Integrand size = 35, antiderivative size = 266 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=-\frac {2 i b d^2 x \sqrt {1+c^2 x^2}}{\sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {b c d^2 x^2 \sqrt {1+c^2 x^2}}{4 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {2 i d^2 \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{c \sqrt {d+i c d x} \sqrt {f-i c f x}}-\frac {d^2 x \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{2 \sqrt {d+i c d x} \sqrt {f-i c f x}}+\frac {3 d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))^2}{4 b c \sqrt {d+i c d x} \sqrt {f-i c f x}} \]
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*arcsinh(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)
Time = 5.55 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.29 \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\frac {-16 i b c d x \sqrt {d+i c d x} \sqrt {f-i c f x}+16 i a d \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-4 a c d x \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2}-4 b d (-4 i+c x) \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2} \text {arcsinh}(c x)+6 b d \sqrt {d+i c d x} \sqrt {f-i c f x} \text {arcsinh}(c x)^2+b d \sqrt {d+i c d x} \sqrt {f-i c f x} \cosh (2 \text {arcsinh}(c x))+12 a d^{3/2} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )}{8 c f \sqrt {1+c^2 x^2}} \]
((-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])
Time = 0.69 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.50, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {6211, 27, 6253, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx\) |
\(\Big \downarrow \) 6211 |
\(\displaystyle \frac {\sqrt {c^2 x^2+1} \int \frac {d^2 (i c x+1)^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {d^2 \sqrt {c^2 x^2+1} \int \frac {(i c x+1)^2 (a+b \text {arcsinh}(c x))}{\sqrt {c^2 x^2+1}}dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
\(\Big \downarrow \) 6253 |
\(\displaystyle \frac {d^2 \sqrt {c^2 x^2+1} \int \left (-\frac {c^2 (a+b \text {arcsinh}(c x)) x^2}{\sqrt {c^2 x^2+1}}+\frac {2 i c (a+b \text {arcsinh}(c x)) x}{\sqrt {c^2 x^2+1}}+\frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}\right )dx}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d^2 \sqrt {c^2 x^2+1} \left (-\frac {1}{2} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {2 i \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}+\frac {3 (a+b \text {arcsinh}(c x))^2}{4 b c}+\frac {1}{4} b c x^2-2 i b x\right )}{\sqrt {d+i c d x} \sqrt {f-i c f x}}\) |
(d^2*Sqrt[1 + c^2*x^2]*((-2*I)*b*x + (b*c*x^2)/4 + ((2*I)*Sqrt[1 + c^2*x^2 ]*(a + b*ArcSinh[c*x]))/c - (x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/2 + (3*(a + b*ArcSinh[c*x])^2)/(4*b*c)))/(Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x] )
3.6.53.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_))^(p_)*((f_ ) + (g_.)*(x_))^(q_), x_Symbol] :> Simp[(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]
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]))
\[\int \frac {\left (i c d x +d \right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}{\sqrt {-i c f x +f}}d x\]
\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {-i \, c f x + f}} \,d x } \]
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)
\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\int \frac {\left (i d \left (c x - i\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{\sqrt {- i f \left (c x + i\right )}}\, dx \]
Exception generated. \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\int { \frac {{\left (i \, c d x + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{\sqrt {-i \, c f x + f}} \,d x } \]
Timed out. \[ \int \frac {(d+i c d x)^{3/2} (a+b \text {arcsinh}(c x))}{\sqrt {f-i c f x}} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{3/2}}{\sqrt {f-c\,f\,x\,1{}\mathrm {i}}} \,d x \]