3.4.43 \(\int \sqrt {b \sec (e+f x)} (d \tan (e+f x))^{4/3} \, dx\) [343]

Optimal. Leaf size=64 \[ \frac {3 \cos ^2(e+f x)^{17/12} \, _2F_1\left (\frac {7}{6},\frac {17}{12};\frac {13}{6};\sin ^2(e+f x)\right ) \sqrt {b \sec (e+f x)} (d \tan (e+f x))^{7/3}}{7 d f} \]

[Out]

3/7*(cos(f*x+e)^2)^(17/12)*hypergeom([7/6, 17/12],[13/6],sin(f*x+e)^2)*(b*sec(f*x+e))^(1/2)*(d*tan(f*x+e))^(7/
3)/d/f

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2697} \begin {gather*} \frac {3 \cos ^2(e+f x)^{17/12} \sqrt {b \sec (e+f x)} (d \tan (e+f x))^{7/3} \, _2F_1\left (\frac {7}{6},\frac {17}{12};\frac {13}{6};\sin ^2(e+f x)\right )}{7 d f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*(Cos[e + f*x]^2)^(17/12)*Hypergeometric2F1[7/6, 17/12, 13/6, Sin[e + f*x]^2]*Sqrt[b*Sec[e + f*x]]*(d*Tan[e
+ f*x])^(7/3))/(7*d*f)

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*} \int \sqrt {b \sec (e+f x)} (d \tan (e+f x))^{4/3} \, dx &=\frac {3 \cos ^2(e+f x)^{17/12} \, _2F_1\left (\frac {7}{6},\frac {17}{12};\frac {13}{6};\sin ^2(e+f x)\right ) \sqrt {b \sec (e+f x)} (d \tan (e+f x))^{7/3}}{7 d f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.16, size = 62, normalized size = 0.97 \begin {gather*} \frac {2 d \, _2F_1\left (-\frac {1}{6},\frac {1}{4};\frac {5}{4};\sec ^2(e+f x)\right ) \sqrt {b \sec (e+f x)} \sqrt [3]{d \tan (e+f x)}}{f \sqrt [6]{-\tan ^2(e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.26, size = 0, normalized size = 0.00 \[\int \sqrt {b \sec \left (f x +e \right )}\, \left (d \tan \left (f x +e \right )\right )^{\frac {4}{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: SystemError >> excessive stack use: stack is 4370 deep

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{4/3}\,\sqrt {\frac {b}{\cos \left (e+f\,x\right )}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________