3.45 \(\int (f+g x) (d+c^2 d x^2)^{5/2} (a+b \sinh ^{-1}(c x)) \, dx\)

Optimal. Leaf size=494 \[ \frac{1}{6} d^2 f x \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{16} d^2 f x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} d^2 f x \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5 d^2 f \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt{c^2 x^2+1}}+\frac{d^2 g \left (c^2 x^2+1\right )^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac{5 b c^3 d^2 f x^4 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{25 b c d^2 f x^2 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{b d^2 f \left (c^2 x^2+1\right )^{5/2} \sqrt{c^2 d x^2+d}}{36 c}-\frac{b c^5 d^2 g x^7 \sqrt{c^2 d x^2+d}}{49 \sqrt{c^2 x^2+1}}-\frac{3 b c^3 d^2 g x^5 \sqrt{c^2 d x^2+d}}{35 \sqrt{c^2 x^2+1}}-\frac{b c d^2 g x^3 \sqrt{c^2 d x^2+d}}{7 \sqrt{c^2 x^2+1}}-\frac{b d^2 g x \sqrt{c^2 d x^2+d}}{7 c \sqrt{c^2 x^2+1}} \]

[Out]

-(b*d^2*g*x*Sqrt[d + c^2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2]) - (25*b*c*d^2*f*x^2*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 +
c^2*x^2]) - (b*c*d^2*g*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*f*x^4*Sqrt[d + c^2*d*x^2
])/(96*Sqrt[1 + c^2*x^2]) - (3*b*c^3*d^2*g*x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g*x^7*
Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) - (b*d^2*f*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5*d^
2*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/16 + (5*d^2*f*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSi
nh[c*x]))/24 + (d^2*f*x*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/6 + (d^2*g*(1 + c^2*x^2)^3*S
qrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^2) + (5*d^2*f*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*
c*Sqrt[1 + c^2*x^2])

________________________________________________________________________________________

Rubi [A]  time = 0.397254, antiderivative size = 494, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 10, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.357, Rules used = {5835, 5821, 5684, 5682, 5675, 30, 14, 261, 5717, 194} \[ \frac{1}{6} d^2 f x \left (c^2 x^2+1\right )^2 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{16} d^2 f x \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} d^2 f x \left (c^2 x^2+1\right ) \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5 d^2 f \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt{c^2 x^2+1}}+\frac{d^2 g \left (c^2 x^2+1\right )^3 \sqrt{c^2 d x^2+d} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}-\frac{5 b c^3 d^2 f x^4 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{25 b c d^2 f x^2 \sqrt{c^2 d x^2+d}}{96 \sqrt{c^2 x^2+1}}-\frac{b d^2 f \left (c^2 x^2+1\right )^{5/2} \sqrt{c^2 d x^2+d}}{36 c}-\frac{b c^5 d^2 g x^7 \sqrt{c^2 d x^2+d}}{49 \sqrt{c^2 x^2+1}}-\frac{3 b c^3 d^2 g x^5 \sqrt{c^2 d x^2+d}}{35 \sqrt{c^2 x^2+1}}-\frac{b c d^2 g x^3 \sqrt{c^2 d x^2+d}}{7 \sqrt{c^2 x^2+1}}-\frac{b d^2 g x \sqrt{c^2 d x^2+d}}{7 c \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Int[(f + g*x)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

-(b*d^2*g*x*Sqrt[d + c^2*d*x^2])/(7*c*Sqrt[1 + c^2*x^2]) - (25*b*c*d^2*f*x^2*Sqrt[d + c^2*d*x^2])/(96*Sqrt[1 +
c^2*x^2]) - (b*c*d^2*g*x^3*Sqrt[d + c^2*d*x^2])/(7*Sqrt[1 + c^2*x^2]) - (5*b*c^3*d^2*f*x^4*Sqrt[d + c^2*d*x^2
])/(96*Sqrt[1 + c^2*x^2]) - (3*b*c^3*d^2*g*x^5*Sqrt[d + c^2*d*x^2])/(35*Sqrt[1 + c^2*x^2]) - (b*c^5*d^2*g*x^7*
Sqrt[d + c^2*d*x^2])/(49*Sqrt[1 + c^2*x^2]) - (b*d^2*f*(1 + c^2*x^2)^(5/2)*Sqrt[d + c^2*d*x^2])/(36*c) + (5*d^
2*f*x*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/16 + (5*d^2*f*x*(1 + c^2*x^2)*Sqrt[d + c^2*d*x^2]*(a + b*ArcSi
nh[c*x]))/24 + (d^2*f*x*(1 + c^2*x^2)^2*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/6 + (d^2*g*(1 + c^2*x^2)^3*S
qrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x]))/(7*c^2) + (5*d^2*f*Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2)/(32*b*
c*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 261

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 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]

Rubi steps

\begin{align*} \int (f+g x) \left (d+c^2 d x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \int (f+g x) \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (d^2 \sqrt{d+c^2 d x^2}\right ) \int \left (f \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )+g x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{\left (d^2 f \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}+\frac{\left (d^2 g \sqrt{d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{\sqrt{1+c^2 x^2}}\\ &=\frac{1}{6} d^2 f x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{d^2 g \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}+\frac{\left (5 d^2 f \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}{6 \sqrt{1+c^2 x^2}}-\frac{\left (b c d^2 f \sqrt{d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right )^2 \, dx}{6 \sqrt{1+c^2 x^2}}-\frac{\left (b d^2 g \sqrt{d+c^2 d x^2}\right ) \int \left (1+c^2 x^2\right )^3 \, dx}{7 c \sqrt{1+c^2 x^2}}\\ &=-\frac{b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2}}{36 c}+\frac{5}{24} d^2 f 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^2 f x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{d^2 g \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}+\frac{\left (5 d^2 f \sqrt{d+c^2 d x^2}\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c d^2 f \sqrt{d+c^2 d x^2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{24 \sqrt{1+c^2 x^2}}-\frac{\left (b d^2 g \sqrt{d+c^2 d x^2}\right ) \int \left (1+3 c^2 x^2+3 c^4 x^4+c^6 x^6\right ) \, dx}{7 c \sqrt{1+c^2 x^2}}\\ &=-\frac{b d^2 g x \sqrt{d+c^2 d x^2}}{7 c \sqrt{1+c^2 x^2}}-\frac{b c d^2 g x^3 \sqrt{d+c^2 d x^2}}{7 \sqrt{1+c^2 x^2}}-\frac{3 b c^3 d^2 g x^5 \sqrt{d+c^2 d x^2}}{35 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 g x^7 \sqrt{d+c^2 d x^2}}{49 \sqrt{1+c^2 x^2}}-\frac{b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} d^2 f 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^2 f x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{d^2 g \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}+\frac{\left (5 d^2 f \sqrt{d+c^2 d x^2}\right ) \int \frac{a+b \sinh ^{-1}(c x)}{\sqrt{1+c^2 x^2}} \, dx}{16 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c d^2 f \sqrt{d+c^2 d x^2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{24 \sqrt{1+c^2 x^2}}-\frac{\left (5 b c d^2 f \sqrt{d+c^2 d x^2}\right ) \int x \, dx}{16 \sqrt{1+c^2 x^2}}\\ &=-\frac{b d^2 g x \sqrt{d+c^2 d x^2}}{7 c \sqrt{1+c^2 x^2}}-\frac{25 b c d^2 f x^2 \sqrt{d+c^2 d x^2}}{96 \sqrt{1+c^2 x^2}}-\frac{b c d^2 g x^3 \sqrt{d+c^2 d x^2}}{7 \sqrt{1+c^2 x^2}}-\frac{5 b c^3 d^2 f x^4 \sqrt{d+c^2 d x^2}}{96 \sqrt{1+c^2 x^2}}-\frac{3 b c^3 d^2 g x^5 \sqrt{d+c^2 d x^2}}{35 \sqrt{1+c^2 x^2}}-\frac{b c^5 d^2 g x^7 \sqrt{d+c^2 d x^2}}{49 \sqrt{1+c^2 x^2}}-\frac{b d^2 f \left (1+c^2 x^2\right )^{5/2} \sqrt{d+c^2 d x^2}}{36 c}+\frac{5}{16} d^2 f x \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{5}{24} d^2 f 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^2 f x \left (1+c^2 x^2\right )^2 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )+\frac{d^2 g \left (1+c^2 x^2\right )^3 \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )}{7 c^2}+\frac{5 d^2 f \sqrt{d+c^2 d x^2} \left (a+b \sinh ^{-1}(c x)\right )^2}{32 b c \sqrt{1+c^2 x^2}}\\ \end{align*}

Mathematica [A]  time = 1.29327, size = 656, normalized size = 1.33 \[ \frac{d^2 \left (94080 a c^6 f x^5 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+305760 a c^4 f x^3 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+388080 a c^2 f x \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+176400 a c \sqrt{d} f \sqrt{c^2 x^2+1} \log \left (\sqrt{d} \sqrt{c^2 d x^2+d}+c d x\right )+80640 a c^6 g x^6 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+241920 a c^4 g x^4 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+241920 a c^2 g x^2 \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+80640 a g \sqrt{c^2 x^2+1} \sqrt{c^2 d x^2+d}+420 b \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x) \left (192 c^6 g x^6 \sqrt{c^2 x^2+1}+576 c^4 g x^4 \sqrt{c^2 x^2+1}+576 c^2 g x^2 \sqrt{c^2 x^2+1}+192 g \sqrt{c^2 x^2+1}+315 c f \sinh \left (2 \sinh ^{-1}(c x)\right )+63 c f \sinh \left (4 \sinh ^{-1}(c x)\right )+7 c f \sinh \left (6 \sinh ^{-1}(c x)\right )\right )+88200 b c f \sqrt{c^2 d x^2+d} \sinh ^{-1}(c x)^2-66150 b c f \sqrt{c^2 d x^2+d} \cosh \left (2 \sinh ^{-1}(c x)\right )-6615 b c f \sqrt{c^2 d x^2+d} \cosh \left (4 \sinh ^{-1}(c x)\right )-490 b c f \sqrt{c^2 d x^2+d} \cosh \left (6 \sinh ^{-1}(c x)\right )-11520 b c^7 g x^7 \sqrt{c^2 d x^2+d}-48384 b c^5 g x^5 \sqrt{c^2 d x^2+d}-80640 b c^3 g x^3 \sqrt{c^2 d x^2+d}-80640 b c g x \sqrt{c^2 d x^2+d}\right )}{564480 c^2 \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(f + g*x)*(d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x]),x]

[Out]

(d^2*(-80640*b*c*g*x*Sqrt[d + c^2*d*x^2] - 80640*b*c^3*g*x^3*Sqrt[d + c^2*d*x^2] - 48384*b*c^5*g*x^5*Sqrt[d +
c^2*d*x^2] - 11520*b*c^7*g*x^7*Sqrt[d + c^2*d*x^2] + 80640*a*g*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 388080*
a*c^2*f*x*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 241920*a*c^2*g*x^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 3
05760*a*c^4*f*x^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 241920*a*c^4*g*x^4*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*
x^2] + 94080*a*c^6*f*x^5*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2] + 80640*a*c^6*g*x^6*Sqrt[1 + c^2*x^2]*Sqrt[d +
c^2*d*x^2] + 88200*b*c*f*Sqrt[d + c^2*d*x^2]*ArcSinh[c*x]^2 - 66150*b*c*f*Sqrt[d + c^2*d*x^2]*Cosh[2*ArcSinh[c
*x]] - 6615*b*c*f*Sqrt[d + c^2*d*x^2]*Cosh[4*ArcSinh[c*x]] - 490*b*c*f*Sqrt[d + c^2*d*x^2]*Cosh[6*ArcSinh[c*x]
] + 176400*a*c*Sqrt[d]*f*Sqrt[1 + c^2*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 420*b*Sqrt[d + c^2*d*x^2
]*ArcSinh[c*x]*(192*g*Sqrt[1 + c^2*x^2] + 576*c^2*g*x^2*Sqrt[1 + c^2*x^2] + 576*c^4*g*x^4*Sqrt[1 + c^2*x^2] +
192*c^6*g*x^6*Sqrt[1 + c^2*x^2] + 315*c*f*Sinh[2*ArcSinh[c*x]] + 63*c*f*Sinh[4*ArcSinh[c*x]] + 7*c*f*Sinh[6*Ar
cSinh[c*x]])))/(564480*c^2*Sqrt[1 + c^2*x^2])

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Maple [A]  time = 0.362, size = 805, normalized size = 1.6 \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)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x)

[Out]

1/7*a*g/c^2/d*(c^2*d*x^2+d)^(7/2)+1/6*a*f*x*(c^2*d*x^2+d)^(5/2)+5/24*a*f*d*x*(c^2*d*x^2+d)^(3/2)+5/16*a*f*d^2*
x*(c^2*d*x^2+d)^(1/2)+5/16*a*f*d^3*ln(x*c^2*d/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/7*b*(d*(c^2*x
^2+1))^(1/2)*g*d^2/c^2/(c^2*x^2+1)*arcsinh(c*x)-299/2304*b*(d*(c^2*x^2+1))^(1/2)*f*d^2/c/(c^2*x^2+1)^(1/2)+1/7
*b*(d*(c^2*x^2+1))^(1/2)*g*d^2*c^6/(c^2*x^2+1)*arcsinh(c*x)*x^8-1/49*b*(d*(c^2*x^2+1))^(1/2)*g*d^2*c^5/(c^2*x^
2+1)^(1/2)*x^7+4/7*b*(d*(c^2*x^2+1))^(1/2)*g*d^2*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^6-3/35*b*(d*(c^2*x^2+1))^(1/2)
*g*d^2*c^3/(c^2*x^2+1)^(1/2)*x^5+6/7*b*(d*(c^2*x^2+1))^(1/2)*g*d^2*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^4-1/7*b*(d*(
c^2*x^2+1))^(1/2)*g*d^2*c/(c^2*x^2+1)^(1/2)*x^3+4/7*b*(d*(c^2*x^2+1))^(1/2)*g*d^2/(c^2*x^2+1)*arcsinh(c*x)*x^2
-1/7*b*(d*(c^2*x^2+1))^(1/2)*g*d^2/c/(c^2*x^2+1)^(1/2)*x+1/6*b*(d*(c^2*x^2+1))^(1/2)*f*d^2*c^6/(c^2*x^2+1)*arc
sinh(c*x)*x^7-1/36*b*(d*(c^2*x^2+1))^(1/2)*f*d^2*c^5/(c^2*x^2+1)^(1/2)*x^6+17/24*b*(d*(c^2*x^2+1))^(1/2)*f*d^2
*c^4/(c^2*x^2+1)*arcsinh(c*x)*x^5-13/96*b*(d*(c^2*x^2+1))^(1/2)*f*d^2*c^3/(c^2*x^2+1)^(1/2)*x^4+59/48*b*(d*(c^
2*x^2+1))^(1/2)*f*d^2*c^2/(c^2*x^2+1)*arcsinh(c*x)*x^3-11/32*b*(d*(c^2*x^2+1))^(1/2)*f*d^2*c/(c^2*x^2+1)^(1/2)
*x^2+11/16*b*(d*(c^2*x^2+1))^(1/2)*f*d^2/(c^2*x^2+1)*arcsinh(c*x)*x+5/32*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(
1/2)/c*f*arcsinh(c*x)^2*d^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)*(c^2*d*x^2+d)^(5/2)*(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^{4} d^{2} g x^{5} + a c^{4} d^{2} f x^{4} + 2 \, a c^{2} d^{2} g x^{3} + 2 \, a c^{2} d^{2} f x^{2} + a d^{2} g x + a d^{2} f +{\left (b c^{4} d^{2} g x^{5} + b c^{4} d^{2} f x^{4} + 2 \, b c^{2} d^{2} g x^{3} + 2 \, b c^{2} d^{2} f x^{2} + b d^{2} g x + b d^{2} f\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)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral((a*c^4*d^2*g*x^5 + a*c^4*d^2*f*x^4 + 2*a*c^2*d^2*g*x^3 + 2*a*c^2*d^2*f*x^2 + a*d^2*g*x + a*d^2*f + (b
*c^4*d^2*g*x^5 + b*c^4*d^2*f*x^4 + 2*b*c^2*d^2*g*x^3 + 2*b*c^2*d^2*f*x^2 + b*d^2*g*x + b*d^2*f)*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)*(c**2*d*x**2+d)**(5/2)*(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{5}{2}}{\left (g x + f\right )}{\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)*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x)),x, algorithm="giac")

[Out]

integrate((c^2*d*x^2 + d)^(5/2)*(g*x + f)*(b*arcsinh(c*x) + a), x)