\(\int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx\) [553]

Optimal result

Integrand size = 35, antiderivative size = 266 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=-\frac {2 i b d^2 x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c d^2 x^2 \sqrt {1+c^2 x^2}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]

[Out]

2*I*d^2*(c^2*x^2+1)*(a+b*arcsinh(c*x))/c/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-1/2*d^2*x*(c^2*x^2+1)*(a+b*arcsin
h(c*x))/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)-2*I*b*d^2*x*(c^2*x^2+1)^(1/2)/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+
1/4*b*c*d^2*x^2*(c^2*x^2+1)^(1/2)/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)+3/4*d^2*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)
^(1/2)/b/c/(d+I*c*d*x)^(1/2)/(f-I*c*f*x)^(1/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5796, 5838, 5783, 5798, 8, 5812, 30} \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\frac {3 d^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {d^2 x \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i d^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c d^2 x^2 \sqrt {c^2 x^2+1}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {2 i b d^2 x \sqrt {c^2 x^2+1}}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \]

[In]

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

[Out]

((-2*I)*b*d^2*x*Sqrt[1 + c^2*x^2])/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (b*c*d^2*x^2*Sqrt[1 + c^2*x^2])/(4*
Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + ((2*I)*d^2*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(c*Sqrt[d + I*c*d*x]*Sqr
t[f - I*c*f*x]) - (d^2*x*(1 + c^2*x^2)*(a + b*ArcSinh[c*x]))/(2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]) + (3*d^2*
Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])^2)/(4*b*c*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x])

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

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

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 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 5812

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

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 {\sqrt {1+c^2 x^2} \int \frac {(d+i c d x)^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {\sqrt {1+c^2 x^2} \int \left (\frac {d^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}+\frac {2 i c d^2 x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}-\frac {c^2 d^2 x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}}\right ) \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {\left (d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (2 i c d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (c^2 d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {x^2 (a+b \text {arcsinh}(c x))}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = \frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{2 b c \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (d^2 \sqrt {1+c^2 x^2}\right ) \int \frac {a+b \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {\left (2 i b d^2 \sqrt {1+c^2 x^2}\right ) \int 1 \, dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {\left (b c d^2 \sqrt {1+c^2 x^2}\right ) \int x \, dx}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ & = -\frac {2 i b d^2 x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c d^2 x^2 \sqrt {1+c^2 x^2}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.55 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.29 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\frac {-16 i b c d x \sqrt {d+i c d x} \sqrt {f-i c f x}+16 i a d \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-4 a c d x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-4 b d (-4 i+c x) \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+6 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2+b d \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+12 a d^{3/2} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{8 c f \sqrt {1+c^2 x^2}} \]

[In]

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

[Out]

((-16*I)*b*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] + (16*I)*a*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 +
c^2*x^2] - 4*a*c*d*x*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2] - 4*b*d*(-4*I + c*x)*Sqrt[d + I*c*
d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2]*ArcSinh[c*x] + 6*b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*ArcSinh[c*x]
^2 + b*d*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Cosh[2*ArcSinh[c*x]] + 12*a*d^(3/2)*Sqrt[f]*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]])/(8*c*f*Sqrt[1 + c^2*x^2])

Maple [F]

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

[In]

int((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))/(f-I*c*f*x)^(1/2),x)

[Out]

int((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))/(f-I*c*f*x)^(1/2),x)

Fricas [F]

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

[In]

integrate((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))/(f-I*c*f*x)^(1/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

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

[In]

integrate((d+I*c*d*x)**(3/2)*(a+b*asinh(c*x))/(f-I*c*f*x)**(1/2),x)

[Out]

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

Maxima [F(-2)]

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

[In]

integrate((d+I*c*d*x)^(3/2)*(a+b*arcsinh(c*x))/(f-I*c*f*x)^(1/2),x, algorithm="maxima")

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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