3.583 \(\int (d+i c d x)^{3/2} (f-i c f x)^{5/2} (a+b \sinh ^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=774 \[ \frac{2 i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \left (c^2 x^2+1\right )^{3/2}}+\frac{4 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \left (c^2 x^2+1\right )^{3/2}}-\frac{3 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (c^2 x^2+1\right )^{3/2}}+\frac{3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (c^2 x^2+1\right )}+\frac{2 i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 \left (c^2 x^2+1\right )^{3/2}}+\frac{f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (c^2 x^2+1\right )^{3/2}}-\frac{i f \left (c^2 x^2+1\right ) (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c}-\frac{b f \sqrt{c^2 x^2+1} (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{15 b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (c^2 x^2+1\right )}-\frac{2 i b^2 f \left (c^2 x^2+1\right ) (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{125 c}-\frac{16 i b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{75 c \left (c^2 x^2+1\right )}-\frac{9 b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sinh ^{-1}(c x)}{64 c \left (c^2 x^2+1\right )^{3/2}}+\frac{1}{32} b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}-\frac{8 i b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{225 c} \]

[Out]

(((-8*I)/225)*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/c + (b^2*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3
/2))/32 - (((16*I)/75)*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(c*(1 + c^2*x^2)) + (15*b^2*f*x*(d + I*c
*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(64*(1 + c^2*x^2)) - (((2*I)/125)*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/
2)*(1 + c^2*x^2))/c - (9*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*ArcSinh[c*x])/(64*c*(1 + c^2*x^2)^(3/2)
) + (((2*I)/5)*b*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) - (3*b*
c*f*x^2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(8*(1 + c^2*x^2)^(3/2)) + (((4*I)/15)*b*
c^2*f*x^3*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) + (((2*I)/25)*b*c^
4*f*x^5*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) - (b*f*(d + I*c*d*x)
^(3/2)*(f - I*c*f*x)^(3/2)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c) + (f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f
*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + (3*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(
8*(1 + c^2*x^2)) - ((I/5)*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/c +
(f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^3)/(8*b*c*(1 + c^2*x^2)^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.833405, antiderivative size = 774, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 15, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.405, Rules used = {5712, 5821, 5684, 5682, 5675, 5661, 321, 215, 5717, 195, 194, 5679, 12, 1247, 698} \[ \frac{2 i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \left (c^2 x^2+1\right )^{3/2}}+\frac{4 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \left (c^2 x^2+1\right )^{3/2}}-\frac{3 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (c^2 x^2+1\right )^{3/2}}+\frac{3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (c^2 x^2+1\right )}+\frac{2 i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 \left (c^2 x^2+1\right )^{3/2}}+\frac{f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (c^2 x^2+1\right )^{3/2}}-\frac{i f \left (c^2 x^2+1\right ) (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c}-\frac{b f \sqrt{c^2 x^2+1} (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{15 b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (c^2 x^2+1\right )}-\frac{2 i b^2 f \left (c^2 x^2+1\right ) (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{125 c}-\frac{16 i b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{75 c \left (c^2 x^2+1\right )}-\frac{9 b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sinh ^{-1}(c x)}{64 c \left (c^2 x^2+1\right )^{3/2}}+\frac{1}{32} b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}-\frac{8 i b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{225 c} \]

Antiderivative was successfully verified.

[In]

Int[(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2,x]

[Out]

(((-8*I)/225)*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/c + (b^2*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3
/2))/32 - (((16*I)/75)*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(c*(1 + c^2*x^2)) + (15*b^2*f*x*(d + I*c
*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(64*(1 + c^2*x^2)) - (((2*I)/125)*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/
2)*(1 + c^2*x^2))/c - (9*b^2*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*ArcSinh[c*x])/(64*c*(1 + c^2*x^2)^(3/2)
) + (((2*I)/5)*b*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) - (3*b*
c*f*x^2*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(8*(1 + c^2*x^2)^(3/2)) + (((4*I)/15)*b*
c^2*f*x^3*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) + (((2*I)/25)*b*c^
4*f*x^5*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/(1 + c^2*x^2)^(3/2) - (b*f*(d + I*c*d*x)
^(3/2)*(f - I*c*f*x)^(3/2)*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8*c) + (f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f
*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/4 + (3*f*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(
8*(1 + c^2*x^2)) - ((I/5)*f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x])^2)/c +
(f*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^3)/(8*b*c*(1 + c^2*x^2)^(3/2))

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 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 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]])

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 5679

Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> With[{u = IntHide[(d + e*x^2
)^p, x]}, Dist[a + b*ArcSinh[c*x], u, x] - Dist[b*c, Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /;
FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 1247

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[
Int[(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x]

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int (d+i c d x)^{3/2} (f-i c f x)^{5/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac{\left ((d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int (f-i c f x) \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{\left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{\left ((d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (f \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-i c f x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2\right ) \, dx}{\left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{\left (f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{\left (1+c^2 x^2\right )^{3/2}}-\frac{\left (i c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{\left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2-\frac{i f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c}+\frac{\left (3 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}+\frac{\left (2 i b f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (b c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{2 \left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{2 i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \left (1+c^2 x^2\right )^{3/2}}+\frac{2 i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \left (1+c^2 x^2\right )^{3/2}}-\frac{b f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (1+c^2 x^2\right )}-\frac{i f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c}+\frac{\left (3 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt{1+c^2 x^2}} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}+\frac{\left (b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^{3/2} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (3 b c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (2 i b^2 c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{x \left (15+10 c^2 x^2+3 c^4 x^4\right )}{15 \sqrt{1+c^2 x^2}} \, dx}{5 \left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{1}{32} b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}+\frac{2 i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac{3 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \left (1+c^2 x^2\right )^{3/2}}+\frac{2 i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \left (1+c^2 x^2\right )^{3/2}}-\frac{b f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (1+c^2 x^2\right )}-\frac{i f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c}+\frac{f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (1+c^2 x^2\right )^{3/2}}+\frac{\left (3 b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \sqrt{1+c^2 x^2} \, dx}{32 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (2 i b^2 c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{x \left (15+10 c^2 x^2+3 c^4 x^4\right )}{\sqrt{1+c^2 x^2}} \, dx}{75 \left (1+c^2 x^2\right )^{3/2}}+\frac{\left (3 b^2 c^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{x^2}{\sqrt{1+c^2 x^2}} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{1}{32} b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}+\frac{15 b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (1+c^2 x^2\right )}+\frac{2 i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac{3 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \left (1+c^2 x^2\right )^{3/2}}+\frac{2 i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \left (1+c^2 x^2\right )^{3/2}}-\frac{b f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (1+c^2 x^2\right )}-\frac{i f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c}+\frac{f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (1+c^2 x^2\right )^{3/2}}+\frac{\left (3 b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{64 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (3 b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac{1}{\sqrt{1+c^2 x^2}} \, dx}{16 \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (i b^2 c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \operatorname{Subst}\left (\int \frac{15+10 c^2 x+3 c^4 x^2}{\sqrt{1+c^2 x}} \, dx,x,x^2\right )}{75 \left (1+c^2 x^2\right )^{3/2}}\\ &=\frac{1}{32} b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}+\frac{15 b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (1+c^2 x^2\right )}-\frac{9 b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sinh ^{-1}(c x)}{64 c \left (1+c^2 x^2\right )^{3/2}}+\frac{2 i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac{3 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \left (1+c^2 x^2\right )^{3/2}}+\frac{2 i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \left (1+c^2 x^2\right )^{3/2}}-\frac{b f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (1+c^2 x^2\right )}-\frac{i f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c}+\frac{f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (1+c^2 x^2\right )^{3/2}}-\frac{\left (i b^2 c f (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \operatorname{Subst}\left (\int \left (\frac{8}{\sqrt{1+c^2 x}}+4 \sqrt{1+c^2 x}+3 \left (1+c^2 x\right )^{3/2}\right ) \, dx,x,x^2\right )}{75 \left (1+c^2 x^2\right )^{3/2}}\\ &=-\frac{8 i b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{225 c}+\frac{1}{32} b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}-\frac{16 i b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{75 c \left (1+c^2 x^2\right )}+\frac{15 b^2 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{64 \left (1+c^2 x^2\right )}-\frac{2 i b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right )}{125 c}-\frac{9 b^2 f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sinh ^{-1}(c x)}{64 c \left (1+c^2 x^2\right )^{3/2}}+\frac{2 i b f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac{3 b c f x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{8 \left (1+c^2 x^2\right )^{3/2}}+\frac{4 i b c^2 f x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{15 \left (1+c^2 x^2\right )^{3/2}}+\frac{2 i b c^4 f x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{25 \left (1+c^2 x^2\right )^{3/2}}-\frac{b f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{8 c}+\frac{1}{4} f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2+\frac{3 f x (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2}{8 \left (1+c^2 x^2\right )}-\frac{i f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2}{5 c}+\frac{f (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^3}{8 b c \left (1+c^2 x^2\right )^{3/2}}\\ \end{align*}

Mathematica [A]  time = 3.1153, size = 1084, normalized size = 1.4 \[ \frac{-57600 i a^2 c^4 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^4+72000 a^2 c^3 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^3-115200 i a^2 c^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x^2+72000 i a b c d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} x+180000 a^2 c d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1} x+36000 b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x)^3-72000 a b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left (2 \sinh ^{-1}(c x)\right )-4000 i b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left (3 \sinh ^{-1}(c x)\right )-4500 a b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left (4 \sinh ^{-1}(c x)\right )-288 i b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \cosh \left (5 \sinh ^{-1}(c x)\right )+108000 a^2 d^{3/2} f^{5/2} \sqrt{c^2 x^2+1} \log \left (c d f x+\sqrt{d} \sqrt{f} \sqrt{i c x d+d} \sqrt{f-i c f x}\right )+36000 b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left (2 \sinh ^{-1}(c x)\right )+12000 i a b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left (3 \sinh ^{-1}(c x)\right )+1125 b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left (4 \sinh ^{-1}(c x)\right )+1800 b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x)^2 \left (60 a-10 i b \cosh \left (3 \sinh ^{-1}(c x)\right )-2 i b \cosh \left (5 \sinh ^{-1}(c x)\right )+40 b \sinh \left (2 \sinh ^{-1}(c x)\right )+5 b \sinh \left (4 \sinh ^{-1}(c x)\right )-20 i b \sqrt{c^2 x^2+1}\right )+1440 i a b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh \left (5 \sinh ^{-1}(c x)\right )+60 b d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sinh ^{-1}(c x) \left (-600 i \cosh \left (3 \sinh ^{-1}(c x)\right ) a-120 i \cosh \left (5 \sinh ^{-1}(c x)\right ) a+2400 \sinh \left (2 \sinh ^{-1}(c x)\right ) a+300 \sinh \left (4 \sinh ^{-1}(c x)\right ) a-1200 i \sqrt{c^2 x^2+1} a+1200 i b c x-1200 b \cosh \left (2 \sinh ^{-1}(c x)\right )-75 b \cosh \left (4 \sinh ^{-1}(c x)\right )+200 i b \sinh \left (3 \sinh ^{-1}(c x)\right )+24 i b \sinh \left (5 \sinh ^{-1}(c x)\right )\right )-57600 i a^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1}-72000 i b^2 d f^2 \sqrt{i c x d+d} \sqrt{f-i c f x} \sqrt{c^2 x^2+1}}{288000 c \sqrt{c^2 x^2+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(5/2)*(a + b*ArcSinh[c*x])^2,x]

[Out]

((72000*I)*a*b*c*d*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] - (57600*I)*a^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f -
I*c*f*x]*Sqrt[1 + c^2*x^2] - (72000*I)*b^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 18000
0*a^2*c*d*f^2*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - (115200*I)*a^2*c^2*d*f^2*x^2*Sqrt[d +
I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 72000*a^2*c^3*d*f^2*x^3*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqr
t[1 + c^2*x^2] - (57600*I)*a^2*c^4*d*f^2*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 36000*b^2
*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^3 - 72000*a*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x
]*Cosh[2*ArcSinh[c*x]] - (4000*I)*b^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[3*ArcSinh[c*x]] - 4500*a*
b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] - (288*I)*b^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f
- I*c*f*x]*Cosh[5*ArcSinh[c*x]] + 108000*a^2*d^(3/2)*f^(5/2)*Sqrt[1 + c^2*x^2]*Log[c*d*f*x + Sqrt[d]*Sqrt[f]*S
qrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]] + 36000*b^2*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[2*ArcSinh[c*x]]
+ (12000*I)*a*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 1125*b^2*d*f^2*Sqrt[d + I*c*
d*x]*Sqrt[f - I*c*f*x]*Sinh[4*ArcSinh[c*x]] + 1800*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2*
(60*a - (20*I)*b*Sqrt[1 + c^2*x^2] - (10*I)*b*Cosh[3*ArcSinh[c*x]] - (2*I)*b*Cosh[5*ArcSinh[c*x]] + 40*b*Sinh[
2*ArcSinh[c*x]] + 5*b*Sinh[4*ArcSinh[c*x]]) + (1440*I)*a*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[5*Ar
cSinh[c*x]] + 60*b*d*f^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]*((1200*I)*b*c*x - (1200*I)*a*Sqrt[1
+ c^2*x^2] - 1200*b*Cosh[2*ArcSinh[c*x]] - (600*I)*a*Cosh[3*ArcSinh[c*x]] - 75*b*Cosh[4*ArcSinh[c*x]] - (120*I
)*a*Cosh[5*ArcSinh[c*x]] + 2400*a*Sinh[2*ArcSinh[c*x]] + (200*I)*b*Sinh[3*ArcSinh[c*x]] + 300*a*Sinh[4*ArcSinh
[c*x]] + (24*I)*b*Sinh[5*ArcSinh[c*x]]))/(288000*c*Sqrt[1 + c^2*x^2])

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Maple [F]  time = 0.269, size = 0, normalized size = 0. \begin{align*} \int \left ( d+icdx \right ) ^{{\frac{3}{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)^(3/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2,x)

[Out]

int((d+I*c*d*x)^(3/2)*(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)^(3/2)*(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 (-i \, b^{2} c^{3} d f^{2} x^{3} + b^{2} c^{2} d f^{2} x^{2} - i \, b^{2} c d f^{2} x + b^{2} d 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} +{\left (-2 i \, a b c^{3} d f^{2} x^{3} + 2 \, a b c^{2} d f^{2} x^{2} - 2 i \, a b c d f^{2} x + 2 \, a b d 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 (-i \, a^{2} c^{3} d f^{2} x^{3} + a^{2} c^{2} d f^{2} x^{2} - i \, a^{2} c d f^{2} x + a^{2} d 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)^(3/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral((-I*b^2*c^3*d*f^2*x^3 + b^2*c^2*d*f^2*x^2 - I*b^2*c*d*f^2*x + b^2*d*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*I*a*b*c^3*d*f^2*x^3 + 2*a*b*c^2*d*f^2*x^2 - 2*I*a*b*c*d*f^2*x +
2*a*b*d*f^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1)) + (-I*a^2*c^3*d*f^2*x^3 + a^2*c
^2*d*f^2*x^2 - I*a^2*c*d*f^2*x + a^2*d*f^2)*sqrt(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)**(3/2)*(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)^(3/2)*(f-I*c*f*x)^(5/2)*(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

Timed out