\(\int \frac {a B+b B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx\) [471]

Optimal result

Integrand size = 38, antiderivative size = 111 \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {B \arctan \left (\frac {\sqrt {i a-b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {i a-b} d}+\frac {B \text {arctanh}\left (\frac {\sqrt {i a+b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {i a+b} d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {21, 3656, 926, 95, 211, 214} \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {B \arctan \left (\frac {\sqrt {-b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {-b+i a}}+\frac {B \text {arctanh}\left (\frac {\sqrt {b+i a} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{d \sqrt {b+i a}} \]

[In]

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

[Out]

(B*ArcTan[(Sqrt[I*a - b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a - b]*d) + (B*ArcTanh[(Sqrt[I
*a + b]*Sqrt[Tan[c + d*x]])/Sqrt[a + b*Tan[c + d*x]]])/(Sqrt[I*a + b]*d)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*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 211

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

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 926

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rule 3656

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

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

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.13 \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\frac {(-1)^{3/4} B \left (-\frac {\arctan \left (\frac {\sqrt [4]{-1} \sqrt {-a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {-a+i b}}-\frac {\arctan \left (\frac {\sqrt [4]{-1} \sqrt {a+i b} \sqrt {\tan (c+d x)}}{\sqrt {a+b \tan (c+d x)}}\right )}{\sqrt {a+i b}}\right )}{d} \]

[In]

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

[Out]

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

Maple [B] (warning: unable to verify)

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

Time = 0.54 (sec) , antiderivative size = 940264, normalized size of antiderivative = 8470.85

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

[In]

int((B*a+b*B*tan(d*x+c))/tan(d*x+c)^(1/2)/(a+b*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. 4469 vs. \(2 (87) = 174\).

Time = 0.77 (sec) , antiderivative size = 4469, normalized size of antiderivative = 40.26 \[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/8*sqrt(-((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2))*log(1/2*(2
*(2*B^3*a^4*b + 4*B^3*a^2*b^3 + (B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c) + (2*(B*a^5*b + 3*B*a^3*b
^3 + 2*B*a*b^5)*d^2*tan(d*x + c) - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2)*sqrt(-B^4*a^2/((a^4 + 2*
a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) + (2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x +
c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d + ((a^7 +
8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x
+ c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^2
+ b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2)))/(tan(d*x + c)^2 + 1)) +
1/8*sqrt(-((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2))*log(-1/2*(
2*(2*B^3*a^4*b + 4*B^3*a^2*b^3 + (B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c) + (2*(B*a^5*b + 3*B*a^3*
b^3 + 2*B*a*b^5)*d^2*tan(d*x + c) - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2)*sqrt(-B^4*a^2/((a^4 + 2
*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) + (2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x
+ c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d + ((a^7
+ 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*
x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^
2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2)))/(tan(d*x + c)^2 + 1)) -
1/8*sqrt(-((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2))*log(1/2*(
2*(2*B^3*a^4*b + 4*B^3*a^2*b^3 + (B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c) + (2*(B*a^5*b + 3*B*a^3*
b^3 + 2*B*a*b^5)*d^2*tan(d*x + c) - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2)*sqrt(-B^4*a^2/((a^4 + 2
*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) - (2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x
+ c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d + ((a^7
+ 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*
x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a^
2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2)))/(tan(d*x + c)^2 + 1)) -
1/8*sqrt(-((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2))*log(-1/2*
(2*(2*B^3*a^4*b + 4*B^3*a^2*b^3 + (B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c) + (2*(B*a^5*b + 3*B*a^3
*b^3 + 2*B*a*b^5)*d^2*tan(d*x + c) - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2)*sqrt(-B^4*a^2/((a^4 +
2*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) - (2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x
+ c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d + ((a^7
+ 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d
*x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(-((a
^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 - B^2*b)/((a^2 + b^2)*d^2)))/(tan(d*x + c)^2 + 1))
+ 1/8*sqrt(((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2))*log(1/2*(
2*(2*B^3*a^4*b + 4*B^3*a^2*b^3 + (B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c) - (2*(B*a^5*b + 3*B*a^3*
b^3 + 2*B*a*b^5)*d^2*tan(d*x + c) - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2)*sqrt(-B^4*a^2/((a^4 + 2
*a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) + (2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x
+ c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d - ((a^7
+ 8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*
x + c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2
+ b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2)))/(tan(d*x + c)^2 + 1)) +
1/8*sqrt(((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2))*log(-1/2*(2
*(2*B^3*a^4*b + 4*B^3*a^2*b^3 + (B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c) - (2*(B*a^5*b + 3*B*a^3*b
^3 + 2*B*a*b^5)*d^2*tan(d*x + c) - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2)*sqrt(-B^4*a^2/((a^4 + 2*
a^2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) + (2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x +
c)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d - ((a^7 +
8*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x
+ c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2
+ b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2)))/(tan(d*x + c)^2 + 1)) - 1
/8*sqrt(((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2))*log(1/2*(2*(
2*B^3*a^4*b + 4*B^3*a^2*b^3 + (B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c) - (2*(B*a^5*b + 3*B*a^3*b^3
+ 2*B*a*b^5)*d^2*tan(d*x + c) - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2)*sqrt(-B^4*a^2/((a^4 + 2*a^
2*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) - (2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x + c
)^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d - ((a^7 + 8
*a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x +
c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2 +
b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2)))/(tan(d*x + c)^2 + 1)) - 1/8
*sqrt(((a^2 + b^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2))*log(-1/2*(2*(2
*B^3*a^4*b + 4*B^3*a^2*b^3 + (B^3*a^5 + 3*B^3*a^3*b^2 + 4*B^3*a*b^4)*tan(d*x + c) - (2*(B*a^5*b + 3*B*a^3*b^3
+ 2*B*a*b^5)*d^2*tan(d*x + c) - (B*a^6 + 4*B*a^4*b^2 + 7*B*a^2*b^4 + 4*B*b^6)*d^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2
*b^2 + b^4)*d^4)))*sqrt(b*tan(d*x + c) + a)*sqrt(tan(d*x + c)) - (2*(B^2*a^3*b^3 + 4*B^2*a*b^5)*d*tan(d*x + c)
^2 + 2*(B^2*a^6 + 5*B^2*a^4*b^2 + 8*B^2*a^2*b^4)*d*tan(d*x + c) + 2*(B^2*a^5*b + 2*B^2*a^3*b^3)*d - ((a^7 + 8*
a^5*b^2 + 19*a^3*b^4 + 12*a*b^6)*d^3*tan(d*x + c)^2 + 2*(a^6*b + 2*a^4*b^3 - 3*a^2*b^5 - 4*b^7)*d^3*tan(d*x +
c) - (a^7 + 4*a^5*b^2 + 7*a^3*b^4 + 4*a*b^6)*d^3)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4)))*sqrt(((a^2 + b
^2)*sqrt(-B^4*a^2/((a^4 + 2*a^2*b^2 + b^4)*d^4))*d^2 + B^2*b)/((a^2 + b^2)*d^2)))/(tan(d*x + c)^2 + 1))

Sympy [F]

\[ \int \frac {a B+b B \tan (c+d x)}{\sqrt {\tan (c+d x)} (a+b \tan (c+d x))^{3/2}} \, dx=B \int \frac {1}{\sqrt {a + b \tan {\left (c + d x \right )}} \sqrt {\tan {\left (c + d x \right )}}}\, dx \]

[In]

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

[Out]

B*Integral(1/(sqrt(a + b*tan(c + d*x))*sqrt(tan(c + d*x))), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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