\(\int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \, dx\) [1272]

Optimal result

Integrand size = 29, antiderivative size = 258 \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \, dx=-\frac {i \sqrt {a-i b} (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i \sqrt {a+i b} (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {d} (3 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} f}+\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f} \]

[Out]

-I*(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))*(a-I*b)^(1
/2)/f+I*(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))*(a+I*
b)^(1/2)/f+(a*d+3*b*c)*arctanh(d^(1/2)*(a+b*tan(f*x+e))^(1/2)/b^(1/2)/(c+d*tan(f*x+e))^(1/2))*d^(1/2)/f/b^(1/2
)+d*(a+b*tan(f*x+e))^(1/2)*(c+d*tan(f*x+e))^(1/2)/f

Rubi [A] (verified)

Time = 2.60 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {3651, 3736, 6857, 65, 223, 212, 95, 214} \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \, dx=-\frac {i \sqrt {a-i b} (c-i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c-i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a-i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {i \sqrt {a+i b} (c+i d)^{3/2} \text {arctanh}\left (\frac {\sqrt {c+i d} \sqrt {a+b \tan (e+f x)}}{\sqrt {a+i b} \sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\sqrt {d} (a d+3 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b} \sqrt {c+d \tan (e+f x)}}\right )}{\sqrt {b} f}+\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f} \]

[In]

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

[Out]

((-I)*Sqrt[a - I*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]])])/f + (I*Sqrt[a + I*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]])])/f + (Sqrt[d]*(3*b*c + a*d)*ArcTanh[(Sqrt[d]*Sqrt[a + b*Tan[e + f*x]])/(Sq
rt[b]*Sqrt[c + d*Tan[e + f*x]])])/(Sqrt[b]*f) + (d*Sqrt[a + b*Tan[e + f*x]]*Sqrt[c + d*Tan[e + f*x]])/f

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 223

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]

Rule 3651

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b*(a + b*Tan[e + f*x])^(m - 1)*((c + d*Tan[e + f*x])^n/(f*(m + n - 1))), x] + Dist[1/(m + n - 1), Int[(a +
b*Tan[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] /; Fre
eQ[{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 3736

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*t
an[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x
]}, Dist[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]

Rule 6857

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\int \frac {\frac {1}{2} \left (2 a c^2-d (b c+a d)\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) \tan (e+f x)+\frac {1}{2} d (3 b c+a d) \tan ^2(e+f x)}{\sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}} \, dx \\ & = \frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\text {Subst}\left (\int \frac {\frac {1}{2} \left (2 a c^2-d (b c+a d)\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) x+\frac {1}{2} d (3 b c+a d) x^2}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\text {Subst}\left (\int \left (\frac {d (3 b c+a d)}{2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {-2 b c d+a \left (c^2-d^2\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )}\right ) \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\text {Subst}\left (\int \frac {-2 b c d+a \left (c^2-d^2\right )+\left (2 a c d+b \left (c^2-d^2\right )\right ) x}{\sqrt {a+b x} \sqrt {c+d x} \left (1+x^2\right )} \, dx,x,\tan (e+f x)\right )}{f}+\frac {(d (3 b c+a d)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f} \\ & = \frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\text {Subst}\left (\int \left (\frac {-2 a c d-b \left (c^2-d^2\right )+i \left (-2 b c d+a \left (c^2-d^2\right )\right )}{2 (i-x) \sqrt {a+b x} \sqrt {c+d x}}+\frac {2 a c d+b \left (c^2-d^2\right )+i \left (-2 b c d+a \left (c^2-d^2\right )\right )}{2 (i+x) \sqrt {a+b x} \sqrt {c+d x}}\right ) \, dx,x,\tan (e+f x)\right )}{f}+\frac {(d (3 b c+a d)) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b \tan (e+f x)}\right )}{b f} \\ & = \frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\left ((i a+b) (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{(i+x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {\left ((i a-b) (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{(i-x) \sqrt {a+b x} \sqrt {c+d x}} \, dx,x,\tan (e+f x)\right )}{2 f}+\frac {(d (3 b c+a d)) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{b f} \\ & = \frac {\sqrt {d} (3 b c+a d) \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} f}+\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f}+\frac {\left ((i a+b) (c-i d)^2\right ) \text {Subst}\left (\int \frac {1}{-a+i b-(-c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f}+\frac {\left ((i a-b) (c+i d)^2\right ) \text {Subst}\left (\int \frac {1}{a+i b-(c+i d) x^2} \, dx,x,\frac {\sqrt {a+b \tan (e+f x)}}{\sqrt {c+d \tan (e+f x)}}\right )}{f} \\ & = -\frac {i \sqrt {a-i b} (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 \sqrt {a+i b} (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 {\sqrt {d} (3 b c+a d) \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} f}+\frac {d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)}}{f} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1526\) vs. \(2(258)=516\).

Time = 6.19 (sec) , antiderivative size = 1526, normalized size of antiderivative = 5.91 \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \, dx=-\frac {i (-a-i b) \left (-\left ((-c-i d) \left (-\frac {2 (-c-i d) \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} \sqrt {c+i d}}-\frac {2 \sqrt {d} \sqrt {b c-a d} \sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right ) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{b^{3/2} \sqrt {c+d \tan (e+f x)}}\right )\right )-\frac {2 d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )^{3/2} \left (\frac {\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right )}{2 \sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)} \left (1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )^{3/2}}+\frac {1}{2 \left (1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )}\right )}{b \sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}\right )}{2 f}-\frac {i (-a+i b) \left (-\left ((-c+i d) \left (-\frac {2 \sqrt {-c+i d} \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 {2 \sqrt {d} \sqrt {b c-a d} \sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right ) \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}{b^{3/2} \sqrt {c+d \tan (e+f x)}}\right )\right )+\frac {2 d \sqrt {a+b \tan (e+f x)} \sqrt {c+d \tan (e+f x)} \left (1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )^{3/2} \left (\frac {\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}} \text {arcsinh}\left (\frac {\sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)}}{\sqrt {b c-a d} \sqrt {\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}}\right )}{2 \sqrt {b} \sqrt {d} \sqrt {a+b \tan (e+f x)} \left (1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )^{3/2}}+\frac {1}{2 \left (1+\frac {b d (a+b \tan (e+f x))}{(b c-a d) \left (\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}\right )}\right )}\right )}{b \sqrt {\frac {b}{\frac {b^2 c}{b c-a d}-\frac {a b d}{b c-a d}}} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}}}\right )}{2 f} \]

[In]

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

[Out]

((-1/2*I)*(-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]*Sqr
t[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)])]*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)*((Sqr
t[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]]
)/(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)])))/f - ((I/2)*(-a + I*b)*(-((
-c + I*d)*((-2*Sqrt[-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] + (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]])/(Sqrt[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)*S
qrt[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)]*ArcSinh[(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)])))/f

Maple [F(-1)]

Timed out.

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

[In]

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

[Out]

int((a+b*tan(f*x+e))^(1/2)*(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. 6924 vs. \(2 (198) = 396\).

Time = 5.49 (sec) , antiderivative size = 13874, normalized size of antiderivative = 53.78 \[ \int \sqrt {a+b \tan (e+f x)} (c+d \tan (e+f x))^{3/2} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Too large to include

Sympy [F]

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

[In]

integrate((a+b*tan(f*x+e))**(1/2)*(c+d*tan(f*x+e))**(3/2),x)

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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