\(\int (d+c^2 d x^2)^2 (a+b \text {arcsinh}(c x))^2 \, dx\) [211]

Optimal result

Integrand size = 23, antiderivative size = 214 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {298}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3+\frac {2}{125} b^2 c^4 d^2 x^5-\frac {16 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{15 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \]

[Out]

298/225*b^2*d^2*x+76/675*b^2*c^2*d^2*x^3+2/125*b^2*c^4*d^2*x^5-8/45*b*d^2*(c^2*x^2+1)^(3/2)*(a+b*arcsinh(c*x))
/c-2/25*b*d^2*(c^2*x^2+1)^(5/2)*(a+b*arcsinh(c*x))/c+8/15*d^2*x*(a+b*arcsinh(c*x))^2+4/15*d^2*x*(c^2*x^2+1)*(a
+b*arcsinh(c*x))^2+1/5*d^2*x*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))^2-16/15*b*d^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/
2)/c

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {5786, 5772, 5798, 8, 200} \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{5} d^2 x \left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2-\frac {2 b d^2 \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}-\frac {8 b d^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {16 b d^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{15 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {2}{125} b^2 c^4 d^2 x^5+\frac {76}{675} b^2 c^2 d^2 x^3+\frac {298}{225} b^2 d^2 x \]

[In]

Int[(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]

[Out]

(298*b^2*d^2*x)/225 + (76*b^2*c^2*d^2*x^3)/675 + (2*b^2*c^4*d^2*x^5)/125 - (16*b*d^2*Sqrt[1 + c^2*x^2]*(a + b*
ArcSinh[c*x]))/(15*c) - (8*b*d^2*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(45*c) - (2*b*d^2*(1 + c^2*x^2)^(5/
2)*(a + b*ArcSinh[c*x]))/(25*c) + (8*d^2*x*(a + b*ArcSinh[c*x])^2)/15 + (4*d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[
c*x])^2)/15 + (d^2*x*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x])^2)/5

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

Rule 5772

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcSinh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcSinh[c*x])^(n - 1)/Sqrt[1 + c^2*x^2]), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5786

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[x*(d + e*x^2)^p*(
(a + b*ArcSinh[c*x])^n/(2*p + 1)), x] + (Dist[2*d*(p/(2*p + 1)), Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^
n, x], x] - Dist[b*c*(n/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*A
rcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]

Rule 5798

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] - Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{5} (4 d) \int \left (d+c^2 d x^2\right ) (a+b \text {arcsinh}(c x))^2 \, dx-\frac {1}{5} \left (2 b c d^2\right ) \int x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{15} \left (8 d^2\right ) \int (a+b \text {arcsinh}(c x))^2 \, dx+\frac {1}{25} \left (2 b^2 d^2\right ) \int \left (1+c^2 x^2\right )^2 \, dx-\frac {1}{15} \left (8 b c d^2\right ) \int x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx \\ & = -\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{25} \left (2 b^2 d^2\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx+\frac {1}{45} \left (8 b^2 d^2\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac {1}{15} \left (16 b c d^2\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx \\ & = \frac {58}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3+\frac {2}{125} b^2 c^4 d^2 x^5-\frac {16 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{15 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2+\frac {1}{15} \left (16 b^2 d^2\right ) \int 1 \, dx \\ & = \frac {298}{225} b^2 d^2 x+\frac {76}{675} b^2 c^2 d^2 x^3+\frac {2}{125} b^2 c^4 d^2 x^5-\frac {16 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{15 c}-\frac {8 b d^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))}{45 c}-\frac {2 b d^2 \left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{25 c}+\frac {8}{15} d^2 x (a+b \text {arcsinh}(c x))^2+\frac {4}{15} d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))^2+\frac {1}{5} d^2 x \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.89 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {d^2 \left (225 a^2 c x \left (15+10 c^2 x^2+3 c^4 x^4\right )-30 a b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )+2 b^2 c x \left (2235+190 c^2 x^2+27 c^4 x^4\right )-30 b \left (-15 a c x \left (15+10 c^2 x^2+3 c^4 x^4\right )+b \sqrt {1+c^2 x^2} \left (149+38 c^2 x^2+9 c^4 x^4\right )\right ) \text {arcsinh}(c x)+225 b^2 c x \left (15+10 c^2 x^2+3 c^4 x^4\right ) \text {arcsinh}(c x)^2\right )}{3375 c} \]

[In]

Integrate[(d + c^2*d*x^2)^2*(a + b*ArcSinh[c*x])^2,x]

[Out]

(d^2*(225*a^2*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) - 30*a*b*Sqrt[1 + c^2*x^2]*(149 + 38*c^2*x^2 + 9*c^4*x^4) + 2*
b^2*c*x*(2235 + 190*c^2*x^2 + 27*c^4*x^4) - 30*b*(-15*a*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) + b*Sqrt[1 + c^2*x^2
]*(149 + 38*c^2*x^2 + 9*c^4*x^4))*ArcSinh[c*x] + 225*b^2*c*x*(15 + 10*c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]^2))/(3
375*c)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.23

method	result	size
derivativedivides	\(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 c x \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\)	\(264\)
default	\(\frac {d^{2} a^{2} \left (\frac {1}{5} c^{5} x^{5}+\frac {2}{3} c^{3} x^{3}+c x \right )+d^{2} b^{2} \left (\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 c x \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )+2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\)	\(264\)
parts	\(d^{2} a^{2} \left (\frac {1}{5} c^{4} x^{5}+\frac {2}{3} x^{3} c^{2}+x \right )+\frac {d^{2} b^{2} \left (\frac {8 \operatorname {arcsinh}\left (c x \right )^{2} x c}{15}+\frac {\operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )^{2}}{5}+\frac {4 \operatorname {arcsinh}\left (c x \right )^{2} c x \left (c^{2} x^{2}+1\right )}{15}-\frac {16 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}}{15}+\frac {4144 c x}{3375}-\frac {2 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{25}+\frac {2 c x \left (c^{2} x^{2}+1\right )^{2}}{125}+\frac {272 c x \left (c^{2} x^{2}+1\right )}{3375}-\frac {8 \,\operatorname {arcsinh}\left (c x \right ) \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{45}\right )}{c}+\frac {2 d^{2} a b \left (\frac {\operatorname {arcsinh}\left (c x \right ) c^{5} x^{5}}{5}+\frac {2 \,\operatorname {arcsinh}\left (c x \right ) c^{3} x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x -\frac {149 \sqrt {c^{2} x^{2}+1}}{225}-\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{25}-\frac {38 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{225}\right )}{c}\)	\(264\)

[In]

int((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x,method=_RETURNVERBOSE)

[Out]

1/c*(d^2*a^2*(1/5*c^5*x^5+2/3*c^3*x^3+c*x)+d^2*b^2*(8/15*arcsinh(c*x)^2*x*c+1/5*arcsinh(c*x)^2*c*x*(c^2*x^2+1)
^2+4/15*arcsinh(c*x)^2*c*x*(c^2*x^2+1)-16/15*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+4144/3375*c*x-2/25*arcsinh(c*x)*(c
^2*x^2+1)^(5/2)+2/125*c*x*(c^2*x^2+1)^2+272/3375*c*x*(c^2*x^2+1)-8/45*arcsinh(c*x)*(c^2*x^2+1)^(3/2))+2*d^2*a*
b*(1/5*arcsinh(c*x)*c^5*x^5+2/3*arcsinh(c*x)*c^3*x^3+arcsinh(c*x)*c*x-149/225*(c^2*x^2+1)^(1/2)-1/25*c^4*x^4*(
c^2*x^2+1)^(1/2)-38/225*c^2*x^2*(c^2*x^2+1)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.30 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} d^{2} x^{5} + 10 \, {\left (225 \, a^{2} + 38 \, b^{2}\right )} c^{3} d^{2} x^{3} + 15 \, {\left (225 \, a^{2} + 298 \, b^{2}\right )} c d^{2} x + 225 \, {\left (3 \, b^{2} c^{5} d^{2} x^{5} + 10 \, b^{2} c^{3} d^{2} x^{3} + 15 \, b^{2} c d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (45 \, a b c^{5} d^{2} x^{5} + 150 \, a b c^{3} d^{2} x^{3} + 225 \, a b c d^{2} x - {\left (9 \, b^{2} c^{4} d^{2} x^{4} + 38 \, b^{2} c^{2} d^{2} x^{2} + 149 \, b^{2} d^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 30 \, {\left (9 \, a b c^{4} d^{2} x^{4} + 38 \, a b c^{2} d^{2} x^{2} + 149 \, a b d^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{3375 \, c} \]

[In]

integrate((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

1/3375*(27*(25*a^2 + 2*b^2)*c^5*d^2*x^5 + 10*(225*a^2 + 38*b^2)*c^3*d^2*x^3 + 15*(225*a^2 + 298*b^2)*c*d^2*x +
225*(3*b^2*c^5*d^2*x^5 + 10*b^2*c^3*d^2*x^3 + 15*b^2*c*d^2*x)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 30*(45*a*b*c^5
*d^2*x^5 + 150*a*b*c^3*d^2*x^3 + 225*a*b*c*d^2*x - (9*b^2*c^4*d^2*x^4 + 38*b^2*c^2*d^2*x^2 + 149*b^2*d^2)*sqrt
(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 30*(9*a*b*c^4*d^2*x^4 + 38*a*b*c^2*d^2*x^2 + 149*a*b*d^2)*sqrt(c
^2*x^2 + 1))/c

Sympy [A] (verification not implemented)

Time = 0.46 (sec) , antiderivative size = 389, normalized size of antiderivative = 1.82 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} \frac {a^{2} c^{4} d^{2} x^{5}}{5} + \frac {2 a^{2} c^{2} d^{2} x^{3}}{3} + a^{2} d^{2} x + \frac {2 a b c^{4} d^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2 a b c^{3} d^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25} + \frac {4 a b c^{2} d^{2} x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {76 a b c d^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{225} + 2 a b d^{2} x \operatorname {asinh}{\left (c x \right )} - \frac {298 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{225 c} + \frac {b^{2} c^{4} d^{2} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {2 b^{2} c^{4} d^{2} x^{5}}{125} - \frac {2 b^{2} c^{3} d^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25} + \frac {2 b^{2} c^{2} d^{2} x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {76 b^{2} c^{2} d^{2} x^{3}}{675} - \frac {76 b^{2} c d^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{225} + b^{2} d^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + \frac {298 b^{2} d^{2} x}{225} - \frac {298 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{225 c} & \text {for}\: c \neq 0 \\a^{2} d^{2} x & \text {otherwise} \end {cases} \]

[In]

integrate((c**2*d*x**2+d)**2*(a+b*asinh(c*x))**2,x)

[Out]

Piecewise((a**2*c**4*d**2*x**5/5 + 2*a**2*c**2*d**2*x**3/3 + a**2*d**2*x + 2*a*b*c**4*d**2*x**5*asinh(c*x)/5 -
2*a*b*c**3*d**2*x**4*sqrt(c**2*x**2 + 1)/25 + 4*a*b*c**2*d**2*x**3*asinh(c*x)/3 - 76*a*b*c*d**2*x**2*sqrt(c**
2*x**2 + 1)/225 + 2*a*b*d**2*x*asinh(c*x) - 298*a*b*d**2*sqrt(c**2*x**2 + 1)/(225*c) + b**2*c**4*d**2*x**5*asi
nh(c*x)**2/5 + 2*b**2*c**4*d**2*x**5/125 - 2*b**2*c**3*d**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/25 + 2*b**2*c*
*2*d**2*x**3*asinh(c*x)**2/3 + 76*b**2*c**2*d**2*x**3/675 - 76*b**2*c*d**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)
/225 + b**2*d**2*x*asinh(c*x)**2 + 298*b**2*d**2*x/225 - 298*b**2*d**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(225*c),
Ne(c, 0)), (a**2*d**2*x, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (190) = 380\).

Time = 0.23 (sec) , antiderivative size = 457, normalized size of antiderivative = 2.14 \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{5} \, b^{2} c^{4} d^{2} x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} c^{4} d^{2} x^{5} + \frac {2}{3} \, b^{2} c^{2} d^{2} x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b c^{4} d^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} c^{4} d^{2} + \frac {2}{3} \, a^{2} c^{2} d^{2} x^{3} + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d^{2} - \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} c^{2} d^{2} + b^{2} d^{2} x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \]

[In]

integrate((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

1/5*b^2*c^4*d^2*x^5*arcsinh(c*x)^2 + 1/5*a^2*c^4*d^2*x^5 + 2/3*b^2*c^2*d^2*x^3*arcsinh(c*x)^2 + 2/75*(15*x^5*a
rcsinh(c*x) - (3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c)*a*b*c^4
*d^2 - 2/1125*(15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcs
inh(c*x) - (9*c^4*x^5 - 20*c^2*x^3 + 120*x)/c^4)*b^2*c^4*d^2 + 2/3*a^2*c^2*d^2*x^3 + 4/9*(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*c^2*d^2 - 4/27*(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*c^2*d^2 + b^2*d^2*x*arcsinh(c*x)^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

Giac [F(-2)]

Exception generated. \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((c^2*d*x^2+d)^2*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \left (d+c^2 d x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \]

[In]

int((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2,x)

[Out]

int((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^2, x)