\(\int \frac {(a+b \tan (e+f x))^{3/2} (A+B \tan (e+f x)+C \tan ^2(e+f x))}{\sqrt {c+d \tan (e+f x)}} \, dx\) [148]

Optimal result

Integrand size = 49, antiderivative size = 381 \[ \int \frac {(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=-\frac {(a-i b)^{3/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {c-i d} f}-\frac {(a+i b)^{3/2} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {c+i d} f}+\frac {\left (3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (3 c^2 C-4 B c d+8 (A-C) d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 \sqrt {b} d^{5/2} f}-\frac {(3 b c C-4 b B d-3 a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 d^2 f}+\frac {C (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d f} \] Output:

-(a-I*b)^(3/2)*(I*A+B-I*C)*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))/(c-I*d)^(1/2)/f-(a+I*b)^(3/2)*(B-I*(A- 
C))*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))/(c+I*d)^(1/2)/f+1/4*(3*a^2*C*d^2-6*a*b*d*(-2*B*d+C*c)+b^2*(3* 
c^2*C-4*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^(1/2)/d^(5/2)/f-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)/d^2/f+1/2*C*(a+b*tan(f*x+e)) 
^(3/2)*(c+d*tan(f*x+e))^(1/2)/d/f
 

Mathematica [A] (verified)

Time = 4.64 (sec) , antiderivative size = 582, normalized size of antiderivative = 1.53 \[ \int \frac {(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\frac {\frac {4 \left (b \left (a^2 B-b^2 B+2 a b (A-C)\right )-\sqrt {-b^2} \left (2 a b B+b^2 (A-C)+a^2 (-A+C)\right )\right ) d^2 \text {arctanh}\left (\frac {\sqrt {-c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{b \sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}-\frac {4 \left (b \left (a^2 B-b^2 B+2 a b (A-C)\right )+\sqrt {-b^2} \left (2 a b B+b^2 (A-C)+a^2 (-A+C)\right )\right ) d^2 \text {arctanh}\left (\frac {\sqrt {c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{b \sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}+(-3 b c C+4 b B d+3 a C d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}+2 C d (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}+\frac {\sqrt {c-\frac {a d}{b}} \left (3 a^2 C d^2+6 a b d (-c C+2 B d)+b^2 \left (3 c^2 C-4 B c d+8 (A-C) d^2\right )\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right ) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{\sqrt {b} \sqrt {d} \sqrt {c+d \tan (e+f x)}}}{4 d^2 f} \] Input:

Integrate[((a + b*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x] 
^2))/Sqrt[c + d*Tan[e + f*x]],x]
 

Output:

((4*(b*(a^2*B - b^2*B + 2*a*b*(A - C)) - Sqrt[-b^2]*(2*a*b*B + b^2*(A - C) 
 + a^2*(-A + C)))*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]])])/(b*Sqrt[-a + 
 Sqrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - (4*(b*(a^2*B - b^2*B + 2*a*b*( 
A - C)) + Sqrt[-b^2]*(2*a*b*B + b^2*(A - C) + a^2*(-A + C)))*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]])])/(b*Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]* 
d)/b]) + (-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*C*d*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x] 
] + (Sqrt[c - (a*d)/b]*(3*a^2*C*d^2 + 6*a*b*d*(-(c*C) + 2*B*d) + b^2*(3*c^ 
2*C - 4*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 + d*Tan[e + f*x]))/(b*c - a*d)])/ 
(Sqrt[b]*Sqrt[d]*Sqrt[c + d*Tan[e + f*x]]))/(4*d^2*f)
 

Rubi [A] (verified)

Time = 3.56 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.204, Rules used = {3042, 4130, 27, 3042, 4130, 27, 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.

\(\displaystyle \int \frac {(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan (e+f x)^2\right )}{\sqrt {c+d \tan (e+f x)}}dx\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {\int -\frac {\sqrt {a+b \tan (e+f x)} \left ((3 b c C-3 a d C-4 b B d) \tan ^2(e+f x)-4 (A b-C b+a B) d \tan (e+f x)+3 b c C-a (4 A-C) d\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{2 d}+\frac {C (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d f}-\frac {\int \frac {\sqrt {a+b \tan (e+f x)} \left ((3 b c C-3 a d C-4 b B d) \tan ^2(e+f x)-4 (A b-C b+a B) d \tan (e+f x)+3 b c C-a (4 A-C) d\right )}{\sqrt {c+d \tan (e+f x)}}dx}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d f}-\frac {\int \frac {\sqrt {a+b \tan (e+f x)} \left ((3 b c C-3 a d C-4 b B d) \tan (e+f x)^2-4 (A b-C b+a B) d \tan (e+f x)+3 b c C-a (4 A-C) d\right )}{\sqrt {c+d \tan (e+f x)}}dx}{4 d}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d f}-\frac {\frac {\int -\frac {c (3 c C-4 B d) b^2-2 a d (3 c C+2 B d) b+a^2 (8 A-5 C) d^2+\left (8 b (A b-C b+a B) d^2+(b c-a d) (3 b c C-3 a d C-4 b B d)\right ) \tan ^2(e+f x)+8 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{d}+\frac {(-3 a C d-4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}}{4 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d f}-\frac {\frac {(-3 a C d-4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}-\frac {\int \frac {c (3 c C-4 B d) b^2-2 a d (3 c C+2 B d) b+a^2 (8 A-5 C) d^2+\left (8 b (A b-C b+a B) d^2+(b c-a d) (3 b c C-3 a d C-4 b B d)\right ) \tan ^2(e+f x)+8 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 d}}{4 d}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d f}-\frac {\frac {(-3 a C d-4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}-\frac {\int \frac {c (3 c C-4 B d) b^2-2 a d (3 c C+2 B d) b+a^2 (8 A-5 C) d^2+\left (8 b (A b-C b+a B) d^2+(b c-a d) (3 b c C-3 a d C-4 b B d)\right ) \tan (e+f x)^2+8 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 d}}{4 d}\)

\(\Big \downarrow \) 4138

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d f}-\frac {\frac {(-3 a C d-4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}-\frac {\int \frac {c (3 c C-4 B d) b^2-2 a d (3 c C+2 B d) b+a^2 (8 A-5 C) d^2+\left (8 b (A b-C b+a B) d^2+(b c-a d) (3 b c C-3 a d C-4 b B d)\right ) \tan ^2(e+f x)+8 \left (B a^2+2 b (A-C) a-b^2 B\right ) d^2 \tan (e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 d f}}{4 d}\)

\(\Big \downarrow \) 2348

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d f}-\frac {\frac {(-3 a C d-4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}-\frac {\int \left (\frac {8 A d^2 b^2-8 C d^2 b^2+3 c^2 C b^2-4 B c d b^2+12 a B d^2 b-6 a c C d b+3 a^2 C d^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-16 a A b d^2-8 a^2 B d^2+8 b^2 B d^2+16 a b C d^2+i \left (-8 A b^2 d^2+8 a^2 A d^2-16 a b B d^2-8 a^2 C d^2+8 b^2 C d^2\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {16 a A b d^2+8 a^2 B d^2-8 b^2 B d^2-16 a b C d^2+i \left (-8 A b^2 d^2+8 a^2 A d^2-16 a b B d^2-8 a^2 C d^2+8 b^2 C d^2\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{2 d f}}{4 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {C (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 d f}-\frac {\frac {(-3 a C d-4 b B d+3 b c C) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{d f}-\frac {\frac {2 \left (3 a^2 C d^2-6 a b d (c C-2 B d)+b^2 \left (8 d^2 (A-C)-4 B c d+3 c^2 C\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} \sqrt {d}}-\frac {8 d^2 (a-i b)^{3/2} (i A+B-i C) \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {c-i d}}-\frac {8 d^2 (a+i b)^{3/2} (B-i (A-C)) \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {c+i d}}}{2 d f}}{4 d}\)

Input:

Int[((a + b*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x] + C*Tan[e + f*x]^2))/S 
qrt[c + d*Tan[e + f*x]],x]
 

Output:

(C*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(2*d*f) - (-1/2*(( 
-8*(a - I*b)^(3/2)*(I*A + B - I*C)*d^2*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b*T 
an[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[c - I*d] - ( 
8*(a + I*b)^(3/2)*(B - I*(A - C))*d^2*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Ta 
n[e + f*x]])/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[c + I*d] + (2 
*(3*a^2*C*d^2 - 6*a*b*d*(c*C - 2*B*d) + b^2*(3*c^2*C - 4*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]))/(d*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]])/(d*f))/(4*d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2348

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])
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4130

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])))
 

rule 4138

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]
 

Maple [F(-1)]

Timed out.

\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{\frac {3}{2}} \left (A +B \tan \left (f x +e \right )+C \tan \left (f x +e \right )^{2}\right )}{\sqrt {c +d \tan \left (f x +e \right )}}d x\]

Input:

int((a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e) 
)^(1/2),x)
 

Output:

int((a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e) 
)^(1/2),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\text {Timed out} \] Input:

integrate((a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan( 
f*x+e))^(1/2),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \left (A + B \tan {\left (e + f x \right )} + C \tan ^{2}{\left (e + f x \right )}\right )}{\sqrt {c + d \tan {\left (e + f x \right )}}}\, dx \] Input:

integrate((a+b*tan(f*x+e))**(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)**2)/(c+d*ta 
n(f*x+e))**(1/2),x)
 

Output:

Integral((a + b*tan(e + f*x))**(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)* 
*2)/sqrt(c + d*tan(e + f*x)), x)
 

Maxima [F]

\[ \int \frac {(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\int { \frac {{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {d \tan \left (f x + e\right ) + c}} \,d x } \] Input:

integrate((a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan( 
f*x+e))^(1/2),x, algorithm="maxima")
 

Output:

integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)*(b*tan(f*x + e) + a)^(3/ 
2)/sqrt(d*tan(f*x + e) + c), x)
 

Giac [F]

\[ \int \frac {(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\int { \frac {{\left (C \tan \left (f x + e\right )^{2} + B \tan \left (f x + e\right ) + A\right )} {\left (b \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {d \tan \left (f x + e\right ) + c}} \,d x } \] Input:

integrate((a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan( 
f*x+e))^(1/2),x, algorithm="giac")
 

Output:

integrate((C*tan(f*x + e)^2 + B*tan(f*x + e) + A)*(b*tan(f*x + e) + a)^(3/ 
2)/sqrt(d*tan(f*x + e) + c), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}\,\left (C\,{\mathrm {tan}\left (e+f\,x\right )}^2+B\,\mathrm {tan}\left (e+f\,x\right )+A\right )}{\sqrt {c+d\,\mathrm {tan}\left (e+f\,x\right )}} \,d x \] Input:

int(((a + b*tan(e + f*x))^(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/( 
c + d*tan(e + f*x))^(1/2),x)
 

Output:

int(((a + b*tan(e + f*x))^(3/2)*(A + B*tan(e + f*x) + C*tan(e + f*x)^2))/( 
c + d*tan(e + f*x))^(1/2), x)
 

Reduce [F]

\[ \int \frac {(a+b \tan (e+f x))^{3/2} \left (A+B \tan (e+f x)+C \tan ^2(e+f x)\right )}{\sqrt {c+d \tan (e+f x)}} \, dx=\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{3}}{d \tan \left (f x +e \right )+c}d x \right ) b c +\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{2}}{d \tan \left (f x +e \right )+c}d x \right ) a c +\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{2}}{d \tan \left (f x +e \right )+c}d x \right ) b^{2}+2 \left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )}{d \tan \left (f x +e \right )+c}d x \right ) a b +\left (\int \frac {\sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}}{d \tan \left (f x +e \right )+c}d x \right ) a^{2} \] Input:

int((a+b*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e)+C*tan(f*x+e)^2)/(c+d*tan(f*x+e) 
)^(1/2),x)
 

Output:

int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**3)/(t 
an(e + f*x)*d + c),x)*b*c + int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x 
)*b + a)*tan(e + f*x)**2)/(tan(e + f*x)*d + c),x)*a*c + int((sqrt(tan(e + 
f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e + f*x)**2)/(tan(e + f*x)*d + c) 
,x)*b**2 + 2*int((sqrt(tan(e + f*x)*d + c)*sqrt(tan(e + f*x)*b + a)*tan(e 
+ f*x))/(tan(e + f*x)*d + c),x)*a*b + int((sqrt(tan(e + f*x)*d + c)*sqrt(t 
an(e + f*x)*b + a))/(tan(e + f*x)*d + c),x)*a**2