\(\int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx\) [47]

Optimal result

Integrand size = 30, antiderivative size = 430 \[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=-\frac {3 b f^2 g x \sqrt {1+c^2 x^2}}{c \sqrt {d+c^2 d x^2}}+\frac {2 b g^3 x \sqrt {1+c^2 x^2}}{3 c^3 \sqrt {d+c^2 d x^2}}-\frac {3 b f g^2 x^2 \sqrt {1+c^2 x^2}}{4 c \sqrt {d+c^2 d x^2}}-\frac {b g^3 x^3 \sqrt {1+c^2 x^2}}{9 c \sqrt {d+c^2 d x^2}}+\frac {3 f^2 g \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c^2 \sqrt {d+c^2 d x^2}}-\frac {2 g^3 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c^4 \sqrt {d+c^2 d x^2}}+\frac {3 f g^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{2 c^2 \sqrt {d+c^2 d x^2}}+\frac {g^3 x^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c^2 \sqrt {d+c^2 d x^2}}+\frac {f^3 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+c^2 d x^2}}-\frac {3 f g^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c^3 \sqrt {d+c^2 d x^2}} \]

[Out]

3*f^2*g*(c^2*x^2+1)*(a+b*arcsinh(c*x))/c^2/(c^2*d*x^2+d)^(1/2)-2/3*g^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))/c^4/(c^2
*d*x^2+d)^(1/2)+3/2*f*g^2*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))/c^2/(c^2*d*x^2+d)^(1/2)+1/3*g^3*x^2*(c^2*x^2+1)*(a+
b*arcsinh(c*x))/c^2/(c^2*d*x^2+d)^(1/2)-3*b*f^2*g*x*(c^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)+2/3*b*g^3*x*(c^2*x
^2+1)^(1/2)/c^3/(c^2*d*x^2+d)^(1/2)-3/4*b*f*g^2*x^2*(c^2*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)-1/9*b*g^3*x^3*(c^2
*x^2+1)^(1/2)/c/(c^2*d*x^2+d)^(1/2)+1/2*f^3*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/b/c/(c^2*d*x^2+d)^(1/2)-3/4
*f*g^2*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)/b/c^3/(c^2*d*x^2+d)^(1/2)

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 430, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {5845, 5838, 5783, 5798, 8, 5812, 30} \[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {f^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {c^2 d x^2+d}}+\frac {3 f^2 g \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{c^2 \sqrt {c^2 d x^2+d}}+\frac {3 f g^2 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{2 c^2 \sqrt {c^2 d x^2+d}}+\frac {g^3 x^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{3 c^2 \sqrt {c^2 d x^2+d}}-\frac {2 g^3 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{3 c^4 \sqrt {c^2 d x^2+d}}-\frac {3 f g^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{4 b c^3 \sqrt {c^2 d x^2+d}}-\frac {3 b f^2 g x \sqrt {c^2 x^2+1}}{c \sqrt {c^2 d x^2+d}}-\frac {3 b f g^2 x^2 \sqrt {c^2 x^2+1}}{4 c \sqrt {c^2 d x^2+d}}-\frac {b g^3 x^3 \sqrt {c^2 x^2+1}}{9 c \sqrt {c^2 d x^2+d}}+\frac {2 b g^3 x \sqrt {c^2 x^2+1}}{3 c^3 \sqrt {c^2 d x^2+d}} \]

[In]

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

[Out]

(-3*b*f^2*g*x*Sqrt[1 + c^2*x^2])/(c*Sqrt[d + c^2*d*x^2]) + (2*b*g^3*x*Sqrt[1 + c^2*x^2])/(3*c^3*Sqrt[d + c^2*d
*x^2]) - (3*b*f*g^2*x^2*Sqrt[1 + c^2*x^2])/(4*c*Sqrt[d + c^2*d*x^2]) - (b*g^3*x^3*Sqrt[1 + c^2*x^2])/(9*c*Sqrt
[d + c^2*d*x^2]) + (3*f^2*g*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c^2*Sqrt[d + c^2*d*x^2]) - (2*g^3*(1 + c^2*x^
2)*(a + b*ArcSinh[c*x]))/(3*c^4*Sqrt[d + c^2*d*x^2]) + (3*f*g^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(2*c^2*S
qrt[d + c^2*d*x^2]) + (g^3*x^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(3*c^2*Sqrt[d + c^2*d*x^2]) + (f^3*Sqrt[1 +
c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(2*b*c*Sqrt[d + c^2*d*x^2]) - (3*f*g^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]
)^2)/(4*b*c^3*Sqrt[d + c^2*d*x^2])

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

Mathematica [A] (verified)

Time = 1.01 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.71 \[ \int \frac {(f+g x)^3 (a+b \text {arcsinh}(c x))}{\sqrt {d+c^2 d x^2}} \, dx=\frac {4 \sqrt {d} g \left (-2 b c x \sqrt {1+c^2 x^2} \left (-6 g^2+c^2 \left (27 f^2+g^2 x^2\right )\right )+3 a \left (1+c^2 x^2\right ) \left (-4 g^2+c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right )\right )+12 b \sqrt {d} g \left (1+c^2 x^2\right ) \left (-4 g^2+c^2 \left (18 f^2+9 f g x+2 g^2 x^2\right )\right ) \text {arcsinh}(c x)+18 b c \sqrt {d} f \left (2 c^2 f^2-3 g^2\right ) \sqrt {1+c^2 x^2} \text {arcsinh}(c x)^2-27 b c \sqrt {d} f g^2 \sqrt {1+c^2 x^2} \cosh (2 \text {arcsinh}(c x))+36 a c f \left (2 c^2 f^2-3 g^2\right ) \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )}{72 c^4 \sqrt {d} \sqrt {d+c^2 d x^2}} \]

[In]

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

[Out]

(4*Sqrt[d]*g*(-2*b*c*x*Sqrt[1 + c^2*x^2]*(-6*g^2 + c^2*(27*f^2 + g^2*x^2)) + 3*a*(1 + c^2*x^2)*(-4*g^2 + c^2*(
18*f^2 + 9*f*g*x + 2*g^2*x^2))) + 12*b*Sqrt[d]*g*(1 + c^2*x^2)*(-4*g^2 + c^2*(18*f^2 + 9*f*g*x + 2*g^2*x^2))*A
rcSinh[c*x] + 18*b*c*Sqrt[d]*f*(2*c^2*f^2 - 3*g^2)*Sqrt[1 + c^2*x^2]*ArcSinh[c*x]^2 - 27*b*c*Sqrt[d]*f*g^2*Sqr
t[1 + c^2*x^2]*Cosh[2*ArcSinh[c*x]] + 36*a*c*f*(2*c^2*f^2 - 3*g^2)*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqr
t[d + c^2*d*x^2]])/(72*c^4*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. \(784\) vs. \(2(382)=764\).

Time = 0.93 (sec) , antiderivative size = 785, normalized size of antiderivative = 1.83

method	result	size
default	\(a \left (\frac {f^{3} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+g^{3} \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+3 f \,g^{2} \left (\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {\ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )+\frac {3 f^{2} g \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (c x \right )^{2} \left (2 c^{2} f^{2}-3 g^{2}\right )}{4 \sqrt {c^{2} x^{2}+1}\, c^{3} d}+\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 ) g^{3} \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \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 \,g^{2} \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (4 \,\operatorname {arcsinh}\left (c x \right ) c^{2} f^{2}-4 c^{2} f^{2}-\operatorname {arcsinh}\left (c x \right ) g^{2}+g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (4 \,\operatorname {arcsinh}\left (c x \right ) c^{2} f^{2}+4 c^{2} f^{2}-\operatorname {arcsinh}\left (c x \right ) g^{2}-g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \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 \,g^{2} \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{3} d \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 ) g^{3} \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\)	\(785\)
parts	\(a \left (\frac {f^{3} \ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+g^{3} \left (\frac {x^{2} \sqrt {c^{2} d \,x^{2}+d}}{3 c^{2} d}-\frac {2 \sqrt {c^{2} d \,x^{2}+d}}{3 d \,c^{4}}\right )+3 f \,g^{2} \left (\frac {x \sqrt {c^{2} d \,x^{2}+d}}{2 c^{2} d}-\frac {\ln \left (\frac {c^{2} d x}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{2 c^{2} \sqrt {c^{2} d}}\right )+\frac {3 f^{2} g \sqrt {c^{2} d \,x^{2}+d}}{c^{2} d}\right )+b \left (\frac {\sqrt {d \left (c^{2} x^{2}+1\right )}\, f \operatorname {arcsinh}\left (c x \right )^{2} \left (2 c^{2} f^{2}-3 g^{2}\right )}{4 \sqrt {c^{2} x^{2}+1}\, c^{3} d}+\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 ) g^{3} \left (-1+3 \,\operatorname {arcsinh}\left (c x \right )\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \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 \,g^{2} \left (-1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{3} d \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}+c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (4 \,\operatorname {arcsinh}\left (c x \right ) c^{2} f^{2}-4 c^{2} f^{2}-\operatorname {arcsinh}\left (c x \right ) g^{2}+g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (c^{2} x^{2}-c x \sqrt {c^{2} x^{2}+1}+1\right ) g \left (4 \,\operatorname {arcsinh}\left (c x \right ) c^{2} f^{2}+4 c^{2} f^{2}-\operatorname {arcsinh}\left (c x \right ) g^{2}-g^{2}\right )}{8 c^{4} d \left (c^{2} x^{2}+1\right )}+\frac {3 \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 \,g^{2} \left (1+2 \,\operatorname {arcsinh}\left (c x \right )\right )}{16 c^{3} d \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 ) g^{3} \left (3 \,\operatorname {arcsinh}\left (c x \right )+1\right )}{72 c^{4} d \left (c^{2} x^{2}+1\right )}\right )\)	\(785\)

[In]

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

[Out]

a*(f^3*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+g^3*(1/3*x^2/c^2/d*(c^2*d*x^2+d)^(1/2)-2/3/
d/c^4*(c^2*d*x^2+d)^(1/2))+3*f*g^2*(1/2*x/c^2/d*(c^2*d*x^2+d)^(1/2)-1/2/c^2*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^
2+d)^(1/2))/(c^2*d)^(1/2))+3*f^2*g/c^2/d*(c^2*d*x^2+d)^(1/2))+b*(1/4*(d*(c^2*x^2+1))^(1/2)*f*arcsinh(c*x)^2*(2
*c^2*f^2-3*g^2)/(c^2*x^2+1)^(1/2)/c^3/d+1/72*(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)*g^3*(-1+3*arcsinh(c*x))/c^4/d/(c^2*x^2+1)+3/16*(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*g^2*(-1+2*arcsinh(c*x))/c^3/d/(c^2*x^2+1)+3/8*(d*(c^
2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*g*(4*arcsinh(c*x)*c^2*f^2-4*c^2*f^2-arcsinh(c*x)*g^2+g^2)/c^
4/d/(c^2*x^2+1)+3/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*g*(4*arcsinh(c*x)*c^2*f^2+4*c^2*f^
2-arcsinh(c*x)*g^2-g^2)/c^4/d/(c^2*x^2+1)+3/16*(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*g^2*(1+2*arcsinh(c*x))/c^3/d/(c^2*x^2+1)+1/72*(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)*g^3*(3*arcsinh(c*x)+1)/c^4/d/(c^2*x^2+1))

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((g*x+f)^3*(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]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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