Integrand size = 23, antiderivative size = 157 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a-i b}}\right )}{(a-i b)^{5/2} d}-\frac {i \text {arctanh}\left (\frac {\sqrt {a+b \tan (c+d x)}}{\sqrt {a+i b}}\right )}{(a+i b)^{5/2} d}-\frac {2 a^2}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}}+\frac {4 a b}{\left (a^2+b^2\right )^2 d \sqrt {a+b \tan (c+d x)}} \]
I*arctanh((a+b*tan(d*x+c))^(1/2)/(a-I*b)^(1/2))/(a-I*b)^(5/2)/d-I*arctanh( (a+b*tan(d*x+c))^(1/2)/(a+I*b)^(1/2))/(a+I*b)^(5/2)/d+4*a*b/(a^2+b^2)^2/d/ (a+b*tan(d*x+c))^(1/2)-2/3*a^2/b/(a^2+b^2)/d/(a+b*tan(d*x+c))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.19 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.78 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\frac {b (-i a+b) \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \tan (c+d x)}{a-i b}\right )-(a-i b) \left (2 a+2 i b-i b \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},\frac {a+b \tan (c+d x)}{a+i b}\right )\right )}{3 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^{3/2}} \]
(b*((-I)*a + b)*Hypergeometric2F1[-3/2, 1, -1/2, (a + b*Tan[c + d*x])/(a - I*b)] - (a - I*b)*(2*a + (2*I)*b - I*b*Hypergeometric2F1[-3/2, 1, -1/2, ( a + b*Tan[c + d*x])/(a + I*b)]))/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^( 3/2))
Time = 0.87 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.522, Rules used = {3042, 4025, 25, 3042, 4012, 3042, 4022, 3042, 4020, 25, 73, 221}
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.
| \(\displaystyle \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\tan (c+d x)^2}{(a+b \tan (c+d x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle \frac {\int -\frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}}dx}{a^2+b^2}-\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}}dx}{a^2+b^2}-\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \frac {a-b \tan (c+d x)}{(a+b \tan (c+d x))^{3/2}}dx}{a^2+b^2}-\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle -\frac {\frac {\int \frac {a^2-2 b \tan (c+d x) a-b^2}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a^2+b^2}-\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\frac {\int \frac {a^2-2 b \tan (c+d x) a-b^2}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}}{a^2+b^2}-\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}\) |
| \(\Big \downarrow \) 4022 |
| \(\displaystyle -\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {1}{2} (a-i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} (a+i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {1}{2} (a-i b)^2 \int \frac {1-i \tan (c+d x)}{\sqrt {a+b \tan (c+d x)}}dx+\frac {1}{2} (a+i b)^2 \int \frac {i \tan (c+d x)+1}{\sqrt {a+b \tan (c+d x)}}dx}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 4020 |
| \(\displaystyle -\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {i (a+i b)^2 \int -\frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}-\frac {i (a-i b)^2 \int -\frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {i (a-i b)^2 \int \frac {1}{(i \tan (c+d x)+1) \sqrt {a+b \tan (c+d x)}}d(-i \tan (c+d x))}{2 d}-\frac {i (a+i b)^2 \int \frac {1}{(1-i \tan (c+d x)) \sqrt {a+b \tan (c+d x)}}d(i \tan (c+d x))}{2 d}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {(a-i b)^2 \int \frac {1}{-\frac {i \tan ^2(c+d x)}{b}-\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}+\frac {(a+i b)^2 \int \frac {1}{\frac {i \tan ^2(c+d x)}{b}+\frac {i a}{b}+1}d\sqrt {a+b \tan (c+d x)}}{b d}}{a^2+b^2}}{a^2+b^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {2 a^2}{3 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^{3/2}}-\frac {-\frac {4 a b}{d \left (a^2+b^2\right ) \sqrt {a+b \tan (c+d x)}}+\frac {\frac {(a-i b)^2 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a+i b}}\right )}{d \sqrt {a+i b}}+\frac {(a+i b)^2 \arctan \left (\frac {\tan (c+d x)}{\sqrt {a-i b}}\right )}{d \sqrt {a-i b}}}{a^2+b^2}}{a^2+b^2}\) |
(-2*a^2)/(3*b*(a^2 + b^2)*d*(a + b*Tan[c + d*x])^(3/2)) - ((((a + I*b)^2*A rcTan[Tan[c + d*x]/Sqrt[a - I*b]])/(Sqrt[a - I*b]*d) + ((a - I*b)^2*ArcTan [Tan[c + d*x]/Sqrt[a + I*b]])/(Sqrt[a + I*b]*d))/(a^2 + b^2) - (4*a*b)/((a ^2 + b^2)*d*Sqrt[a + b*Tan[c + d*x]]))/(a^2 + b^2)
3.6.49.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x] )^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[c*(d/f) Subst[Int[(a + (b/d)*x)^m/(d^2 + c*x), x], x, d*Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[c^2 + d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c + I*d)/2 Int[(a + b*Tan[e + f*x])^m*( 1 - I*Tan[e + f*x]), x], x] + Simp[(c - I*d)/2 Int[(a + b*Tan[e + f*x])^m *(1 + I*Tan[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !IntegerQ[m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2322\) vs. \(2(133)=266\).
Time = 0.13 (sec) , antiderivative size = 2323, normalized size of antiderivative = 14.80
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(2323\) |
| default | \(\text {Expression too large to display}\) | \(2323\) |
4*a*b/(a^2+b^2)^2/d/(a+b*tan(d*x+c))^(1/2)-2/3*a^2/b/(a^2+b^2)/d/(a+b*tan( d*x+c))^(3/2)-1/d/b/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arctan(( (2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1/2) -2*a)^(1/2))*a^4+1/d/b/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)*arcta n(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2+b^2)^(1 /2)-2*a)^(1/2))*a^6-1/d*b^3/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/2)* arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+b^ 2)^(1/2)-2*a)^(1/2))*a^2+1/d/b/(a^2+b^2)^(5/2)/(2*(a^2+b^2)^(1/2)-2*a)^(1/ 2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2 +b^2)^(1/2)-2*a)^(1/2))*a^4-1/d/b/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^ (1/2)*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*( a^2+b^2)^(1/2)-2*a)^(1/2))*a^6-3/4/d*b^3/(a^2+b^2)^(7/2)*ln(b*tan(d*x+c)+a +(a+b*tan(d*x+c))^(1/2)*(2*(a^2+b^2)^(1/2)+2*a)^(1/2)+(a^2+b^2)^(1/2))*(2* (a^2+b^2)^(1/2)+2*a)^(1/2)*a-2/d*b/(a^2+b^2)^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/ 2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a^2 +b^2)^(1/2)-2*a)^(1/2))*a^3+2/d*b/(a^2+b^2)^3/(2*(a^2+b^2)^(1/2)-2*a)^(1/2 )*arctan((2*(a+b*tan(d*x+c))^(1/2)+(2*(a^2+b^2)^(1/2)+2*a)^(1/2))/(2*(a^2+ b^2)^(1/2)-2*a)^(1/2))*a^3+4/d*b/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2)-2*a)^( 1/2)*arctan(((2*(a^2+b^2)^(1/2)+2*a)^(1/2)-2*(a+b*tan(d*x+c))^(1/2))/(2*(a ^2+b^2)^(1/2)-2*a)^(1/2))*a^4+1/d*b^3/(a^2+b^2)^(7/2)/(2*(a^2+b^2)^(1/2...
Leaf count of result is larger than twice the leaf count of optimal. 3320 vs. \(2 (127) = 254\).
Time = 0.31 (sec) , antiderivative size = 3320, normalized size of antiderivative = 21.15 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
-1/6*(3*((a^4*b^3 + 2*a^2*b^5 + b^7)*d*tan(d*x + c)^2 + 2*(a^5*b^2 + 2*a^3 *b^4 + a*b^6)*d*tan(d*x + c) + (a^6*b + 2*a^4*b^3 + a^2*b^5)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5* a^2*b^8 + b^10)*d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2 *b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12* b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b ^18 + b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^ 8 + b^10)*d^2))*log((5*a^4*b - 10*a^2*b^3 + b^5)*sqrt(b*tan(d*x + c) + a) + ((a^13 + 2*a^11*b^2 - 5*a^9*b^4 - 20*a^7*b^6 - 25*a^5*b^8 - 14*a^3*b^10 - 3*a*b^12)*d^3*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2*b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12*b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b^18 + b^20)*d^4)) + (15*a^6*b^2 - 35*a^4*b^4 + 13*a^2*b^6 - b^8)*d)*sqrt(-(a^5 - 10*a^3*b^2 + 5*a*b^4 + (a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5* a^2*b^8 + b^10)*d^2*sqrt(-(25*a^8*b^2 - 100*a^6*b^4 + 110*a^4*b^6 - 20*a^2 *b^8 + b^10)/((a^20 + 10*a^18*b^2 + 45*a^16*b^4 + 120*a^14*b^6 + 210*a^12* b^8 + 252*a^10*b^10 + 210*a^8*b^12 + 120*a^6*b^14 + 45*a^4*b^16 + 10*a^2*b ^18 + b^20)*d^4)))/((a^10 + 5*a^8*b^2 + 10*a^6*b^4 + 10*a^4*b^6 + 5*a^2*b^ 8 + b^10)*d^2))) - 3*((a^4*b^3 + 2*a^2*b^5 + b^7)*d*tan(d*x + c)^2 + 2*(a^ 5*b^2 + 2*a^3*b^4 + a*b^6)*d*tan(d*x + c) + (a^6*b + 2*a^4*b^3 + a^2*b^...
\[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int \frac {\tan ^{2}{\left (c + d x \right )}}{\left (a + b \tan {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
\[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\int { \frac {\tan \left (d x + c\right )^{2}}{{\left (b \tan \left (d x + c\right ) + a\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Timed out} \]
Time = 9.97 (sec) , antiderivative size = 3708, normalized size of antiderivative = 23.62 \[ \int \frac {\tan ^2(c+d x)}{(a+b \tan (c+d x))^{5/2}} \, dx=\text {Too large to display} \]
(log(((-1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2 *10i - 10*a^3*b^2*d^2))^(1/2)*(((-1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(896*a^6*b^15*d^4 - 160*a^2*b^19*d^4 - 128*a^4*b^17*d^4 - 32*b^21*d^4 + 3136*a^8*b^13*d^4 + 4928*a^10*b^11*d^4 + 4480*a^12*b^9*d^4 + 2432*a^14*b^7*d^4 + 736*a^16*b^5* d^4 + 96*a^18*b^3*d^4 - ((-1/(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d ^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2))^(1/2)*(a + b*tan(c + d*x))^(1/2 )*(64*a*b^22*d^5 + 640*a^3*b^20*d^5 + 2880*a^5*b^18*d^5 + 7680*a^7*b^16*d^ 5 + 13440*a^9*b^14*d^5 + 16128*a^11*b^12*d^5 + 13440*a^13*b^10*d^5 + 7680* a^15*b^8*d^5 + 2880*a^17*b^6*d^5 + 640*a^19*b^4*d^5 + 64*a^21*b^2*d^5))/2) )/2 + (a + b*tan(c + d*x))^(1/2)*(320*a^4*b^14*d^3 - 16*b^18*d^3 + 1024*a^ 6*b^12*d^3 + 1440*a^8*b^10*d^3 + 1024*a^10*b^8*d^3 + 320*a^12*b^6*d^3 - 16 *a^16*b^2*d^3)))/2 - 16*a*b^15*d^2 - 96*a^3*b^13*d^2 - 240*a^5*b^11*d^2 - 320*a^7*b^9*d^2 - 240*a^9*b^7*d^2 - 96*a^11*b^5*d^2 - 16*a^13*b^3*d^2)*(-1 /(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10 *a^3*b^2*d^2))^(1/2))/2 - log(((-1/(4*(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 - a^4*b*d^2*5i + a^2*b^3*d^2*10i - 10*a^3*b^2*d^2)))^(1/2)*(896*a^6*b^15*d ^4 - 160*a^2*b^19*d^4 - 128*a^4*b^17*d^4 - 32*b^21*d^4 + 3136*a^8*b^13*d^4 + 4928*a^10*b^11*d^4 + 4480*a^12*b^9*d^4 + 2432*a^14*b^7*d^4 + 736*a^16*b ^5*d^4 + 96*a^18*b^3*d^4 + (-1/(4*(a^5*d^2 - b^5*d^2*1i + 5*a*b^4*d^2 -...