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

Optimal. Leaf size=355 \[ \frac {a^3 \sinh ^{-1}(a+b x)^3}{3 b^3}-\frac {6 a^2 \sqrt {(a+b x)^2+1}}{b^3}+\frac {6 a^2 (a+b x) \sinh ^{-1}(a+b x)}{b^3}-\frac {3 a^2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{b^3}+\frac {3 a \sqrt {(a+b x)^2+1} (a+b x)}{4 b^3}-\frac {2 \left ((a+b x)^2+1\right )^{3/2}}{27 b^3}+\frac {14 \sqrt {(a+b x)^2+1}}{9 b^3}+\frac {2 (a+b x)^3 \sinh ^{-1}(a+b x)}{9 b^3}-\frac {\sqrt {(a+b x)^2+1} (a+b x)^2 \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac {3 a (a+b x)^2 \sinh ^{-1}(a+b x)}{2 b^3}+\frac {3 a \sqrt {(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)^2}{2 b^3}-\frac {4 (a+b x) \sinh ^{-1}(a+b x)}{3 b^3}-\frac {a \sinh ^{-1}(a+b x)^3}{2 b^3}+\frac {2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac {3 a \sinh ^{-1}(a+b x)}{4 b^3}+\frac {1}{3} x^3 \sinh ^{-1}(a+b x)^3 \]

[Out]

-2/27*(1+(b*x+a)^2)^(3/2)/b^3-3/4*a*arcsinh(b*x+a)/b^3-4/3*(b*x+a)*arcsinh(b*x+a)/b^3+6*a^2*(b*x+a)*arcsinh(b*
x+a)/b^3-3/2*a*(b*x+a)^2*arcsinh(b*x+a)/b^3+2/9*(b*x+a)^3*arcsinh(b*x+a)/b^3-1/2*a*arcsinh(b*x+a)^3/b^3+1/3*a^
3*arcsinh(b*x+a)^3/b^3+1/3*x^3*arcsinh(b*x+a)^3+14/9*(1+(b*x+a)^2)^(1/2)/b^3-6*a^2*(1+(b*x+a)^2)^(1/2)/b^3+3/4
*a*(b*x+a)*(1+(b*x+a)^2)^(1/2)/b^3+2/3*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)/b^3-3*a^2*arcsinh(b*x+a)^2*(1+(b*x
+a)^2)^(1/2)/b^3+3/2*a*(b*x+a)*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)/b^3-1/3*(b*x+a)^2*arcsinh(b*x+a)^2*(1+(b*x
+a)^2)^(1/2)/b^3

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Rubi [A]  time = 0.45, antiderivative size = 355, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 11, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.917, Rules used = {5865, 5801, 5831, 3317, 3296, 2638, 3311, 30, 2635, 8, 2633} \[ -\frac {6 a^2 \sqrt {(a+b x)^2+1}}{b^3}+\frac {6 a^2 (a+b x) \sinh ^{-1}(a+b x)}{b^3}+\frac {a^3 \sinh ^{-1}(a+b x)^3}{3 b^3}-\frac {3 a^2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{b^3}+\frac {3 a \sqrt {(a+b x)^2+1} (a+b x)}{4 b^3}-\frac {2 \left ((a+b x)^2+1\right )^{3/2}}{27 b^3}+\frac {14 \sqrt {(a+b x)^2+1}}{9 b^3}+\frac {2 (a+b x)^3 \sinh ^{-1}(a+b x)}{9 b^3}-\frac {\sqrt {(a+b x)^2+1} (a+b x)^2 \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac {3 a (a+b x)^2 \sinh ^{-1}(a+b x)}{2 b^3}+\frac {3 a \sqrt {(a+b x)^2+1} (a+b x) \sinh ^{-1}(a+b x)^2}{2 b^3}-\frac {4 (a+b x) \sinh ^{-1}(a+b x)}{3 b^3}-\frac {a \sinh ^{-1}(a+b x)^3}{2 b^3}+\frac {2 \sqrt {(a+b x)^2+1} \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac {3 a \sinh ^{-1}(a+b x)}{4 b^3}+\frac {1}{3} x^3 \sinh ^{-1}(a+b x)^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcSinh[a + b*x]^3,x]

[Out]

(14*Sqrt[1 + (a + b*x)^2])/(9*b^3) - (6*a^2*Sqrt[1 + (a + b*x)^2])/b^3 + (3*a*(a + b*x)*Sqrt[1 + (a + b*x)^2])
/(4*b^3) - (2*(1 + (a + b*x)^2)^(3/2))/(27*b^3) - (3*a*ArcSinh[a + b*x])/(4*b^3) - (4*(a + b*x)*ArcSinh[a + b*
x])/(3*b^3) + (6*a^2*(a + b*x)*ArcSinh[a + b*x])/b^3 - (3*a*(a + b*x)^2*ArcSinh[a + b*x])/(2*b^3) + (2*(a + b*
x)^3*ArcSinh[a + b*x])/(9*b^3) + (2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(3*b^3) - (3*a^2*Sqrt[1 + (a + b
*x)^2]*ArcSinh[a + b*x]^2)/b^3 + (3*a*(a + b*x)*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(2*b^3) - ((a + b*x)
^2*Sqrt[1 + (a + b*x)^2]*ArcSinh[a + b*x]^2)/(3*b^3) - (a*ArcSinh[a + b*x]^3)/(2*b^3) + (a^3*ArcSinh[a + b*x]^
3)/(3*b^3) + (x^3*ArcSinh[a + b*x]^3)/3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3296

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> -Simp[((c + d*x)^m*Cos[e + f*x])/f, x] +
Dist[(d*m)/f, Int[(c + d*x)^(m - 1)*Cos[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 3311

Int[((c_.) + (d_.)*(x_))^(m_)*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*m*(c + d*x)^(m - 1)*(
b*Sin[e + f*x])^n)/(f^2*n^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)^m*(b*Sin[e + f*x])^(n - 2), x], x] - D
ist[(d^2*m*(m - 1))/(f^2*n^2), Int[(c + d*x)^(m - 2)*(b*Sin[e + f*x])^n, x], x] - Simp[(b*(c + d*x)^m*Cos[e +
f*x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1] && GtQ[m, 1]

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[
(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||
IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rule 5801

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

Rule 5831

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

Rule 5865

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

Rubi steps

\begin {align*} \int x^2 \sinh ^{-1}(a+b x)^3 \, dx &=\frac {\operatorname {Subst}\left (\int \left (-\frac {a}{b}+\frac {x}{b}\right )^2 \sinh ^{-1}(x)^3 \, dx,x,a+b x\right )}{b}\\ &=\frac {1}{3} x^3 \sinh ^{-1}(a+b x)^3-\operatorname {Subst}\left (\int \frac {\left (-\frac {a}{b}+\frac {x}{b}\right )^3 \sinh ^{-1}(x)^2}{\sqrt {1+x^2}} \, dx,x,a+b x\right )\\ &=\frac {1}{3} x^3 \sinh ^{-1}(a+b x)^3-\operatorname {Subst}\left (\int x^2 \left (-\frac {a}{b}+\frac {\sinh (x)}{b}\right )^3 \, dx,x,\sinh ^{-1}(a+b x)\right )\\ &=\frac {1}{3} x^3 \sinh ^{-1}(a+b x)^3-\operatorname {Subst}\left (\int \left (-\frac {a^3 x^2}{b^3}+\frac {3 a^2 x^2 \sinh (x)}{b^3}-\frac {3 a x^2 \sinh ^2(x)}{b^3}+\frac {x^2 \sinh ^3(x)}{b^3}\right ) \, dx,x,\sinh ^{-1}(a+b x)\right )\\ &=\frac {a^3 \sinh ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sinh ^{-1}(a+b x)^3-\frac {\operatorname {Subst}\left (\int x^2 \sinh ^3(x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}+\frac {(3 a) \operatorname {Subst}\left (\int x^2 \sinh ^2(x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}-\frac {\left (3 a^2\right ) \operatorname {Subst}\left (\int x^2 \sinh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=-\frac {3 a (a+b x)^2 \sinh ^{-1}(a+b x)}{2 b^3}+\frac {2 (a+b x)^3 \sinh ^{-1}(a+b x)}{9 b^3}-\frac {3 a^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{2 b^3}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{3 b^3}+\frac {a^3 \sinh ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sinh ^{-1}(a+b x)^3-\frac {2 \operatorname {Subst}\left (\int \sinh ^3(x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{9 b^3}+\frac {2 \operatorname {Subst}\left (\int x^2 \sinh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 b^3}-\frac {(3 a) \operatorname {Subst}\left (\int x^2 \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3}+\frac {(3 a) \operatorname {Subst}\left (\int \sinh ^2(x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{2 b^3}+\frac {\left (6 a^2\right ) \operatorname {Subst}\left (\int x \cosh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {3 a (a+b x) \sqrt {1+(a+b x)^2}}{4 b^3}+\frac {6 a^2 (a+b x) \sinh ^{-1}(a+b x)}{b^3}-\frac {3 a (a+b x)^2 \sinh ^{-1}(a+b x)}{2 b^3}+\frac {2 (a+b x)^3 \sinh ^{-1}(a+b x)}{9 b^3}+\frac {2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac {3 a^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{2 b^3}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac {a \sinh ^{-1}(a+b x)^3}{2 b^3}+\frac {a^3 \sinh ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sinh ^{-1}(a+b x)^3+\frac {2 \operatorname {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\sqrt {1+(a+b x)^2}\right )}{9 b^3}-\frac {4 \operatorname {Subst}\left (\int x \cosh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 b^3}-\frac {(3 a) \operatorname {Subst}\left (\int 1 \, dx,x,\sinh ^{-1}(a+b x)\right )}{4 b^3}-\frac {\left (6 a^2\right ) \operatorname {Subst}\left (\int \sinh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{b^3}\\ &=\frac {2 \sqrt {1+(a+b x)^2}}{9 b^3}-\frac {6 a^2 \sqrt {1+(a+b x)^2}}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2}}{4 b^3}-\frac {2 \left (1+(a+b x)^2\right )^{3/2}}{27 b^3}-\frac {3 a \sinh ^{-1}(a+b x)}{4 b^3}-\frac {4 (a+b x) \sinh ^{-1}(a+b x)}{3 b^3}+\frac {6 a^2 (a+b x) \sinh ^{-1}(a+b x)}{b^3}-\frac {3 a (a+b x)^2 \sinh ^{-1}(a+b x)}{2 b^3}+\frac {2 (a+b x)^3 \sinh ^{-1}(a+b x)}{9 b^3}+\frac {2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac {3 a^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{2 b^3}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac {a \sinh ^{-1}(a+b x)^3}{2 b^3}+\frac {a^3 \sinh ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sinh ^{-1}(a+b x)^3+\frac {4 \operatorname {Subst}\left (\int \sinh (x) \, dx,x,\sinh ^{-1}(a+b x)\right )}{3 b^3}\\ &=\frac {14 \sqrt {1+(a+b x)^2}}{9 b^3}-\frac {6 a^2 \sqrt {1+(a+b x)^2}}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2}}{4 b^3}-\frac {2 \left (1+(a+b x)^2\right )^{3/2}}{27 b^3}-\frac {3 a \sinh ^{-1}(a+b x)}{4 b^3}-\frac {4 (a+b x) \sinh ^{-1}(a+b x)}{3 b^3}+\frac {6 a^2 (a+b x) \sinh ^{-1}(a+b x)}{b^3}-\frac {3 a (a+b x)^2 \sinh ^{-1}(a+b x)}{2 b^3}+\frac {2 (a+b x)^3 \sinh ^{-1}(a+b x)}{9 b^3}+\frac {2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac {3 a^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{b^3}+\frac {3 a (a+b x) \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{2 b^3}-\frac {(a+b x)^2 \sqrt {1+(a+b x)^2} \sinh ^{-1}(a+b x)^2}{3 b^3}-\frac {a \sinh ^{-1}(a+b x)^3}{2 b^3}+\frac {a^3 \sinh ^{-1}(a+b x)^3}{3 b^3}+\frac {1}{3} x^3 \sinh ^{-1}(a+b x)^3\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 175, normalized size = 0.49 \[ \frac {18 \left (2 a^3-3 a+2 b^3 x^3\right ) \sinh ^{-1}(a+b x)^3+\left (-575 a^2+65 a b x-8 b^2 x^2+160\right ) \sqrt {a^2+2 a b x+b^2 x^2+1}-18 \sqrt {a^2+2 a b x+b^2 x^2+1} \left (11 a^2-5 a b x+2 b^2 x^2-4\right ) \sinh ^{-1}(a+b x)^2+3 \left (170 a^3+132 a^2 b x-15 a \left (2 b^2 x^2+5\right )+8 b x \left (b^2 x^2-6\right )\right ) \sinh ^{-1}(a+b x)}{108 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcSinh[a + b*x]^3,x]

[Out]

((160 - 575*a^2 + 65*a*b*x - 8*b^2*x^2)*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2] + 3*(170*a^3 + 132*a^2*b*x + 8*b*x*(
-6 + b^2*x^2) - 15*a*(5 + 2*b^2*x^2))*ArcSinh[a + b*x] - 18*Sqrt[1 + a^2 + 2*a*b*x + b^2*x^2]*(-4 + 11*a^2 - 5
*a*b*x + 2*b^2*x^2)*ArcSinh[a + b*x]^2 + 18*(-3*a + 2*a^3 + 2*b^3*x^3)*ArcSinh[a + b*x]^3)/(108*b^3)

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fricas [A]  time = 0.49, size = 225, normalized size = 0.63 \[ \frac {18 \, {\left (2 \, b^{3} x^{3} + 2 \, a^{3} - 3 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - 18 \, {\left (2 \, b^{2} x^{2} - 5 \, a b x + 11 \, a^{2} - 4\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2} + 3 \, {\left (8 \, b^{3} x^{3} - 30 \, a b^{2} x^{2} + 170 \, a^{3} + 12 \, {\left (11 \, a^{2} - 4\right )} b x - 75 \, a\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right ) - {\left (8 \, b^{2} x^{2} - 65 \, a b x + 575 \, a^{2} - 160\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{108 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/108*(18*(2*b^3*x^3 + 2*a^3 - 3*a)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - 18*(2*b^2*x^2 - 5*a*b
*x + 11*a^2 - 4)*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 + 3*(8*b
^3*x^3 - 30*a*b^2*x^2 + 170*a^3 + 12*(11*a^2 - 4)*b*x - 75*a)*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))
- (8*b^2*x^2 - 65*a*b*x + 575*a^2 - 160)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/b^3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^2*arcsinh(b*x + a)^3, x)

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maple [A]  time = 0.10, size = 311, normalized size = 0.88 \[ \frac {-\frac {a \left (4 \arcsinh \left (b x +a \right )^{3} \left (b x +a \right )^{2}-6 \arcsinh \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}\, \left (b x +a \right )+2 \arcsinh \left (b x +a \right )^{3}+6 \arcsinh \left (b x +a \right ) \left (b x +a \right )^{2}-3 \left (b x +a \right ) \sqrt {1+\left (b x +a \right )^{2}}+3 \arcsinh \left (b x +a \right )\right )}{4}-\frac {\arcsinh \left (b x +a \right )^{3} \left (b x +a \right )}{3}+\frac {\arcsinh \left (b x +a \right )^{3} \left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right )}{3}+\arcsinh \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}-\frac {14 \left (b x +a \right ) \arcsinh \left (b x +a \right )}{9}+\frac {14 \sqrt {1+\left (b x +a \right )^{2}}}{9}-\frac {\arcsinh \left (b x +a \right )^{2} \left (1+\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3}+\frac {2 \left (b x +a \right ) \left (1+\left (b x +a \right )^{2}\right ) \arcsinh \left (b x +a \right )}{9}-\frac {2 \left (1+\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{27}+a^{2} \left (\arcsinh \left (b x +a \right )^{3} \left (b x +a \right )-3 \arcsinh \left (b x +a \right )^{2} \sqrt {1+\left (b x +a \right )^{2}}+6 \left (b x +a \right ) \arcsinh \left (b x +a \right )-6 \sqrt {1+\left (b x +a \right )^{2}}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^3*(-1/4*a*(4*arcsinh(b*x+a)^3*(b*x+a)^2-6*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)*(b*x+a)+2*arcsinh(b*x+a)^3+
6*arcsinh(b*x+a)*(b*x+a)^2-3*(b*x+a)*(1+(b*x+a)^2)^(1/2)+3*arcsinh(b*x+a))-1/3*arcsinh(b*x+a)^3*(b*x+a)+1/3*ar
csinh(b*x+a)^3*(b*x+a)*(1+(b*x+a)^2)+arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)-14/9*(b*x+a)*arcsinh(b*x+a)+14/9*(1+
(b*x+a)^2)^(1/2)-1/3*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(3/2)+2/9*(b*x+a)*(1+(b*x+a)^2)*arcsinh(b*x+a)-2/27*(1+(b*
x+a)^2)^(3/2)+a^2*(arcsinh(b*x+a)^3*(b*x+a)-3*arcsinh(b*x+a)^2*(1+(b*x+a)^2)^(1/2)+6*(b*x+a)*arcsinh(b*x+a)-6*
(1+(b*x+a)^2)^(1/2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{3} \, x^{3} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{3} - \int \frac {{\left (b^{3} x^{5} + 2 \, a b^{2} x^{4} + {\left (a^{2} b + b\right )} x^{3} + {\left (b^{2} x^{4} + a b x^{3}\right )} \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \log \left (b x + a + \sqrt {b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )^{2}}{b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} + {\left (3 \, a^{2} b + b\right )} x + {\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^3*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))^3 - integrate((b^3*x^5 + 2*a*b^2*x^4 + (a^2*b + b)*x^
3 + (b^2*x^4 + a*b*x^3)*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/
(b^3*x^3 + 3*a*b^2*x^2 + a^3 + (3*a^2*b + b)*x + (b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*asinh(a + b*x)^3,x)

[Out]

int(x^2*asinh(a + b*x)^3, x)

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sympy [A]  time = 3.04, size = 432, normalized size = 1.22 \[ \begin {cases} \frac {a^{3} \operatorname {asinh}^{3}{\left (a + b x \right )}}{3 b^{3}} + \frac {85 a^{3} \operatorname {asinh}{\left (a + b x \right )}}{18 b^{3}} + \frac {11 a^{2} x \operatorname {asinh}{\left (a + b x \right )}}{3 b^{2}} - \frac {11 a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{6 b^{3}} - \frac {575 a^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{108 b^{3}} - \frac {5 a x^{2} \operatorname {asinh}{\left (a + b x \right )}}{6 b} + \frac {5 a x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{6 b^{2}} + \frac {65 a x \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{108 b^{2}} - \frac {a \operatorname {asinh}^{3}{\left (a + b x \right )}}{2 b^{3}} - \frac {25 a \operatorname {asinh}{\left (a + b x \right )}}{12 b^{3}} + \frac {x^{3} \operatorname {asinh}^{3}{\left (a + b x \right )}}{3} + \frac {2 x^{3} \operatorname {asinh}{\left (a + b x \right )}}{9} - \frac {x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{3 b} - \frac {2 x^{2} \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{27 b} - \frac {4 x \operatorname {asinh}{\left (a + b x \right )}}{3 b^{2}} + \frac {2 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (a + b x \right )}}{3 b^{3}} + \frac {40 \sqrt {a^{2} + 2 a b x + b^{2} x^{2} + 1}}{27 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \operatorname {asinh}^{3}{\relax (a )}}{3} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((a**3*asinh(a + b*x)**3/(3*b**3) + 85*a**3*asinh(a + b*x)/(18*b**3) + 11*a**2*x*asinh(a + b*x)/(3*b*
*2) - 11*a**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2/(6*b**3) - 575*a**2*sqrt(a**2 + 2*a*b*x +
b**2*x**2 + 1)/(108*b**3) - 5*a*x**2*asinh(a + b*x)/(6*b) + 5*a*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(
a + b*x)**2/(6*b**2) + 65*a*x*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(108*b**2) - a*asinh(a + b*x)**3/(2*b**3) -
25*a*asinh(a + b*x)/(12*b**3) + x**3*asinh(a + b*x)**3/3 + 2*x**3*asinh(a + b*x)/9 - x**2*sqrt(a**2 + 2*a*b*x
+ b**2*x**2 + 1)*asinh(a + b*x)**2/(3*b) - 2*x**2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)/(27*b) - 4*x*asinh(a +
b*x)/(3*b**2) + 2*sqrt(a**2 + 2*a*b*x + b**2*x**2 + 1)*asinh(a + b*x)**2/(3*b**3) + 40*sqrt(a**2 + 2*a*b*x +
b**2*x**2 + 1)/(27*b**3), Ne(b, 0)), (x**3*asinh(a)**3/3, True))

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