\(\int (1+a^2+2 a b x+b^2 x^2)^{3/2} \text {arcsinh}(a+b x)^2 \, dx\) [267]

Optimal result

Integrand size = 30, antiderivative size = 189 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2 \, dx=\frac {15 (a+b x) \sqrt {1+(a+b x)^2}}{64 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2}}{32 b}-\frac {9 \text {arcsinh}(a+b x)}{64 b}-\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{8 b}-\frac {\left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^2}{4 b}+\frac {\text {arcsinh}(a+b x)^3}{8 b} \]

[Out]

1/32*(b*x+a)*(1+(b*x+a)^2)^(3/2)/b-9/64*arcsinh(b*x+a)/b-3/8*(b*x+a)^2*arcsinh(b*x+a)/b-1/8*(1+(b*x+a)^2)^2*ar
csinh(b*x+a)/b+1/4*(b*x+a)*(1+(b*x+a)^2)^(3/2)*arcsinh(b*x+a)^2/b+1/8*arcsinh(b*x+a)^3/b+15/64*(b*x+a)*(1+(b*x
+a)^2)^(1/2)/b+3/8*(b*x+a)*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)/b

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5860, 5786, 5785, 5783, 5776, 327, 221, 5798, 201} \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2 \, dx=\frac {\text {arcsinh}(a+b x)^3}{8 b}+\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2}{4 b}+\frac {3 (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2}{8 b}-\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{8 b}-\frac {\left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)}{8 b}-\frac {9 \text {arcsinh}(a+b x)}{64 b}+\frac {(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{32 b}+\frac {15 (a+b x) \sqrt {(a+b x)^2+1}}{64 b} \]

[In]

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

[Out]

(15*(a + b*x)*Sqrt[1 + (a + b*x)^2])/(64*b) + ((a + b*x)*(1 + (a + b*x)^2)^(3/2))/(32*b) - (9*ArcSinh[a + b*x]
)/(64*b) - (3*(a + b*x)^2*ArcSinh[a + b*x])/(8*b) - ((1 + (a + b*x)^2)^2*ArcSinh[a + b*x])/(8*b) + (3*(a + b*x
)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(8*b) + ((a + b*x)*(1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x]^2)/(4*
b) + ArcSinh[a + b*x]^3/(8*b)

Rule 201

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(n*p + 1)), x] + Dist[a*n*(p/(n*p + 1)),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

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 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)^2 \, 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)^2}{4 b}-\frac {\text {Subst}\left (\int x \left (1+x^2\right ) \text {arcsinh}(x) \, dx,x,a+b x\right )}{2 b}+\frac {3 \text {Subst}\left (\int \sqrt {1+x^2} \text {arcsinh}(x)^2 \, dx,x,a+b x\right )}{4 b} \\ & = -\frac {\left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^2}{4 b}+\frac {\text {Subst}\left (\int \left (1+x^2\right )^{3/2} \, dx,x,a+b x\right )}{8 b}+\frac {3 \text {Subst}\left (\int \frac {\text {arcsinh}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 b}-\frac {3 \text {Subst}(\int x \text {arcsinh}(x) \, dx,x,a+b x)}{4 b} \\ & = \frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2}}{32 b}-\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{8 b}-\frac {\left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^2}{4 b}+\frac {\text {arcsinh}(a+b x)^3}{8 b}+\frac {3 \text {Subst}\left (\int \sqrt {1+x^2} \, dx,x,a+b x\right )}{32 b}+\frac {3 \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{8 b} \\ & = \frac {15 (a+b x) \sqrt {1+(a+b x)^2}}{64 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2}}{32 b}-\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{8 b}-\frac {\left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^2}{4 b}+\frac {\text {arcsinh}(a+b x)^3}{8 b}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{64 b}-\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,a+b x\right )}{16 b} \\ & = \frac {15 (a+b x) \sqrt {1+(a+b x)^2}}{64 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2}}{32 b}-\frac {9 \text {arcsinh}(a+b x)}{64 b}-\frac {3 (a+b x)^2 \text {arcsinh}(a+b x)}{8 b}-\frac {\left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)}{8 b}+\frac {3 (a+b x) \sqrt {1+(a+b x)^2} \text {arcsinh}(a+b x)^2}{8 b}+\frac {(a+b x) \left (1+(a+b x)^2\right )^{3/2} \text {arcsinh}(a+b x)^2}{4 b}+\frac {\text {arcsinh}(a+b x)^3}{8 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.12 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2 \, dx=\frac {\sqrt {1+a^2+2 a b x+b^2 x^2} \left (17 a+2 a^3+17 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right )-\left (17+40 a^2+8 a^4\right ) \text {arcsinh}(a+b x)-8 b x \left (10 a+4 a^3+5 b x+6 a^2 b x+4 a b^2 x^2+b^3 x^3\right ) \text {arcsinh}(a+b x)+8 \sqrt {1+a^2+2 a b x+b^2 x^2} \left (5 a+2 a^3+5 b x+6 a^2 b x+6 a b^2 x^2+2 b^3 x^3\right ) \text {arcsinh}(a+b x)^2+8 \text {arcsinh}(a+b x)^3}{64 b} \]

[In]

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

[Out]

(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) - (17 + 40*a^
2 + 8*a^4)*ArcSinh[a + b*x] - 8*b*x*(10*a + 4*a^3 + 5*b*x + 6*a^2*b*x + 4*a*b^2*x^2 + b^3*x^3)*ArcSinh[a + b*x
] + 8*Sqrt[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]^2 + 8*ArcSinh[a + b*x]^3)/(64*b)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(478\) vs. \(2(165)=330\).

Time = 0.89 (sec) , antiderivative size = 479, normalized size of antiderivative = 2.53

method	result	size
default	\(\frac {16 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{3} x^{3}-8 \,\operatorname {arcsinh}\left (b x +a \right ) b^{4} x^{4}+48 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a \,b^{2} x^{2}-32 \,\operatorname {arcsinh}\left (b x +a \right ) a \,b^{3} x^{3}+48 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b x +2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b^{3} x^{3}-48 \,\operatorname {arcsinh}\left (b x +a \right ) a^{2} b^{2} x^{2}+16 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}+6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a \,b^{2} x^{2}-32 \,\operatorname {arcsinh}\left (b x +a \right ) a^{3} b x +40 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x +6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{2} b x -8 \,\operatorname {arcsinh}\left (b x +a \right ) a^{4}-40 \,\operatorname {arcsinh}\left (b x +a \right ) b^{2} x^{2}+40 \operatorname {arcsinh}\left (b x +a \right )^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a +2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, a^{3}-80 \,\operatorname {arcsinh}\left (b x +a \right ) a b x +17 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, b x +8 \operatorname {arcsinh}\left (b x +a \right )^{3}-40 a^{2} \operatorname {arcsinh}\left (b x +a \right )+17 a \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}-17 \,\operatorname {arcsinh}\left (b x +a \right )}{64 b}\)	\(479\)

[In]

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

[Out]

1/64*(16*arcsinh(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*b^3*x^3-8*arcsinh(b*x+a)*b^4*x^4+48*arcsinh(b*x+a)^2*(
b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a*b^2*x^2-32*arcsinh(b*x+a)*a*b^3*x^3+48*arcsinh(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2+1)
^(1/2)*a^2*b*x+2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*b^3*x^3-48*arcsinh(b*x+a)*a^2*b^2*x^2+16*arcsinh(b*x+a)^2*(b^2*
x^2+2*a*b*x+a^2+1)^(1/2)*a^3+6*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a*b^2*x^2-32*arcsinh(b*x+a)*a^3*b*x+40*arcsinh(b*
x+a)^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*b*x+6*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^2*b*x-8*arcsinh(b*x+a)*a^4-40*arcsi
nh(b*x+a)*b^2*x^2+40*arcsinh(b*x+a)^2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a+2*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*a^3-80*a
rcsinh(b*x+a)*a*b*x+17*(b^2*x^2+2*a*b*x+a^2+1)^(1/2)*b*x+8*arcsinh(b*x+a)^3-40*a^2*arcsinh(b*x+a)+17*a*(b^2*x^
2+2*a*b*x+a^2+1)^(1/2)-17*arcsinh(b*x+a))/b

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.37 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2 \, dx=\frac {8 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 5\right )} b x + 5 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 8 \, \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - {\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \, {\left (6 \, a^{2} + 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \, {\left (2 \, a^{3} + 5 \, a\right )} b x + 40 \, a^{2} + 17\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) + {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 17\right )} b x + 17 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{64 \, b} \]

[In]

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

[Out]

1/64*(8*(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))^2 + 8*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^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*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))/b

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 568 vs. \(2 (173) = 346\).

Time = 0.74 (sec) , antiderivative size = 568, normalized size of antiderivative = 3.01 \[ \int \left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2 \, dx=\begin {cases} - \frac {a^{4} \operatorname {asinh}{\left (a + b x \right )}}{8 b} - \frac {a^{3} x \operatorname {asinh}{\left (a + b x \right )}}{2} + \frac {a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4 b} + \frac {a^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32 b} - \frac {3 a^{2} b x^{2} \operatorname {asinh}{\left (a + b x \right )}}{4} + \frac {3 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} + \frac {3 a^{2} x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32} - \frac {5 a^{2} \operatorname {asinh}{\left (a + b x \right )}}{8 b} - \frac {a b^{2} x^{3} \operatorname {asinh}{\left (a + b x \right )}}{2} + \frac {3 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} + \frac {3 a b x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32} - \frac {5 a x \operatorname {asinh}{\left (a + b x \right )}}{4} + \frac {5 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{8 b} + \frac {17 a \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{64 b} - \frac {b^{3} x^{4} \operatorname {asinh}{\left (a + b x \right )}}{8} + \frac {b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{4} + \frac {b^{2} x^{3} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{32} - \frac {5 b x^{2} \operatorname {asinh}{\left (a + b x \right )}}{8} + \frac {5 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{8} + \frac {17 x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{64} + \frac {\operatorname {asinh}^{3}{\left (a + b x \right )}}{8 b} - \frac {17 \operatorname {asinh}{\left (a + b x \right )}}{64 b} & \text {for}\: b \neq 0 \\x \left (a^{2} + 1\right )^{\frac {3}{2}} \operatorname {asinh}^{2}{\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)**2,x)

[Out]

Piecewise((-a**4*asinh(a + b*x)/(8*b) - a**3*x*asinh(a + b*x)/2 + a**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*as
inh(a + b*x)**2/(4*b) + a**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(32*b) - 3*a**2*b*x**2*asinh(a + b*x)/4 + 3*
a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2/4 + 3*a**2*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1
)/32 - 5*a**2*asinh(a + b*x)/(8*b) - a*b**2*x**3*asinh(a + b*x)/2 + 3*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2
+ 1)*asinh(a + b*x)**2/4 + 3*a*b*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/32 - 5*a*x*asinh(a + b*x)/4 + 5*a*
sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2/(8*b) + 17*a*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(64*b
) - b**3*x**4*asinh(a + b*x)/8 + b**2*x**3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2/4 + b**2*x**
3*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/32 - 5*b*x**2*asinh(a + b*x)/8 + 5*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 +
1)*asinh(a + b*x)**2/8 + 17*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/64 + asinh(a + b*x)**3/(8*b) - 17*asinh(a +
b*x)/(64*b), Ne(b, 0)), (x*(a**2 + 1)**(3/2)*asinh(a)**2, True))

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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