\(\int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx\) [540]

Optimal result

Integrand size = 35, antiderivative size = 459 \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {i b d x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c d x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (1+c^2 x^2\right )^{3/2}}-\frac {2 i b c^2 d x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c^3 d x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (1+c^2 x^2\right )^{3/2}}-\frac {i b c^4 d x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left (1+c^2 x^2\right )^{3/2}}+\frac {1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))+\frac {3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 \left (1+c^2 x^2\right )}+\frac {i d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{5 c}+\frac {3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{16 b c \left (1+c^2 x^2\right )^{3/2}} \]

[Out]

-1/5*I*b*d*x*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)/(c^2*x^2+1)^(3/2)-5/16*b*c*d*x^2*(d+I*c*d*x)^(3/2)*(f-I*c*f*x
)^(3/2)/(c^2*x^2+1)^(3/2)-2/15*I*b*c^2*d*x^3*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)/(c^2*x^2+1)^(3/2)-1/16*b*c^3*
d*x^4*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)/(c^2*x^2+1)^(3/2)-1/25*I*b*c^4*d*x^5*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(
3/2)/(c^2*x^2+1)^(3/2)+1/4*d*x*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x))+3/8*d*x*(d+I*c*d*x)^(3/2
)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x))/(c^2*x^2+1)+1/5*I*d*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)*(c^2*x^2+1)*(a+
b*arcsinh(c*x))/c+3/16*d*(d+I*c*d*x)^(3/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x))^2/b/c/(c^2*x^2+1)^(3/2)

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 459, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.257, Rules used = {5796, 5838, 5786, 5785, 5783, 30, 14, 5798, 200} \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 \left (c^2 x^2+1\right )}+\frac {3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{16 b c \left (c^2 x^2+1\right )^{3/2}}+\frac {i d \left (c^2 x^2+1\right ) (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{5 c}+\frac {1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))-\frac {5 b c d x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (c^2 x^2+1\right )^{3/2}}-\frac {i b d x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left (c^2 x^2+1\right )^{3/2}}-\frac {2 i b c^2 d x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left (c^2 x^2+1\right )^{3/2}}-\frac {i b c^4 d x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left (c^2 x^2+1\right )^{3/2}}-\frac {b c^3 d x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (c^2 x^2+1\right )^{3/2}} \]

[In]

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

[Out]

((-1/5*I)*b*d*x*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(1 + c^2*x^2)^(3/2) - (5*b*c*d*x^2*(d + I*c*d*x)^(3/2
)*(f - I*c*f*x)^(3/2))/(16*(1 + c^2*x^2)^(3/2)) - (((2*I)/15)*b*c^2*d*x^3*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3
/2))/(1 + c^2*x^2)^(3/2) - (b*c^3*d*x^4*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(16*(1 + c^2*x^2)^(3/2)) - ((
I/25)*b*c^4*d*x^5*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2))/(1 + c^2*x^2)^(3/2) + (d*x*(d + I*c*d*x)^(3/2)*(f -
I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x]))/4 + (3*d*x*(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)) + ((I/5)*d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/c +
(3*d*(d + I*c*d*x)^(3/2)*(f - I*c*f*x)^(3/2)*(a + b*ArcSinh[c*x])^2)/(16*b*c*(1 + c^2*x^2)^(3/2))

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 30

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

Rule 200

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 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 5785

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[(1/2)*Simp[Sqrt[d + e*x^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/2)*Simp[Sqrt[d + e*x^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 5786

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/(2*p + 1))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p], Int[x*(1 + c^2*x^2)^(p - 1/2)*(a + b*A
rcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0]

Rule 5796

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 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 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]))

Rubi steps \begin{align*} \text {integral}& = \frac {\left ((d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int (d+i c d x) \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, 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 (d \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))+i c d x \left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))\right ) \, dx}{\left (1+c^2 x^2\right )^{3/2}} \\ & = \frac {\left (d (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} (a+b \text {arcsinh}(c x)) \, dx}{\left (1+c^2 x^2\right )^{3/2}}+\frac {\left (i c d (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} (a+b \text {arcsinh}(c x)) \, dx}{\left (1+c^2 x^2\right )^{3/2}} \\ & = \frac {1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))+\frac {i d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{5 c}+\frac {\left (3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x)) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (i b d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+c^2 x^2\right )^2 \, dx}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (b c d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \left (1+c^2 x^2\right ) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}} \\ & = \frac {1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))+\frac {3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 \left (1+c^2 x^2\right )}+\frac {i d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{5 c}+\frac {\left (3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{8 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (i b d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (1+2 c^2 x^2+c^4 x^4\right ) \, dx}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (b c d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int \left (x+c^2 x^3\right ) \, dx}{4 \left (1+c^2 x^2\right )^{3/2}}-\frac {\left (3 b c d (d+i c d x)^{3/2} (f-i c f x)^{3/2}\right ) \int x \, dx}{8 \left (1+c^2 x^2\right )^{3/2}} \\ & = -\frac {i b d x (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{5 \left (1+c^2 x^2\right )^{3/2}}-\frac {5 b c d x^2 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (1+c^2 x^2\right )^{3/2}}-\frac {2 i b c^2 d x^3 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{15 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c^3 d x^4 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{16 \left (1+c^2 x^2\right )^{3/2}}-\frac {i b c^4 d x^5 (d+i c d x)^{3/2} (f-i c f x)^{3/2}}{25 \left (1+c^2 x^2\right )^{3/2}}+\frac {1}{4} d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))+\frac {3 d x (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))}{8 \left (1+c^2 x^2\right )}+\frac {i d (d+i c d x)^{3/2} (f-i c f x)^{3/2} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{5 c}+\frac {3 d (d+i c d x)^{3/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x))^2}{16 b c \left (1+c^2 x^2\right )^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.17 (sec) , antiderivative size = 683, normalized size of antiderivative = 1.49 \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-1200 i b c d^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x}+1920 i a d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+6000 a c d^2 f x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+3840 i a c^2 d^2 f x^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+2400 a c^3 d^2 f x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1920 i a c^4 d^2 f x^4 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}+1800 b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2-1200 b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))-75 b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (4 \text {arcsinh}(c x))+3600 a d^{5/2} f^{3/2} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )-200 i b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (3 \text {arcsinh}(c x))+60 b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x) \left (10 i \cosh (3 \text {arcsinh}(c x))+2 i \cosh (5 \text {arcsinh}(c x))+5 \left (4 i \sqrt {1+c^2 x^2}+8 \sinh (2 \text {arcsinh}(c x))+\sinh (4 \text {arcsinh}(c x))\right )\right )-24 i b d^2 f \sqrt {d+i c d x} \sqrt {f-i c f x} \sinh (5 \text {arcsinh}(c x))}{9600 c \sqrt {1+c^2 x^2}} \]

[In]

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

[Out]

((-1200*I)*b*c*d^2*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (1920*I)*a*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f
*x]*Sqrt[1 + c^2*x^2] + 6000*a*c*d^2*f*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (3840*I)*a*c^
2*d^2*f*x^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + 2400*a*c^3*d^2*f*x^3*Sqrt[d + I*c*d*x]*Sqr
t[f - I*c*f*x]*Sqrt[1 + c^2*x^2] + (1920*I)*a*c^4*d^2*f*x^4*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x
^2] + 1800*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]^2 - 1200*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f
- I*c*f*x]*Cosh[2*ArcSinh[c*x]] - 75*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[4*ArcSinh[c*x]] + 3600*a
*d^(5/2)*f^(3/2)*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]] - (200*I
)*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[3*ArcSinh[c*x]] + 60*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c
*f*x]*ArcSinh[c*x]*((10*I)*Cosh[3*ArcSinh[c*x]] + (2*I)*Cosh[5*ArcSinh[c*x]] + 5*((4*I)*Sqrt[1 + c^2*x^2] + 8*
Sinh[2*ArcSinh[c*x]] + Sinh[4*ArcSinh[c*x]])) - (24*I)*b*d^2*f*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sinh[5*ArcS
inh[c*x]])/(9600*c*Sqrt[1 + c^2*x^2])

Maple [F]

\[\int \left (i c d x +d \right )^{\frac {5}{2}} \left (-i c f x +f \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )d x\]

[In]

int((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x)),x)

[Out]

int((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x)),x)

Fricas [F]

\[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (i \, c d x + d\right )}^{\frac {5}{2}} {\left (-i \, c f x + f\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]

[In]

integrate((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="fricas")

[Out]

integral((I*b*c^3*d^2*f*x^3 + b*c^2*d^2*f*x^2 + I*b*c*d^2*f*x + b*d^2*f)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f)*
log(c*x + sqrt(c^2*x^2 + 1)) + (I*a*c^3*d^2*f*x^3 + a*c^2*d^2*f*x^2 + I*a*c*d^2*f*x + a*d^2*f)*sqrt(I*c*d*x +
d)*sqrt(-I*c*f*x + f), x)

Sympy [F(-1)]

Timed out. \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Timed out} \]

[In]

integrate((d+I*c*d*x)**(5/2)*(f-I*c*f*x)**(3/2)*(a+b*asinh(c*x)),x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(3/2)*(a+b*arcsinh(c*x)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [F(-2)]

Exception generated. \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Error: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeError: Bad Argument TypeDone

Mupad [F(-1)]

Timed out. \[ \int (d+i c d x)^{5/2} (f-i c f x)^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}\,{\left (f-c\,f\,x\,1{}\mathrm {i}\right )}^{3/2} \,d x \]

[In]

int((a + b*asinh(c*x))*(d + c*d*x*1i)^(5/2)*(f - c*f*x*1i)^(3/2),x)

[Out]

int((a + b*asinh(c*x))*(d + c*d*x*1i)^(5/2)*(f - c*f*x*1i)^(3/2), x)