Integrand size = 35, antiderivative size = 416 \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 i b d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x}}{3 \sqrt {1+c^2 x^2}}-\frac {3 b c d^2 x^2 \sqrt {d+i c d x} \sqrt {f-i c f x}}{16 \sqrt {1+c^2 x^2}}-\frac {2 i b c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x}}{9 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x^4 \sqrt {d+i c d x} \sqrt {f-i c f x}}{16 \sqrt {1+c^2 x^2}}+\frac {3}{8} d^2 x \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))-\frac {1}{4} c^2 d^2 x^3 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))+\frac {2 i d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}{3 c}+\frac {5 d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}} \] Output:
-2/3*I*b*d^2*x*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)-3/16* b*c*d^2*x^2*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)-2/9*I*b* c^2*d^2*x^3*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+1/16*b*c ^3*d^2*x^4*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)/(c^2*x^2+1)^(1/2)+3/8*d^2*x *(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))-1/4*c^2*d^2*x^3*(d +I*c*d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))+2/3*I*d^2*(d+I*c*d*x) ^(1/2)*(f-I*c*f*x)^(1/2)*(c^2*x^2+1)*(a+b*arcsinh(c*x))/c+5/16*d^2*(d+I*c* d*x)^(1/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x))^2/b/c/(c^2*x^2+1)^(1/2)
Time = 2.58 (sec) , antiderivative size = 361, normalized size of antiderivative = 0.87 \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=\frac {48 a d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \sqrt {1+c^2 x^2} \left (16 i+9 c x+16 i c^2 x^2-6 c^3 x^3\right )+720 a d^{5/2} \sqrt {f} \sqrt {1+c^2 x^2} \log \left (c d f x+\sqrt {d} \sqrt {f} \sqrt {d+i c d x} \sqrt {f-i c f x}\right )+144 b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (2 \text {arcsinh}(c x)^2-\cosh (2 \text {arcsinh}(c x))+2 \text {arcsinh}(c x) \sinh (2 \text {arcsinh}(c x))\right )-64 i b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (9 c x-3 \text {arcsinh}(c x) \left (3 \sqrt {1+c^2 x^2}+\cosh (3 \text {arcsinh}(c x))\right )+\sinh (3 \text {arcsinh}(c x))\right )+9 b d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (8 \text {arcsinh}(c x)^2+\cosh (4 \text {arcsinh}(c x))-4 \text {arcsinh}(c x) \sinh (4 \text {arcsinh}(c x))\right )}{1152 c \sqrt {1+c^2 x^2}} \] Input:
Integrate[(d + I*c*d*x)^(5/2)*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]),x]
Output:
(48*a*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*Sqrt[1 + c^2*x^2]*(16*I + 9* c*x + (16*I)*c^2*x^2 - 6*c^3*x^3) + 720*a*d^(5/2)*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]] + 144 *b*d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(2*ArcSinh[c*x]^2 - Cosh[2*ArcS inh[c*x]] + 2*ArcSinh[c*x]*Sinh[2*ArcSinh[c*x]]) - (64*I)*b*d^2*Sqrt[d + I *c*d*x]*Sqrt[f - I*c*f*x]*(9*c*x - 3*ArcSinh[c*x]*(3*Sqrt[1 + c^2*x^2] + C osh[3*ArcSinh[c*x]]) + Sinh[3*ArcSinh[c*x]]) + 9*b*d^2*Sqrt[d + I*c*d*x]*S qrt[f - I*c*f*x]*(8*ArcSinh[c*x]^2 + Cosh[4*ArcSinh[c*x]] - 4*ArcSinh[c*x] *Sinh[4*ArcSinh[c*x]]))/(1152*c*Sqrt[1 + c^2*x^2])
Time = 1.35 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.46, 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.
| \(\displaystyle \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6211 |
| \(\displaystyle \frac {\sqrt {d+i c d x} \sqrt {f-i c f x} \int d^2 (i c x+1)^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \int (i c x+1)^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \int \left (-c^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x^2+2 i c \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) x+\sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))\right )dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d^2 \sqrt {d+i c d x} \sqrt {f-i c f x} \left (\frac {3}{8} x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {2 i \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 c}-\frac {1}{4} c^2 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {5 (a+b \text {arcsinh}(c x))^2}{16 b c}+\frac {1}{16} b c^3 x^4-\frac {2}{9} i b c^2 x^3-\frac {3}{16} b c x^2-\frac {2 i b x}{3}\right )}{\sqrt {c^2 x^2+1}}\) |
Input:
Int[(d + I*c*d*x)^(5/2)*Sqrt[f - I*c*f*x]*(a + b*ArcSinh[c*x]),x]
Output:
(d^2*Sqrt[d + I*c*d*x]*Sqrt[f - I*c*f*x]*(((-2*I)/3)*b*x - (3*b*c*x^2)/16 - ((2*I)/9)*b*c^2*x^3 + (b*c^3*x^4)/16 + (3*x*Sqrt[1 + c^2*x^2]*(a + b*Arc Sinh[c*x]))/8 - (c^2*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/4 + (((2* I)/3)*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/c + (5*(a + b*ArcSinh[c*x] )^2)/(16*b*c)))/Sqrt[1 + c^2*x^2]
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]))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1052 vs. \(2 (339 ) = 678\).
Time = 5.95 (sec) , antiderivative size = 1053, normalized size of antiderivative = 2.53
| method | result | size |
| default | \(\frac {i a \left (i c d x +d \right )^{\frac {5}{2}} \left (-i c f x +f \right )^{\frac {3}{2}}}{4 c f}+\frac {5 i a d \left (i c d x +d \right )^{\frac {3}{2}} \left (-i c f x +f \right )^{\frac {3}{2}}}{12 c f}+\frac {5 i a \,d^{2} \sqrt {i c d x +d}\, \left (-i c f x +f \right )^{\frac {3}{2}}}{8 c f}-\frac {5 i a \,d^{2} \sqrt {-i c f x +f}\, \sqrt {i c d x +d}}{8 c}+\frac {5 a \,d^{3} f \sqrt {\left (-i c f x +f \right ) \left (i c d x +d \right )}\, \ln \left (\frac {c^{2} d f x}{\sqrt {c^{2} d f}}+\sqrt {c^{2} d f \,x^{2}+d f}\right )}{8 \sqrt {-i c f x +f}\, \sqrt {i c d x +d}\, \sqrt {c^{2} d f}}+b \left (\frac {5 \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \operatorname {arcsinh}\left (x c \right )^{2} d^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c}-\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{256 c \left (c^{2} x^{2}+1\right )}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{36 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{16 c \left (c^{2} x^{2}+1\right )}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right ) d^{2}}{4 c \left (c^{2} x^{2}+1\right )}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right ) d^{2}}{4 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{16 c \left (c^{2} x^{2}+1\right )}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{36 c \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{256 c \left (c^{2} x^{2}+1\right )}\right )\) | \(1053\) |
| parts | \(\frac {i a \left (i c d x +d \right )^{\frac {5}{2}} \left (-i c f x +f \right )^{\frac {3}{2}}}{4 c f}+\frac {5 i a d \left (i c d x +d \right )^{\frac {3}{2}} \left (-i c f x +f \right )^{\frac {3}{2}}}{12 c f}+\frac {5 i a \,d^{2} \sqrt {i c d x +d}\, \left (-i c f x +f \right )^{\frac {3}{2}}}{8 c f}-\frac {5 i a \,d^{2} \sqrt {-i c f x +f}\, \sqrt {i c d x +d}}{8 c}+\frac {5 a \,d^{3} f \sqrt {\left (-i c f x +f \right ) \left (i c d x +d \right )}\, \ln \left (\frac {c^{2} d f x}{\sqrt {c^{2} d f}}+\sqrt {c^{2} d f \,x^{2}+d f}\right )}{8 \sqrt {-i c f x +f}\, \sqrt {i c d x +d}\, \sqrt {c^{2} d f}}+b \left (\frac {5 \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \operatorname {arcsinh}\left (x c \right )^{2} d^{2}}{16 \sqrt {c^{2} x^{2}+1}\, c}-\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (8 x^{5} c^{5}+8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}+8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{256 c \left (c^{2} x^{2}+1\right )}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (4 c^{4} x^{4}+4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}+3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (-1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{36 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (2 x^{3} c^{3}+2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c +\sqrt {c^{2} x^{2}+1}\right ) \left (-1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{16 c \left (c^{2} x^{2}+1\right )}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (c^{2} x^{2}+\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )-1\right ) d^{2}}{4 c \left (c^{2} x^{2}+1\right )}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (c^{2} x^{2}-\sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (\operatorname {arcsinh}\left (x c \right )+1\right ) d^{2}}{4 c \left (c^{2} x^{2}+1\right )}+\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (2 x^{3} c^{3}-2 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+2 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+2 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{16 c \left (c^{2} x^{2}+1\right )}+\frac {i \sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (4 c^{4} x^{4}-4 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+5 c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, x c +1\right ) \left (1+3 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{36 c \left (c^{2} x^{2}+1\right )}-\frac {\sqrt {i \left (x c -i\right ) d}\, \sqrt {-i \left (x c +i\right ) f}\, \left (8 x^{5} c^{5}-8 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}+12 x^{3} c^{3}-8 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+4 x c -\sqrt {c^{2} x^{2}+1}\right ) \left (1+4 \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{256 c \left (c^{2} x^{2}+1\right )}\right )\) | \(1053\) |
Input:
int((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(x*c)),x,method=_RETUR NVERBOSE)
Output:
1/4*I*a/c/f*(d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(3/2)+5/12*I*a*d/c/f*(d+I*c*d*x) ^(3/2)*(f-I*c*f*x)^(3/2)+5/8*I*a*d^2/c/f*(d+I*c*d*x)^(1/2)*(f-I*c*f*x)^(3/ 2)-5/8*I*a*d^2/c*(f-I*c*f*x)^(1/2)*(d+I*c*d*x)^(1/2)+5/8*a*d^3*f*((f-I*c*f *x)*(d+I*c*d*x))^(1/2)/(f-I*c*f*x)^(1/2)/(d+I*c*d*x)^(1/2)*ln(c^2*d*f*x/(c ^2*d*f)^(1/2)+(c^2*d*f*x^2+d*f)^(1/2))/(c^2*d*f)^(1/2)+b*(5/16*(I*(x*c-I)* d)^(1/2)*(-I*(I+x*c)*f)^(1/2)/(c^2*x^2+1)^(1/2)/c*arcsinh(x*c)^2*d^2-1/256 *(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(8*x^5*c^5+8*x^4*c^4*(c^2*x^2+1) ^(1/2)+12*x^3*c^3+8*x^2*c^2*(c^2*x^2+1)^(1/2)+4*x*c+(c^2*x^2+1)^(1/2))*(-1 +4*arcsinh(x*c))*d^2/c/(c^2*x^2+1)+1/36*I*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)* f)^(1/2)*(4*c^4*x^4+4*(c^2*x^2+1)^(1/2)*c^3*x^3+5*c^2*x^2+3*(c^2*x^2+1)^(1 /2)*x*c+1)*(-1+3*arcsinh(x*c))*d^2/c/(c^2*x^2+1)+1/16*(I*(x*c-I)*d)^(1/2)* (-I*(I+x*c)*f)^(1/2)*(2*x^3*c^3+2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x*c+(c^2*x^2 +1)^(1/2))*(-1+2*arcsinh(x*c))*d^2/c/(c^2*x^2+1)+1/4*I*(I*(x*c-I)*d)^(1/2) *(-I*(I+x*c)*f)^(1/2)*(c^2*x^2+(c^2*x^2+1)^(1/2)*x*c+1)*(arcsinh(x*c)-1)*d ^2/c/(c^2*x^2+1)+1/4*I*(I*(x*c-I)*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(c^2*x^2-( c^2*x^2+1)^(1/2)*x*c+1)*(arcsinh(x*c)+1)*d^2/c/(c^2*x^2+1)+1/16*(I*(x*c-I) *d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(2*x^3*c^3-2*x^2*c^2*(c^2*x^2+1)^(1/2)+2*x* c-(c^2*x^2+1)^(1/2))*(1+2*arcsinh(x*c))*d^2/c/(c^2*x^2+1)+1/36*I*(I*(x*c-I )*d)^(1/2)*(-I*(I+x*c)*f)^(1/2)*(4*c^4*x^4-4*(c^2*x^2+1)^(1/2)*c^3*x^3+5*c ^2*x^2-3*(c^2*x^2+1)^(1/2)*x*c+1)*(1+3*arcsinh(x*c))*d^2/c/(c^2*x^2+1)-...
\[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (i \, c d x + d\right )}^{\frac {5}{2}} \sqrt {-i \, c f x + f} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \] Input:
integrate((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x)),x, algori thm="fricas")
Output:
integral(-(b*c^2*d^2*x^2 - 2*I*b*c*d^2*x - b*d^2)*sqrt(I*c*d*x + d)*sqrt(- I*c*f*x + f)*log(c*x + sqrt(c^2*x^2 + 1)) - (a*c^2*d^2*x^2 - 2*I*a*c*d^2*x - a*d^2)*sqrt(I*c*d*x + d)*sqrt(-I*c*f*x + f), x)
Timed out. \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=\text {Timed out} \] Input:
integrate((d+I*c*d*x)**(5/2)*(f-I*c*f*x)**(1/2)*(a+b*asinh(c*x)),x)
Output:
Timed out
Exception generated. \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x)),x, algori thm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Exception generated. \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(1/2)*(a+b*arcsinh(c*x)),x, algori thm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=\int \left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^{5/2}\,\sqrt {f-c\,f\,x\,1{}\mathrm {i}} \,d x \] Input:
int((a + b*asinh(c*x))*(d + c*d*x*1i)^(5/2)*(f - c*f*x*1i)^(1/2),x)
Output:
int((a + b*asinh(c*x))*(d + c*d*x*1i)^(5/2)*(f - c*f*x*1i)^(1/2), x)
\[ \int (d+i c d x)^{5/2} \sqrt {f-i c f x} (a+b \text {arcsinh}(c x)) \, dx=\frac {\sqrt {f}\, \sqrt {d}\, d^{2} \left (30 \mathit {asin} \left (\frac {\sqrt {-c i x +1}}{\sqrt {2}}\right ) a i -6 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a \,c^{3} x^{3}+16 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a \,c^{2} i \,x^{2}+9 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a c x +16 \sqrt {c i x +1}\, \sqrt {-c i x +1}\, a i -24 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right ) x^{2}d x \right ) b \,c^{3}+48 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right ) x d x \right ) b \,c^{2} i +24 \left (\int \sqrt {c i x +1}\, \sqrt {-c i x +1}\, \mathit {asinh} \left (c x \right )d x \right ) b c \right )}{24 c} \] Input:
int((d+I*c*d*x)^(5/2)*(f-I*c*f*x)^(1/2)*(a+b*asinh(c*x)),x)
Output:
(sqrt(f)*sqrt(d)*d**2*(30*asin(sqrt( - c*i*x + 1)/sqrt(2))*a*i - 6*sqrt(c* i*x + 1)*sqrt( - c*i*x + 1)*a*c**3*x**3 + 16*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a*c**2*i*x**2 + 9*sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*a*c*x + 16*sqrt (c*i*x + 1)*sqrt( - c*i*x + 1)*a*i - 24*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asinh(c*x)*x**2,x)*b*c**3 + 48*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1) *asinh(c*x)*x,x)*b*c**2*i + 24*int(sqrt(c*i*x + 1)*sqrt( - c*i*x + 1)*asin h(c*x),x)*b*c))/(24*c)