**2))/(16*d)