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

Optimal. Leaf size=64 \[ \frac {6 \cos ^2(e+f x)^{7/6} \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{7/3} \, _2F_1\left (\frac {7}{6},\frac {17}{12};\frac {29}{12};\sin ^2(e+f x)\right )}{17 d f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.09, antiderivative size = 64, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2602, 2577} \[ \frac {6 \cos ^2(e+f x)^{7/6} \sqrt {b \sin (e+f x)} (d \tan (e+f x))^{7/3} \, _2F_1\left (\frac {7}{6},\frac {17}{12};\frac {29}{12};\sin ^2(e+f x)\right )}{17 d f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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)

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]

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]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.43, size = 65, normalized size = 1.02 \[ -\frac {3 d \sqrt {b \sin (e+f x)} \sqrt [3]{d \tan (e+f x)} \left (\sqrt [4]{\sec ^2(e+f x)} \, _2F_1\left (\frac {5}{12},\frac {5}{4};\frac {17}{12};-\tan ^2(e+f x)\right )-1\right )}{f} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-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

________________________________________________________________________________________

fricas [F]  time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {b \sin \left (f x + e\right )} \left (d \tan \left (f x + e\right )\right )^{\frac {1}{3}} d \tan \left (f x + e\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

maple [F]  time = 0.42, size = 0, normalized size = 0.00 \[ \int \sqrt {b \sin \left (f x +e \right )}\, \left (d \tan \left (f x +e \right )\right )^{\frac {4}{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {b\,\sin \left (e+f\,x\right )}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{4/3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________