\(\int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx\) [153]

Optimal result

Integrand size = 25, antiderivative size = 64 \[ \int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx=\frac {6 \cos ^2(e+f x)^{7/6} \operatorname {Hypergeometric2F1}\left (\frac {7}{6},\frac {17}{12},\frac {29}{12},\sin ^2(e+f x)\right ) \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{7/3}}{17 d f} \] Output:

6/17*(cos(f*x+e)^2)^(7/6)*hypergeom([7/6, 17/12],[29/12],sin(f*x+e)^2)*(b* 
sin(f*x+e))^(1/2)*(d*tan(f*x+e))^(7/3)/d/f
 

Mathematica [A] (verified)

Time = 11.14 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.02 \[ \int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx=-\frac {3 d \left (-1+\operatorname {Hypergeometric2F1}\left (\frac {5}{12},\frac {5}{4},\frac {17}{12},-\tan ^2(e+f x)\right ) \sqrt [4]{\sec ^2(e+f x)}\right ) \sqrt {b \sin (e+f x)} \sqrt [3]{d \tan (e+f x)}}{f} \] Input:

Integrate[Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(4/3),x]
 

Output:

(-3*d*(-1 + Hypergeometric2F1[5/12, 5/4, 17/12, -Tan[e + f*x]^2]*(Sec[e + 
f*x]^2)^(1/4))*Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(1/3))/f
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3042, 3082, 3042, 3057}

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.

\(\displaystyle \int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3}dx\)

\(\Big \downarrow \) 3082

\(\displaystyle \frac {b \cos ^{\frac {7}{3}}(e+f x) (d \tan (e+f x))^{7/3} \int \frac {(b \sin (e+f x))^{11/6}}{\cos ^{\frac {4}{3}}(e+f x)}dx}{d (b \sin (e+f x))^{7/3}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {b \cos ^{\frac {7}{3}}(e+f x) (d \tan (e+f x))^{7/3} \int \frac {(b \sin (e+f x))^{11/6}}{\cos (e+f x)^{4/3}}dx}{d (b \sin (e+f x))^{7/3}}\)

\(\Big \downarrow \) 3057

\(\displaystyle \frac {6 \cos ^2(e+f x)^{7/6} \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{7/3} \operatorname {Hypergeometric2F1}\left (\frac {7}{6},\frac {17}{12},\frac {29}{12},\sin ^2(e+f x)\right )}{17 d f}\)

Input:

Int[Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(4/3),x]
 

Output:

(6*(Cos[e + f*x]^2)^(7/6)*Hypergeometric2F1[7/6, 17/12, 29/12, Sin[e + f*x 
]^2]*Sqrt[b*Sin[e + f*x]]*(d*Tan[e + f*x])^(7/3))/(17*d*f)
 

Defintions of rubi rules used

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3057

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_), x_Symbol] :> Simp[b^(2*IntPart[(n - 1)/2] + 1)*(b*Cos[e + f*x])^(2*Frac 
Part[(n - 1)/2])*((a*Sin[e + f*x])^(m + 1)/(a*f*(m + 1)*(Cos[e + f*x]^2)^Fr 
acPart[(n - 1)/2]))*Hypergeometric2F1[(1 + m)/2, (1 - n)/2, (3 + m)/2, Sin[ 
e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x]
 

rule 3082

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( 
n_), x_Symbol] :> Simp[a*Cos[e + f*x]^(n + 1)*((b*Tan[e + f*x])^(n + 1)/(b* 
(a*Sin[e + f*x])^(n + 1)))   Int[(a*Sin[e + f*x])^(m + n)/Cos[e + f*x]^n, x 
], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[n]
 

Maple [F]

\[\int \sqrt {b \sin \left (f x +e \right )}\, \left (d \tan \left (f x +e \right )\right )^{\frac {4}{3}}d x\]

Input:

int((b*sin(f*x+e))^(1/2)*(d*tan(f*x+e))^(4/3),x)
 

Output:

int((b*sin(f*x+e))^(1/2)*(d*tan(f*x+e))^(4/3),x)
 

Fricas [F]

\[ \int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx=\int { \sqrt {b \sin \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac {4}{3}} \,d x } \] Input:

integrate((b*sin(f*x+e))^(1/2)*(d*tan(f*x+e))^(4/3),x, algorithm="fricas")
 

Output:

integral(sqrt(b*sin(f*x + e))*(d*tan(f*x + e))^(1/3)*d*tan(f*x + e), x)
 

Sympy [F(-1)]

Timed out. \[ \int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx=\text {Timed out} \] Input:

integrate((b*sin(f*x+e))**(1/2)*(d*tan(f*x+e))**(4/3),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx=\int { \sqrt {b \sin \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac {4}{3}} \,d x } \] Input:

integrate((b*sin(f*x+e))^(1/2)*(d*tan(f*x+e))^(4/3),x, algorithm="maxima")
 

Output:

integrate(sqrt(b*sin(f*x + e))*(d*tan(f*x + e))^(4/3), x)
 

Giac [F]

\[ \int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx=\int { \sqrt {b \sin \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac {4}{3}} \,d x } \] Input:

integrate((b*sin(f*x+e))^(1/2)*(d*tan(f*x+e))^(4/3),x, algorithm="giac")
 

Output:

integrate(sqrt(b*sin(f*x + e))*(d*tan(f*x + e))^(4/3), x)
 

Mupad [F(-1)]

Timed out. \[ \int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx=\int \sqrt {b\,\sin \left (e+f\,x\right )}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{4/3} \,d x \] Input:

int((b*sin(e + f*x))^(1/2)*(d*tan(e + f*x))^(4/3),x)
 

Output:

int((b*sin(e + f*x))^(1/2)*(d*tan(e + f*x))^(4/3), x)
 

Reduce [F]

\[ \int \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, dx=\frac {d^{\frac {4}{3}} \sqrt {b}\, \left (6 \tan \left (f x +e \right )^{\frac {1}{3}} \sqrt {\sin \left (f x +e \right )}-3 \left (\int \frac {\tan \left (f x +e \right )^{\frac {1}{3}} \sqrt {\sin \left (f x +e \right )}\, \cos \left (f x +e \right )}{\sin \left (f x +e \right )}d x \right ) f -2 \left (\int \frac {\sqrt {\sin \left (f x +e \right )}}{\tan \left (f x +e \right )^{\frac {2}{3}}}d x \right ) f \right )}{2 f} \] Input:

int((b*sin(f*x+e))^(1/2)*(d*tan(f*x+e))^(4/3),x)
 

Output:

(d**(1/3)*sqrt(b)*d*(6*tan(e + f*x)**(1/3)*sqrt(sin(e + f*x)) - 3*int((tan 
(e + f*x)**(1/3)*sqrt(sin(e + f*x))*cos(e + f*x))/sin(e + f*x),x)*f - 2*in 
t((tan(e + f*x)**(1/3)*sqrt(sin(e + f*x)))/tan(e + f*x),x)*f))/(2*f)