Integrand size = 41, antiderivative size = 284 \[ \int (a+b \tan (d+e x)) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2} \, dx=\frac {b \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}{3 e}+\frac {\left (a^4-b^4\right ) \log (\cos (d+e x)) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}{e (b+a \tan (d+e x))^3}+\frac {\left (a^2+b^2\right ) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}{2 e (b+a \tan (d+e x))}-\frac {2 a^4 b \left (a^2+b^2\right ) x \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}{\left (a b+a^2 \tan (d+e x)\right )^3}+\frac {a^4 b \left (a^2+b^2\right ) \tan (d+e x) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}{e \left (a b+a^2 \tan (d+e x)\right )^3} \]
1/3*b*(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2)/e+(a^4-b^4)*ln(cos(e*x +d))*(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2)/e/(b+a*tan(e*x+d))^3+1/ 2*(a^2+b^2)*(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2)/e/(b+a*tan(e*x+d ))-2*a^4*b*(a^2+b^2)*x*(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2)/(a*b+ a^2*tan(e*x+d))^3+a^4*b*(a^2+b^2)*tan(e*x+d)*(b^2+2*a*b*tan(e*x+d)+a^2*tan (e*x+d)^2)^(3/2)/e/(a*b+a^2*tan(e*x+d))^3
Result contains complex when optimal does not.
Time = 1.15 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.52 \[ \int (a+b \tan (d+e x)) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2} \, dx=\frac {\sqrt {(b+a \tan (d+e x))^2} \left (-3 \left (a^2+b^2\right ) \left ((a-i b)^2 \log (i-\tan (d+e x))+(a+i b)^2 \log (i+\tan (d+e x))\right )+6 a b \left (2 a^2+3 b^2\right ) \tan (d+e x)+3 a^2 \left (a^2+3 b^2\right ) \tan ^2(d+e x)+2 a^3 b \tan ^3(d+e x)\right )}{6 e (b+a \tan (d+e x))} \]
(Sqrt[(b + a*Tan[d + e*x])^2]*(-3*(a^2 + b^2)*((a - I*b)^2*Log[I - Tan[d + e*x]] + (a + I*b)^2*Log[I + Tan[d + e*x]]) + 6*a*b*(2*a^2 + 3*b^2)*Tan[d + e*x] + 3*a^2*(a^2 + 3*b^2)*Tan[d + e*x]^2 + 2*a^3*b*Tan[d + e*x]^3))/(6* e*(b + a*Tan[d + e*x]))
Time = 0.80 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.52, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.293, Rules used = {3042, 4193, 27, 3042, 4011, 27, 3042, 4011, 3042, 4008, 3042, 3956}
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.
| \(\displaystyle \int (a+b \tan (d+e x)) \left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \tan (d+e x)) \left (a^2 \tan (d+e x)^2+2 a b \tan (d+e x)+b^2\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4193 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \int 8 \left (\tan (d+e x) a^2+b a\right )^3 (a+b \tan (d+e x))dx}{8 \left (a^2 \tan (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \int \left (\tan (d+e x) a^2+b a\right )^3 (a+b \tan (d+e x))dx}{\left (a^2 \tan (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \int \left (\tan (d+e x) a^2+b a\right )^3 (a+b \tan (d+e x))dx}{\left (a^2 \tan (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \left (\int a \left (a^2+b^2\right ) \tan (d+e x) \left (\tan (d+e x) a^2+b a\right )^2dx+\frac {b \left (a^2 \tan (d+e x)+a b\right )^3}{3 e}\right )}{\left (a^2 \tan (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \left (a \left (a^2+b^2\right ) \int \tan (d+e x) \left (\tan (d+e x) a^2+b a\right )^2dx+\frac {b \left (a^2 \tan (d+e x)+a b\right )^3}{3 e}\right )}{\left (a^2 \tan (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \left (a \left (a^2+b^2\right ) \int \tan (d+e x) \left (\tan (d+e x) a^2+b a\right )^2dx+\frac {b \left (a^2 \tan (d+e x)+a b\right )^3}{3 e}\right )}{\left (a^2 \tan (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \left (a \left (a^2+b^2\right ) \left (\int \left (\tan (d+e x) a^2+b a\right ) \left (a b \tan (d+e x)-a^2\right )dx+\frac {\left (a^2 \tan (d+e x)+a b\right )^2}{2 e}\right )+\frac {b \left (a^2 \tan (d+e x)+a b\right )^3}{3 e}\right )}{\left (a^2 \tan (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \left (a \left (a^2+b^2\right ) \left (\int \left (\tan (d+e x) a^2+b a\right ) \left (a b \tan (d+e x)-a^2\right )dx+\frac {\left (a^2 \tan (d+e x)+a b\right )^2}{2 e}\right )+\frac {b \left (a^2 \tan (d+e x)+a b\right )^3}{3 e}\right )}{\left (a^2 \tan (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 4008 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \left (a \left (a^2+b^2\right ) \left (-a^2 \left (a^2-b^2\right ) \int \tan (d+e x)dx+\frac {a^3 b \tan (d+e x)}{e}-2 a^3 b x+\frac {\left (a^2 \tan (d+e x)+a b\right )^2}{2 e}\right )+\frac {b \left (a^2 \tan (d+e x)+a b\right )^3}{3 e}\right )}{\left (a^2 \tan (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \left (a \left (a^2+b^2\right ) \left (-a^2 \left (a^2-b^2\right ) \int \tan (d+e x)dx+\frac {a^3 b \tan (d+e x)}{e}-2 a^3 b x+\frac {\left (a^2 \tan (d+e x)+a b\right )^2}{2 e}\right )+\frac {b \left (a^2 \tan (d+e x)+a b\right )^3}{3 e}\right )}{\left (a^2 \tan (d+e x)+a b\right )^3}\) |
| \(\Big \downarrow \) 3956 |
| \(\displaystyle \frac {\left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2} \left (\frac {b \left (a^2 \tan (d+e x)+a b\right )^3}{3 e}+a \left (a^2+b^2\right ) \left (\frac {a^3 b \tan (d+e x)}{e}-2 a^3 b x+\frac {a^2 \left (a^2-b^2\right ) \log (\cos (d+e x))}{e}+\frac {\left (a^2 \tan (d+e x)+a b\right )^2}{2 e}\right )\right )}{\left (a^2 \tan (d+e x)+a b\right )^3}\) |
((b^2 + 2*a*b*Tan[d + e*x] + a^2*Tan[d + e*x]^2)^(3/2)*((b*(a*b + a^2*Tan[ d + e*x])^3)/(3*e) + a*(a^2 + b^2)*(-2*a^3*b*x + (a^2*(a^2 - b^2)*Log[Cos[ d + e*x]])/e + (a^3*b*Tan[d + e*x])/e + (a*b + a^2*Tan[d + e*x])^2/(2*e))) )/(a*b + a^2*Tan[d + e*x])^3
3.6.14.3.1 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[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d *x], x]]/d, x] /; FreeQ[{c, d}, x]
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_.) + (f_.) *(x_)]), x_Symbol] :> Simp[(a*c - b*d)*x, x] + (Simp[b*d*(Tan[e + f*x]/f), x] + Simp[(b*c + a*d) Int[Tan[e + f*x], x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[b*c + a*d, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((A_) + (B_.)*tan[(d_.) + (e_.)*(x_)])*((a_) + (b_.)*tan[(d_.) + (e_.)* (x_)] + (c_.)*tan[(d_.) + (e_.)*(x_)]^2)^(n_), x_Symbol] :> Simp[(a + b*Tan [d + e*x] + c*Tan[d + e*x]^2)^n/(b + 2*c*Tan[d + e*x])^(2*n) Int[(A + B*T an[d + e*x])*(b + 2*c*Tan[d + e*x])^(2*n), x], x] /; FreeQ[{a, b, c, d, e, A, B}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[n]
Time = 1.34 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.56
| method | result | size |
| derivativedivides | \(-\frac {\left (b^{2}+2 a b \tan \left (e x +d \right )+a^{2} \tan \left (e x +d \right )^{2}\right )^{\frac {3}{2}} \left (-2 \tan \left (e x +d \right )^{3} a^{3} b -3 a^{4} \tan \left (e x +d \right )^{2}-9 \tan \left (e x +d \right )^{2} a^{2} b^{2}+3 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{4}-3 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) b^{4}+12 \arctan \left (\tan \left (e x +d \right )\right ) a^{3} b +12 \arctan \left (\tan \left (e x +d \right )\right ) a \,b^{3}-12 \tan \left (e x +d \right ) a^{3} b -18 \tan \left (e x +d \right ) a \,b^{3}\right )}{6 e \left (b +a \tan \left (e x +d \right )\right )^{3}}\) | \(158\) |
| default | \(-\frac {\left (b^{2}+2 a b \tan \left (e x +d \right )+a^{2} \tan \left (e x +d \right )^{2}\right )^{\frac {3}{2}} \left (-2 \tan \left (e x +d \right )^{3} a^{3} b -3 a^{4} \tan \left (e x +d \right )^{2}-9 \tan \left (e x +d \right )^{2} a^{2} b^{2}+3 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{4}-3 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) b^{4}+12 \arctan \left (\tan \left (e x +d \right )\right ) a^{3} b +12 \arctan \left (\tan \left (e x +d \right )\right ) a \,b^{3}-12 \tan \left (e x +d \right ) a^{3} b -18 \tan \left (e x +d \right ) a \,b^{3}\right )}{6 e \left (b +a \tan \left (e x +d \right )\right )^{3}}\) | \(158\) |
| parts | \(-\frac {a \left (b^{2}+2 a b \tan \left (e x +d \right )+a^{2} \tan \left (e x +d \right )^{2}\right )^{\frac {3}{2}} \left (-\tan \left (e x +d \right )^{2} a^{3}+\ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{3}-3 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a \,b^{2}+6 \arctan \left (\tan \left (e x +d \right )\right ) a^{2} b -2 \arctan \left (\tan \left (e x +d \right )\right ) b^{3}-6 \tan \left (e x +d \right ) a^{2} b \right )}{2 e \left (b +a \tan \left (e x +d \right )\right )^{3}}-\frac {b \left (b^{2}+2 a b \tan \left (e x +d \right )+a^{2} \tan \left (e x +d \right )^{2}\right )^{\frac {3}{2}} \left (-2 \tan \left (e x +d \right )^{3} a^{3}-9 a^{2} b \tan \left (e x +d \right )^{2}+9 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) a^{2} b -3 \ln \left (1+\tan \left (e x +d \right )^{2}\right ) b^{3}-6 \arctan \left (\tan \left (e x +d \right )\right ) a^{3}+18 \arctan \left (\tan \left (e x +d \right )\right ) a \,b^{2}+6 \tan \left (e x +d \right ) a^{3}-18 \tan \left (e x +d \right ) a \,b^{2}\right )}{6 e \left (b +a \tan \left (e x +d \right )\right )^{3}}\) | \(258\) |
-1/6/e*((b+a*tan(e*x+d))^2)^(3/2)*(-2*tan(e*x+d)^3*a^3*b-3*a^4*tan(e*x+d)^ 2-9*tan(e*x+d)^2*a^2*b^2+3*ln(1+tan(e*x+d)^2)*a^4-3*ln(1+tan(e*x+d)^2)*b^4 +12*arctan(tan(e*x+d))*a^3*b+12*arctan(tan(e*x+d))*a*b^3-12*tan(e*x+d)*a^3 *b-18*tan(e*x+d)*a*b^3)/(b+a*tan(e*x+d))^3
Time = 0.25 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.36 \[ \int (a+b \tan (d+e x)) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2} \, dx=\frac {2 \, a^{3} b \tan \left (e x + d\right )^{3} - 12 \, {\left (a^{3} b + a b^{3}\right )} e x + 3 \, {\left (a^{4} + 3 \, a^{2} b^{2}\right )} \tan \left (e x + d\right )^{2} + 3 \, {\left (a^{4} - b^{4}\right )} \log \left (\frac {1}{\tan \left (e x + d\right )^{2} + 1}\right ) + 6 \, {\left (2 \, a^{3} b + 3 \, a b^{3}\right )} \tan \left (e x + d\right )}{6 \, e} \]
integrate((a+b*tan(e*x+d))*(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2),x , algorithm="fricas")
1/6*(2*a^3*b*tan(e*x + d)^3 - 12*(a^3*b + a*b^3)*e*x + 3*(a^4 + 3*a^2*b^2) *tan(e*x + d)^2 + 3*(a^4 - b^4)*log(1/(tan(e*x + d)^2 + 1)) + 6*(2*a^3*b + 3*a*b^3)*tan(e*x + d))/e
\[ \int (a+b \tan (d+e x)) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2} \, dx=\int \left (a + b \tan {\left (d + e x \right )}\right ) \left (\left (a \tan {\left (d + e x \right )} + b\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Time = 0.31 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.58 \[ \int (a+b \tan (d+e x)) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2} \, dx=\frac {3 \, {\left (a^{3} \tan \left (e x + d\right )^{2} + 6 \, a^{2} b \tan \left (e x + d\right ) - 2 \, {\left (3 \, a^{2} b - b^{3}\right )} {\left (e x + d\right )} - {\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (e x + d\right )^{2} + 1\right )\right )} a + {\left (2 \, a^{3} \tan \left (e x + d\right )^{3} + 9 \, a^{2} b \tan \left (e x + d\right )^{2} + 6 \, {\left (a^{3} - 3 \, a b^{2}\right )} {\left (e x + d\right )} - 3 \, {\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (e x + d\right )^{2} + 1\right ) - 6 \, {\left (a^{3} - 3 \, a b^{2}\right )} \tan \left (e x + d\right )\right )} b}{6 \, e} \]
integrate((a+b*tan(e*x+d))*(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2),x , algorithm="maxima")
1/6*(3*(a^3*tan(e*x + d)^2 + 6*a^2*b*tan(e*x + d) - 2*(3*a^2*b - b^3)*(e*x + d) - (a^3 - 3*a*b^2)*log(tan(e*x + d)^2 + 1))*a + (2*a^3*tan(e*x + d)^3 + 9*a^2*b*tan(e*x + d)^2 + 6*(a^3 - 3*a*b^2)*(e*x + d) - 3*(3*a^2*b - b^3 )*log(tan(e*x + d)^2 + 1) - 6*(a^3 - 3*a*b^2)*tan(e*x + d))*b)/e
Leaf count of result is larger than twice the leaf count of optimal. 1535 vs. \(2 (270) = 540\).
Time = 1.16 (sec) , antiderivative size = 1535, normalized size of antiderivative = 5.40 \[ \int (a+b \tan (d+e x)) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2} \, dx=\text {Too large to display} \]
integrate((a+b*tan(e*x+d))*(b^2+2*a*b*tan(e*x+d)+a^2*tan(e*x+d)^2)^(3/2),x , algorithm="giac")
-1/6*(12*a^3*b*e*x*sgn(a*tan(e*x + d) + b)*tan(e*x)^3*tan(d)^3 + 12*a*b^3* e*x*sgn(a*tan(e*x + d) + b)*tan(e*x)^3*tan(d)^3 - 3*a^4*log(4*(tan(e*x)^2* tan(d)^2 - 2*tan(e*x)*tan(d) + 1)/(tan(e*x)^2*tan(d)^2 + tan(e*x)^2 + tan( d)^2 + 1))*sgn(a*tan(e*x + d) + b)*tan(e*x)^3*tan(d)^3 + 3*b^4*log(4*(tan( e*x)^2*tan(d)^2 - 2*tan(e*x)*tan(d) + 1)/(tan(e*x)^2*tan(d)^2 + tan(e*x)^2 + tan(d)^2 + 1))*sgn(a*tan(e*x + d) + b)*tan(e*x)^3*tan(d)^3 - 36*a^3*b*e *x*sgn(a*tan(e*x + d) + b)*tan(e*x)^2*tan(d)^2 - 36*a*b^3*e*x*sgn(a*tan(e* x + d) + b)*tan(e*x)^2*tan(d)^2 - 3*a^4*sgn(a*tan(e*x + d) + b)*tan(e*x)^3 *tan(d)^3 - 9*a^2*b^2*sgn(a*tan(e*x + d) + b)*tan(e*x)^3*tan(d)^3 + 9*a^4* log(4*(tan(e*x)^2*tan(d)^2 - 2*tan(e*x)*tan(d) + 1)/(tan(e*x)^2*tan(d)^2 + tan(e*x)^2 + tan(d)^2 + 1))*sgn(a*tan(e*x + d) + b)*tan(e*x)^2*tan(d)^2 - 9*b^4*log(4*(tan(e*x)^2*tan(d)^2 - 2*tan(e*x)*tan(d) + 1)/(tan(e*x)^2*tan (d)^2 + tan(e*x)^2 + tan(d)^2 + 1))*sgn(a*tan(e*x + d) + b)*tan(e*x)^2*tan (d)^2 + 12*a^3*b*sgn(a*tan(e*x + d) + b)*tan(e*x)^3*tan(d)^2 + 18*a*b^3*sg n(a*tan(e*x + d) + b)*tan(e*x)^3*tan(d)^2 + 12*a^3*b*sgn(a*tan(e*x + d) + b)*tan(e*x)^2*tan(d)^3 + 18*a*b^3*sgn(a*tan(e*x + d) + b)*tan(e*x)^2*tan(d )^3 + 36*a^3*b*e*x*sgn(a*tan(e*x + d) + b)*tan(e*x)*tan(d) + 36*a*b^3*e*x* sgn(a*tan(e*x + d) + b)*tan(e*x)*tan(d) - 3*a^4*sgn(a*tan(e*x + d) + b)*ta n(e*x)^3*tan(d) - 9*a^2*b^2*sgn(a*tan(e*x + d) + b)*tan(e*x)^3*tan(d) + 3* a^4*sgn(a*tan(e*x + d) + b)*tan(e*x)^2*tan(d)^2 + 9*a^2*b^2*sgn(a*tan(e...
Timed out. \[ \int (a+b \tan (d+e x)) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2} \, dx=\int \left (a+b\,\mathrm {tan}\left (d+e\,x\right )\right )\,{\left (a^2\,{\mathrm {tan}\left (d+e\,x\right )}^2+2\,a\,b\,\mathrm {tan}\left (d+e\,x\right )+b^2\right )}^{3/2} \,d x \]