\(\int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx\) [337]

Optimal result

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

[Out]

2/3*(cos(f*x+e)^2)^(7/12)*hypergeom([7/12, 3/4],[7/4],sin(f*x+e)^2)*(d*tan(f*x+e))^(3/2)/d/f/(b*sec(f*x+e))^(1
/3)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2697} \[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\frac {2 \cos ^2(e+f x)^{7/12} (d \tan (e+f x))^{3/2} \operatorname {Hypergeometric2F1}\left (\frac {7}{12},\frac {3}{4},\frac {7}{4},\sin ^2(e+f x)\right )}{3 d f \sqrt [3]{b \sec (e+f x)}} \]

[In]

Int[Sqrt[d*Tan[e + f*x]]/(b*Sec[e + f*x])^(1/3),x]

[Out]

(2*(Cos[e + f*x]^2)^(7/12)*Hypergeometric2F1[7/12, 3/4, 7/4, Sin[e + f*x]^2]*(d*Tan[e + f*x])^(3/2))/(3*d*f*(b
*Sec[e + f*x])^(1/3))

Rule 2697

Int[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*Sec[e + f
*x])^m*(b*Tan[e + f*x])^(n + 1)*((Cos[e + f*x]^2)^((m + n + 1)/2)/(b*f*(n + 1)))*Hypergeometric2F1[(n + 1)/2,
(m + n + 1)/2, (n + 3)/2, Sin[e + f*x]^2], x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \cos ^2(e+f x)^{7/12} \operatorname {Hypergeometric2F1}\left (\frac {7}{12},\frac {3}{4},\frac {7}{4},\sin ^2(e+f x)\right ) (d \tan (e+f x))^{3/2}}{3 d f \sqrt [3]{b \sec (e+f x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.97 \[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=-\frac {3 d \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{4},\frac {5}{6},\sec ^2(e+f x)\right ) \sqrt [4]{-\tan ^2(e+f x)}}{f \sqrt [3]{b \sec (e+f x)} \sqrt {d \tan (e+f x)}} \]

[In]

Integrate[Sqrt[d*Tan[e + f*x]]/(b*Sec[e + f*x])^(1/3),x]

[Out]

(-3*d*Hypergeometric2F1[-1/6, 1/4, 5/6, Sec[e + f*x]^2]*(-Tan[e + f*x]^2)^(1/4))/(f*(b*Sec[e + f*x])^(1/3)*Sqr
t[d*Tan[e + f*x]])

Maple [F]

\[\int \frac {\sqrt {d \tan \left (f x +e \right )}}{\left (b \sec \left (f x +e \right )\right )^{\frac {1}{3}}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {d \tan \left (f x + e\right )}}{\left (b \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

integral((b*sec(f*x + e))^(2/3)*sqrt(d*tan(f*x + e))/(b*sec(f*x + e)), x)

Sympy [F]

\[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\int \frac {\sqrt {d \tan {\left (e + f x \right )}}}{\sqrt [3]{b \sec {\left (e + f x \right )}}}\, dx \]

[In]

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

[Out]

Integral(sqrt(d*tan(e + f*x))/(b*sec(e + f*x))**(1/3), x)

Maxima [F]

\[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {d \tan \left (f x + e\right )}}{\left (b \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\int { \frac {\sqrt {d \tan \left (f x + e\right )}}{\left (b \sec \left (f x + e\right )\right )^{\frac {1}{3}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d \tan (e+f x)}}{\sqrt [3]{b \sec (e+f x)}} \, dx=\int \frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{{\left (\frac {b}{\cos \left (e+f\,x\right )}\right )}^{1/3}} \,d x \]

[In]

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

[Out]

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