\(\int (f+g x)^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx\) [35]

Optimal result

Integrand size = 30, antiderivative size = 431 \[ \int (f+g x)^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b f g x \sqrt {d+c^2 d x^2}}{3 c \sqrt {1+c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {b g^2 x^2 \sqrt {d+c^2 d x^2}}{16 c \sqrt {1+c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 f g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^2}+\frac {f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}}-\frac {g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c^3 \sqrt {1+c^2 x^2}} \]

[Out]

1/2*f^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/8*g^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+1/4*g^2*
x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+2/3*f*g*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2-2/3*
b*f*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-1/4*b*c*f^2*x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/16*b*g
^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-2/9*b*c*f*g*x^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/16*b*c*
g^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+1/4*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)-1/16*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 = 0.38 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {5845, 5838, 5785, 5783, 30, 5798, 5806, 5812} \[ \int (f+g x)^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {1}{2} f^2 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))+\frac {f^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {c^2 x^2+1}}+\frac {2 f g \left (c^2 x^2+1\right ) \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{3 c^2}+\frac {g^2 x \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))-\frac {g^2 \sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{16 b c^3 \sqrt {c^2 x^2+1}}-\frac {b c f^2 x^2 \sqrt {c^2 d x^2+d}}{4 \sqrt {c^2 x^2+1}}-\frac {2 b f g x \sqrt {c^2 d x^2+d}}{3 c \sqrt {c^2 x^2+1}}-\frac {2 b c f g x^3 \sqrt {c^2 d x^2+d}}{9 \sqrt {c^2 x^2+1}}-\frac {b g^2 x^2 \sqrt {c^2 d x^2+d}}{16 c \sqrt {c^2 x^2+1}}-\frac {b c g^2 x^4 \sqrt {c^2 d x^2+d}}{16 \sqrt {c^2 x^2+1}} \]

[In]

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

[Out]

(-2*b*f*g*x*Sqrt[d + c^2*d*x^2])/(3*c*Sqrt[1 + c^2*x^2]) - (b*c*f^2*x^2*Sqrt[d + c^2*d*x^2])/(4*Sqrt[1 + c^2*x
^2]) - (b*g^2*x^2*Sqrt[d + c^2*d*x^2])/(16*c*Sqrt[1 + c^2*x^2]) - (2*b*c*f*g*x^3*Sqrt[d + c^2*d*x^2])/(9*Sqrt[
1 + c^2*x^2]) - (b*c*g^2*x^4*Sqrt[d + c^2*d*x^2])/(16*Sqrt[1 + c^2*x^2]) + (f^2*x*Sqrt[d + c^2*d*x^2]*(a + b*A
rcSinh[c*x]))/2 + (g^2*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(8*c^2) + (g^2*x^3*Sqrt[d + c^2*d*x^2]*(a +
b*ArcSinh[c*x]))/4 + (2*f*g*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(3*c^2) + (f^2*Sqrt[d + c
^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[1 + c^2*x^2]) - (g^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)
/(16*b*c^3*Sqrt[1 + c^2*x^2])

Rule 30

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

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 {\sqrt {d+c^2 d x^2} \int (f+g x)^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\sqrt {d+c^2 d x^2} \int \left (f^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+2 f g x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))+g^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))\right ) \, dx}{\sqrt {1+c^2 x^2}} \\ & = \frac {\left (f^2 \sqrt {d+c^2 d x^2}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (2 f g \sqrt {d+c^2 d x^2}\right ) \int x \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{\sqrt {1+c^2 x^2}}+\frac {\left (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}{\sqrt {1+c^2 x^2}} \\ & = \frac {1}{2} f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 f g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^2}+\frac {\left (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}{2 \sqrt {1+c^2 x^2}}-\frac {\left (b c f^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{2 \sqrt {1+c^2 x^2}}-\frac {\left (2 b f g \sqrt {d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right ) \, dx}{3 c \sqrt {1+c^2 x^2}}+\frac {\left (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}{4 \sqrt {1+c^2 x^2}}-\frac {\left (b c g^2 \sqrt {d+c^2 d x^2}\right ) \int x^3 \, dx}{4 \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b f g x \sqrt {d+c^2 d x^2}}{3 c \sqrt {1+c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 f g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^2}+\frac {f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}}-\frac {\left (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}{8 c^2 \sqrt {1+c^2 x^2}}-\frac {\left (b g^2 \sqrt {d+c^2 d x^2}\right ) \int x \, dx}{8 c \sqrt {1+c^2 x^2}} \\ & = -\frac {2 b f g x \sqrt {d+c^2 d x^2}}{3 c \sqrt {1+c^2 x^2}}-\frac {b c f^2 x^2 \sqrt {d+c^2 d x^2}}{4 \sqrt {1+c^2 x^2}}-\frac {b g^2 x^2 \sqrt {d+c^2 d x^2}}{16 c \sqrt {1+c^2 x^2}}-\frac {2 b c f g x^3 \sqrt {d+c^2 d x^2}}{9 \sqrt {1+c^2 x^2}}-\frac {b c g^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}+\frac {1}{2} f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 c^2}+\frac {1}{4} g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 f g \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 c^2}+\frac {f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {1+c^2 x^2}}-\frac {g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c^3 \sqrt {1+c^2 x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.77 (sec) , antiderivative size = 301, normalized size of antiderivative = 0.70 \[ \int (f+g x)^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\frac {48 a c \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \left (12 c^2 f^2 x+3 g^2 x \left (1+2 c^2 x^2\right )+16 f \left (g+c^2 g x^2\right )\right )-256 b c f g \sqrt {d+c^2 d x^2} \left (3 c x+c^3 x^3-3 \left (1+c^2 x^2\right )^{3/2} \text {arcsinh}(c x)\right )+144 a \sqrt {d} (2 c f-g) (2 c f+g) \sqrt {1+c^2 x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )-144 b c^2 f^2 \sqrt {d+c^2 d x^2} (\cosh (2 \text {arcsinh}(c x))-2 \text {arcsinh}(c x) (\text {arcsinh}(c x)+\sinh (2 \text {arcsinh}(c x))))-9 b g^2 \sqrt {d+c^2 d x^2} \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{1152 c^3 \sqrt {1+c^2 x^2}} \]

[In]

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

[Out]

(48*a*c*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*(12*c^2*f^2*x + 3*g^2*x*(1 + 2*c^2*x^2) + 16*f*(g + c^2*g*x^2))
- 256*b*c*f*g*Sqrt[d + c^2*d*x^2]*(3*c*x + c^3*x^3 - 3*(1 + c^2*x^2)^(3/2)*ArcSinh[c*x]) + 144*a*Sqrt[d]*(2*c*
f - g)*(2*c*f + g)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] - 144*b*c^2*f^2*Sqrt[d + c^2*d*x
^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSinh[c*x] + Sinh[2*ArcSinh[c*x]])) - 9*b*g^2*Sqrt[d + c^2*d*x^2
]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x]*Sinh[4*ArcSinh[c*x]]))/(1152*c^3*Sqrt[1 + c^2*x^2]
)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(912\) vs. \(2(373)=746\).

Time = 0.72 (sec) , antiderivative size = 913, normalized size of antiderivative = 2.12

method	result	size
default	\(a \left (f^{2} \left (\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (\frac {x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}-\frac {\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )+\frac {2 f g \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} \left (4 c^{2} f^{2}-g^{2}\right )}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\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 ) g^{2} \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) f g \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{36 c^{2} \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 ) f^{2} \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) f g \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{4 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) f g \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{4 c^{2} \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 ) f^{2} \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) f g \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{36 c^{2} \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 ) g^{2} \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(913\)
parts	\(a \left (f^{2} \left (\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}\right )+g^{2} \left (\frac {x \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{4 c^{2} d}-\frac {\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2}+\frac {d \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 \sqrt {c^{2} d}}}{4 c^{2}}\right )+\frac {2 f g \left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3 c^{2} d}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (c x \right )^{2} \left (4 c^{2} f^{2}-g^{2}\right )}{16 \sqrt {c^{2} x^{2}+1}\, c^{3}}+\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 ) g^{2} \left (-1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}+4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}+3 c x \sqrt {c^{2} x^{2}+1}+1\right ) f g \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{36 c^{2} \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 ) f^{2} \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) f g \left (-1+\operatorname {arcsinh}\left (c x \right )\right )}{4 c^{2} \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) f g \left (\operatorname {arcsinh}\left (c x \right )+1\right )}{4 c^{2} \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 ) f^{2} \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (4 c^{4} x^{4}-4 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}+5 c^{2} x^{2}-3 c x \sqrt {c^{2} x^{2}+1}+1\right ) f g \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{36 c^{2} \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 ) g^{2} \left (1+4 \,\operatorname {arcsinh}\left (c x \right )\right )}{256 c^{3} \left (c^{2} x^{2}+1\right )}\right )\)	\(913\)

[In]

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

[Out]

a*(f^2*(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*(1/4*
x*(c^2*d*x^2+d)^(3/2)/c^2/d-1/4/c^2*(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)))+2/3*f*g/c^2/d*(c^2*d*x^2+d)^(3/2))+b*(1/16*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2*(4*c^2*f^
2-g^2)/(c^2*x^2+1)^(1/2)/c^3+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))*g^2*(-1+4*arcsinh(c*x))/c^3/(c^2*x^2+1)+1/36*(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))/c^2
/(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))*f^2*(-
1+2*arcsinh(c*x))/c/(c^2*x^2+1)+1/4*(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+arcsinh(c*
x))/c^2/(c^2*x^2+1)+1/4*(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)/c^2/(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))*f^2*(1+2*arc
sinh(c*x))/c/(c^2*x^2+1)+1/36*(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)/c^2/(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))*g^2*(1+4*arcsinh(c*x))/c^3/(c^2*
x^2+1))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((g*x+f)^2*(a+b*arcsinh(c*x))*(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(-2)]

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

[In]

integrate((g*x+f)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2),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. \[ \int (f+g x)^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,\sqrt {d\,c^2\,x^2+d} \,d x \]

[In]

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

[Out]

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