\(\int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx\) [417]

Optimal result

Integrand size = 22, antiderivative size = 139 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=-\frac {320 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}-\frac {16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e}-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}{5 e} \]

[Out]

-2/5*(3*cos(e*x+d)-4*sin(e*x+d))*(5+4*cos(e*x+d)+3*sin(e*x+d))^(3/2)/e-320/3*(3*cos(e*x+d)-4*sin(e*x+d))/e/(5+
4*cos(e*x+d)+3*sin(e*x+d))^(1/2)-16/3*(3*cos(e*x+d)-4*sin(e*x+d))*(5+4*cos(e*x+d)+3*sin(e*x+d))^(1/2)/e

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {3192, 3191} \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (3 \sin (d+e x)+4 \cos (d+e x)+5)^{3/2}}{5 e}-\frac {16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}}{3 e}-\frac {320 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {3 \sin (d+e x)+4 \cos (d+e x)+5}} \]

[In]

Int[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2),x]

[Out]

(-320*(3*Cos[d + e*x] - 4*Sin[d + e*x]))/(3*e*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]]) - (16*(3*Cos[d + e*x]
- 4*Sin[d + e*x])*Sqrt[5 + 4*Cos[d + e*x] + 3*Sin[d + e*x]])/(3*e) - (2*(3*Cos[d + e*x] - 4*Sin[d + e*x])*(5
+ 4*Cos[d + e*x] + 3*Sin[d + e*x])^(3/2))/(5*e)

Rule 3191

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Simp[-2*((c*Cos[d
+ e*x] - b*Sin[d + e*x])/(e*Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]])), x] /; FreeQ[{a, b, c, d, e}, x] && E
qQ[a^2 - b^2 - c^2, 0]

Rule 3192

Int[(cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(c*Cos[d
+ e*x] - b*Sin[d + e*x]))*((a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1)/(e*n)), x] + Dist[a*((2*n - 1)/n), In
t[(a + b*Cos[d + e*x] + c*Sin[d + e*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[a^2 - b^2 - c^2, 0
] && GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}{5 e}+8 \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2} \, dx \\ & = -\frac {16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e}-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}{5 e}+\frac {160}{3} \int \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)} \, dx \\ & = -\frac {320 (3 \cos (d+e x)-4 \sin (d+e x))}{3 e \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}-\frac {16 (3 \cos (d+e x)-4 \sin (d+e x)) \sqrt {5+4 \cos (d+e x)+3 \sin (d+e x)}}{3 e}-\frac {2 (3 \cos (d+e x)-4 \sin (d+e x)) (5+4 \cos (d+e x)+3 \sin (d+e x))^{3/2}}{5 e} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.25 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.94 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=-\frac {(5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \left (3750 \cos \left (\frac {1}{2} (d+e x)\right )+1625 \cos \left (\frac {3}{2} (d+e x)\right )+3 \left (79 \cos \left (\frac {5}{2} (d+e x)\right )-3750 \sin \left (\frac {1}{2} (d+e x)\right )-375 \sin \left (\frac {3}{2} (d+e x)\right )+3 \sin \left (\frac {5}{2} (d+e x)\right )\right )\right )}{30 e \left (3 \cos \left (\frac {1}{2} (d+e x)\right )+\sin \left (\frac {1}{2} (d+e x)\right )\right )^5} \]

[In]

Integrate[(5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2),x]

[Out]

-1/30*((5 + 4*Cos[d + e*x] + 3*Sin[d + e*x])^(5/2)*(3750*Cos[(d + e*x)/2] + 1625*Cos[(3*(d + e*x))/2] + 3*(79*
Cos[(5*(d + e*x))/2] - 3750*Sin[(d + e*x)/2] - 375*Sin[(3*(d + e*x))/2] + 3*Sin[(5*(d + e*x))/2])))/(e*(3*Cos[
(d + e*x)/2] + Sin[(d + e*x)/2])^5)

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.53

method	result	size
default	\(\frac {50 \left (1+\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )\right ) \left (\sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )-1\right ) \left (3 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )^{2}+14 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )+43\right )}{3 \cos \left (e x +d +\arctan \left (\frac {4}{3}\right )\right ) \sqrt {5+5 \sin \left (e x +d +\arctan \left (\frac {4}{3}\right )\right )}\, e}\)	\(74\)

[In]

int((5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2),x,method=_RETURNVERBOSE)

[Out]

50/3*(1+sin(e*x+d+arctan(4/3)))*(sin(e*x+d+arctan(4/3))-1)*(3*sin(e*x+d+arctan(4/3))^2+14*sin(e*x+d+arctan(4/3
))+43)/cos(e*x+d+arctan(4/3))/(5+5*sin(e*x+d+arctan(4/3)))^(1/2)/e

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.73 \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=-\frac {2 \, {\left (237 \, \cos \left (e x + d\right )^{3} + 931 \, \cos \left (e x + d\right )^{2} + 9 \, {\left (\cos \left (e x + d\right )^{2} - 62 \, \cos \left (e x + d\right ) - 344\right )} \sin \left (e x + d\right ) + 1166 \, \cos \left (e x + d\right ) + 472\right )} \sqrt {4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5}}{15 \, {\left (3 \, e \cos \left (e x + d\right ) + e \sin \left (e x + d\right ) + 3 \, e\right )}} \]

[In]

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2),x, algorithm="fricas")

[Out]

-2/15*(237*cos(e*x + d)^3 + 931*cos(e*x + d)^2 + 9*(cos(e*x + d)^2 - 62*cos(e*x + d) - 344)*sin(e*x + d) + 116
6*cos(e*x + d) + 472)*sqrt(4*cos(e*x + d) + 3*sin(e*x + d) + 5)/(3*e*cos(e*x + d) + e*sin(e*x + d) + 3*e)

Sympy [F(-1)]

Timed out. \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))**(5/2),x)

[Out]

Timed out

Maxima [F]

\[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\int { {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2),x, algorithm="maxima")

[Out]

integrate((4*cos(e*x + d) + 3*sin(e*x + d) + 5)^(5/2), x)

Giac [F]

\[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\int { {\left (4 \, \cos \left (e x + d\right ) + 3 \, \sin \left (e x + d\right ) + 5\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((5+4*cos(e*x+d)+3*sin(e*x+d))^(5/2),x, algorithm="giac")

[Out]

integrate((4*cos(e*x + d) + 3*sin(e*x + d) + 5)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int (5+4 \cos (d+e x)+3 \sin (d+e x))^{5/2} \, dx=\int {\left (4\,\cos \left (d+e\,x\right )+3\,\sin \left (d+e\,x\right )+5\right )}^{5/2} \,d x \]

[In]

int((4*cos(d + e*x) + 3*sin(d + e*x) + 5)^(5/2),x)

[Out]

int((4*cos(d + e*x) + 3*sin(d + e*x) + 5)^(5/2), x)