Integrand size = 16, antiderivative size = 170 \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\frac {(a-b)^{5/2} \arctan \left (\frac {\sqrt {a-b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{d}+\frac {\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )}{8 d}+\frac {(7 a-4 b) b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{8 d}+\frac {b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{4 d} \]
(a-b)^(5/2)*arctan((a-b)^(1/2)*tan(d*x+c)/(a+b*tan(d*x+c)^2)^(1/2))/d+1/8* (15*a^2-20*a*b+8*b^2)*arctanh(b^(1/2)*tan(d*x+c)/(a+b*tan(d*x+c)^2)^(1/2)) *b^(1/2)/d+1/8*(7*a-4*b)*b*(a+b*tan(d*x+c)^2)^(1/2)*tan(d*x+c)/d+1/4*b*tan (d*x+c)*(a+b*tan(d*x+c)^2)^(3/2)/d
Time = 1.23 (sec) , antiderivative size = 169, normalized size of antiderivative = 0.99 \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\frac {-8 (a-b)^{5/2} \arctan \left (\frac {\sqrt {b}+\sqrt {b} \tan ^2(c+d x)-\tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}}{\sqrt {a-b}}\right )-\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \log \left (-\sqrt {b} \tan (c+d x)+\sqrt {a+b \tan ^2(c+d x)}\right )+b \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)} \left (9 a-4 b+2 b \tan ^2(c+d x)\right )}{8 d} \]
(-8*(a - b)^(5/2)*ArcTan[(Sqrt[b] + Sqrt[b]*Tan[c + d*x]^2 - Tan[c + d*x]* Sqrt[a + b*Tan[c + d*x]^2])/Sqrt[a - b]] - Sqrt[b]*(15*a^2 - 20*a*b + 8*b^ 2)*Log[-(Sqrt[b]*Tan[c + d*x]) + Sqrt[a + b*Tan[c + d*x]^2]] + b*Tan[c + d *x]*Sqrt[a + b*Tan[c + d*x]^2]*(9*a - 4*b + 2*b*Tan[c + d*x]^2))/(8*d)
Time = 0.34 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.562, Rules used = {3042, 4144, 318, 403, 398, 224, 219, 291, 216}
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.
| \(\displaystyle \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \tan (c+d x)^2\right )^{5/2}dx\) |
| \(\Big \downarrow \) 4144 |
| \(\displaystyle \frac {\int \frac {\left (b \tan ^2(c+d x)+a\right )^{5/2}}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle \frac {\frac {1}{4} \int \frac {\sqrt {b \tan ^2(c+d x)+a} \left ((7 a-4 b) b \tan ^2(c+d x)+a (4 a-b)\right )}{\tan ^2(c+d x)+1}d\tan (c+d x)+\frac {1}{4} b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{d}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \int \frac {b \left (15 a^2-20 b a+8 b^2\right ) \tan ^2(c+d x)+a \left (8 a^2-9 b a+4 b^2\right )}{\left (\tan ^2(c+d x)+1\right ) \sqrt {b \tan ^2(c+d x)+a}}d\tan (c+d x)+\frac {1}{2} b (7 a-4 b) \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}\right )+\frac {1}{4} b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{d}\) |
| \(\Big \downarrow \) 398 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (b \left (15 a^2-20 a b+8 b^2\right ) \int \frac {1}{\sqrt {b \tan ^2(c+d x)+a}}d\tan (c+d x)+8 (a-b)^3 \int \frac {1}{\left (\tan ^2(c+d x)+1\right ) \sqrt {b \tan ^2(c+d x)+a}}d\tan (c+d x)\right )+\frac {1}{2} b (7 a-4 b) \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}\right )+\frac {1}{4} b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{d}\) |
| \(\Big \downarrow \) 224 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (b \left (15 a^2-20 a b+8 b^2\right ) \int \frac {1}{1-\frac {b \tan ^2(c+d x)}{b \tan ^2(c+d x)+a}}d\frac {\tan (c+d x)}{\sqrt {b \tan ^2(c+d x)+a}}+8 (a-b)^3 \int \frac {1}{\left (\tan ^2(c+d x)+1\right ) \sqrt {b \tan ^2(c+d x)+a}}d\tan (c+d x)\right )+\frac {1}{2} b (7 a-4 b) \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}\right )+\frac {1}{4} b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (8 (a-b)^3 \int \frac {1}{\left (\tan ^2(c+d x)+1\right ) \sqrt {b \tan ^2(c+d x)+a}}d\tan (c+d x)+\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )\right )+\frac {1}{2} b (7 a-4 b) \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}\right )+\frac {1}{4} b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{d}\) |
| \(\Big \downarrow \) 291 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (8 (a-b)^3 \int \frac {1}{1-\frac {(b-a) \tan ^2(c+d x)}{b \tan ^2(c+d x)+a}}d\frac {\tan (c+d x)}{\sqrt {b \tan ^2(c+d x)+a}}+\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )\right )+\frac {1}{2} b (7 a-4 b) \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}\right )+\frac {1}{4} b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{d}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {1}{4} \left (\frac {1}{2} \left (\sqrt {b} \left (15 a^2-20 a b+8 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )+8 (a-b)^{5/2} \arctan \left (\frac {\sqrt {a-b} \tan (c+d x)}{\sqrt {a+b \tan ^2(c+d x)}}\right )\right )+\frac {1}{2} b (7 a-4 b) \tan (c+d x) \sqrt {a+b \tan ^2(c+d x)}\right )+\frac {1}{4} b \tan (c+d x) \left (a+b \tan ^2(c+d x)\right )^{3/2}}{d}\) |
((b*Tan[c + d*x]*(a + b*Tan[c + d*x]^2)^(3/2))/4 + ((8*(a - b)^(5/2)*ArcTa n[(Sqrt[a - b]*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]^2]] + Sqrt[b]*(15*a^2 - 20*a*b + 8*b^2)*ArcTanh[(Sqrt[b]*Tan[c + d*x])/Sqrt[a + b*Tan[c + d*x]^ 2]])/2 + ((7*a - 4*b)*b*Tan[c + d*x]*Sqrt[a + b*Tan[c + d*x]^2])/2)/4)/d
3.4.19.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] && !GtQ[a, 0]
Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst [Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) , x_Symbol] :> Simp[f/b Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ b Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} , x]
Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
Int[((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(a + b* (ff*x)^n)^p/(c^2 + ff^2*x^2), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, e, f, n, p}, x] && (IntegersQ[n, p] || IGtQ[p, 0] || EqQ[n^2, 4] || EqQ[n^2, 16])
Leaf count of result is larger than twice the leaf count of optimal. \(460\) vs. \(2(148)=296\).
Time = 0.16 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.71
| method | result | size |
| derivativedivides | \(\frac {b^{\frac {5}{2}} \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{d}+\frac {b^{2} \tan \left (d x +c \right )^{3} \sqrt {a +b \tan \left (d x +c \right )^{2}}}{4 d}+\frac {9 b a \tan \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )^{2}}}{8 d}+\frac {15 \sqrt {b}\, a^{2} \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{8 d}-\frac {b^{2} \tan \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )^{2}}}{2 d}-\frac {5 b^{\frac {3}{2}} a \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{2 d}-\frac {b \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {3 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}-\frac {3 a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}+\frac {a^{3} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\) | \(461\) |
| default | \(\frac {b^{\frac {5}{2}} \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{d}+\frac {b^{2} \tan \left (d x +c \right )^{3} \sqrt {a +b \tan \left (d x +c \right )^{2}}}{4 d}+\frac {9 b a \tan \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )^{2}}}{8 d}+\frac {15 \sqrt {b}\, a^{2} \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{8 d}-\frac {b^{2} \tan \left (d x +c \right ) \sqrt {a +b \tan \left (d x +c \right )^{2}}}{2 d}-\frac {5 b^{\frac {3}{2}} a \ln \left (\sqrt {b}\, \tan \left (d x +c \right )+\sqrt {a +b \tan \left (d x +c \right )^{2}}\right )}{2 d}-\frac {b \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}+\frac {3 a \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \left (a -b \right )}-\frac {3 a^{2} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d b \left (a -b \right )}+\frac {a^{3} \sqrt {b^{4} \left (a -b \right )}\, \arctan \left (\frac {b^{2} \left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {b^{4} \left (a -b \right )}\, \sqrt {a +b \tan \left (d x +c \right )^{2}}}\right )}{d \,b^{2} \left (a -b \right )}\) | \(461\) |
1/d*b^(5/2)*ln(b^(1/2)*tan(d*x+c)+(a+b*tan(d*x+c)^2)^(1/2))+1/4/d*b^2*tan( d*x+c)^3*(a+b*tan(d*x+c)^2)^(1/2)+9/8/d*b*a*tan(d*x+c)*(a+b*tan(d*x+c)^2)^ (1/2)+15/8/d*b^(1/2)*a^2*ln(b^(1/2)*tan(d*x+c)+(a+b*tan(d*x+c)^2)^(1/2))-1 /2/d*b^2*tan(d*x+c)*(a+b*tan(d*x+c)^2)^(1/2)-5/2/d*b^(3/2)*a*ln(b^(1/2)*ta n(d*x+c)+(a+b*tan(d*x+c)^2)^(1/2))-1/d*b*(b^4*(a-b))^(1/2)/(a-b)*arctan(b^ 2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*tan(d*x+c)^2)^(1/2)*tan(d*x+c))+3/d*a*(b^4* (a-b))^(1/2)/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*tan(d*x+c)^2)^( 1/2)*tan(d*x+c))-3/d*a^2/b*(b^4*(a-b))^(1/2)/(a-b)*arctan(b^2*(a-b)/(b^4*( a-b))^(1/2)/(a+b*tan(d*x+c)^2)^(1/2)*tan(d*x+c))+1/d*a^3*(b^4*(a-b))^(1/2) /b^2/(a-b)*arctan(b^2*(a-b)/(b^4*(a-b))^(1/2)/(a+b*tan(d*x+c)^2)^(1/2)*tan (d*x+c))
Time = 1.30 (sec) , antiderivative size = 703, normalized size of antiderivative = 4.14 \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\left [\frac {{\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {b} \log \left (2 \, b \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {b} \tan \left (d x + c\right ) + a\right ) + 8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-a + b} \tan \left (d x + c\right ) - a}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{16 \, d}, \frac {16 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a}}{\sqrt {a - b} \tan \left (d x + c\right )}\right ) + {\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {b} \log \left (2 \, b \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {b} \tan \left (d x + c\right ) + a\right ) + 2 \, {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{16 \, d}, -\frac {{\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-b}}{b \tan \left (d x + c\right )}\right ) - 4 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {-a + b} \log \left (-\frac {{\left (a - 2 \, b\right )} \tan \left (d x + c\right )^{2} + 2 \, \sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-a + b} \tan \left (d x + c\right ) - a}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{8 \, d}, \frac {8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \sqrt {a - b} \arctan \left (-\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a}}{\sqrt {a - b} \tan \left (d x + c\right )}\right ) - {\left (15 \, a^{2} - 20 \, a b + 8 \, b^{2}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {b \tan \left (d x + c\right )^{2} + a} \sqrt {-b}}{b \tan \left (d x + c\right )}\right ) + {\left (2 \, b^{2} \tan \left (d x + c\right )^{3} + {\left (9 \, a b - 4 \, b^{2}\right )} \tan \left (d x + c\right )\right )} \sqrt {b \tan \left (d x + c\right )^{2} + a}}{8 \, d}\right ] \]
[1/16*((15*a^2 - 20*a*b + 8*b^2)*sqrt(b)*log(2*b*tan(d*x + c)^2 + 2*sqrt(b *tan(d*x + c)^2 + a)*sqrt(b)*tan(d*x + c) + a) + 8*(a^2 - 2*a*b + b^2)*sqr t(-a + b)*log(-((a - 2*b)*tan(d*x + c)^2 + 2*sqrt(b*tan(d*x + c)^2 + a)*sq rt(-a + b)*tan(d*x + c) - a)/(tan(d*x + c)^2 + 1)) + 2*(2*b^2*tan(d*x + c) ^3 + (9*a*b - 4*b^2)*tan(d*x + c))*sqrt(b*tan(d*x + c)^2 + a))/d, 1/16*(16 *(a^2 - 2*a*b + b^2)*sqrt(a - b)*arctan(-sqrt(b*tan(d*x + c)^2 + a)/(sqrt( a - b)*tan(d*x + c))) + (15*a^2 - 20*a*b + 8*b^2)*sqrt(b)*log(2*b*tan(d*x + c)^2 + 2*sqrt(b*tan(d*x + c)^2 + a)*sqrt(b)*tan(d*x + c) + a) + 2*(2*b^2 *tan(d*x + c)^3 + (9*a*b - 4*b^2)*tan(d*x + c))*sqrt(b*tan(d*x + c)^2 + a) )/d, -1/8*((15*a^2 - 20*a*b + 8*b^2)*sqrt(-b)*arctan(sqrt(b*tan(d*x + c)^2 + a)*sqrt(-b)/(b*tan(d*x + c))) - 4*(a^2 - 2*a*b + b^2)*sqrt(-a + b)*log( -((a - 2*b)*tan(d*x + c)^2 + 2*sqrt(b*tan(d*x + c)^2 + a)*sqrt(-a + b)*tan (d*x + c) - a)/(tan(d*x + c)^2 + 1)) - (2*b^2*tan(d*x + c)^3 + (9*a*b - 4* b^2)*tan(d*x + c))*sqrt(b*tan(d*x + c)^2 + a))/d, 1/8*(8*(a^2 - 2*a*b + b^ 2)*sqrt(a - b)*arctan(-sqrt(b*tan(d*x + c)^2 + a)/(sqrt(a - b)*tan(d*x + c ))) - (15*a^2 - 20*a*b + 8*b^2)*sqrt(-b)*arctan(sqrt(b*tan(d*x + c)^2 + a) *sqrt(-b)/(b*tan(d*x + c))) + (2*b^2*tan(d*x + c)^3 + (9*a*b - 4*b^2)*tan( d*x + c))*sqrt(b*tan(d*x + c)^2 + a))/d]
\[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\int \left (a + b \tan ^{2}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
\[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\int { {\left (b \tan \left (d x + c\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\text {Timed out} \]
Timed out. \[ \int \left (a+b \tan ^2(c+d x)\right )^{5/2} \, dx=\int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^2+a\right )}^{5/2} \,d x \]