3.40 \(\int (f+g x)^2 (d+c^2 d x^2)^{3/2} (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=651 \[ \frac{3}{8} d f^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d f^2 x \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d f^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt{c^2 x^2+1}}+\frac{2 d f g \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{d g^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}-\frac{d g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{c^2 x^2+1}}-\frac{b c^3 d f^2 x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{5 b c d f^2 x^2 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{2 b c^3 d f g x^5 \sqrt{c^2 d x^2+d}}{25 \sqrt{c^2 x^2+1}}-\frac{4 b c d f g x^3 \sqrt{c^2 d x^2+d}}{15 \sqrt{c^2 x^2+1}}-\frac{2 b d f g x \sqrt{c^2 d x^2+d}}{5 c \sqrt{c^2 x^2+1}}-\frac{b c^3 d g^2 x^6 \sqrt{c^2 d x^2+d}}{36 \sqrt{c^2 x^2+1}}-\frac{7 b c d g^2 x^4 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{b d g^2 x^2 \sqrt{c^2 d x^2+d}}{32 c \sqrt{c^2 x^2+1}} \]

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.730868, antiderivative size = 651, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {5835, 5821, 5684, 5682, 5675, 30, 14, 5717, 194, 5744, 5742, 5758} \[ \frac{3}{8} d f^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d f^2 x \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{3 d f^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt{c^2 x^2+1}}+\frac{2 d f g \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{d g^2 x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}-\frac{d g^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{c^2 x^2+1}}-\frac{b c^3 d f^2 x^4 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{5 b c d f^2 x^2 \sqrt{c^2 d x^2+d}}{16 \sqrt{c^2 x^2+1}}-\frac{2 b c^3 d f g x^5 \sqrt{c^2 d x^2+d}}{25 \sqrt{c^2 x^2+1}}-\frac{4 b c d f g x^3 \sqrt{c^2 d x^2+d}}{15 \sqrt{c^2 x^2+1}}-\frac{2 b d f g x \sqrt{c^2 d x^2+d}}{5 c \sqrt{c^2 x^2+1}}-\frac{b c^3 d g^2 x^6 \sqrt{c^2 d x^2+d}}{36 \sqrt{c^2 x^2+1}}-\frac{7 b c d g^2 x^4 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{b d g^2 x^2 \sqrt{c^2 d x^2+d}}{32 c \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[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 5835

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol]
:> Dist[(d^IntPart[p]*(d + e*x^2)^FracPart[p])/(1 + c^2*x^2)^FracPart[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]

Rule 5821

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 5684

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/((2*p + 1)*(1 + c^2*x^2)^FracPart[p]), Int[x*(1
+ c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && Gt
Q[n, 0] && GtQ[p, 0]

Rule 5682

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[Sqrt[d + e*x^2]/(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*Sqrt[d + e*x^2])/(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 5675

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

Rule 30

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

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 5717

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*d^IntPart[p]*(d + e*x^2)^FracPart[p])/(2*c*(p +
1)*(1 + c^2*x^2)^FracPart[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 194

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 5744

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*d^IntPart[p]*(d + e*x^2)^FracPart[p]
)/(f*(m + 2*p + 1)*(1 + c^2*x^2)^FracPart[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]
&& (RationalQ[m] || EqQ[n, 1])

Rule 5742

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[Sqrt[d + e*x^2]/((m + 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*Sqrt[d + e*x^2])/(f*
(m + 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] && GtQ[n, 0] &&  !LtQ[m, -1] && (RationalQ[m] || EqQ[n, 1])

Rule 5758

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

Rubi steps

\begin{align*} \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+2 f g x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+g^2 x^2 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )\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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{2 d f g \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac{\left (3 d f^2 \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right ) \, 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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{8} d g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d f^2 x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{2 d f g \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac{\left (3 d f^2 \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(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 \left (a+b \sinh ^{-1}(c x)\right )}{\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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{d g^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d f^2 x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{2 d f g \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac{3 d f^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^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 \sinh ^{-1}(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} \left (a+b \sinh ^{-1}(c x)\right )+\frac{d g^2 x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{16 c^2}+\frac{1}{8} d g^2 x^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{4} d f^2 x \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{2 d f g \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{5 c^2}+\frac{3 d f^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 b c \sqrt{1+c^2 x^2}}-\frac{d g^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c^3 \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 2.19319, size = 546, normalized size = 0.84 \[ \frac{3600 a d^{3/2} \sqrt{c^2 x^2+1} \left (6 c^2 f^2-g^2\right ) \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+240 a c d \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d} \left (30 c^2 f^2 x \left (2 c^2 x^2+5\right )+96 f g \left (c^2 x^2+1\right )^2+5 g^2 x \left (8 c^4 x^4+14 c^2 x^2+3\right )\right )-7200 b c^2 d f^2 \sqrt{c^2 d x^2+d} \left (\cosh \left (2 \sinh ^{-1}(c x)\right )-2 \sinh ^{-1}(c x) \left (\sinh ^{-1}(c x)+\sinh \left (2 \sinh ^{-1}(c x)\right )\right )\right )-450 b c^2 d f^2 \sqrt{c^2 d x^2+d} \left (8 \sinh ^{-1}(c x)^2-4 \sinh \left (4 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )-12800 b c d f g \sqrt{c^2 d x^2+d} \left (c^3 x^3-3 \left (c^2 x^2+1\right )^{3/2} \sinh ^{-1}(c x)+3 c x\right )-512 b c d f g \sqrt{c^2 d x^2+d} \left (c x \left (9 c^4 x^4+5 c^2 x^2-30\right )-15 \sqrt{c^2 x^2+1} \left (3 c^4 x^4+c^2 x^2-2\right ) \sinh ^{-1}(c x)\right )-450 b d g^2 \sqrt{c^2 d x^2+d} \left (8 \sinh ^{-1}(c x)^2-4 \sinh \left (4 \sinh ^{-1}(c x)\right ) \sinh ^{-1}(c x)+\cosh \left (4 \sinh ^{-1}(c x)\right )\right )+25 b d g^2 \sqrt{c^2 d x^2+d} \left (72 \sinh ^{-1}(c x)^2+12 \left (-3 \sinh \left (2 \sinh ^{-1}(c x)\right )-3 \sinh \left (4 \sinh ^{-1}(c x)\right )+\sinh \left (6 \sinh ^{-1}(c x)\right )\right ) \sinh ^{-1}(c x)+18 \cosh \left (2 \sinh ^{-1}(c x)\right )+9 \cosh \left (4 \sinh ^{-1}(c x)\right )-2 \cosh \left (6 \sinh ^{-1}(c x)\right )\right )}{57600 c^3 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(240*a*c*d*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*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)) - 12800*b*c*d*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]) - 512*b*c*d*f*g*Sqrt[d + c^2*d*x^2]*(c*x*(-30 + 5*c^2*x^2 + 9*c^4*x^4) - 15*Sqrt[1 + c^2*x^2]
*(-2 + c^2*x^2 + 3*c^4*x^4)*ArcSinh[c*x]) + 3600*a*d^(3/2)*(6*c^2*f^2 - g^2)*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqr
t[d]*Sqrt[d + c^2*d*x^2]] - 7200*b*c^2*d*f^2*Sqrt[d + c^2*d*x^2]*(Cosh[2*ArcSinh[c*x]] - 2*ArcSinh[c*x]*(ArcSi
nh[c*x] + Sinh[2*ArcSinh[c*x]])) - 450*b*c^2*d*f^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]]) - 450*b*d*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]]) + 25*b*d*g^2*Sqrt[d + c^2*d*x^2]*(72*ArcSinh[c*x]^2 + 18*Cosh[2*A
rcSinh[c*x]] + 9*Cosh[4*ArcSinh[c*x]] - 2*Cosh[6*ArcSinh[c*x]] + 12*ArcSinh[c*x]*(-3*Sinh[2*ArcSinh[c*x]] - 3*
Sinh[4*ArcSinh[c*x]] + Sinh[6*ArcSinh[c*x]])))/(57600*c^3*Sqrt[1 + c^2*x^2])

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Maple [A]  time = 0.388, size = 1087, normalized size = 1.7 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

17/48*b*(d*(c^2*x^2+1))^(1/2)*g^2*d/(c^2*x^2+1)*arcsinh(c*x)*x^3+5/8*b*(d*(c^2*x^2+1))^(1/2)*d/(c^2*x^2+1)*arc
sinh(c*x)*x*f^2-1/32*b*(d*(c^2*x^2+1))^(1/2)*arcsinh(c*x)^2*d/(c^2*x^2+1)^(1/2)/c^3*g^2+3/16*b*(d*(c^2*x^2+1))
^(1/2)*arcsinh(c*x)^2*d/(c^2*x^2+1)^(1/2)/c*f^2-1/36*b*(d*(c^2*x^2+1))^(1/2)*g^2*d*c^3/(c^2*x^2+1)^(1/2)*x^6-7
/96*b*(d*(c^2*x^2+1))^(1/2)*g^2*d*c/(c^2*x^2+1)^(1/2)*x^4-1/32*b*(d*(c^2*x^2+1))^(1/2)*g^2*d/c/(c^2*x^2+1)^(1/
2)*x^2-1/16*b*(d*(c^2*x^2+1))^(1/2)*d*c^3/(c^2*x^2+1)^(1/2)*x^4*f^2-5/16*b*(d*(c^2*x^2+1))^(1/2)*d*c/(c^2*x^2+
1)^(1/2)*x^2*f^2-4/15*b*(d*(c^2*x^2+1))^(1/2)*f*g*d*c/(c^2*x^2+1)^(1/2)*x^3+6/5*b*(d*(c^2*x^2+1))^(1/2)*f*g*d/
(c^2*x^2+1)*arcsinh(c*x)*x^2+1/6*b*(d*(c^2*x^2+1))^(1/2)*g^2*d*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^7+11/24*b*(d*(c^
2*x^2+1))^(1/2)*g^2*d*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^5+1/16*b*(d*(c^2*x^2+1))^(1/2)*g^2*d/c^2/(c^2*x^2+1)*arcs
inh(c*x)*x+2/5*b*(d*(c^2*x^2+1))^(1/2)*f*g*d/c^2/(c^2*x^2+1)*arcsinh(c*x)+1/4*b*(d*(c^2*x^2+1))^(1/2)*d*c^4/(c
^2*x^2+1)*arcsinh(c*x)*x^5*f^2+7/8*b*(d*(c^2*x^2+1))^(1/2)*d*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^3*f^2-2/5*b*(d*(c^
2*x^2+1))^(1/2)*f*g*d/c/(c^2*x^2+1)^(1/2)*x-2/25*b*(d*(c^2*x^2+1))^(1/2)*f*g*d*c^3/(c^2*x^2+1)^(1/2)*x^5+2/5*b
*(d*(c^2*x^2+1))^(1/2)*f*g*d*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^6+6/5*b*(d*(c^2*x^2+1))^(1/2)*f*g*d*c^2/(c^2*x^2+1
)*arcsinh(c*x)*x^4-1/24*a*g^2/c^2*x*(c^2*d*x^2+d)^(3/2)+3/8*a*f^2*d*x*(c^2*d*x^2+d)^(1/2)+3/8*a*f^2*d^2*ln(x*c
^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/4*a*f^2*x*(c^2*d*x^2+d)^(3/2)+7/2304*b*(d*(c^2*x^2+1))
^(1/2)*g^2*d/c^3/(c^2*x^2+1)^(1/2)-17/128*b*(d*(c^2*x^2+1))^(1/2)*d/c/(c^2*x^2+1)^(1/2)*f^2+1/6*a*g^2*x*(c^2*d
*x^2+d)^(5/2)/c^2/d-1/16*a*g^2/c^2*d*x*(c^2*d*x^2+d)^(1/2)-1/16*a*g^2/c^2*d^2*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*
x^2+d)^(1/2))/(c^2*d)^(1/2)+2/5*a*f*g/c^2/d*(c^2*d*x^2+d)^(5/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (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} +{\left (a c^{2} d f^{2} + a d g^{2}\right )} x^{2} +{\left (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} +{\left (b c^{2} d f^{2} + b d g^{2}\right )} x^{2}\right )} \operatorname{arsinh}\left (c x\right )\right )} \sqrt{c^{2} d x^{2} + d}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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