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

Optimal. Leaf size=64 \[ \frac{6 \cos ^2(e+f x)^{2/3} \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, _2F_1\left (\frac{2}{3},\frac{11}{12};\frac{23}{12};\sin ^2(e+f x)\right )}{11 d f} \]

[Out]

(6*(Cos[e + f*x]^2)^(2/3)*Hypergeometric2F1[2/3, 11/12, 23/12, Sin[e + f*x]^2]*Sqrt[b*Sin[e + f*x]]*(d*Tan[e +
f*x])^(4/3))/(11*d*f)

________________________________________________________________________________________

Rubi [A]  time = 0.0865125, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2602, 2577} \[ \frac{6 \cos ^2(e+f x)^{2/3} \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, _2F_1\left (\frac{2}{3},\frac{11}{12};\frac{23}{12};\sin ^2(e+f x)\right )}{11 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*(Cos[e + f*x]^2)^(2/3)*Hypergeometric2F1[2/3, 11/12, 23/12, Sin[e + f*x]^2]*Sqrt[b*Sin[e + f*x]]*(d*Tan[e +
f*x])^(4/3))/(11*d*f)

Rule 2602

Int[((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[(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]

Rule 2577

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*FracPart[(n - 1)/2])*(a*Sin[e + f*x])^(m + 1)*Hypergeometric2F1[(1 + m)/2
, (1 - n)/2, (3 + m)/2, Sin[e + f*x]^2])/(a*f*(m + 1)*(Cos[e + f*x]^2)^FracPart[(n - 1)/2]), x] /; FreeQ[{a, b
, e, f, m, n}, x]

Rubi steps

\begin{align*} \int \sqrt{b \sin (e+f x)} \sqrt [3]{d \tan (e+f x)} \, dx &=\frac{\left (b \cos ^{\frac{4}{3}}(e+f x) (d \tan (e+f x))^{4/3}\right ) \int \frac{(b \sin (e+f x))^{5/6}}{\sqrt [3]{\cos (e+f x)}} \, dx}{d (b \sin (e+f x))^{4/3}}\\ &=\frac{6 \cos ^2(e+f x)^{2/3} \, _2F_1\left (\frac{2}{3},\frac{11}{12};\frac{23}{12};\sin ^2(e+f x)\right ) \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{4/3}}{11 d f}\\ \end{align*}

Mathematica [A]  time = 0.335994, size = 66, normalized size = 1.03 \[ \frac{6 \sqrt [4]{\sec ^2(e+f x)} \sqrt{b \sin (e+f x)} (d \tan (e+f x))^{4/3} \, _2F_1\left (\frac{11}{12},\frac{5}{4};\frac{23}{12};-\tan ^2(e+f x)\right )}{11 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Hypergeometric2F1[11/12, 5/4, 23/12, -Tan[e + f*x]^2]*(Sec[e + f*x]^2)^(1/4)*Sqrt[b*Sin[e + f*x]]*(d*Tan[e
+ f*x])^(4/3))/(11*d*f)

________________________________________________________________________________________

Maple [F]  time = 0.211, size = 0, normalized size = 0. \begin{align*} \int \sqrt{b\sin \left ( fx+e \right ) }\sqrt [3]{d\tan \left ( fx+e \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sin \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*sin(f*x+e))^(1/2)*(d*tan(f*x+e))^(1/3),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError