3.267 \(\int (1+a^2+2 a b x+b^2 x^2)^{3/2} \sinh ^{-1}(a+b x)^2 \, dx\)

Optimal. Leaf size=189 \[ \frac{(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{32 b}+\frac{15 (a+b x) \sqrt{(a+b x)^2+1}}{64 b}+\frac{\sinh ^{-1}(a+b x)^3}{8 b}+\frac{(a+b x) \left ((a+b x)^2+1\right )^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}+\frac{3 (a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{8 b}-\frac{3 (a+b x)^2 \sinh ^{-1}(a+b x)}{8 b}-\frac{\left ((a+b x)^2+1\right )^2 \sinh ^{-1}(a+b x)}{8 b}-\frac{9 \sinh ^{-1}(a+b x)}{64 b} \]

[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)

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Rubi [A]  time = 0.190467, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {5867, 5684, 5682, 5675, 5661, 321, 215, 5717, 195} \[ \frac{(a+b x) \left ((a+b x)^2+1\right )^{3/2}}{32 b}+\frac{15 (a+b x) \sqrt{(a+b x)^2+1}}{64 b}+\frac{\sinh ^{-1}(a+b x)^3}{8 b}+\frac{(a+b x) \left ((a+b x)^2+1\right )^{3/2} \sinh ^{-1}(a+b x)^2}{4 b}+\frac{3 (a+b x) \sqrt{(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{8 b}-\frac{3 (a+b x)^2 \sinh ^{-1}(a+b x)}{8 b}-\frac{\left ((a+b x)^2+1\right )^2 \sinh ^{-1}(a+b x)}{8 b}-\frac{9 \sinh ^{-1}(a+b x)}{64 b} \]

Antiderivative was successfully verified.

[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 5867

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*x^2)/d^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]

Rule 5684

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 + c^2*x^2)^FracPart[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] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 5682

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[Sqrt[d + e*x^2]/(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*Sqrt[d + e*x^2])/(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 5675

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

Rule 5661

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 321

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 215

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 5717

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[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 195

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]])

Rubi steps

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

Mathematica [A]  time = 0.155967, size = 211, normalized size = 1.12 \[ \frac{\sqrt{a^2+2 a b x+b^2 x^2+1} \left (6 a^2 b x+2 a^3+6 a b^2 x^2+17 a+2 b^3 x^3+17 b x\right )+8 \sqrt{a^2+2 a b x+b^2 x^2+1} \left (6 a^2 b x+2 a^3+6 a b^2 x^2+5 a+2 b^3 x^3+5 b x\right ) \sinh ^{-1}(a+b x)^2-8 b x \left (6 a^2 b x+4 a^3+4 a b^2 x^2+10 a+b^3 x^3+5 b x\right ) \sinh ^{-1}(a+b x)-\left (8 a^4+40 a^2+17\right ) \sinh ^{-1}(a+b x)+8 \sinh ^{-1}(a+b x)^3}{64 b} \]

Antiderivative was successfully verified.

[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)

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Maple [B]  time = 0.063, size = 479, normalized size = 2.5 \begin{align*}{\frac{1}{64\,b} \left ( 16\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}{x}^{3}{b}^{3}-8\,{\it Arcsinh} \left ( bx+a \right ){x}^{4}{b}^{4}+48\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}{x}^{2}a{b}^{2}-32\,{\it Arcsinh} \left ( bx+a \right ){x}^{3}a{b}^{3}+48\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}x{a}^{2}b+2\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}{x}^{3}{b}^{3}-48\,{\it Arcsinh} \left ( bx+a \right ){x}^{2}{a}^{2}{b}^{2}+16\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}{a}^{3}+6\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}{x}^{2}a{b}^{2}-32\,{\it Arcsinh} \left ( bx+a \right ) x{a}^{3}b+40\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}xb+6\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}x{a}^{2}b-40\,{\it Arcsinh} \left ( bx+a \right ){x}^{2}{b}^{2}-8\,{\it Arcsinh} \left ( bx+a \right ){a}^{4}+40\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}a+2\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}{a}^{3}-80\,{\it Arcsinh} \left ( bx+a \right ) xab+17\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}xb+8\, \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{3}-40\,{\it Arcsinh} \left ( bx+a \right ){a}^{2}+17\,\sqrt{{b}^{2}{x}^{2}+2\,xab+{a}^{2}+1}a-17\,{\it Arcsinh} \left ( bx+a \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.81868, size = 612, normalized size = 3.24 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 16.2136, size = 568, normalized size = 3.01 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)