\(\int (1+a^2+2 a b x+b^2 x^2)^{3/2} \text {arcsinh}(a+b x)^3 \, dx\) [266]

Optimal result

Integrand size = 30, antiderivative size = 235 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=-\frac {51 (a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}+\frac {45 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{64 b}+\frac {3 (a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)}{32 b}-\frac {27 \text {arcsinh}(a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \text {arcsinh}(a+b x)^2}{16 b}-\frac {3 \left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^3}{4 b}+\frac {3 \text {arcsinh}(a+b x)^4}{32 b} \]

[Out]

-51/128*(b*x+a)^2/b-3/128*(b*x+a)^4/b+3/32*(b*x+a)*(1+(b*x+a)^2)^(3/2)*arcsinh(b*x+a)/b-27/128*arcsinh(b*x+a)^
2/b-9/16*(b*x+a)^2*arcsinh(b*x+a)^2/b-3/16*(1+(b*x+a)^2)^2*arcsinh(b*x+a)^2/b+1/4*(b*x+a)*(1+(b*x+a)^2)^(3/2)*
arcsinh(b*x+a)^3/b+3/32*arcsinh(b*x+a)^4/b+45/64*(b*x+a)*arcsinh(b*x+a)*(1+(b*x+a)^2)^(1/2)/b+3/8*(b*x+a)*arcs
inh(b*x+a)^3*(1+(b*x+a)^2)^(1/2)/b

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5860, 5786, 5785, 5783, 5776, 5812, 30, 5798, 14} \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=-\frac {9 (a+b x)^2 \text {arcsinh}(a+b x)^2}{16 b}+\frac {\left ((a+b x)^2+1\right )^{3/2} (a+b x) \text {arcsinh}(a+b x)^3}{4 b}+\frac {3 \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)^3}{8 b}+\frac {3 \left ((a+b x)^2+1\right )^{3/2} (a+b x) \text {arcsinh}(a+b x)}{32 b}+\frac {45 \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)}{64 b}+\frac {3 \text {arcsinh}(a+b x)^4}{32 b}-\frac {3 \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)^2}{16 b}-\frac {27 \text {arcsinh}(a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}-\frac {51 (a+b x)^2}{128 b} \]

[In]

Int[(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]^3,x]

[Out]

(-51*(a + b*x)^2)/(128*b) - (3*(a + b*x)^4)/(128*b) + (45*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x])/(6
4*b) + (3*(a + b*x)*(1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x])/(32*b) - (27*ArcSinh[a + b*x]^2)/(128*b) - (9*(a
+ b*x)^2*ArcSinh[a + b*x]^2)/(16*b) - (3*(1 + (a + b*x)^2)^2*ArcSinh[a + b*x]^2)/(16*b) + (3*(a + b*x)*Sqrt[1
+ (a + b*x)^2]*ArcSinh[a + b*x]^3)/(8*b) + ((a + b*x)*(1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x]^3)/(4*b) + (3*
ArcSinh[a + b*x]^4)/(32*b)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 5776

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcS
inh[c*x])^n/(d*(m + 1))), x] - Dist[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]

Rule 5783

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(1/(b*c*(n + 1)))*S
imp[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]

Rule 5785

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

Rule 5812

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] + (-Dist[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] - Dist[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
]

Rule 5860

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)^(p_.), x_Symbol] :> D
ist[1/d, Subst[Int[(C/d^2 + (C/d^2)*x^2)^p*(a + b*ArcSinh[x])^n, x], x, c + d*x], x] /; FreeQ[{a, b, c, d, A,
B, C, n, p}, x] && EqQ[B*(1 + c^2) - 2*A*c*d, 0] && EqQ[2*c*C - B*d, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (1+x^2\right )^{3/2} \text {arcsinh}(x)^3 \, dx,x,a+b x\right )}{b} \\ & = \frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^3}{4 b}-\frac {3 \text {Subst}\left (\int x \left (1+x^2\right ) \text {arcsinh}(x)^2 \, dx,x,a+b x\right )}{4 b}+\frac {3 \text {Subst}\left (\int \sqrt {1+x^2} \text {arcsinh}(x)^3 \, dx,x,a+b x\right )}{4 b} \\ & = -\frac {3 \left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^3}{4 b}+\frac {3 \text {Subst}\left (\int \left (1+x^2\right )^{3/2} \text {arcsinh}(x) \, dx,x,a+b x\right )}{8 b}+\frac {3 \text {Subst}\left (\int \frac {\text {arcsinh}(x)^3}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 b}-\frac {9 \text {Subst}\left (\int x \text {arcsinh}(x)^2 \, dx,x,a+b x\right )}{8 b} \\ & = \frac {3 (a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)}{32 b}-\frac {9 (a+b x)^2 \text {arcsinh}(a+b x)^2}{16 b}-\frac {3 \left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^3}{4 b}+\frac {3 \text {arcsinh}(a+b x)^4}{32 b}-\frac {3 \text {Subst}\left (\int x \left (1+x^2\right ) \, dx,x,a+b x\right )}{32 b}+\frac {9 \text {Subst}\left (\int \sqrt {1+x^2} \text {arcsinh}(x) \, dx,x,a+b x\right )}{32 b}+\frac {9 \text {Subst}\left (\int \frac {x^2 \text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 b} \\ & = \frac {45 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{64 b}+\frac {3 (a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)}{32 b}-\frac {9 (a+b x)^2 \text {arcsinh}(a+b x)^2}{16 b}-\frac {3 \left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^3}{4 b}+\frac {3 \text {arcsinh}(a+b x)^4}{32 b}-\frac {3 \text {Subst}\left (\int \left (x+x^3\right ) \, dx,x,a+b x\right )}{32 b}-\frac {9 \text {Subst}(\int x \, dx,x,a+b x)}{64 b}+\frac {9 \text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{64 b}-\frac {9 \text {Subst}(\int x \, dx,x,a+b x)}{16 b}-\frac {9 \text {Subst}\left (\int \frac {\text {arcsinh}(x)}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{16 b} \\ & = -\frac {51 (a+b x)^2}{128 b}-\frac {3 (a+b x)^4}{128 b}+\frac {45 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)}{64 b}+\frac {3 (a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)}{32 b}-\frac {27 \text {arcsinh}(a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \text {arcsinh}(a+b x)^2}{16 b}-\frac {3 \left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)^2}{16 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^3}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^3}{4 b}+\frac {3 \text {arcsinh}(a+b x)^4}{32 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.13 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=-\frac {6 a \left (17+2 a^2\right ) b x+3 \left (17+6 a^2\right ) b^2 x^2+12 a b^3 x^3+3 b^4 x^4-6 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (17 a+2 a^3+17 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \text {arcsinh}(a+b x)+3 \left (17+8 a^4+32 a^3 b x+40 b^2 x^2+8 b^4 x^4+16 a b x \left (5+2 b^2 x^2\right )+8 a^2 \left (5+6 b^2 x^2\right )\right ) \text {arcsinh}(a+b x)^2-16 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (5 a+2 a^3+5 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \text {arcsinh}(a+b x)^3-12 \text {arcsinh}(a+b x)^4}{128 b} \]

[In]

Integrate[(1 + a^2 + 2*a*b*x + b^2*x^2)^(3/2)*ArcSinh[a + b*x]^3,x]

[Out]

-1/128*(6*a*(17 + 2*a^2)*b*x + 3*(17 + 6*a^2)*b^2*x^2 + 12*a*b^3*x^3 + 3*b^4*x^4 - 6*Sqrt[1 + a^2 + 2*a*b*x +
b^2*x^2]*(17*a + 2*a^3 + 17*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSinh[a + b*x] + 3*(17 + 8*a^4 + 32*a
^3*b*x + 40*b^2*x^2 + 8*b^4*x^4 + 16*a*b*x*(5 + 2*b^2*x^2) + 8*a^2*(5 + 6*b^2*x^2))*ArcSinh[a + b*x]^2 - 16*Sq
rt[1 + a^2 + 2*a*b*x + b^2*x^2]*(5*a + 2*a^3 + 5*b*x + 6*a^2*b*x + 6*a*b^2*x^2 + 2*b^3*x^3)*ArcSinh[a + b*x]^3
- 12*ArcSinh[a + b*x]^4)/b

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(591\) vs. \(2(207)=414\).

Time = 0.73 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.52

method	result	size
default	\(\frac {-48-102 a b x -51 a^{2}-51 b^{2} x^{2}+12 \operatorname {arcsinh}\left (b x +a \right )^{4}-51 \operatorname {arcsinh}\left (b x +a \right )^{2}-12 a \,b^{3} x^{3}-3 b^{4} x^{4}+96 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a \,b^{2} x^{2}+96 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b x +36 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a \,b^{2} x^{2}+36 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b x -3 a^{4}+12 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{3} x^{3}+32 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{3} x^{3}-96 \operatorname {arcsinh}\left (b x +a \right )^{2} a \,b^{3} x^{3}-144 \operatorname {arcsinh}\left (b x +a \right )^{2} a^{2} b^{2} x^{2}-96 \operatorname {arcsinh}\left (b x +a \right )^{2} a^{3} b x +80 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x -240 \operatorname {arcsinh}\left (b x +a \right )^{2} a b x +102 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x -24 a^{4} \operatorname {arcsinh}\left (b x +a \right )^{2}-120 a^{2} \operatorname {arcsinh}\left (b x +a \right )^{2}-120 \operatorname {arcsinh}\left (b x +a \right )^{2} b^{2} x^{2}+80 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a +102 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a +12 \,\operatorname {arcsinh}\left (b x +a \right ) \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}+32 \operatorname {arcsinh}\left (b x +a \right )^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}-24 \operatorname {arcsinh}\left (b x +a \right )^{2} b^{4} x^{4}-18 a^{2} b^{2} x^{2}-12 a^{3} b x}{128 b}\)	\(592\)

[In]

int((b^2*x^2+2*a*b*x+a^2+1)^(3/2)*arcsinh(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/128*(-48-102*a*b*x-51*a^2-51*b^2*x^2+12*arcsinh(b*x+a)^4-51*arcsinh(b*x+a)^2-12*a*b^3*x^3-3*b^4*x^4+96*arcsi
nh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a*b^2*x^2+96*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^2*b*x+
36*arcsinh(b*x+a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a*b^2*x^2+36*arcsinh(b*x+a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^2*
b*x-3*a^4+12*arcsinh(b*x+a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*b^3*x^3+32*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^
(1/2)*b^3*x^3-96*arcsinh(b*x+a)^2*a*b^3*x^3-144*arcsinh(b*x+a)^2*a^2*b^2*x^2-96*arcsinh(b*x+a)^2*a^3*b*x+80*ar
csinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*b*x-240*arcsinh(b*x+a)^2*a*b*x+102*arcsinh(b*x+a)*(b^2*x^2+2*a*b*
x+a^2+1)^(1/2)*b*x-24*a^4*arcsinh(b*x+a)^2-120*a^2*arcsinh(b*x+a)^2-120*arcsinh(b*x+a)^2*b^2*x^2+80*arcsinh(b*
x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a+102*arcsinh(b*x+a)*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a+12*arcsinh(b*x+a)*(b
^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^3+32*arcsinh(b*x+a)^3*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^3-24*arcsinh(b*x+a)^2*b^4*
x^4-18*a^2*b^2*x^2-12*a^3*b*x)/b

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.41 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=-\frac {3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} + 3 \, {\left (6 \, a^{2} + 17\right )} b^{2} x^{2} - 16 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 5\right )} b x + 5 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 12 \, \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{4} + 6 \, {\left (2 \, a^{3} + 17 \, a\right )} b x + 3 \, {\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \, {\left (6 \, a^{2} + 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \, {\left (2 \, a^{3} + 5 \, a\right )} b x + 40 \, a^{2} + 17\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} - 6 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 17\right )} b x + 17 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{128 \, b} \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2+1)^(3/2)*arcsinh(b*x+a)^3,x, algorithm="fricas")

[Out]

-1/128*(3*b^4*x^4 + 12*a*b^3*x^3 + 3*(6*a^2 + 17)*b^2*x^2 - 16*(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 + 5)*
b*x + 5*a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - 12*log(b*x +
a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^4 + 6*(2*a^3 + 17*a)*b*x + 3*(8*b^4*x^4 + 32*a*b^3*x^3 + 8*(6*a^2 + 5)
*b^2*x^2 + 8*a^4 + 16*(2*a^3 + 5*a)*b*x + 40*a^2 + 17)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^2 - 6*
(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2 + 17)*b*x + 17*a)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1)*log(b*x + a + sq
rt(b^2*x^2 + 2*a*b*x + a^2 + 1)))/b

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (223) = 446\).

Time = 1.15 (sec) , antiderivative size = 694, normalized size of antiderivative = 2.95 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=\begin {cases} - \frac {3 a^{4} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16 b} - \frac {3 a^{3} x \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} - \frac {3 a^{3} x}{32} + \frac {a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4 b} + \frac {3 a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32 b} - \frac {9 a^{2} b x^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{8} - \frac {9 a^{2} b x^{2}}{64} + \frac {3 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4} + \frac {9 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32} - \frac {15 a^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16 b} - \frac {3 a b^{2} x^{3} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} - \frac {3 a b^{2} x^{3}}{32} + \frac {3 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4} + \frac {9 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32} - \frac {15 a x \operatorname {asinh}^{2}{\left (a + b x \right )}}{8} - \frac {51 a x}{64} + \frac {5 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{8 b} + \frac {51 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{64 b} - \frac {3 b^{3} x^{4} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16} - \frac {3 b^{3} x^{4}}{128} + \frac {b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{4} + \frac {3 b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{32} - \frac {15 b x^{2} \operatorname {asinh}^{2}{\left (a + b x \right )}}{16} - \frac {51 b x^{2}}{128} + \frac {5 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (a + b x \right )}}{8} + \frac {51 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}{\left (a + b x \right )}}{64} + \frac {3 \operatorname {asinh}^{4}{\left (a + b x \right )}}{32 b} - \frac {51 \operatorname {asinh}^{2}{\left (a + b x \right )}}{128 b} & \text {for}\: b \neq 0 \\x \left (a^{2} + 1\right )^{\frac {3}{2}} \operatorname {asinh}^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate((b**2*x**2+2*a*b*x+a**2+1)**(3/2)*asinh(b*x+a)**3,x)

[Out]

Piecewise((-3*a**4*asinh(a + b*x)**2/(16*b) - 3*a**3*x*asinh(a + b*x)**2/4 - 3*a**3*x/32 + a**3*sqrt(a**2 + 2*
a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**3/(4*b) + 3*a**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(3
2*b) - 9*a**2*b*x**2*asinh(a + b*x)**2/8 - 9*a**2*b*x**2/64 + 3*a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*as
inh(a + b*x)**3/4 + 9*a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/32 - 15*a**2*asinh(a + b*x)**
2/(16*b) - 3*a*b**2*x**3*asinh(a + b*x)**2/4 - 3*a*b**2*x**3/32 + 3*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 +
1)*asinh(a + b*x)**3/4 + 9*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/32 - 15*a*x*asinh(a +
b*x)**2/8 - 51*a*x/64 + 5*a*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**3/(8*b) + 51*a*sqrt(a**2 + 2
*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/(64*b) - 3*b**3*x**4*asinh(a + b*x)**2/16 - 3*b**3*x**4/128 + b**2*x**3
*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**3/4 + 3*b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a
sinh(a + b*x)/32 - 15*b*x**2*asinh(a + b*x)**2/16 - 51*b*x**2/128 + 5*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*a
sinh(a + b*x)**3/8 + 51*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)/64 + 3*asinh(a + b*x)**4/(32*b)
- 51*asinh(a + b*x)**2/(128*b), Ne(b, 0)), (x*(a**2 + 1)**(3/2)*asinh(a)**3, True))

Maxima [F]

\[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=\int { {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2+1)^(3/2)*arcsinh(b*x+a)^3,x, algorithm="maxima")

[Out]

integrate((b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*arcsinh(b*x + a)^3, x)

Giac [F]

\[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=\int { {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{3} \,d x } \]

[In]

integrate((b^2*x^2+2*a*b*x+a^2+1)^(3/2)*arcsinh(b*x+a)^3,x, algorithm="giac")

[Out]

integrate((b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2)*arcsinh(b*x + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^3 \, dx=\int {\mathrm {asinh}\left (a+b\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2} \,d x \]

[In]

int(asinh(a + b*x)^3*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(3/2),x)

[Out]

int(asinh(a + b*x)^3*(a^2 + b^2*x^2 + 2*a*b*x + 1)^(3/2), x)