\(\int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2} \, dx\) [1287]

Optimal result

Integrand size = 29, antiderivative size = 330 \[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2} \, dx=-\frac {i (a-i b)^{3/2} (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i (a+i b)^{3/2} (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\left (18 a b c d+3 a^2 d^2+b^2 \left (3 c^2-8 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{4 \sqrt {b} \sqrt {d} f}+\frac {(3 b c+5 a d) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{4 f}+\frac {b \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f} \] Output:

-I*(a-I*b)^(3/2)*(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+I*(a+I*b)^(3/2)*(c+I*d)^(3/2)*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*(18*a*b*c*d+3*a^2*d^2+b^2*(3*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^(1/2)/d^(1/2)/f+1/4*( 
5*a*d+3*b*c)*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/f+1/2*b*(a+b*ta 
n(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(3/2)/f
 

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2566\) vs. \(2(330)=660\).

Time = 6.30 (sec) , antiderivative size = 2566, normalized size of antiderivative = 7.78 \[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2} \, dx=\text {Result too large to show} \] Input:

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

Output:

((-1/2*I)*(-a - I*b)*(-((-a - I*b)*(-((-c - I*d)*((-2*(-c - I*d)*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]*Sqrt[c + I*d]) - (2*Sqrt[d]*Sqrt[b*c - a*d]*Sqrt[b 
/((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b^2*c)/(b*c - a*d) - ( 
a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sq 
rt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])]*Sqrt[(b*(c 
 + d*Tan[e + f*x]))/(b*c - a*d)])/(b^(3/2)*Sqrt[c + d*Tan[e + f*x]]))) - ( 
2*d*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]]*(1 + (b*d*(a + b*Tan 
[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))))^(3/ 
2)*((Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)]*ArcSi 
nh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[b*c - a*d]*Sqrt[(b^2*c 
)/(b*c - a*d) - (a*b*d)/(b*c - a*d)])])/(2*Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[ 
e + f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d 
) - (a*b*d)/(b*c - a*d))))^(3/2)) + 1/(2*(1 + (b*d*(a + b*Tan[e + f*x]))/( 
(b*c - a*d)*((b^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d)))))))/(b*Sqrt[b/((b 
^2*c)/(b*c - a*d) - (a*b*d)/(b*c - a*d))]*Sqrt[(b*(c + d*Tan[e + f*x]))/(b 
*c - a*d)]))) - (2*(b*c - a*d)*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + 
 f*x]]*(1 + (b*d*(a + b*Tan[e + f*x]))/((b*c - a*d)*((b^2*c)/(b*c - a*d) - 
 (a*b*d)/(b*c - a*d))))^(5/2)*((3*Sqrt[b*c - a*d]*Sqrt[(b^2*c)/(b*c - a*d) 
 - (a*b*d)/(b*c - a*d)]*ArcSinh[(Sqrt[b]*Sqrt[d]*Sqrt[a + b*Tan[e + f*x...
 

Rubi [A] (verified)

Time = 1.79 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.01, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.345, Rules used = {3042, 4053, 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.

\(\displaystyle \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2}dx\)

\(\Big \downarrow \) 4053

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 3042

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

\(\Big \downarrow \) 4130

\(\displaystyle \frac {1}{4} \left (\frac {\int -\frac {-b \left (\left (3 c^2-8 d^2\right ) b^2+18 a c d b+3 a^2 d^2\right ) \tan ^2(e+f x)-16 b (b c+a d) (a c-b d) \tan (e+f x)+b \left (-\left (\left (8 c^2-5 d^2\right ) a^2\right )+14 b c d a+5 b^2 c^2\right )}{2 \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{b}+\frac {(5 a d+3 b c) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}\right )+\frac {b \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{4} \left (\frac {(5 a d+3 b c) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {\int \frac {-b \left (\left (3 c^2-8 d^2\right ) b^2+18 a c d b+3 a^2 d^2\right ) \tan ^2(e+f x)-16 b (b c+a d) (a c-b d) \tan (e+f x)+b \left (-\left (\left (8 c^2-5 d^2\right ) a^2\right )+14 b c d a+5 b^2 c^2\right )}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}\right )+\frac {b \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {1}{4} \left (\frac {(5 a d+3 b c) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {\int \frac {-b \left (\left (3 c^2-8 d^2\right ) b^2+18 a c d b+3 a^2 d^2\right ) \tan (e+f x)^2-16 b (b c+a d) (a c-b d) \tan (e+f x)+b \left (-\left (\left (8 c^2-5 d^2\right ) a^2\right )+14 b c d a+5 b^2 c^2\right )}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}dx}{2 b}\right )+\frac {b \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 4138

\(\displaystyle \frac {1}{4} \left (\frac {(5 a d+3 b c) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {\int \frac {-b \left (\left (3 c^2-8 d^2\right ) b^2+18 a c d b+3 a^2 d^2\right ) \tan ^2(e+f x)-16 b (b c+a d) (a c-b d) \tan (e+f x)+b \left (-\left (\left (8 c^2-5 d^2\right ) a^2\right )+14 b c d a+5 b^2 c^2\right )}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{2 b f}\right )+\frac {b \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}\)

\(\Big \downarrow \) 2348

\(\displaystyle \frac {b \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}+\frac {1}{4} \left (\frac {(5 a d+3 b c) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {\int \left (\frac {b \left (-\left (\left (3 c^2-8 d^2\right ) b^2\right )-18 a c d b-3 a^2 d^2\right )}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {-16 c d b^3+16 a c^2 b^2-16 a d^2 b^2+16 a^2 c d b+i \left (8 c^2 b^3-8 d^2 b^3+32 a c d b^2-8 a^2 c^2 b+8 a^2 d^2 b\right )}{2 (i-\tan (e+f x)) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}+\frac {16 c d b^3-16 a c^2 b^2+16 a d^2 b^2-16 a^2 c d b+i \left (8 c^2 b^3-8 d^2 b^3+32 a c d b^2-8 a^2 c^2 b+8 a^2 d^2 b\right )}{2 (\tan (e+f x)+i) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}\right )d\tan (e+f x)}{2 b f}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2}}{2 f}+\frac {1}{4} \left (\frac {(5 a d+3 b c) \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}-\frac {-\frac {2 \sqrt {b} \left (3 a^2 d^2+18 a b c d+b^2 \left (3 c^2-8 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {d}}+8 i b (a-i b)^{3/2} (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )-8 i b (a+i b)^{3/2} (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{2 b f}\right )\)

Input:

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

Output:

(b*Sqrt[a + b*Tan[e + f*x]]*(c + d*Tan[e + f*x])^(3/2))/(2*f) + (-1/2*((8* 
I)*(a - I*b)^(3/2)*b*(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]])] - (8*I)*(a + I*b)^(3 
/2)*b*(c + I*d)^(3/2)*ArcTanh[(Sqrt[c + I*d]*Sqrt[a + b*Tan[e + f*x]])/(Sq 
rt[a + I*b]*Sqrt[c + d*Tan[e + f*x]])] - (2*Sqrt[b]*(18*a*b*c*d + 3*a^2*d^ 
2 + b^2*(3*c^2 - 8*d^2))*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sqrt[ 
b]*Sqrt[c + d*Tan[e + f*x]])])/Sqrt[d])/(b*f) + ((3*b*c + 5*a*d)*Sqrt[a + 
b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/f)/4
 

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 4053

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

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.

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

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 9848 vs. \(2 (260) = 520\).

Time = 8.48 (sec) , antiderivative size = 19712, normalized size of antiderivative = 59.73 \[ \int (a+b \tan (e+f x))^{3/2} (c+d \tan (e+f x))^{3/2} \, dx=\text {Too large to display} \] Input:

integrate((a+b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(3/2),x, algorithm="fric 
as")
                                                                                    
                                                                                    
 

Output:

Too large to include
 

Sympy [F]

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

integrate((a+b*tan(f*x+e))**(3/2)*(c+d*tan(f*x+e))**(3/2),x)
 

Output:

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

Maxima [F]

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

integrate((a+b*tan(f*x+e))^(3/2)*(c+d*tan(f*x+e))^(3/2),x, algorithm="maxi 
ma")
 

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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