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

Optimal. Leaf size=548 \[ \frac{5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (c^2 x^2+1\right )^{5/2}}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (c^2 x^2+1\right )}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (c^2 x^2+1\right )^2}-\frac{b \sqrt{c^2 x^2+1} (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt{c^2 x^2+1}}-\frac{5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (c^2 x^2+1\right )^{5/2}}+\frac{1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (c^2 x^2+1\right )}+\frac{245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left (c^2 x^2+1\right )^2}-\frac{115 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sinh ^{-1}(c x)}{1152 c \left (c^2 x^2+1\right )^{5/2}}+\frac{1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \]

[Out]

(b^2*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/108 + (245*b^2*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/(115
2*(1 + c^2*x^2)^2) + (65*b^2*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/(1728*(1 + c^2*x^2)) - (115*b^2*(d + I
*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*ArcSinh[c*x])/(1152*c*(1 + c^2*x^2)^(5/2)) - (5*b*c*x^2*(d + I*c*d*x)^(5/2)*
(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/(16*(1 + c^2*x^2)^(5/2)) - (5*b*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5
/2)*(a + b*ArcSinh[c*x]))/(48*c*Sqrt[1 + c^2*x^2]) - (b*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*Sqrt[1 + c^2*x
^2]*(a + b*ArcSinh[c*x]))/(18*c) + (x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/6 + (5*x
*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(16*(1 + c^2*x^2)^2) + (5*x*(d + I*c*d*x)^(5/
2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(24*(1 + c^2*x^2)) + (5*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)
*(a + b*ArcSinh[c*x])^3)/(48*b*c*(1 + c^2*x^2)^(5/2))

________________________________________________________________________________________

Rubi [A]  time = 0.611743, antiderivative size = 548, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.243, Rules used = {5712, 5684, 5682, 5675, 5661, 321, 215, 5717, 195} \[ \frac{5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (c^2 x^2+1\right )^{5/2}}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (c^2 x^2+1\right )}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (c^2 x^2+1\right )^2}-\frac{b \sqrt{c^2 x^2+1} (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}-\frac{5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt{c^2 x^2+1}}-\frac{5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (c^2 x^2+1\right )^{5/2}}+\frac{1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (c^2 x^2+1\right )}+\frac{245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left (c^2 x^2+1\right )^2}-\frac{115 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sinh ^{-1}(c x)}{1152 c \left (c^2 x^2+1\right )^{5/2}}+\frac{1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b^2*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/108 + (245*b^2*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/(115
2*(1 + c^2*x^2)^2) + (65*b^2*x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2))/(1728*(1 + c^2*x^2)) - (115*b^2*(d + I
*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*ArcSinh[c*x])/(1152*c*(1 + c^2*x^2)^(5/2)) - (5*b*c*x^2*(d + I*c*d*x)^(5/2)*
(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x]))/(16*(1 + c^2*x^2)^(5/2)) - (5*b*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5
/2)*(a + b*ArcSinh[c*x]))/(48*c*Sqrt[1 + c^2*x^2]) - (b*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*Sqrt[1 + c^2*x
^2]*(a + b*ArcSinh[c*x]))/(18*c) + (x*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/6 + (5*x
*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(16*(1 + c^2*x^2)^2) + (5*x*(d + I*c*d*x)^(5/
2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2)/(24*(1 + c^2*x^2)) + (5*(d + I*c*d*x)^(5/2)*(f - I*c*f*x)^(5/2)
*(a + b*ArcSinh[c*x])^3)/(48*b*c*(1 + c^2*x^2)^(5/2))

Rule 5712

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

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 5661

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*ArcS
inh[c*x])^n)/(d*(m + 1)), x] - Dist[(b*c*n)/(d*(m + 1)), Int[((d*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1))/Sqrt
[1 + c^2*x^2], x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && NeQ[m, -1]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rubi steps

\begin{align*} \int (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{\left ((d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{\left (1+c^2 x^2\right )^{5/2}}\\ &=\frac{1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{\left (5 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{6 \left (1+c^2 x^2\right )^{5/2}}-\frac{\left (b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int x \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{3 \left (1+c^2 x^2\right )^{5/2}}\\ &=-\frac{b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac{\left (5 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{8 \left (1+c^2 x^2\right )^{5/2}}+\frac{\left (b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \left (1+c^2 x^2\right )^{5/2} \, dx}{18 \left (1+c^2 x^2\right )^{5/2}}-\frac{\left (5 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{12 \left (1+c^2 x^2\right )^{5/2}}\\ &=\frac{1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}-\frac{5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt{1+c^2 x^2}}-\frac{b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (1+c^2 x^2\right )^2}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac{\left (5 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{16 \left (1+c^2 x^2\right )^{5/2}}+\frac{\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{108 \left (1+c^2 x^2\right )^{5/2}}+\frac{\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{48 \left (1+c^2 x^2\right )^{5/2}}-\frac{\left (5 b c (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{8 \left (1+c^2 x^2\right )^{5/2}}\\ &=\frac{1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}+\frac{65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (1+c^2 x^2\right )}-\frac{5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (1+c^2 x^2\right )^{5/2}}-\frac{5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt{1+c^2 x^2}}-\frac{b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (1+c^2 x^2\right )^2}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac{5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (1+c^2 x^2\right )^{5/2}}+\frac{\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{144 \left (1+c^2 x^2\right )^{5/2}}+\frac{\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{64 \left (1+c^2 x^2\right )^{5/2}}+\frac{\left (5 b^2 c^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{16 \left (1+c^2 x^2\right )^{5/2}}\\ &=\frac{1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}+\frac{245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left (1+c^2 x^2\right )^2}+\frac{65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (1+c^2 x^2\right )}-\frac{5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (1+c^2 x^2\right )^{5/2}}-\frac{5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt{1+c^2 x^2}}-\frac{b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (1+c^2 x^2\right )^2}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac{5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (1+c^2 x^2\right )^{5/2}}+\frac{\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{288 \left (1+c^2 x^2\right )^{5/2}}+\frac{\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{128 \left (1+c^2 x^2\right )^{5/2}}-\frac{\left (5 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{32 \left (1+c^2 x^2\right )^{5/2}}\\ &=\frac{1}{108} b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}+\frac{245 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1152 \left (1+c^2 x^2\right )^2}+\frac{65 b^2 x (d+i c d x)^{5/2} (f-i c f x)^{5/2}}{1728 \left (1+c^2 x^2\right )}-\frac{115 b^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sinh ^{-1}(c x)}{1152 c \left (1+c^2 x^2\right )^{5/2}}-\frac{5 b c x^2 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{16 \left (1+c^2 x^2\right )^{5/2}}-\frac{5 b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{48 c \sqrt{1+c^2 x^2}}-\frac{b (d+i c d x)^{5/2} (f-i c f x)^{5/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{18 c}+\frac{1}{6} x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{16 \left (1+c^2 x^2\right )^2}+\frac{5 x (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{24 \left (1+c^2 x^2\right )}+\frac{5 (d+i c d x)^{5/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{48 b c \left (1+c^2 x^2\right )^{5/2}}\\ \end{align*}

Mathematica [A]  time = 2.24734, size = 735, normalized size = 1.34 \[ \frac{2304 a^2 c^5 d^2 f^2 x^5 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+7488 a^2 c^3 d^2 f^2 x^3 \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+9504 a^2 c d^2 f^2 x \sqrt{c^2 x^2+1} \sqrt{d+i c d x} \sqrt{f-i c f x}+4320 a^2 d^{5/2} f^{5/2} \sqrt{c^2 x^2+1} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{d+i c d x} \sqrt{f-i c f x}\right )+72 b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left (60 a+45 b \sinh \left (2 \sinh ^{-1}(c x)\right )+9 b \sinh \left (4 \sinh ^{-1}(c x)\right )+b \sinh \left (6 \sinh ^{-1}(c x)\right )\right )-3240 a b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-324 a b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (4 \sinh ^{-1}(c x)\right )-24 a b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \cosh \left (6 \sinh ^{-1}(c x)\right )-12 b d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left (-540 a \sinh \left (2 \sinh ^{-1}(c x)\right )-108 a \sinh \left (4 \sinh ^{-1}(c x)\right )-12 a \sinh \left (6 \sinh ^{-1}(c x)\right )+270 b \cosh \left (2 \sinh ^{-1}(c x)\right )+27 b \cosh \left (4 \sinh ^{-1}(c x)\right )+2 b \cosh \left (6 \sinh ^{-1}(c x)\right )\right )+1440 b^2 d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3+1620 b^2 d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )+81 b^2 d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (4 \sinh ^{-1}(c x)\right )+4 b^2 d^2 f^2 \sqrt{d+i c d x} \sqrt{f-i c f x} \sinh \left (6 \sinh ^{-1}(c x)\right )}{13824 c \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9504*a^2*c*d^2*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 7488*a^2*c^3*d^2*f^2*x^3*Sqrt[d
+ I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2304*a^2*c^5*d^2*f^2*x^5*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*
Sqrt[1 + c^2*x^2] + 1440*b^2*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 3240*a*b*d^2*f^2*Sqr
t[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 324*a*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Co
sh[4*ArcSinh[c*x]] - 24*a*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[6*ArcSinh[c*x]] + 4320*a^2*d^(5/2
)*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 1620*b^2*d^2*
f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]] + 81*b^2*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f
*x]*Sinh[4*ArcSinh[c*x]] + 4*b^2*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[6*ArcSinh[c*x]] - 12*b*d^2*f
^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*(270*b*Cosh[2*ArcSinh[c*x]] + 27*b*Cosh[4*ArcSinh[c*x]] +
2*b*Cosh[6*ArcSinh[c*x]] - 540*a*Sinh[2*ArcSinh[c*x]] - 108*a*Sinh[4*ArcSinh[c*x]] - 12*a*Sinh[6*ArcSinh[c*x]]
) + 72*b*d^2*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*(60*a + 45*b*Sinh[2*ArcSinh[c*x]] + 9*b*Si
nh[4*ArcSinh[c*x]] + b*Sinh[6*ArcSinh[c*x]]))/(13824*c*Sqrt[1 + c^2*x^2])

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Maple [F]  time = 0.266, size = 0, normalized size = 0. \begin{align*} \int \left ( d+icdx \right ) ^{{\frac{5}{2}}} \left ( f-icfx \right ) ^{{\frac{5}{2}}} \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2,x)

[Out]

int((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2,x)

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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((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2,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 (b^{2} c^{4} d^{2} f^{2} x^{4} + 2 \, b^{2} c^{2} d^{2} f^{2} x^{2} + b^{2} d^{2} f^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right )^{2} + 2 \,{\left (a b c^{4} d^{2} f^{2} x^{4} + 2 \, a b c^{2} d^{2} f^{2} x^{2} + a b d^{2} f^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f} \log \left (c x + \sqrt{c^{2} x^{2} + 1}\right ) +{\left (a^{2} c^{4} d^{2} f^{2} x^{4} + 2 \, a^{2} c^{2} d^{2} f^{2} x^{2} + a^{2} d^{2} f^{2}\right )} \sqrt{i \, c d x + d} \sqrt{-i \, c f x + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral((b^2*c^4*d^2*f^2*x^4 + 2*b^2*c^2*d^2*f^2*x^2 + b^2*d^2*f^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(
c*x + sqrt(c^2*x^2 + 1))^2 + 2*(a*b*c^4*d^2*f^2*x^4 + 2*a*b*c^2*d^2*f^2*x^2 + a*b*d^2*f^2)*sqrt(I*c*d*x + d)*s
qrt(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1)) + (a^2*c^4*d^2*f^2*x^4 + 2*a^2*c^2*d^2*f^2*x^2 + a^2*d^2*f^2)*s
qrt(I*c*d*x + d)*sqrt(-I*c*f*x + f), 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((d+I*c*d*x)**(5/2)*(f-I*c*f*x)**(5/2)*(a+b*asinh(c*x))**2,x)

[Out]

Timed out

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Giac [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((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

Timed out