\(\int (f+g x)^2 (d+c^2 d x^2)^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) [40]

Optimal result

Integrand size = 30, antiderivative size = 651 \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b d f g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {5 b c d f^2 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b d g^2 x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {4 b c d f g x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {d g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {3 d f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {d g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c^3 \sqrt {1+c^2 x^2}} \]

[Out]

3/8*d*f^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/16*d*g^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+1/8
*d*g^2*x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/4*d*f^2*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/
2)+1/6*d*g^2*x^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+2/5*d*f*g*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))
*(c^2*d*x^2+d)^(1/2)/c^2-2/5*b*d*f*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-5/16*b*c*d*f^2*x^2*(c^2*d*x^2+d
)^(1/2)/(c^2*x^2+1)^(1/2)-1/32*b*d*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-4/15*b*c*d*f*g*x^3*(c^2*d*x
^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/16*b*c^3*d*f^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-7/96*b*c*d*g^2*x^4*(c
^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-2/25*b*c^3*d*f*g*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/36*b*c^3*d*g^
2*x^6*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+3/16*d*f^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c/(c^2*x^2+1
)^(1/2)-1/32*d*g^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c^3/(c^2*x^2+1)^(1/2)

Rubi [A] (verified)

Time = 0.53 (sec) , antiderivative size = 651, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5845, 5838, 5786, 5785, 5783, 30, 14, 5798, 200, 5808, 5806, 5812} \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {3}{8} d f^2 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^2 x \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {3 d f^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {c^2 x^2+1}}+\frac {2 d f g \left (c^2 x^2+1\right )^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {d g^2 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {1}{6} d g^2 x^3 \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {d g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{32 b c^3 \sqrt {c^2 x^2+1}}-\frac {5 b c d f^2 x^2 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}-\frac {2 b d f g x \sqrt {c^2 d x^2+d}}{5 c \sqrt {c^2 x^2+1}}-\frac {4 b c d f g x^3 \sqrt {c^2 d x^2+d}}{15 \sqrt {c^2 x^2+1}}-\frac {b d g^2 x^2 \sqrt {c^2 d x^2+d}}{32 c \sqrt {c^2 x^2+1}}-\frac {7 b c d g^2 x^4 \sqrt {c^2 d x^2+d}}{96 \sqrt {c^2 x^2+1}}-\frac {b c^3 d f^2 x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}}-\frac {2 b c^3 d f g x^5 \sqrt {c^2 d x^2+d}}{25 \sqrt {c^2 x^2+1}}-\frac {b c^3 d g^2 x^6 \sqrt {c^2 d x^2+d}}{36 \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

(-2*b*d*f*g*x*Sqrt[d + c^2*d*x^2])/(5*c*Sqrt[1 + c^2*x^2]) - (5*b*c*d*f^2*x^2*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1
+ c^2*x^2]) - (b*d*g^2*x^2*Sqrt[d + c^2*d*x^2])/(32*c*Sqrt[1 + c^2*x^2]) - (4*b*c*d*f*g*x^3*Sqrt[d + c^2*d*x^2
])/(15*Sqrt[1 + c^2*x^2]) - (b*c^3*d*f^2*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) - (7*b*c*d*g^2*x^4*Sq
rt[d + c^2*d*x^2])/(96*Sqrt[1 + c^2*x^2]) - (2*b*c^3*d*f*g*x^5*Sqrt[d + c^2*d*x^2])/(25*Sqrt[1 + c^2*x^2]) - (
b*c^3*d*g^2*x^6*Sqrt[d + c^2*d*x^2])/(36*Sqrt[1 + c^2*x^2]) + (3*d*f^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*
x]))/8 + (d*g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(16*c^2) + (d*g^2*x^3*Sqrt[d + c^2*d*x^2]*(a + b*A
rcSinh[c*x]))/8 + (d*f^2*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/4 + (d*g^2*x^3*(1 + c^2*x^2
)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/6 + (2*d*f*g*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*
x]))/(5*c^2) + (3*d*f^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(16*b*c*Sqrt[1 + c^2*x^2]) - (d*g^2*Sqrt[d
+ c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*c^3*Sqrt[1 + c^2*x^2])

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 200

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^n)^p, x], x] /; FreeQ[{a, b}, x]
&& IGtQ[n, 0] && IGtQ[p, 0]

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 5785

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[(1/2)*Simp[Sqrt[d + e*x^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/2)*Simp[Sqrt[d + e*x^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 5786

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/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*A
rcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]

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 5808

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

Rule 5838

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

Rule 5845

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int (f+g x)^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d \sqrt {d+c^2 d x^2}\right ) \int \left (f^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+2 f g x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+g^2 x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (d f^2 \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (2 d f g \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (d g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{4} d f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {\left (3 d f^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (b c d f^2 \sqrt {d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (2 b d f g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx}{5 c \sqrt {1+c^2 x^2}}+\frac {\left (d g^2 \sqrt {d+c^2 d x^2}\right ) \int x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b c d g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \left (1+c^2 x^2\right ) \, dx}{6 \sqrt {1+c^2 x^2}} \\ & = \frac {3}{8} d f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{8} d g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {\left (3 d f^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d f^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{4 \sqrt {1+c^2 x^2}}-\frac {\left (3 b c d f^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (2 b d f g \sqrt {d+c^2 d x^2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx}{5 c \sqrt {1+c^2 x^2}}+\frac {\left (d g^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{8 \sqrt {1+c^2 x^2}}-\frac {\left (b c d g^2 \sqrt {d+c^2 d x^2}\right ) \int \left (x^3+c^2 x^5\right ) \, dx}{6 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b d f g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {5 b c d f^2 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {4 b c d f g x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {d g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {3 d f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {\left (d g^2 \sqrt {d+c^2 d x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{16 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b d g^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{16 c \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b d f g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {5 b c d f^2 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b d g^2 x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {4 b c d f g x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {d g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {3 d f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {d g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.16 (sec) , antiderivative size = 416, normalized size of antiderivative = 0.64 \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^2 \left (1+c^2 x^2\right ) \left (b \left (-175 g^2+90 c^2 \left (85 f^2+256 f g x+20 g^2 x^2\right )+120 c^4 x^2 \left (150 f^2+128 f g x+35 g^2 x^2\right )+16 c^6 x^4 \left (225 f^2+288 f g x+100 g^2 x^2\right )\right )-240 a c \sqrt {1+c^2 x^2} \left (96 f g \left (1+c^2 x^2\right )^2+30 c^2 f^2 x \left (5+2 c^2 x^2\right )+5 g^2 x \left (3+14 c^2 x^2+8 c^4 x^4\right )\right )\right )+1800 b d^2 \left (6 c^2 f^2-g^2\right ) \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+3600 a d^{3/2} \left (6 c^2 f^2-g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+60 b d^2 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (15 \left (16 c^2 f^2-g^2\right ) \sinh (2 \text {arcsinh}(c x))+15 \left (2 c^2 f^2+g^2\right ) \sinh (4 \text {arcsinh}(c x))+g \left (384 c f \left (1+c^2 x^2\right )^{5/2}+5 g \sinh (6 \text {arcsinh}(c x))\right )\right )}{57600 c^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \]

[In]

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

[Out]

(-(d^2*(1 + c^2*x^2)*(b*(-175*g^2 + 90*c^2*(85*f^2 + 256*f*g*x + 20*g^2*x^2) + 120*c^4*x^2*(150*f^2 + 128*f*g*
x + 35*g^2*x^2) + 16*c^6*x^4*(225*f^2 + 288*f*g*x + 100*g^2*x^2)) - 240*a*c*Sqrt[1 + c^2*x^2]*(96*f*g*(1 + c^2
*x^2)^2 + 30*c^2*f^2*x*(5 + 2*c^2*x^2) + 5*g^2*x*(3 + 14*c^2*x^2 + 8*c^4*x^4)))) + 1800*b*d^2*(6*c^2*f^2 - g^2
)*(1 + c^2*x^2)*ArcSinh[c*x]^2 + 3600*a*d^(3/2)*(6*c^2*f^2 - g^2)*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*Log[c*
d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 60*b*d^2*(1 + c^2*x^2)*ArcSinh[c*x]*(15*(16*c^2*f^2 - g^2)*Sinh[2*ArcSinh
[c*x]] + 15*(2*c^2*f^2 + g^2)*Sinh[4*ArcSinh[c*x]] + g*(384*c*f*(1 + c^2*x^2)^(5/2) + 5*g*Sinh[6*ArcSinh[c*x]]
)))/(57600*c^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2])

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1567\) vs. \(2(567)=1134\).

Time = 0.82 (sec) , antiderivative size = 1568, normalized size of antiderivative = 2.41

method	result	size
default	\(\text {Expression too large to display}\)	\(1568\)
parts	\(\text {Expression too large to display}\)	\(1568\)

[In]

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

[Out]

a*(f^2*(1/4*x*(c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(c^2*d*x^2+d)^(1/2)+1/2*d*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d
)^(1/2))/(c^2*d)^(1/2)))+g^2*(1/6*x*(c^2*d*x^2+d)^(5/2)/c^2/d-1/6/c^2*(1/4*x*(c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*
(c^2*d*x^2+d)^(1/2)+1/2*d*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2))))+2/5*f*g/c^2/d*(c^2*d*
x^2+d)^(5/2))+b*(1/32*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2*(6*c^2*f^2-g^2)*d/(c^2*x^2+1)^(1/2)/c^3+1/2304*(d*(
c^2*x^2+1))^(1/2)*(32*c^7*x^7+32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5+48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3+
18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x+(c^2*x^2+1)^(1/2))*g^2*(-1+6*arcsinh(c*x))*d/c^3/(c^2*x^2+1)+1/400*(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)*f*g*(-1+5*arcsinh(c*x))*d/c^2/(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)*c^2*f^2-
2*c^2*f^2+4*arcsinh(c*x)*g^2-g^2)*d/c^3/(c^2*x^2+1)+1/48*(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)*f*g*(-1+3*arcsinh(c*x))*d/c^2/(c^2*x^2+1)+1/256*(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))*(32*arcsinh(c*x)*c^2*f^2-16*c^2*f^2-2*arc
sinh(c*x)*g^2+g^2)*d/c^3/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*f*g*(-1+arcsi
nh(c*x))*d/c^2/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*f*g*(arcsinh(c*x)+1)*d/
c^2/(c^2*x^2+1)+1/256*(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))*(3
2*arcsinh(c*x)*c^2*f^2+16*c^2*f^2-2*arcsinh(c*x)*g^2-g^2)*d/c^3/(c^2*x^2+1)+1/48*(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)*f*g*(3*arcsinh(c*x)+1)*d/c^2/(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)*c^2*f^2+2*c^2*f^2+4*arcsinh(c*x)*g^2+g^2)*d/c^3/(c^2*x^2+1)+1/400*(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)*f*g*(1+5*arcsinh(c*x))*d/c^2/(c^2*x^2+1)+1/2304*(d*(c^2*x^2+1))^(1/2)*(32*c^7*x^7-32*c
^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5-48*c^4*x^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3-18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x
-(c^2*x^2+1)^(1/2))*g^2*(1+6*arcsinh(c*x))*d/c^3/(c^2*x^2+1))

Fricas [F]

\[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]

[In]

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

[Out]

integral((a*c^2*d*g^2*x^4 + 2*a*c^2*d*f*g*x^3 + 2*a*d*f*g*x + a*d*f^2 + (a*c^2*d*f^2 + a*d*g^2)*x^2 + (b*c^2*d
*g^2*x^4 + 2*b*c^2*d*f*g*x^3 + 2*b*d*f*g*x + b*d*f^2 + (b*c^2*d*f^2 + b*d*g^2)*x^2)*arcsinh(c*x))*sqrt(c^2*d*x
^2 + d), x)

Sympy [F]

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

[In]

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

[Out]

Integral((d*(c**2*x**2 + 1))**(3/2)*(a + b*asinh(c*x))*(f + g*x)**2, x)

Maxima [F(-2)]

Exception generated. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> an error occurred running a Giac command:INPUT:sage2OUTPUT:sym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

Mupad [F(-1)]

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

[In]

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

[Out]

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