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

3.3.67.1 Optimal result

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

output

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*arc 
sinh(b*x+a)/b-1/8*(1+(b*x+a)^2)^2*arcsinh(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
 

3.3.67.2 Mathematica [A] (verified)

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

input

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

output

(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*(1 
0*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)
 

3.3.67.3 Rubi [A] (verified)

Time = 0.76 (sec) , antiderivative size = 226, normalized size of antiderivative = 1.20, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {6275, 6201, 6200, 6191, 262, 222, 6198, 6213, 211, 211, 222}

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.

\(\displaystyle \int \left (a^2+2 a b x+b^2 x^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2 \, dx\)

\(\Big \downarrow \) 6275

\(\displaystyle \frac {\int \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2d(a+b x)}{b}\)

\(\Big \downarrow \) 6201

\(\displaystyle \frac {-\frac {1}{2} \int (a+b x) \left ((a+b x)^2+1\right ) \text {arcsinh}(a+b x)d(a+b x)+\frac {3}{4} \int \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2d(a+b x)+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2}{b}\)

\(\Big \downarrow \) 6200

\(\displaystyle \frac {-\frac {1}{2} \int (a+b x) \left ((a+b x)^2+1\right ) \text {arcsinh}(a+b x)d(a+b x)+\frac {3}{4} \left (-\int (a+b x) \text {arcsinh}(a+b x)d(a+b x)+\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)^2}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2}{b}\)

\(\Big \downarrow \) 6191

\(\displaystyle \frac {-\frac {1}{2} \int (a+b x) \left ((a+b x)^2+1\right ) \text {arcsinh}(a+b x)d(a+b x)+\frac {3}{4} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)^2}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} \int \frac {(a+b x)^2}{\sqrt {(a+b x)^2+1}}d(a+b x)-\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)+\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)^2\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2}{b}\)

\(\Big \downarrow \) 262

\(\displaystyle \frac {-\frac {1}{2} \int (a+b x) \left ((a+b x)^2+1\right ) \text {arcsinh}(a+b x)d(a+b x)+\frac {3}{4} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)^2}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} \left (\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1}-\frac {1}{2} \int \frac {1}{\sqrt {(a+b x)^2+1}}d(a+b x)\right )-\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)+\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)^2\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2}{b}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {-\frac {1}{2} \int (a+b x) \left ((a+b x)^2+1\right ) \text {arcsinh}(a+b x)d(a+b x)+\frac {3}{4} \left (\frac {1}{2} \int \frac {\text {arcsinh}(a+b x)^2}{\sqrt {(a+b x)^2+1}}d(a+b x)-\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)+\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x) \text {arcsinh}(a+b x)^2+\frac {1}{2} \left (\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1}-\frac {1}{2} \text {arcsinh}(a+b x)\right )\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2}{b}\)

\(\Big \downarrow \) 6198

\(\displaystyle \frac {-\frac {1}{2} \int (a+b x) \left ((a+b x)^2+1\right ) \text {arcsinh}(a+b x)d(a+b x)+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2+\frac {3}{4} \left (\frac {1}{6} \text {arcsinh}(a+b x)^3+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2-\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)+\frac {1}{2} \left (\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1}-\frac {1}{2} \text {arcsinh}(a+b x)\right )\right )}{b}\)

\(\Big \downarrow \) 6213

\(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \int \left ((a+b x)^2+1\right )^{3/2}d(a+b x)-\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2+\frac {3}{4} \left (\frac {1}{6} \text {arcsinh}(a+b x)^3+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2-\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)+\frac {1}{2} \left (\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1}-\frac {1}{2} \text {arcsinh}(a+b x)\right )\right )}{b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {3}{4} \int \sqrt {(a+b x)^2+1}d(a+b x)+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2}\right )-\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2+\frac {3}{4} \left (\frac {1}{6} \text {arcsinh}(a+b x)^3+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2-\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)+\frac {1}{2} \left (\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1}-\frac {1}{2} \text {arcsinh}(a+b x)\right )\right )}{b}\)

\(\Big \downarrow \) 211

\(\displaystyle \frac {\frac {1}{2} \left (\frac {1}{4} \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {1}{\sqrt {(a+b x)^2+1}}d(a+b x)+\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x)\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2}\right )-\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2+\frac {3}{4} \left (\frac {1}{6} \text {arcsinh}(a+b x)^3+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2-\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)+\frac {1}{2} \left (\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1}-\frac {1}{2} \text {arcsinh}(a+b x)\right )\right )}{b}\)

\(\Big \downarrow \) 222

\(\displaystyle \frac {\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2} \text {arcsinh}(a+b x)^2+\frac {1}{2} \left (\frac {1}{4} \left (\frac {3}{4} \left (\frac {1}{2} \text {arcsinh}(a+b x)+\frac {1}{2} \sqrt {(a+b x)^2+1} (a+b x)\right )+\frac {1}{4} (a+b x) \left ((a+b x)^2+1\right )^{3/2}\right )-\frac {1}{4} \left ((a+b x)^2+1\right )^2 \text {arcsinh}(a+b x)\right )+\frac {3}{4} \left (\frac {1}{6} \text {arcsinh}(a+b x)^3+\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1} \text {arcsinh}(a+b x)^2-\frac {1}{2} (a+b x)^2 \text {arcsinh}(a+b x)+\frac {1}{2} \left (\frac {1}{2} (a+b x) \sqrt {(a+b x)^2+1}-\frac {1}{2} \text {arcsinh}(a+b x)\right )\right )}{b}\)

input

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

output

(((a + b*x)*(1 + (a + b*x)^2)^(3/2)*ArcSinh[a + b*x]^2)/4 + ((((a + b*x)*( 
1 + (a + b*x)^2)^(3/2))/4 + (3*(((a + b*x)*Sqrt[1 + (a + b*x)^2])/2 + ArcS 
inh[a + b*x]/2))/4)/4 - ((1 + (a + b*x)^2)^2*ArcSinh[a + b*x])/4)/2 + (3*( 
(((a + b*x)*Sqrt[1 + (a + b*x)^2])/2 - ArcSinh[a + b*x]/2)/2 - ((a + b*x)^ 
2*ArcSinh[a + b*x])/2 + ((a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^ 
2)/2 + ArcSinh[a + b*x]^3/6))/4)/b
 

3.3.67.3.1 Defintions of rubi rules used

rule 211

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 
)), x] + Simp[2*a*(p/(2*p + 1))   Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ 
{a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
 

rule 222

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 262

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) 
^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ 
(b*(m + 2*p + 1)))   Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b 
, c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c 
, 2, m, p, x]
 

rule 6191

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]
 

rule 6198

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]
 

rule 6200

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] + (Simp[(1 
/2)*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]]   Int[(a + b*ArcSinh[c*x])^n/Sq 
rt[1 + c^2*x^2], x], x] - Simp[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 6201

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] + 
(Simp[2*d*(p/(2*p + 1))   Int[(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x 
], x] - Simp[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*ArcSinh[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 6213

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] - Simp[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 6275

Int[((a_.) + ArcSinh[(c_) + (d_.)*(x_)]*(b_.))^(n_.)*((A_.) + (B_.)*(x_) + 
(C_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[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]
 

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

input

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

output

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*arcsinh(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*arcsinh 
(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
 

3.3.67.5 Fricas [A] (verification not implemented)

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

input

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

output

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 + 3 
2*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
 

3.3.67.6 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} \]

input

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

output

Piecewise((-a**4*asinh(a + b*x)/(8*b) - a**3*x*asinh(a + b*x)/2 + a**3*sqr 
t(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(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*s 
qrt(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))
 

3.3.67.7 Maxima [F]

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

input

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

output

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

3.3.67.8 Giac [F]

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

input

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

output

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

3.3.67.9 Mupad [F(-1)]

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

input

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

output

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