Integrand size = 49, antiderivative size = 384 \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx=-\frac {(i A+B-i C) (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a-i b} f}+\frac {(i A-B-i C) (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b} f}+\frac {\left (3 a^2 C d^2-2 a b d (3 c C+2 B d)+b^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{5/2} \sqrt {d} f}+\frac {(3 b c C+4 b B d-3 a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b^2 f}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f} \]
-(I*A+B-I*C)*(c-I*d)^(3/2)*arctanh((c-I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a -I*b)^(1/2)/(c+d*tan(f*x+e))^(1/2))/f/(a-I*b)^(1/2)+(I*A-B-I*C)*(c+I*d)^(3 /2)*arctanh((c+I*d)^(1/2)*(a+b*tan(f*x+e))^(1/2)/(a+I*b)^(1/2)/(c+d*tan(f* x+e))^(1/2))/f/(a+I*b)^(1/2)+1/4*(3*a^2*C*d^2-2*a*b*d*(2*B*d+3*C*c)+b^2*(3 *c^2*C+12*B*c*d+8*(A-C)*d^2))*arctanh(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^(1/ 2)/(c+d*tan(f*x+e))^(1/2))/b^(5/2)/f/d^(1/2)+1/4*(4*B*b*d-3*C*a*d+3*C*b*c) *(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/b^2/f+1/2*C*(a+b*tan(f*x+e) )^(1/2)*(c+d*tan(f*x+e))^(3/2)/b/f
Time = 7.86 (sec) , antiderivative size = 613, normalized size of antiderivative = 1.60 \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx=\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\frac {(3 b c C+4 b B d-3 a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {-\frac {2 b^2 \left (\sqrt {-b^2} \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )-b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {-c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}-\frac {2 b^2 \left (\sqrt {-b^2} \left (c^2 C+2 B c d-C d^2-A \left (c^2-d^2\right )\right )+b \left (2 c (A-C) d+B \left (c^2-d^2\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}+\frac {\sqrt {b} \sqrt {c-\frac {a d}{b}} \left (3 a^2 C d^2-2 a b d (3 c C+2 B d)+b^2 \left (3 c^2 C+12 B c d+8 (A-C) d^2\right )\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right ) \sqrt {\frac {b c+b d \tan (e+f x)}{b c-a d}}}{2 \sqrt {d} \sqrt {c+d \tan (e+f x)}}}{b^2 f}}{2 b} \]
Integrate[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x] ^2))/Sqrt[a + b*Tan[e + f*x]],x]
(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*b*f) + (((3*b*c *C + 4*b*B*d - 3*a*C*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]) /(2*b*f) + ((-2*b^2*(Sqrt[-b^2]*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) - b*(2*c*(A - C)*d + B*(c^2 - d^2)))*ArcTanh[(Sqrt[-c + (Sqrt[-b^2]*d)/b]* Sqrt[a + b*Tan[e + f*x]])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]]) ])/(Sqrt[-a + Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - (2*b^2*(Sqrt[-b^2 ]*(c^2*C + 2*B*c*d - C*d^2 - A*(c^2 - d^2)) + b*(2*c*(A - C)*d + B*(c^2 - d^2)))*ArcTanh[(Sqrt[c + (Sqrt[-b^2]*d)/b]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt [a + Sqrt[-b^2]]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d)/b]) + (Sqrt[b]*Sqrt[c - (a*d)/b]*(3*a^2*C*d^2 - 2*a*b*d*( 3*c*C + 2*B*d) + b^2*(3*c^2*C + 12*B*c*d + 8*(A - C)*d^2))*ArcSinh[(Sqrt[d ]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b])]*Sqrt[(b*c + b*d*T an[e + f*x])/(b*c - a*d)])/(2*Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(b^2*f))/ (2*b)
Time = 2.37 (sec) , antiderivative size = 393, normalized size of antiderivative = 1.02, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.224, Rules used = {3042, 4130, 27, 3042, 4130, 27, 25, 3042, 4138, 2348, 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.
\(\displaystyle \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{\sqrt {a+b \tan (e+f x)}}dx\) |
\(\Big \downarrow \) 4130 |
\(\displaystyle \frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left ((3 b c C-3 a d C+4 b B d) \tan ^2(e+f x)+4 b (B c+(A-C) d) \tan (e+f x)+4 A b c-C (b c+3 a d)\right )}{2 \sqrt {a+b \tan (e+f x)}}dx}{2 b}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left ((3 b c C-3 a d C+4 b B d) \tan ^2(e+f x)+4 b (B c+(A-C) d) \tan (e+f x)+4 A b c-b c C-3 a C d\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 b}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \frac {\sqrt {c+d \tan (e+f x)} \left ((3 b c C-3 a d C+4 b B d) \tan (e+f x)^2+4 b (B c+(A-C) d) \tan (e+f x)+4 A b c-b c C-3 a C d\right )}{\sqrt {a+b \tan (e+f x)}}dx}{4 b}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}\) |
\(\Big \downarrow \) 4130 |
\(\displaystyle \frac {\frac {\int -\frac {-8 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x) b^2-2 c (4 A b c-C (b c+3 a d)) b-\left (8 d (B c+(A-C) d) b^2+(b c-a d) (3 b c C-3 a d C+4 b B d)\right ) \tan ^2(e+f x)+(b c+a d) (3 b c C-3 a d C+4 b B d)}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{b}+\frac {(-3 a C d+4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}}{4 b}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\frac {(-3 a C d+4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}-\frac {\int -\frac {8 A c^2 b^2-c (5 c C+4 B d) b^2+8 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x) b^2-2 a d (3 c C+2 B d) b+3 a^2 C d^2+\left (8 d (B c+(A-C) d) b^2+(b c-a d) (3 b c C-3 a d C+4 b B d)\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}}{4 b}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\frac {\int \frac {8 A c^2 b^2-c (5 c C+4 B d) b^2+8 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x) b^2-2 a d (3 c C+2 B d) b+3 a^2 C d^2+\left (8 d (B c+(A-C) d) b^2+(b c-a d) (3 b c C-3 a d C+4 b B d)\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}+\frac {(-3 a C d+4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}}{4 b}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\frac {\int \frac {8 A c^2 b^2-c (5 c C+4 B d) b^2+8 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x) b^2-2 a d (3 c C+2 B d) b+3 a^2 C d^2+\left (8 d (B c+(A-C) d) b^2+(b c-a d) (3 b c C-3 a d C+4 b B d)\right ) \tan (e+f x)^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}+\frac {(-3 a C d+4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}}{4 b}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}\) |
\(\Big \downarrow \) 4138 |
\(\displaystyle \frac {\frac {\int \frac {8 A c^2 b^2-c (5 c C+4 B d) b^2+8 \left (2 c (A-C) d+B \left (c^2-d^2\right )\right ) \tan (e+f x) b^2-2 a d (3 c C+2 B d) b+3 a^2 C d^2+\left (8 d (B c+(A-C) d) b^2+(b c-a d) (3 b c C-3 a d C+4 b B d)\right ) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 b f}+\frac {(-3 a C d+4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}}{4 b}+\frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}\) |
\(\Big \downarrow \) 2348 |
\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\frac {(-3 a C d+4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}+\frac {\int \left (\frac {8 A d^2 b^2-8 C d^2 b^2+3 c^2 C b^2+12 B c d b^2-4 a B d^2 b-6 a c C d b+3 a^2 C d^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-8 B c^2 b^2+8 B d^2 b^2-16 A c d b^2+16 c C d b^2+i \left (8 A c^2 b^2-8 A d^2 b^2+8 C d^2 b^2-8 c^2 C b^2-16 B c d b^2\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {8 B c^2 b^2-8 B d^2 b^2+16 A c d b^2-16 c C d b^2+i \left (8 A c^2 b^2-8 A d^2 b^2+8 C d^2 b^2-8 c^2 C b^2-16 B c d b^2\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{2 b f}}{4 b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {C \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 b f}+\frac {\frac {(-3 a C d+4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{b f}+\frac {\frac {2 \left (3 a^2 C d^2-2 a b d (2 B d+3 c C)+b^2 \left (8 d^2 (A-C)+12 B c d+3 c^2 C\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} \sqrt {d}}-\frac {8 b^2 (c-i d)^{3/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a-i b}}-\frac {8 b^2 (c+i d)^{3/2} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+i b}}}{2 b f}}{4 b}\) |
Int[((c + d*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/S qrt[a + b*Tan[e + f*x]],x]
(C*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*b*f) + (((-8*b^ 2*(I*A + B - I*C)*(c - I*d)^(3/2)*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[a - I*b] - (8*b^2 *(B - I*(A - C))*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[a + I*b] + (2*(3*a ^2*C*d^2 - 2*a*b*d*(3*c*C + 2*B*d) + b^2*(3*c^2*C + 12*B*c*d + 8*(A - C)*d ^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[b]*Sqrt[d]))/(2*b*f) + ((3*b*c*C + 4*b*B*d - 3*a*C*d)*Sqr t[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/(b*f))/(4*b)
3.2.37.3.1 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[(Px_)*((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_. )*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(c + d*x)^m*(e + f*x)^ n*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && PolyQ[P x, x] && (IntegerQ[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0])) && !(IGtQ[m, 0] && IGtQ[n, 0])
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_. ) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*(a + b*Tan[e + f*x])^m*((c + d*Tan[ e + f*x])^(n + 1)/(d*f*(m + n + 1))), x] + Simp[1/(d*(m + n + 1)) Int[(a + b*Tan[e + f*x])^(m - 1)*(c + d*Tan[e + f*x])^n*Simp[a*A*d*(m + n + 1) - C *(b*c*m + a*d*(n + 1)) + d*(A*b + a*B - b*C)*(m + n + 1)*Tan[e + f*x] - (C* m*(b*c - a*d) - b*B*d*(m + n + 1))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 0] && !(IGtQ[n, 0] && ( !IntegerQ[m] || (EqQ[ c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, S imp[ff/f Subst[Int[(a + b*ff*x)^m*(c + d*ff*x)^n*((A + B*ff*x + C*ff^2*x^ 2)/(1 + ff^2*x^2)), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, c, d, e, f , A, B, C, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0]
Timed out.
\[\int \frac {\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\sqrt {a +b \tan \left (f x +e \right )}}d x\]
Leaf count of result is larger than twice the leaf count of optimal. 57709 vs. \(2 (311) = 622\).
Time = 196.39 (sec) , antiderivative size = 115434, normalized size of antiderivative = 300.61 \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx=\text {Too large to display} \]
integrate((c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(1/2),x, algorithm="fricas")
\[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx=\int \frac {\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\sqrt {a + b \tan {\left (e + f x \right )}}}\, dx \]
integrate((c+d*tan(f*x+e))**(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(a+b*ta n(f*x+e))**(1/2),x)
Integral((c + d*tan(e + f*x))**(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)* *2)/sqrt(a + b*tan(e + f*x)), x)
\[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx=\int { \frac {{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{\sqrt {b \tan \left (f x + e\right ) + a}} \,d x } \]
integrate((c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(1/2),x, algorithm="maxima")
integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)*(d*tan(f*x + e) + c)^(3/ 2)/sqrt(b*tan(f*x + e) + a), x)
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx=\text {Timed out} \]
integrate((c+d*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(a+b*tan( f*x+e))^(1/2),x, algorithm="giac")
Timed out. \[ \int \frac {(c+d \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {a+b \tan (e+f x)}} \, dx=\text {Hanged} \]