Integrand size = 25, antiderivative size = 165 \[ \int \frac {1}{\tan ^{\frac {4}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx=-\frac {3 \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b \tan (c+d x)}{a},-i \tan (c+d x)\right ) \sqrt {\frac {a+b \tan (c+d x)}{a}}}{2 d \sqrt [3]{\tan (c+d x)} \sqrt {a+b \tan (c+d x)}}-\frac {3 \operatorname {AppellF1}\left (-\frac {1}{3},\frac {1}{2},1,\frac {2}{3},-\frac {b \tan (c+d x)}{a},i \tan (c+d x)\right ) \sqrt {\frac {a+b \tan (c+d x)}{a}}}{2 d \sqrt [3]{\tan (c+d x)} \sqrt {a+b \tan (c+d x)}} \] Output:
-3/2*AppellF1(-1/3,1,1/2,2/3,-I*tan(d*x+c),-b*tan(d*x+c)/a)*((a+b*tan(d*x+ c))/a)^(1/2)/d/tan(d*x+c)^(1/3)/(a+b*tan(d*x+c))^(1/2)-3/2*AppellF1(-1/3,1 ,1/2,2/3,I*tan(d*x+c),-b*tan(d*x+c)/a)*((a+b*tan(d*x+c))/a)^(1/2)/d/tan(d* x+c)^(1/3)/(a+b*tan(d*x+c))^(1/2)
Timed out. \[ \int \frac {1}{\tan ^{\frac {4}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx=\text {\$Aborted} \] Input:
Integrate[1/(Tan[c + d*x]^(4/3)*Sqrt[a + b*Tan[c + d*x]]),x]
Output:
$Aborted
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[1/(Tan[c + d*x]^(4/3)*Sqrt[a + b*Tan[c + d*x]]),x]
Output:
$Aborted
\[\int \frac {1}{\tan \left (d x +c \right )^{\frac {4}{3}} \sqrt {a +b \tan \left (d x +c \right )}}d x\]
Input:
int(1/tan(d*x+c)^(4/3)/(a+b*tan(d*x+c))^(1/2),x)
Output:
int(1/tan(d*x+c)^(4/3)/(a+b*tan(d*x+c))^(1/2),x)
Timed out. \[ \int \frac {1}{\tan ^{\frac {4}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate(1/tan(d*x+c)^(4/3)/(a+b*tan(d*x+c))^(1/2),x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {1}{\tan ^{\frac {4}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {1}{\sqrt {a + b \tan {\left (c + d x \right )}} \tan ^{\frac {4}{3}}{\left (c + d x \right )}}\, dx \] Input:
integrate(1/tan(d*x+c)**(4/3)/(a+b*tan(d*x+c))**(1/2),x)
Output:
Integral(1/(sqrt(a + b*tan(c + d*x))*tan(c + d*x)**(4/3)), x)
\[ \int \frac {1}{\tan ^{\frac {4}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx=\int { \frac {1}{\sqrt {b \tan \left (d x + c\right ) + a} \tan \left (d x + c\right )^{\frac {4}{3}}} \,d x } \] Input:
integrate(1/tan(d*x+c)^(4/3)/(a+b*tan(d*x+c))^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b*tan(d*x + c) + a)*tan(d*x + c)^(4/3)), x)
Timed out. \[ \int \frac {1}{\tan ^{\frac {4}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx=\text {Timed out} \] Input:
integrate(1/tan(d*x+c)^(4/3)/(a+b*tan(d*x+c))^(1/2),x, algorithm="giac")
Output:
Timed out
Timed out. \[ \int \frac {1}{\tan ^{\frac {4}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {1}{{\mathrm {tan}\left (c+d\,x\right )}^{4/3}\,\sqrt {a+b\,\mathrm {tan}\left (c+d\,x\right )}} \,d x \] Input:
int(1/(tan(c + d*x)^(4/3)*(a + b*tan(c + d*x))^(1/2)),x)
Output:
int(1/(tan(c + d*x)^(4/3)*(a + b*tan(c + d*x))^(1/2)), x)
\[ \int \frac {1}{\tan ^{\frac {4}{3}}(c+d x) \sqrt {a+b \tan (c+d x)}} \, dx=\int \frac {\sqrt {a +\tan \left (d x +c \right ) b}}{\tan \left (d x +c \right )^{\frac {7}{3}} b +\tan \left (d x +c \right )^{\frac {4}{3}} a}d x \] Input:
int(1/tan(d*x+c)^(4/3)/(a+b*tan(d*x+c))^(1/2),x)
Output:
int(sqrt(tan(c + d*x)*b + a)/(tan(c + d*x)**(1/3)*tan(c + d*x)**2*b + tan( c + d*x)**(1/3)*tan(c + d*x)*a),x)