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

Optimal. Leaf size=235 \[ -\frac {3 (a+b x)^4}{128 b}-\frac {51 (a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{16 b}+\frac {\left ((a+b x)^2+1\right )^{3/2} (a+b x) \sinh ^{-1}(a+b x)^3}{4 b}+\frac {3 \sqrt {(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)^3}{8 b}+\frac {3 \left ((a+b x)^2+1\right )^{3/2} (a+b x) \sinh ^{-1}(a+b x)}{32 b}+\frac {45 \sqrt {(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)}{64 b}+\frac {3 \sinh ^{-1}(a+b x)^4}{32 b}-\frac {3 \left ((a+b x)^2+1\right )^2 \sinh ^{-1}(a+b x)^2}{16 b}-\frac {27 \sinh ^{-1}(a+b x)^2}{128 b} \]

[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

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Rubi [A]  time = 0.31, antiderivative size = 235, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 9, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {5867, 5684, 5682, 5675, 5661, 5758, 30, 5717, 14} \[ -\frac {3 (a+b x)^4}{128 b}-\frac {51 (a+b x)^2}{128 b}-\frac {9 (a+b x)^2 \sinh ^{-1}(a+b x)^2}{16 b}+\frac {\left ((a+b x)^2+1\right )^{3/2} (a+b x) \sinh ^{-1}(a+b x)^3}{4 b}+\frac {3 \sqrt {(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)^3}{8 b}+\frac {3 \left ((a+b x)^2+1\right )^{3/2} (a+b x) \sinh ^{-1}(a+b x)}{32 b}+\frac {45 \sqrt {(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)}{64 b}+\frac {3 \sinh ^{-1}(a+b x)^4}{32 b}-\frac {3 \left ((a+b x)^2+1\right )^2 \sinh ^{-1}(a+b x)^2}{16 b}-\frac {27 \sinh ^{-1}(a+b x)^2}{128 b} \]

Antiderivative was successfully verified.

[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 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 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 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 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 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 5758

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

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]

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

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Mathematica [A]  time = 0.24, size = 266, normalized size = 1.13 \[ -\frac {3 \left (6 a^2+17\right ) b^2 x^2+6 a \left (2 a^2+17\right ) b x-16 \sqrt {a^2+2 a b x+b^2 x^2+1} \left (2 a^3+6 a^2 b x+6 a b^2 x^2+5 a+2 b^3 x^3+5 b x\right ) \sinh ^{-1}(a+b x)^3-6 \sqrt {a^2+2 a b x+b^2 x^2+1} \left (2 a^3+6 a^2 b x+6 a b^2 x^2+17 a+2 b^3 x^3+17 b x\right ) \sinh ^{-1}(a+b x)+3 \left (8 a^4+32 a^3 b x+8 a^2 \left (6 b^2 x^2+5\right )+16 a b x \left (2 b^2 x^2+5\right )+8 b^4 x^4+40 b^2 x^2+17\right ) \sinh ^{-1}(a+b x)^2+12 a b^3 x^3-12 \sinh ^{-1}(a+b x)^4+3 b^4 x^4}{128 b} \]

Antiderivative was successfully verified.

[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

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fricas [A]  time = 0.46, size = 332, normalized size = 1.41 \[ -\frac {3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} + 3 \, {\left (6 \, a^{2} + 17\right )} b^{2} x^{2} - 16 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 5\right )} b x + 5 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 12 \, \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{4} + 6 \, {\left (2 \, a^{3} + 17 \, a\right )} b x + 3 \, {\left (8 \, b^{4} x^{4} + 32 \, a b^{3} x^{3} + 8 \, {\left (6 \, a^{2} + 5\right )} b^{2} x^{2} + 8 \, a^{4} + 16 \, {\left (2 \, a^{3} + 5 \, a\right )} b x + 40 \, a^{2} + 17\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} - 6 \, {\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} + {\left (6 \, a^{2} + 17\right )} b x + 17 \, a\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}{128 \, b} \]

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)^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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{3}\,{d x} \]

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)^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)

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maple [B]  time = 0.14, size = 592, normalized size = 2.52 \[ \frac {-48-102 a b x +80 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} x b -240 \arcsinh \left (b x +a \right )^{2} x a b +102 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) x b -51 b^{2} x^{2}-51 a^{2}-24 \arcsinh \left (b x +a \right )^{2} a^{4}-3 x^{4} b^{4}-120 \arcsinh \left (b x +a \right )^{2} a^{2}-96 \arcsinh \left (b x +a \right )^{2} x^{3} a \,b^{3}+32 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} x^{3} b^{3}+12 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) x^{3} b^{3}-144 \arcsinh \left (b x +a \right )^{2} x^{2} a^{2} b^{2}-96 \arcsinh \left (b x +a \right )^{2} x \,a^{3} b -3 a^{4}-51 \arcsinh \left (b x +a \right )^{2}+32 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} a^{3}+12 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) a^{3}-24 \arcsinh \left (b x +a \right )^{2} x^{4} b^{4}-12 x^{3} a \,b^{3}-18 x^{2} a^{2} b^{2}-12 x \,a^{3} b +12 \arcsinh \left (b x +a \right )^{4}-120 \arcsinh \left (b x +a \right )^{2} x^{2} b^{2}+80 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} a +102 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) a +36 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) x \,a^{2} b +96 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} x^{2} a \,b^{2}+96 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right )^{3} x \,a^{2} b +36 \sqrt {b^{2} x^{2}+2 a b x +a^{2}+1}\, \arcsinh \left (b x +a \right ) x^{2} a \,b^{2}}{128 b} \]

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)^3,x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{3}\,{d x} \]

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)^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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\mathrm {asinh}\left (a+b\,x\right )}^3\,{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 17.37, size = 694, normalized size = 2.95 \[ \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}{\relax (a )} & \text {otherwise} \end {cases} \]

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)**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))

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