\(\int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx\) [676]

Optimal result

Integrand size = 25, antiderivative size = 165 \[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {3 \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \tan ^{\frac {5}{3}}(c+d x) \sqrt {\frac {a+b \tan (c+d x)}{a}}}{10 d \sqrt {a+b \tan (c+d x)}}+\frac {3 \operatorname {AppellF1}\left (\frac {5}{3},\frac {1}{2},1,\frac {8}{3},-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right ) \tan ^{\frac {5}{3}}(c+d x) \sqrt {\frac {a+b \tan (c+d x)}{a}}}{10 d \sqrt {a+b \tan (c+d x)}} \] Output:

3/10*AppellF1(5/3,1,1/2,8/3,-I*tan(d*x+c),-b*tan(d*x+c)/a)*tan(d*x+c)^(5/3 
)*((a+b*tan(d*x+c))/a)^(1/2)/d/(a+b*tan(d*x+c))^(1/2)+3/10*AppellF1(5/3,1, 
1/2,8/3,I*tan(d*x+c),-b*tan(d*x+c)/a)*tan(d*x+c)^(5/3)*((a+b*tan(d*x+c))/a 
)^(1/2)/d/(a+b*tan(d*x+c))^(1/2)
 

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(7362\) vs. \(2(165)=330\).

Time = 102.50 (sec) , antiderivative size = 7362, normalized size of antiderivative = 44.62 \[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [F]

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.

Failed to integrate

Input:

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

Output:

$Aborted
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\tan ^{\frac {2}{3}}(c+d x)}{\sqrt {a+b \tan (c+d x)}} \, dx=\frac {6 \tan \left (d x +c \right )^{\frac {2}{3}} \sqrt {a +\tan \left (d x +c \right ) b}-7 \left (\int \frac {\tan \left (d x +c \right )^{\frac {8}{3}} \sqrt {a +\tan \left (d x +c \right ) b}}{a +\tan \left (d x +c \right ) b}d x \right ) b d -4 \left (\int \frac {\tan \left (d x +c \right )^{\frac {5}{3}} \sqrt {a +\tan \left (d x +c \right ) b}}{a +\tan \left (d x +c \right ) b}d x \right ) a d -4 \left (\int \frac {\tan \left (d x +c \right )^{\frac {2}{3}} \sqrt {a +\tan \left (d x +c \right ) b}}{\tan \left (d x +c \right )^{2} b +\tan \left (d x +c \right ) a}d x \right ) a d}{7 b d} \] Input:

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

Output:

(6*tan(c + d*x)**(2/3)*sqrt(tan(c + d*x)*b + a) - 7*int((tan(c + d*x)**(2/ 
3)*sqrt(tan(c + d*x)*b + a)*tan(c + d*x)**2)/(tan(c + d*x)*b + a),x)*b*d - 
 4*int((tan(c + d*x)**(2/3)*sqrt(tan(c + d*x)*b + a)*tan(c + d*x))/(tan(c 
+ d*x)*b + a),x)*a*d - 4*int((tan(c + d*x)**(2/3)*sqrt(tan(c + d*x)*b + a) 
)/(tan(c + d*x)**2*b + tan(c + d*x)*a),x)*a*d)/(7*b*d)