Integrand size = 27, antiderivative size = 242 \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx=-\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-m,1,\frac {1}{2},\frac {b (c+d \tan (e+f x))}{b c-a d},\frac {c+d \tan (e+f x)}{c-i d}\right ) (a+b \tan (e+f x))^m \left (-\frac {d (a+b \tan (e+f x))}{b c-a d}\right )^{-m}}{(i c+d) f \sqrt {c+d \tan (e+f x)}}+\frac {\operatorname {AppellF1}\left (-\frac {1}{2},-m,1,\frac {1}{2},\frac {b (c+d \tan (e+f x))}{b c-a d},\frac {c+d \tan (e+f x)}{c+i d}\right ) (a+b \tan (e+f x))^m \left (-\frac {d (a+b \tan (e+f x))}{b c-a d}\right )^{-m}}{(i c-d) f \sqrt {c+d \tan (e+f x)}} \] Output:
-AppellF1(-1/2,1,-m,1/2,(c+d*tan(f*x+e))/(c-I*d),b*(c+d*tan(f*x+e))/(-a*d+ b*c))*(a+b*tan(f*x+e))^m/(I*c+d)/f/((-d*(a+b*tan(f*x+e))/(-a*d+b*c))^m)/(c +d*tan(f*x+e))^(1/2)+AppellF1(-1/2,1,-m,1/2,(c+d*tan(f*x+e))/(c+I*d),b*(c+ d*tan(f*x+e))/(-a*d+b*c))*(a+b*tan(f*x+e))^m/(I*c-d)/f/((-d*(a+b*tan(f*x+e ))/(-a*d+b*c))^m)/(c+d*tan(f*x+e))^(1/2)
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx \] Input:
Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(3/2),x]
Output:
Integrate[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(3/2), x]
Time = 0.51 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.16, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {3042, 4058, 662, 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.
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}}dx\) |
| \(\Big \downarrow \) 4058 |
| \(\displaystyle \frac {\int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 662 |
| \(\displaystyle \frac {\int \left (\frac {i (a+b \tan (e+f x))^m}{2 (i-\tan (e+f x)) (c+d \tan (e+f x))^{3/2}}+\frac {i (a+b \tan (e+f x))^m}{2 (\tan (e+f x)+i) (c+d \tan (e+f x))^{3/2}}\right )d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b (a+b \tan (e+f x))^{m+1} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \operatorname {AppellF1}\left (m+1,\frac {3}{2},1,m+2,-\frac {d (a+b \tan (e+f x))}{b c-a d},\frac {a+b \tan (e+f x)}{a-i b}\right )}{2 (m+1) (b+i a) (b c-a d) \sqrt {c+d \tan (e+f x)}}-\frac {b (a+b \tan (e+f x))^{m+1} \sqrt {\frac {b (c+d \tan (e+f x))}{b c-a d}} \operatorname {AppellF1}\left (m+1,\frac {3}{2},1,m+2,-\frac {d (a+b \tan (e+f x))}{b c-a d},\frac {a+b \tan (e+f x)}{a+i b}\right )}{2 (m+1) (-b+i a) (b c-a d) \sqrt {c+d \tan (e+f x)}}}{f}\) |
Input:
Int[(a + b*Tan[e + f*x])^m/(c + d*Tan[e + f*x])^(3/2),x]
Output:
((b*AppellF1[1 + m, 3/2, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)) , (a + b*Tan[e + f*x])/(a - I*b)]*(a + b*Tan[e + f*x])^(1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(2*(I*a + b)*(b*c - a*d)*(1 + m)*Sqrt[c + d*Tan[e + f*x]]) - (b*AppellF1[1 + m, 3/2, 1, 2 + m, -((d*(a + b*Tan[e + f*x]))/(b*c - a*d)), (a + b*Tan[e + f*x])/(a + I*b)]*(a + b*Tan[e + f*x])^ (1 + m)*Sqrt[(b*(c + d*Tan[e + f*x]))/(b*c - a*d)])/(2*(I*a - b)*(b*c - a* d)*(1 + m)*Sqrt[c + d*Tan[e + f*x]]))/f
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_ )^2), x_Symbol] :> Int[ExpandIntegrand[(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] && !IntegerQ[m] && !Inte gerQ[n]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), 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/(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]
\[\int \frac {\left (a +b \tan \left (f x +e \right )\right )^{m}}{\left (c +d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}d x\]
Input:
int((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e))^(3/2),x)
Output:
int((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e))^(3/2),x)
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e))^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(d*tan(f*x + e) + c)*(b*tan(f*x + e) + a)^m/(d^2*tan(f*x + e) ^2 + 2*c*d*tan(f*x + e) + c^2), x)
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{m}}{\left (c + d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*tan(f*x+e))**m/(c+d*tan(f*x+e))**(3/2),x)
Output:
Integral((a + b*tan(e + f*x))**m/(c + d*tan(e + f*x))**(3/2), x)
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e))^(3/2),x, algorithm="maxima")
Output:
integrate((b*tan(f*x + e) + a)^m/(d*tan(f*x + e) + c)^(3/2), x)
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{m}}{{\left (d \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e))^(3/2),x, algorithm="giac")
Output:
integrate((b*tan(f*x + e) + a)^m/(d*tan(f*x + e) + c)^(3/2), x)
Timed out. \[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^m}{{\left (c+d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}} \,d x \] Input:
int((a + b*tan(e + f*x))^m/(c + d*tan(e + f*x))^(3/2),x)
Output:
int((a + b*tan(e + f*x))^m/(c + d*tan(e + f*x))^(3/2), x)
\[ \int \frac {(a+b \tan (e+f x))^m}{(c+d \tan (e+f x))^{3/2}} \, dx=\text {too large to display} \] Input:
int((a+b*tan(f*x+e))^m/(c+d*tan(f*x+e))^(3/2),x)
Output:
( - 4*sqrt(tan(e + f*x)*d + c)*(tan(e + f*x)*b + a)**m*a*d*m + 4*sqrt(tan( e + f*x)*d + c)*(tan(e + f*x)*b + a)**m*b*c*m + 8*int((sqrt(tan(e + f*x)*d + c)*(tan(e + f*x)*b + a)**m*tan(e + f*x)**3)/(2*tan(e + f*x)**3*a*b*d**3 *m - tan(e + f*x)**3*a*b*d**3 - 4*tan(e + f*x)**3*b**2*c*d**2*m**2 + 2*tan (e + f*x)**3*b**2*c*d**2*m + 2*tan(e + f*x)**2*a**2*d**3*m - tan(e + f*x)* *2*a**2*d**3 - 4*tan(e + f*x)**2*a*b*c*d**2*m**2 + 6*tan(e + f*x)**2*a*b*c *d**2*m - 2*tan(e + f*x)**2*a*b*c*d**2 - 8*tan(e + f*x)**2*b**2*c**2*d*m** 2 + 4*tan(e + f*x)**2*b**2*c**2*d*m + 4*tan(e + f*x)*a**2*c*d**2*m - 2*tan (e + f*x)*a**2*c*d**2 - 8*tan(e + f*x)*a*b*c**2*d*m**2 + 6*tan(e + f*x)*a* b*c**2*d*m - tan(e + f*x)*a*b*c**2*d - 4*tan(e + f*x)*b**2*c**3*m**2 + 2*t an(e + f*x)*b**2*c**3*m + 2*a**2*c**2*d*m - a**2*c**2*d - 4*a*b*c**3*m**2 + 2*a*b*c**3*m),x)*tan(e + f*x)*a**2*b*d**4*f*m**3 - 4*int((sqrt(tan(e + f *x)*d + c)*(tan(e + f*x)*b + a)**m*tan(e + f*x)**3)/(2*tan(e + f*x)**3*a*b *d**3*m - tan(e + f*x)**3*a*b*d**3 - 4*tan(e + f*x)**3*b**2*c*d**2*m**2 + 2*tan(e + f*x)**3*b**2*c*d**2*m + 2*tan(e + f*x)**2*a**2*d**3*m - tan(e + f*x)**2*a**2*d**3 - 4*tan(e + f*x)**2*a*b*c*d**2*m**2 + 6*tan(e + f*x)**2* a*b*c*d**2*m - 2*tan(e + f*x)**2*a*b*c*d**2 - 8*tan(e + f*x)**2*b**2*c**2* d*m**2 + 4*tan(e + f*x)**2*b**2*c**2*d*m + 4*tan(e + f*x)*a**2*c*d**2*m - 2*tan(e + f*x)*a**2*c*d**2 - 8*tan(e + f*x)*a*b*c**2*d*m**2 + 6*tan(e + f* x)*a*b*c**2*d*m - tan(e + f*x)*a*b*c**2*d - 4*tan(e + f*x)*b**2*c**3*m*...