\(\int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx\) [291]

Optimal result

Integrand size = 28, antiderivative size = 323 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {15 b^2 x \left (1+c^2 x^2\right )}{64 c^4 \sqrt {d+c^2 d x^2}}+\frac {b^2 x^3 \left (1+c^2 x^2\right )}{32 c^2 \sqrt {d+c^2 d x^2}}+\frac {15 b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{64 c^5 \sqrt {d+c^2 d x^2}}+\frac {3 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 c^2 d}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{8 b c^5 \sqrt {d+c^2 d x^2}} \]

[Out]

-15/64*b^2*x*(c^2*x^2+1)/c^4/(c^2*d*x^2+d)^(1/2)+1/32*b^2*x^3*(c^2*x^2+1)/c^2/(c^2*d*x^2+d)^(1/2)+15/64*b^2*ar
csinh(c*x)*(c^2*x^2+1)^(1/2)/c^5/(c^2*d*x^2+d)^(1/2)+3/8*b*x^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^3/(c^2*d
*x^2+d)^(1/2)-1/8*b*x^4*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)+1/8*(a+b*arcsinh(c*x))^3*(c
^2*x^2+1)^(1/2)/b/c^5/(c^2*d*x^2+d)^(1/2)-3/8*x*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^4/d+1/4*x^3*(a+b*ar
csinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^2/d

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5812, 5783, 5776, 327, 221} \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {b x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{8 c \sqrt {c^2 d x^2+d}}+\frac {x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{4 c^2 d}+\frac {\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^3}{8 b c^5 \sqrt {c^2 d x^2+d}}-\frac {3 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{8 c^4 d}+\frac {3 b x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{8 c^3 \sqrt {c^2 d x^2+d}}+\frac {15 b^2 \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{64 c^5 \sqrt {c^2 d x^2+d}}+\frac {b^2 x^3 \left (c^2 x^2+1\right )}{32 c^2 \sqrt {c^2 d x^2+d}}-\frac {15 b^2 x \left (c^2 x^2+1\right )}{64 c^4 \sqrt {c^2 d x^2+d}} \]

[In]

Int[(x^4*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]

[Out]

(-15*b^2*x*(1 + c^2*x^2))/(64*c^4*Sqrt[d + c^2*d*x^2]) + (b^2*x^3*(1 + c^2*x^2))/(32*c^2*Sqrt[d + c^2*d*x^2])
+ (15*b^2*Sqrt[1 + c^2*x^2]*ArcSinh[c*x])/(64*c^5*Sqrt[d + c^2*d*x^2]) + (3*b*x^2*Sqrt[1 + c^2*x^2]*(a + b*Arc
Sinh[c*x]))/(8*c^3*Sqrt[d + c^2*d*x^2]) - (b*x^4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c*Sqrt[d + c^2*d*x
^2]) - (3*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(8*c^4*d) + (x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*
x])^2)/(4*c^2*d) + (Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^3)/(8*b*c^5*Sqrt[d + c^2*d*x^2])

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 5812

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

Rubi steps \begin{align*} \text {integral}& = \frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 c^2 d}-\frac {3 \int \frac {x^2 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{4 c^2}-\frac {\left (b \sqrt {1+c^2 x^2}\right ) \int x^3 (a+b \text {arcsinh}(c x)) \, dx}{2 c \sqrt {d+c^2 d x^2}} \\ & = -\frac {b x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 c^2 d}+\frac {3 \int \frac {(a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx}{8 c^4}+\frac {\left (b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {d+c^2 d x^2}}+\frac {\left (3 b \sqrt {1+c^2 x^2}\right ) \int x (a+b \text {arcsinh}(c x)) \, dx}{4 c^3 \sqrt {d+c^2 d x^2}} \\ & = \frac {b^2 x^3 \left (1+c^2 x^2\right )}{32 c^2 \sqrt {d+c^2 d x^2}}+\frac {3 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 c^2 d}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{8 b c^5 \sqrt {d+c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{32 c^2 \sqrt {d+c^2 d x^2}}-\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{8 c^2 \sqrt {d+c^2 d x^2}} \\ & = -\frac {15 b^2 x \left (1+c^2 x^2\right )}{64 c^4 \sqrt {d+c^2 d x^2}}+\frac {b^2 x^3 \left (1+c^2 x^2\right )}{32 c^2 \sqrt {d+c^2 d x^2}}+\frac {3 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 c^2 d}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{8 b c^5 \sqrt {d+c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{64 c^4 \sqrt {d+c^2 d x^2}}+\frac {\left (3 b^2 \sqrt {1+c^2 x^2}\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{16 c^4 \sqrt {d+c^2 d x^2}} \\ & = -\frac {15 b^2 x \left (1+c^2 x^2\right )}{64 c^4 \sqrt {d+c^2 d x^2}}+\frac {b^2 x^3 \left (1+c^2 x^2\right )}{32 c^2 \sqrt {d+c^2 d x^2}}+\frac {15 b^2 \sqrt {1+c^2 x^2} \text {arcsinh}(c x)}{64 c^5 \sqrt {d+c^2 d x^2}}+\frac {3 b x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c^3 \sqrt {d+c^2 d x^2}}-\frac {b x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{8 c \sqrt {d+c^2 d x^2}}-\frac {3 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{8 c^4 d}+\frac {x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 c^2 d}+\frac {\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^3}{8 b c^5 \sqrt {d+c^2 d x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.05 (sec) , antiderivative size = 268, normalized size of antiderivative = 0.83 \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\frac {32 a^2 c \sqrt {d} x \left (1+c^2 x^2\right ) \left (-3+2 c^2 x^2\right )+96 a^2 \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+b^2 \sqrt {d} \sqrt {1+c^2 x^2} \left (32 \text {arcsinh}(c x)^3-4 \text {arcsinh}(c x) (-16 \cosh (2 \text {arcsinh}(c x))+\cosh (4 \text {arcsinh}(c x)))-32 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))+8 \text {arcsinh}(c x)^2 (-8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x)))\right )+4 a b \sqrt {d} \sqrt {1+c^2 x^2} (16 \cosh (2 \text {arcsinh}(c x))-\cosh (4 \text {arcsinh}(c x))+4 \text {arcsinh}(c x) (6 \text {arcsinh}(c x)-8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))))}{256 c^5 \sqrt {d} \sqrt {d+c^2 d x^2}} \]

[In]

Integrate[(x^4*(a + b*ArcSinh[c*x])^2)/Sqrt[d + c^2*d*x^2],x]

[Out]

(32*a^2*c*Sqrt[d]*x*(1 + c^2*x^2)*(-3 + 2*c^2*x^2) + 96*a^2*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c
^2*d*x^2]] + b^2*Sqrt[d]*Sqrt[1 + c^2*x^2]*(32*ArcSinh[c*x]^3 - 4*ArcSinh[c*x]*(-16*Cosh[2*ArcSinh[c*x]] + Cos
h[4*ArcSinh[c*x]]) - 32*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]] + 8*ArcSinh[c*x]^2*(-8*Sinh[2*ArcSinh[c*x]
] + Sinh[4*ArcSinh[c*x]])) + 4*a*b*Sqrt[d]*Sqrt[1 + c^2*x^2]*(16*Cosh[2*ArcSinh[c*x]] - Cosh[4*ArcSinh[c*x]] +
4*ArcSinh[c*x]*(6*ArcSinh[c*x] - 8*Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])))/(256*c^5*Sqrt[d]*Sqrt[d + c
^2*d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(991\) vs. \(2(283)=566\).

Time = 0.26 (sec) , antiderivative size = 992, normalized size of antiderivative = 3.07

method	result	size
default	\(\frac {a^{2} x^{3} \sqrt {c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{4} \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{3}}{8 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (c x \right )^{2}-4 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{512 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (c x \right )^{2}+4 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{512 c^{5} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}\right )\)	\(992\)
parts	\(\frac {a^{2} x^{3} \sqrt {c^{2} d \,x^{2}+d}}{4 c^{2} d}-\frac {3 a^{2} x \sqrt {c^{2} d \,x^{2}+d}}{8 c^{4} d}+\frac {3 a^{2} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{8 c^{4} \sqrt {c^{2} d}}+b^{2} \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{3}}{8 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (c x \right )^{2}-4 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{512 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (c x \right )^{2}-2 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (2 \operatorname {arcsinh}\left (c x \right )^{2}+2 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (8 \operatorname {arcsinh}\left (c x \right )^{2}+4 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{512 c^{5} d \left (c^{2} x^{2}+1\right )}\right )+2 a b \left (\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c^{5} d}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}+8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}+8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}+2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (2 c^{3} x^{3}-2 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+2 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{5} d \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (8 c^{5} x^{5}-8 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}+12 c^{3} x^{3}-8 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+4 c x -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{5} d \left (c^{2} x^{2}+1\right )}\right )\)	\(992\)

[In]

int(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/4*a^2*x^3/c^2/d*(c^2*d*x^2+d)^(1/2)-3/8*a^2/c^4*x/d*(c^2*d*x^2+d)^(1/2)+3/8*a^2/c^4*ln(c^2*d*x/(c^2*d)^(1/2)
+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+b^2*(1/8*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d*arcsinh(c*x)^3+1/51
2*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5+8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3+8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x+(c
^2*x^2+1)^(1/2))*(8*arcsinh(c*x)^2-4*arcsinh(c*x)+1)/c^5/d/(c^2*x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3+2
*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2+1)^(1/2))*(2*arcsinh(c*x)^2-2*arcsinh(c*x)+1)/c^5/d/(c^2*x^2+1)-1/16
*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x-(c^2*x^2+1)^(1/2))*(2*arcsinh(c*x)^2+2*arc
sinh(c*x)+1)/c^5/d/(c^2*x^2+1)+1/512*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3-8
*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*x^2+1)^(1/2))*(8*arcsinh(c*x)^2+4*arcsinh(c*x)+1)/c^5/d/(c^2*x^2+1))+2*a
*b*(3/16*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c^5/d*arcsinh(c*x)^2+1/256*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5+8
*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3+8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x+(c^2*x^2+1)^(1/2))*(-1+4*arcsinh(c*x))
/c^5/d/(c^2*x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x^2+1)^(1/2))*
(-1+2*arcsinh(c*x))/c^5/d/(c^2*x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x-
(c^2*x^2+1)^(1/2))*(1+2*arcsinh(c*x))/c^5/d/(c^2*x^2+1)+1/256*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5-8*c^4*x^4*(c^2*
x^2+1)^(1/2)+12*c^3*x^3-8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*x^2+1)^(1/2))*(1+4*arcsinh(c*x))/c^5/d/(c^2*x^2
+1))

Fricas [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{\sqrt {c^{2} d x^{2} + d}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="fricas")

[Out]

integral((b^2*x^4*arcsinh(c*x)^2 + 2*a*b*x^4*arcsinh(c*x) + a^2*x^4)/sqrt(c^2*d*x^2 + d), x)

Sympy [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^{4} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \]

[In]

integrate(x**4*(a+b*asinh(c*x))**2/(c**2*d*x**2+d)**(1/2),x)

[Out]

Integral(x**4*(a + b*asinh(c*x))**2/sqrt(d*(c**2*x**2 + 1)), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F]

\[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2} x^{4}}{\sqrt {c^{2} d x^{2} + d}} \,d x } \]

[In]

integrate(x^4*(a+b*arcsinh(c*x))^2/(c^2*d*x^2+d)^(1/2),x, algorithm="giac")

[Out]

integrate((b*arcsinh(c*x) + a)^2*x^4/sqrt(c^2*d*x^2 + d), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 (a+b \text {arcsinh}(c x))^2}{\sqrt {d+c^2 d x^2}} \, dx=\int \frac {x^4\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \]

[In]

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(1/2),x)

[Out]

int((x^4*(a + b*asinh(c*x))^2)/(d + c^2*d*x^2)^(1/2), x)