\(\int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx\) [258]

Optimal result

Integrand size = 28, antiderivative size = 358 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=-\frac {52 b^2 \sqrt {d+c^2 d x^2}}{225 c^4}+\frac {4 a b x \sqrt {d+c^2 d x^2}}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {26 b^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{675 c^4}+\frac {2 b^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}}{125 c^4}+\frac {4 b^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {2 b x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \]

[Out]

-52/225*b^2*(c^2*d*x^2+d)^(1/2)/c^4-26/675*b^2*(c^2*x^2+1)*(c^2*d*x^2+d)^(1/2)/c^4+2/125*b^2*(c^2*x^2+1)^2*(c^
2*d*x^2+d)^(1/2)/c^4-2/15*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/c^4+1/15*x^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^
2+d)^(1/2)/c^2+1/5*x^4*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)+4/15*a*b*x*(c^2*d*x^2+d)^(1/2)/c^3/(c^2*x^2+1)
^(1/2)+4/15*b^2*x*arcsinh(c*x)*(c^2*d*x^2+d)^(1/2)/c^3/(c^2*x^2+1)^(1/2)-2/45*b*x^3*(a+b*arcsinh(c*x))*(c^2*d*
x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-2/25*b*c*x^5*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5806, 5812, 5798, 5772, 267, 5776, 272, 45} \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {x^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{15 c^2}-\frac {2 b c x^5 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{25 \sqrt {c^2 x^2+1}}+\frac {1}{5} x^4 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{45 c \sqrt {c^2 x^2+1}}-\frac {2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{15 c^4}+\frac {4 a b x \sqrt {c^2 d x^2+d}}{15 c^3 \sqrt {c^2 x^2+1}}+\frac {4 b^2 x \text {arcsinh}(c x) \sqrt {c^2 d x^2+d}}{15 c^3 \sqrt {c^2 x^2+1}}+\frac {2 b^2 \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d}}{125 c^4}-\frac {26 b^2 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d}}{675 c^4}-\frac {52 b^2 \sqrt {c^2 d x^2+d}}{225 c^4} \]

[In]

Int[x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]

[Out]

(-52*b^2*Sqrt[d + c^2*d*x^2])/(225*c^4) + (4*a*b*x*Sqrt[d + c^2*d*x^2])/(15*c^3*Sqrt[1 + c^2*x^2]) - (26*b^2*(
1 + c^2*x^2)*Sqrt[d + c^2*d*x^2])/(675*c^4) + (2*b^2*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2])/(125*c^4) + (4*b^2*x
*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x])/(15*c^3*Sqrt[1 + c^2*x^2]) - (2*b*x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*
x]))/(45*c*Sqrt[1 + c^2*x^2]) - (2*b*c*x^5*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(25*Sqrt[1 + c^2*x^2]) -
(2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(15*c^4) + (x^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(15
*c^2) + (x^4*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/5

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 267

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 5772

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

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 5798

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/(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 5806

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(
f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*ArcSinh[c*x])^n/(f*(m + 2))), x] + (Dist[(1/(m + 2))*Simp[Sqrt[d + e*x^2]
/Sqrt[1 + c^2*x^2]], Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x], x] - Dist[b*c*(n/(f*(m + 2)))
*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]], Int[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a,
b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[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 {1}{5} x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {\sqrt {d+c^2 d x^2} \int \frac {x^3 (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{5 \sqrt {1+c^2 x^2}}-\frac {\left (2 b c \sqrt {d+c^2 d x^2}\right ) \int x^4 (a+b \text {arcsinh}(c x)) \, dx}{5 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b c x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2-\frac {\left (2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))^2}{\sqrt {1+c^2 x^2}} \, dx}{15 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (2 b \sqrt {d+c^2 d x^2}\right ) \int x^2 (a+b \text {arcsinh}(c x)) \, dx}{15 c \sqrt {1+c^2 x^2}}+\frac {\left (2 b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^5}{\sqrt {1+c^2 x^2}} \, dx}{25 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {\left (2 b^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^3}{\sqrt {1+c^2 x^2}} \, dx}{45 \sqrt {1+c^2 x^2}}+\frac {\left (4 b \sqrt {d+c^2 d x^2}\right ) \int (a+b \text {arcsinh}(c x)) \, dx}{15 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{25 \sqrt {1+c^2 x^2}} \\ & = \frac {4 a b x \sqrt {d+c^2 d x^2}}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {2 b x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {\left (b^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{45 \sqrt {1+c^2 x^2}}+\frac {\left (4 b^2 \sqrt {d+c^2 d x^2}\right ) \int \text {arcsinh}(c x) \, dx}{15 c^3 \sqrt {1+c^2 x^2}}+\frac {\left (b^2 c^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{c^4 \sqrt {1+c^2 x}}-\frac {2 \sqrt {1+c^2 x}}{c^4}+\frac {\left (1+c^2 x\right )^{3/2}}{c^4}\right ) \, dx,x,x^2\right )}{25 \sqrt {1+c^2 x^2}} \\ & = \frac {2 b^2 \sqrt {d+c^2 d x^2}}{25 c^4}+\frac {4 a b x \sqrt {d+c^2 d x^2}}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {4 b^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{75 c^4}+\frac {2 b^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}}{125 c^4}+\frac {4 b^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {2 b x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {\left (b^2 \sqrt {d+c^2 d x^2}\right ) \text {Subst}\left (\int \left (-\frac {1}{c^2 \sqrt {1+c^2 x}}+\frac {\sqrt {1+c^2 x}}{c^2}\right ) \, dx,x,x^2\right )}{45 \sqrt {1+c^2 x^2}}-\frac {\left (4 b^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x}{\sqrt {1+c^2 x^2}} \, dx}{15 c^2 \sqrt {1+c^2 x^2}} \\ & = -\frac {52 b^2 \sqrt {d+c^2 d x^2}}{225 c^4}+\frac {4 a b x \sqrt {d+c^2 d x^2}}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {26 b^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2}}{675 c^4}+\frac {2 b^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2}}{125 c^4}+\frac {4 b^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{15 c^3 \sqrt {1+c^2 x^2}}-\frac {2 b x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{45 c \sqrt {1+c^2 x^2}}-\frac {2 b c x^5 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{25 \sqrt {1+c^2 x^2}}-\frac {2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^4}+\frac {x^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{15 c^2}+\frac {1}{5} x^4 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.62 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {\sqrt {d+c^2 d x^2} \left (225 \left (-2+3 c^2 x^2\right ) \left (a+a c^2 x^2\right )^2-30 a b c x \sqrt {1+c^2 x^2} \left (-30+5 c^2 x^2+9 c^4 x^4\right )+2 b^2 \left (-428-439 c^2 x^2+16 c^4 x^4+27 c^6 x^6\right )-30 b \left (-15 a \left (1+c^2 x^2\right )^2 \left (-2+3 c^2 x^2\right )+b c x \sqrt {1+c^2 x^2} \left (-30+5 c^2 x^2+9 c^4 x^4\right )\right ) \text {arcsinh}(c x)+225 \left (-2+3 c^2 x^2\right ) \left (b+b c^2 x^2\right )^2 \text {arcsinh}(c x)^2\right )}{3375 c^4 \left (1+c^2 x^2\right )} \]

[In]

Integrate[x^3*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2,x]

[Out]

(Sqrt[d + c^2*d*x^2]*(225*(-2 + 3*c^2*x^2)*(a + a*c^2*x^2)^2 - 30*a*b*c*x*Sqrt[1 + c^2*x^2]*(-30 + 5*c^2*x^2 +
9*c^4*x^4) + 2*b^2*(-428 - 439*c^2*x^2 + 16*c^4*x^4 + 27*c^6*x^6) - 30*b*(-15*a*(1 + c^2*x^2)^2*(-2 + 3*c^2*x
^2) + b*c*x*Sqrt[1 + c^2*x^2]*(-30 + 5*c^2*x^2 + 9*c^4*x^4))*ArcSinh[c*x] + 225*(-2 + 3*c^2*x^2)*(b + b*c^2*x^
2)^2*ArcSinh[c*x]^2))/(3375*c^4*(1 + c^2*x^2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1161\) vs. \(2(310)=620\).

Time = 0.38 (sec) , antiderivative size = 1162, normalized size of antiderivative = 3.25

method	result	size
default	\(\text {Expression too large to display}\)	\(1162\)
parts	\(\text {Expression too large to display}\)	\(1162\)

[In]

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

[Out]

a^2*(1/5*x^2*(c^2*d*x^2+d)^(3/2)/c^2/d-2/15/d/c^4*(c^2*d*x^2+d)^(3/2))+b^2*(1/4000*(d*(c^2*x^2+1))^(1/2)*(16*c
^6*x^6+16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4+20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^2+1)^(1/2)
+1)*(25*arcsinh(c*x)^2-10*arcsinh(c*x)+2)/c^4/(c^2*x^2+1)-1/864*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^
2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(9*arcsinh(c*x)^2-6*arcsinh(c*x)+2)/c^4/(c^2*x^2+1)-1/16*(
d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)^2-2*arcsinh(c*x)+2)/c^4/(c^2*x^2+1)-1/16*
(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)^2+2*arcsinh(c*x)+2)/c^4/(c^2*x^2+1)-1/86
4*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*(9*arcsinh
(c*x)^2+6*arcsinh(c*x)+2)/c^4/(c^2*x^2+1)+1/4000*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6-16*c^5*x^5*(c^2*x^2+1)^(1/2
)+28*c^4*x^4-20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2-5*c*x*(c^2*x^2+1)^(1/2)+1)*(25*arcsinh(c*x)^2+10*arcsinh(
c*x)+2)/c^4/(c^2*x^2+1))+2*a*b*(1/800*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6+16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^
4+20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+5*arcsinh(c*x))/c^4/(c^2*x^2+1)-1/288
*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*(-1+3*arcsi
nh(c*x))/c^4/(c^2*x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*(-1+arcsinh(c*x))/c^4/(c
^2*x^2+1)-1/16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*(arcsinh(c*x)+1)/c^4/(c^2*x^2+1)-1/288*
(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*(3*arcsinh(c
*x)+1)/c^4/(c^2*x^2+1)+1/800*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6-16*c^5*x^5*(c^2*x^2+1)^(1/2)+28*c^4*x^4-20*c^3*
x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2-5*c*x*(c^2*x^2+1)^(1/2)+1)*(1+5*arcsinh(c*x))/c^4/(c^2*x^2+1))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 316, normalized size of antiderivative = 0.88 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {225 \, {\left (3 \, b^{2} c^{6} x^{6} + 4 \, b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2} - 2 \, b^{2}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 30 \, {\left (45 \, a b c^{6} x^{6} + 60 \, a b c^{4} x^{4} - 15 \, a b c^{2} x^{2} - 30 \, a b - {\left (9 \, b^{2} c^{5} x^{5} + 5 \, b^{2} c^{3} x^{3} - 30 \, b^{2} c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) + {\left (27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{6} x^{6} + 4 \, {\left (225 \, a^{2} + 8 \, b^{2}\right )} c^{4} x^{4} - {\left (225 \, a^{2} + 878 \, b^{2}\right )} c^{2} x^{2} - 450 \, a^{2} - 856 \, b^{2} - 30 \, {\left (9 \, a b c^{5} x^{5} + 5 \, a b c^{3} x^{3} - 30 \, a b c x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \sqrt {c^{2} d x^{2} + d}}{3375 \, {\left (c^{6} x^{2} + c^{4}\right )}} \]

[In]

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

[Out]

1/3375*(225*(3*b^2*c^6*x^6 + 4*b^2*c^4*x^4 - b^2*c^2*x^2 - 2*b^2)*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 +
1))^2 + 30*(45*a*b*c^6*x^6 + 60*a*b*c^4*x^4 - 15*a*b*c^2*x^2 - 30*a*b - (9*b^2*c^5*x^5 + 5*b^2*c^3*x^3 - 30*b
^2*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d)*log(c*x + sqrt(c^2*x^2 + 1)) + (27*(25*a^2 + 2*b^2)*c^6*x^6 + 4
*(225*a^2 + 8*b^2)*c^4*x^4 - (225*a^2 + 878*b^2)*c^2*x^2 - 450*a^2 - 856*b^2 - 30*(9*a*b*c^5*x^5 + 5*a*b*c^3*x
^3 - 30*a*b*c*x)*sqrt(c^2*x^2 + 1))*sqrt(c^2*d*x^2 + d))/(c^6*x^2 + c^4)

Sympy [F]

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

[In]

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

[Out]

Integral(x**3*sqrt(d*(c**2*x**2 + 1))*(a + b*asinh(c*x))**2, x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 302, normalized size of antiderivative = 0.84 \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{15} \, b^{2} {\left (\frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{15} \, a b {\left (\frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} \operatorname {arsinh}\left (c x\right ) + \frac {1}{15} \, a^{2} {\left (\frac {3 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} x^{2}}{c^{2} d} - \frac {2 \, {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}}}{c^{4} d}\right )} + \frac {2}{3375} \, b^{2} {\left (\frac {27 \, \sqrt {c^{2} x^{2} + 1} c^{2} \sqrt {d} x^{4} - 11 \, \sqrt {c^{2} x^{2} + 1} \sqrt {d} x^{2} - \frac {428 \, \sqrt {c^{2} x^{2} + 1} \sqrt {d}}{c^{2}}}{c^{2}} - \frac {15 \, {\left (9 \, c^{4} \sqrt {d} x^{5} + 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} \operatorname {arsinh}\left (c x\right )}{c^{3}}\right )} - \frac {2 \, {\left (9 \, c^{4} \sqrt {d} x^{5} + 5 \, c^{2} \sqrt {d} x^{3} - 30 \, \sqrt {d} x\right )} a b}{225 \, c^{3}} \]

[In]

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

[Out]

1/15*b^2*(3*(c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(3/2)/(c^4*d))*arcsinh(c*x)^2 + 2/15*a*b*(3*
(c^2*d*x^2 + d)^(3/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(3/2)/(c^4*d))*arcsinh(c*x) + 1/15*a^2*(3*(c^2*d*x^2 + d
)^(3/2)*x^2/(c^2*d) - 2*(c^2*d*x^2 + d)^(3/2)/(c^4*d)) + 2/3375*b^2*((27*sqrt(c^2*x^2 + 1)*c^2*sqrt(d)*x^4 - 1
1*sqrt(c^2*x^2 + 1)*sqrt(d)*x^2 - 428*sqrt(c^2*x^2 + 1)*sqrt(d)/c^2)/c^2 - 15*(9*c^4*sqrt(d)*x^5 + 5*c^2*sqrt(
d)*x^3 - 30*sqrt(d)*x)*arcsinh(c*x)/c^3) - 2/225*(9*c^4*sqrt(d)*x^5 + 5*c^2*sqrt(d)*x^3 - 30*sqrt(d)*x)*a*b/c^
3

Giac [F(-2)]

Exception generated. \[ \int x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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