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

Optimal. Leaf size=64 \[ \frac{3 \cos ^2(e+f x)^{7/12} \sqrt{b \sec (e+f x)} (d \tan (e+f x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{7}{12};\frac{4}{3};\sin ^2(e+f x)\right )}{2 d f} \]

[Out]

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

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Rubi [A]  time = 0.0466348, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2617} \[ \frac{3 \cos ^2(e+f x)^{7/12} \sqrt{b \sec (e+f x)} (d \tan (e+f x))^{2/3} \, _2F_1\left (\frac{1}{3},\frac{7}{12};\frac{4}{3};\sin ^2(e+f x)\right )}{2 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2617

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)*Hypergeometric2F1[(n + 1)/2, (m + n + 1)/2,
(n + 3)/2, Sin[e + f*x]^2])/(b*f*(n + 1)), x] /; FreeQ[{a, b, e, f, m, n}, x] &&  !IntegerQ[(n - 1)/2] &&  !In
tegerQ[m/2]

Rubi steps

\begin{align*} \int \frac{\sqrt{b \sec (e+f x)}}{\sqrt [3]{d \tan (e+f x)}} \, dx &=\frac{3 \cos ^2(e+f x)^{7/12} \, _2F_1\left (\frac{1}{3},\frac{7}{12};\frac{4}{3};\sin ^2(e+f x)\right ) \sqrt{b \sec (e+f x)} (d \tan (e+f x))^{2/3}}{2 d f}\\ \end{align*}

Mathematica [A]  time = 0.112473, size = 62, normalized size = 0.97 \[ \frac{2 d \left (-\tan ^2(e+f x)\right )^{2/3} \sqrt{b \sec (e+f x)} \, _2F_1\left (\frac{1}{4},\frac{2}{3};\frac{5}{4};\sec ^2(e+f x)\right )}{f (d \tan (e+f x))^{4/3}} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.221, size = 0, normalized size = 0. \begin{align*} \int{\sqrt{b\sec \left ( fx+e \right ) }{\frac{1}{\sqrt [3]{d\tan \left ( fx+e \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (f x + e\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b \sec \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac{2}{3}}}{d \tan \left (f x + e\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec{\left (e + f x \right )}}}{\sqrt [3]{d \tan{\left (e + f x \right )}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{b \sec \left (f x + e\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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