\(\int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx\) [1294]

Optimal result

Integrand size = 29, antiderivative size = 339 \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=-\frac {i \sqrt {a-i b} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i \sqrt {a+i b} (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {d} \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 b^{3/2} f}+\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 b f}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f} \] Output:

-I*(a-I*b)^(1/2)*(c-I*d)^(5/2)*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+I*(a+I*b)^(1/2)*(c+I*d)^(5/2)*ar 
ctanh((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+1/4*d^(1/2)*(10*a*b*c*d-a^2*d^2+b^2*(15*c^2-8*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^(3/2)/f+1/4*d* 
(-a*d+9*b*c)*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/b/f+1/2*d^2*(a+ 
b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(1/2)/b/f
 

Mathematica [A] (verified)

Time = 4.35 (sec) , antiderivative size = 550, normalized size of antiderivative = 1.62 \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\frac {\frac {4 \left (a \sqrt {-b^2} c \left (c^2-3 d^2\right )+b \left (-a+\sqrt {-b^2}\right ) d \left (-3 c^2+d^2\right )+b^2 \left (c^3-3 c d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {-c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {-a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {-a+\sqrt {-b^2}} \sqrt {-c+\frac {\sqrt {-b^2} d}{b}}}-\frac {4 \left (-a \sqrt {-b^2} c \left (c^2-3 d^2\right )-b \left (a+\sqrt {-b^2}\right ) d \left (-3 c^2+d^2\right )+b^2 \left (c^3-3 c d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c+\frac {\sqrt {-b^2} d}{b}} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {a+\sqrt {-b^2}} \sqrt {c+\frac {\sqrt {-b^2} d}{b}}}+d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}+2 d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}+\frac {\sqrt {d} \sqrt {c-\frac {a d}{b}} \left (10 a b c d-a^2 d^2+b^2 \left (15 c^2-8 d^2\right )\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c-\frac {a d}{b}}}\right ) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}}{4 b f} \] Input:

Integrate[Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

((4*(a*Sqrt[-b^2]*c*(c^2 - 3*d^2) + b*(-a + Sqrt[-b^2])*d*(-3*c^2 + d^2) + 
 b^2*(c^3 - 3*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]])])/(Sqrt[-a + S 
qrt[-b^2]]*Sqrt[-c + (Sqrt[-b^2]*d)/b]) - (4*(-(a*Sqrt[-b^2]*c*(c^2 - 3*d^ 
2)) - b*(a + Sqrt[-b^2])*d*(-3*c^2 + d^2) + b^2*(c^3 - 3*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]])])/(Sqrt[a + Sqrt[-b^2]]*Sqrt[c + (Sqrt[-b^2]*d) 
/b]) + d*(9*b*c - a*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]] + 
 2*d^2*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]] + (Sqrt[d]*Sqrt 
[c - (a*d)/b]*(10*a*b*c*d - a^2*d^2 + b^2*(15*c^2 - 8*d^2))*ArcSinh[(Sqrt[ 
d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b]*Sqrt[c - (a*d)/b])]*Sqrt[(b*(c + d*T 
an[e + f*x]))/(b*c - a*d)])/(Sqrt[b]*Sqrt[c + d*Tan[e + f*x]]))/(4*b*f)
 

Rubi [A] (verified)

Time = 1.88 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.02, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 4049, 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.

\(\displaystyle \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2}dx\)

\(\Big \downarrow \) 4049

\(\displaystyle \frac {\int \frac {\sqrt {a+b \tan (e+f x)} \left (4 b c^3-3 b d^2 c-a d^3+d^2 (9 b c-a d) \tan ^2(e+f x)+4 b d \left (3 c^2-d^2\right ) \tan (e+f x)\right )}{2 \sqrt {c+d \tan (e+f x)}}dx}{2 b}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {\sqrt {a+b \tan (e+f x)} \left (4 b c^3-3 b d^2 c-a d^3+d^2 (9 b c-a d) \tan ^2(e+f x)+4 b d \left (3 c^2-d^2\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{4 b}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int \frac {\sqrt {a+b \tan (e+f x)} \left (4 b c^3-3 b d^2 c-a d^3+d^2 (9 b c-a d) \tan (e+f x)^2+4 b d \left (3 c^2-d^2\right ) \tan (e+f x)\right )}{\sqrt {c+d \tan (e+f x)}}dx}{4 b}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}\)

\(\Big \downarrow \) 4130

\(\displaystyle \frac {\frac {\int -\frac {-d^2 \left (\left (15 c^2-8 d^2\right ) b^2+10 a c d b-a^2 d^2\right ) \tan ^2(e+f x)-8 b d \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right ) \tan (e+f x)+d \left (a^2 d^3+9 b^2 c^2 d-a b \left (8 c^3-14 c d^2\right )\right )}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{d}+\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}}{4 b}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {\int \frac {-d^2 \left (\left (15 c^2-8 d^2\right ) b^2+10 a c d b-a^2 d^2\right ) \tan ^2(e+f x)-8 b d \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right ) \tan (e+f x)+d \left (a^2 d^3+9 b^2 c^2 d-a b \left (8 c^3-14 c d^2\right )\right )}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 d}}{4 b}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {\int \frac {-d^2 \left (\left (15 c^2-8 d^2\right ) b^2+10 a c d b-a^2 d^2\right ) \tan (e+f x)^2-8 b d \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right ) \tan (e+f x)+d \left (a^2 d^3+9 b^2 c^2 d-a b \left (8 c^3-14 c d^2\right )\right )}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 d}}{4 b}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}\)

\(\Big \downarrow \) 4138

\(\displaystyle \frac {\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {\int \frac {-d^2 \left (\left (15 c^2-8 d^2\right ) b^2+10 a c d b-a^2 d^2\right ) \tan ^2(e+f x)-8 b d \left (b c^3+3 a d c^2-3 b d^2 c-a d^3\right ) \tan (e+f x)+d \left (a^2 d^3+9 b^2 c^2 d-a b \left (8 c^3-14 c d^2\right )\right )}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 d f}}{4 b}+\frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}\)

\(\Big \downarrow \) 2348

\(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {\int \left (\frac {\left (-\left (\left (15 c^2-8 d^2\right ) b^2\right )-10 a c d b+a^2 d^2\right ) d^2}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-8 a b d^4-24 b^2 c d^3+24 a b c^2 d^2+8 b^2 c^3 d+i \left (-8 b^2 d^4+24 a b c d^3+24 b^2 c^2 d^2-8 a b c^3 d\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {8 a b d^4+24 b^2 c d^3-24 a b c^2 d^2-8 b^2 c^3 d+i \left (-8 b^2 d^4+24 a b c d^3+24 b^2 c^2 d^2-8 a b c^3 d\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{2 d f}}{4 b}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 (a+b \tan (e+f x))^{3/2} \sqrt {c+d \tan (e+f x)}}{2 b f}+\frac {\frac {d (9 b c-a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {-\frac {2 d^{3/2} \left (-a^2 d^2+10 a b c d+b^2 \left (15 c^2-8 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b}}+8 i b d \sqrt {a-i b} (c-i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )-8 i b d \sqrt {a+i b} (c+i d)^{5/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{2 d f}}{4 b}\)

Input:

Int[Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(5/2),x]
 

Output:

(d^2*(a + b*Tan[e + f*x])^(3/2)*Sqrt[c + d*Tan[e + f*x]])/(2*b*f) + (-1/2* 
((8*I)*Sqrt[a - I*b]*b*(c - I*d)^(5/2)*d*ArcTanh[(Sqrt[c - I*d]*Sqrt[a + b 
*Tan[e + f*x]])/(Sqrt[a - I*b]*Sqrt[c + d*Tan[e + f*x]])] - (8*I)*Sqrt[a + 
 I*b]*b*(c + I*d)^(5/2)*d*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]]) 
/(Sqrt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])] - (2*d^(3/2)*(10*a*b*c*d - a^2* 
d^2 + b^2*(15*c^2 - 8*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sq 
rt[b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[b])/(d*f) + (d*(9*b*c - a*d)*Sqrt[a 
 + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/f)/(4*b)
 

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 4049

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + 
 (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((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 - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n 
- 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ 
e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], 
x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 
, 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I 
ntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) 
)
 

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.

\[\int \sqrt {a +b \tan \left (f x +e \right )}\, \left (c +d \tan \left (f x +e \right )\right )^{\frac {5}{2}}d x\]

Input:

int((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x)
 

Output:

int((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 10538 vs. \(2 (269) = 538\).

Time = 10.68 (sec) , antiderivative size = 21137, normalized size of antiderivative = 62.35 \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \] Input:

integrate((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="fric 
as")
                                                                                    
                                                                                    
 

Output:

Too large to include
 

Sympy [F]

\[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\int \sqrt {a + b \tan {\left (e + f x \right )}} \left (c + d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx \] Input:

integrate((a+b*tan(f*x+e))**(1/2)*(c+d*tan(f*x+e))**(5/2),x)
 

Output:

Integral(sqrt(a + b*tan(e + f*x))*(c + d*tan(e + f*x))**(5/2), x)
 

Maxima [F]

\[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\int { \sqrt {b \tan \left (f x + e\right ) + a} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="maxi 
ma")
 

Output:

integrate(sqrt(b*tan(f*x + e) + a)*(d*tan(f*x + e) + c)^(5/2), x)
 

Giac [F]

\[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\int { \sqrt {b \tan \left (f x + e\right ) + a} {\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \] Input:

integrate((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x, algorithm="giac 
")
 

Output:

integrate(sqrt(b*tan(f*x + e) + a)*(d*tan(f*x + e) + c)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\int \sqrt {a+b\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2} \,d x \] Input:

int((a + b*tan(e + f*x))^(1/2)*(c + d*tan(e + f*x))^(5/2),x)
 

Output:

int((a + b*tan(e + f*x))^(1/2)*(c + d*tan(e + f*x))^(5/2), x)
 

Reduce [F]

\[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{5/2} \, dx=\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )^{2}d x \right ) d^{2}+2 \left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}\, \tan \left (f x +e \right )d x \right ) c d +\left (\int \sqrt {d \tan \left (f x +e \right )+c}\, \sqrt {\tan \left (f x +e \right ) b +a}d x \right ) c^{2} \] Input:

int((a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(5/2),x)
 

Output:

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