\(\int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx\) [619]

Optimal result

Integrand size = 25, antiderivative size = 145 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\frac {i (i a-b)^{3/2} \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {i (i a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {3648, 3697, 3696, 95, 209, 212} \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\frac {i (-b+i a)^{3/2} \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {i (b+i a)^{3/2} \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d}-\frac {2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}} \]

[In]

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

[Out]

(I*(I*a - b)^(3/2)*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (I*(I*a + b)^(3/2)
*ArcTanh[(Sqrt[I*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/d - (2*a*Sqrt[a + b*Tan[c + d*x]])/(d*S
qrt[Tan[c + d*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 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

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 3648

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n - 1)/(f*(m + 1)*(a^2 + b^2))), x] + Dist[
1/((m + 1)*(a^2 + b^2)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 2)*Simp[a*c^2*(m + 1) + a*
d^2*(n - 1) + b*c*d*(m - n + 2) - (b*c^2 - 2*a*c*d - b*d^2)*(m + 1)*Tan[e + f*x] - d*(b*c - a*d)*(m + n)*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] && LtQ[m, -1] && LtQ[1, n, 2] && IntegerQ[2*m]

Rule 3696

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e
_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[A^2/f, Subst[Int[(a + b*x)^m*((c + d*x)^n/(A - B*x)), x], x, Tan[e
+ f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && EqQ[A^2 +
B^2, 0]

Rule 3697

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

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

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\frac {\sqrt [4]{-1} (-a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}-\sqrt [4]{-1} (a+i b)^{3/2} \arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right ) \sqrt {\tan (c+d x)}-2 a \sqrt {a+b \tan (c+d x)}}{d \sqrt {\tan (c+d x)}} \]

[In]

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

[Out]

((-1)^(1/4)*(-a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[-a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]]*S
qrt[Tan[c + d*x]] - (-1)^(1/4)*(a + I*b)^(3/2)*ArcTan[((-1)^(1/4)*Sqrt[a + I*b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b
*Tan[c + d*x]]]*Sqrt[Tan[c + d*x]] - 2*a*Sqrt[a + b*Tan[c + d*x]])/(d*Sqrt[Tan[c + d*x]])

Maple [B] (warning: unable to verify)

result has leaf size over 500,000. Avoiding possible recursion issues.

Time = 1.16 (sec) , antiderivative size = 1344189, normalized size of antiderivative = 9270.27

\[\text {output too large to display}\]

[In]

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

[Out]

result too large to display

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3461 vs. \(2 (113) = 226\).

Time = 0.60 (sec) , antiderivative size = 3461, normalized size of antiderivative = 23.87 \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*(d*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log((2*(2*a^6*b - 2*a^4*b^3 -
12*a^2*b^5 + (a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c) - (a^4 + 3*a^2*
b^2 + 4*b^4)*d^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) + ((a^
6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*
b^2 + 12*a^2*b^4)*d - 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 -
6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2))/(tan(d
*x + c)^2 + 1))*tan(d*x + c) + d*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(
-(2*(2*a^6*b - 2*a^4*b^3 - 12*a^2*b^5 + (a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*t
an(d*x + c) - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a
)*sqrt(tan(d*x + c)) + ((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*t
an(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d - 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*t
an(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*
a^2*b^4)/d^4))/d^2))/(tan(d*x + c)^2 + 1))*tan(d*x + c) - d*sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 +
9*a^2*b^4)/d^4))/d^2)*log((2*(2*a^6*b - 2*a^4*b^3 - 12*a^2*b^5 + (a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c) +
(2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c) - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4
))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) - ((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b -
5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d - 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d
^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 + d^2*s
qrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2))/(tan(d*x + c)^2 + 1))*tan(d*x + c) - d*sqrt((3*a^2*b - b^3 + d^
2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(2*(2*a^6*b - 2*a^4*b^3 - 12*a^2*b^5 + (a^7 - 5*a^3*b^4
- 12*a*b^6)*tan(d*x + c) + (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c) - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2)*sqrt(-(a^6 -
6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) - ((a^6 + a^4*b^2 - 12*a^2*b^4)*d*ta
n(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d - 2*(a^3*d
^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*
sqrt((3*a^2*b - b^3 + d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2))/(tan(d*x + c)^2 + 1))*tan(d*x + c) +
d*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log((2*(2*a^6*b - 2*a^4*b^3 - 12*a
^2*b^5 + (a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c) - (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c) - (a^4 + 3*a^2*b^2
+ 4*b^4)*d^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) + ((a^6 +
a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2
+ 12*a^2*b^4)*d + 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a
^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2))/(tan(d*x +
c)^2 + 1))*tan(d*x + c) + d*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(2*
(2*a^6*b - 2*a^4*b^3 - 12*a^2*b^5 + (a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c) - (2*(a^3*b + 2*a*b^3)*d^2*tan(d
*x + c) - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)*sq
rt(tan(d*x + c)) + ((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d
*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d
*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*
b^4)/d^4))/d^2))/(tan(d*x + c)^2 + 1))*tan(d*x + c) - d*sqrt((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a
^2*b^4)/d^4))/d^2)*log((2*(2*a^6*b - 2*a^4*b^3 - 12*a^2*b^5 + (a^7 - 5*a^3*b^4 - 12*a*b^6)*tan(d*x + c) - (2*(
a^3*b + 2*a*b^3)*d^2*tan(d*x + c) - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*s
qrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) - ((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*x + c)^2 + 2*(3*a^5*b - 5*a
^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*(a^3*d^3*tan(d*x + c) + 2*a^2*b*d^3 -
(a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt((3*a^2*b - b^3 - d^2*sqrt(
-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2))/(tan(d*x + c)^2 + 1))*tan(d*x + c) - d*sqrt((3*a^2*b - b^3 - d^2*sq
rt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2)*log(-(2*(2*a^6*b - 2*a^4*b^3 - 12*a^2*b^5 + (a^7 - 5*a^3*b^4 - 12
*a*b^6)*tan(d*x + c) - (2*(a^3*b + 2*a*b^3)*d^2*tan(d*x + c) - (a^4 + 3*a^2*b^2 + 4*b^4)*d^2)*sqrt(-(a^6 - 6*a
^4*b^2 + 9*a^2*b^4)/d^4))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) - ((a^6 + a^4*b^2 - 12*a^2*b^4)*d*tan(d*
x + c)^2 + 2*(3*a^5*b - 5*a^3*b^3 - 12*a*b^5)*d*tan(d*x + c) - (a^6 - 7*a^4*b^2 + 12*a^2*b^4)*d + 2*(a^3*d^3*t
an(d*x + c) + 2*a^2*b*d^3 - (a^2*b + 4*b^3)*d^3*tan(d*x + c)^2)*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))*sqrt
((3*a^2*b - b^3 - d^2*sqrt(-(a^6 - 6*a^4*b^2 + 9*a^2*b^4)/d^4))/d^2))/(tan(d*x + c)^2 + 1))*tan(d*x + c) - 16*
sqrt(b*tan(d*x + c) + a)*a*sqrt(tan(d*x + c)))/(d*tan(d*x + c))

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

Timed out. \[ \int \frac {(a+b \tan (c+d x))^{3/2}}{\tan ^{\frac {3}{2}}(c+d x)} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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