\(\int (3+3 \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx\) [581]

Optimal result

Integrand size = 29, antiderivative size = 230 \[ \int (3+3 \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx=-\frac {9 \sqrt {3} (c+d) \left (c^2-6 c d+25 d^2\right ) \arctan \left (\frac {\sqrt {3} \sqrt {d} \cos (e+f x)}{\sqrt {3+3 \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{5/2} f}-\frac {27 \left (c^2-6 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d^2 f \sqrt {3+3 \sin (e+f x)}}+\frac {9 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{4 d^2 f \sqrt {3+3 \sin (e+f x)}}-\frac {3 \cos (e+f x) \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{d f} \]

[Out]

-1/8*a^(5/2)*(c+d)*(c^2-6*c*d+25*d^2)*arctan(cos(f*x+e)*a^(1/2)*d^(1/2)/(a+a*sin(f*x+e))^(1/2)/(c+d*sin(f*x+e)
)^(1/2))/d^(5/2)/f+1/12*a^3*(3*c-13*d)*cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)/d^2/f/(a+a*sin(f*x+e))^(1/2)-1/3*a^2*
cos(f*x+e)*(c+d*sin(f*x+e))^(3/2)*(a+a*sin(f*x+e))^(1/2)/d/f-1/8*a^3*(c^2-6*c*d+25*d^2)*cos(f*x+e)*(c+d*sin(f*
x+e))^(1/2)/d^2/f/(a+a*sin(f*x+e))^(1/2)

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2842, 3060, 2849, 2854, 211} \[ \int (3+3 \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx=-\frac {a^{5/2} (c+d) \left (c^2-6 c d+25 d^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{5/2} f}-\frac {a^3 \left (c^2-6 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d^2 f \sqrt {a \sin (e+f x)+a}}+\frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a \sin (e+f x)+a}}-\frac {a^2 \cos (e+f x) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^{3/2}}{3 d f} \]

[In]

Int[(a + a*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]],x]

[Out]

-1/8*(a^(5/2)*(c + d)*(c^2 - 6*c*d + 25*d^2)*ArcTan[(Sqrt[a]*Sqrt[d]*Cos[e + f*x])/(Sqrt[a + a*Sin[e + f*x]]*S
qrt[c + d*Sin[e + f*x]])])/(d^(5/2)*f) - (a^3*(c^2 - 6*c*d + 25*d^2)*Cos[e + f*x]*Sqrt[c + d*Sin[e + f*x]])/(8
*d^2*f*Sqrt[a + a*Sin[e + f*x]]) + (a^3*(3*c - 13*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^(3/2))/(12*d^2*f*Sqrt[a
+ a*Sin[e + f*x]]) - (a^2*Cos[e + f*x]*Sqrt[a + a*Sin[e + f*x]]*(c + d*Sin[e + f*x])^(3/2))/(3*d*f)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 2842

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Sim
p[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 2)*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(m + n))), x] + Dist[1/(
d*(m + n)), Int[(a + b*Sin[e + f*x])^(m - 2)*(c + d*Sin[e + f*x])^n*Simp[a*b*c*(m - 2) + b^2*d*(n + 1) + a^2*d
*(m + n) - b*(b*c*(m - 1) - a*d*(3*m + 2*n - 2))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] &
& NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && GtQ[m, 1] &&  !LtQ[n, -1] && (IntegersQ[2*m,
2*n] || IntegerQ[m + 1/2] || (IntegerQ[m] && EqQ[c, 0]))

Rule 2849

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

Rule 2854

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/Sqrt[(c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[
-2*(b/f), Subst[Int[1/(b + d*x^2), x], x, b*(Cos[e + f*x]/(Sqrt[a + b*Sin[e + f*x]]*Sqrt[c + d*Sin[e + f*x]]))
], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3060

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[-2*b*B*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(d*f*(2*n + 3)*Sqrt
[a + b*Sin[e + f*x]])), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\int \sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a^2 (c+9 d)-\frac {1}{2} a^2 (3 c-13 d) \sin (e+f x)\right ) \sqrt {c+d \sin (e+f x)} \, dx}{3 d} \\ & = \frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\left (a^2 \left (c^2-6 c d+25 d^2\right )\right ) \int \sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)} \, dx}{8 d^2} \\ & = -\frac {a^3 \left (c^2-6 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}+\frac {\left (a^2 (c+d) \left (c^2-6 c d+25 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{16 d^2} \\ & = -\frac {a^3 \left (c^2-6 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f}-\frac {\left (a^3 (c+d) \left (c^2-6 c d+25 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^2 f} \\ & = -\frac {a^{5/2} (c+d) \left (c^2-6 c d+25 d^2\right ) \arctan \left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{8 d^{5/2} f}-\frac {a^3 \left (c^2-6 c d+25 d^2\right ) \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{8 d^2 f \sqrt {a+a \sin (e+f x)}}+\frac {a^3 (3 c-13 d) \cos (e+f x) (c+d \sin (e+f x))^{3/2}}{12 d^2 f \sqrt {a+a \sin (e+f x)}}-\frac {a^2 \cos (e+f x) \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^{3/2}}{3 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.38 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.25 \[ \int (3+3 \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx=\frac {9 \sqrt {3} (1+\sin (e+f x))^{5/2} \left (\frac {(c+d) \left (c^2-6 c d+25 d^2\right ) \left (2 \arctan \left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )+\text {arctanh}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )}{\sqrt {c+d \sin (e+f x)}}\right )-\log \left (\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e-\pi +2 f x)\right )+\sqrt {c+d \sin (e+f x)}\right )\right )}{d^{5/2}}+\frac {2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sqrt {c+d \sin (e+f x)} \left (3 c^2-16 c d-79 d^2+4 d^2 \cos (2 (e+f x))-2 d (c+17 d) \sin (e+f x)\right )}{3 d^2}\right )}{16 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]

[In]

Integrate[(3 + 3*Sin[e + f*x])^(5/2)*Sqrt[c + d*Sin[e + f*x]],x]

[Out]

(9*Sqrt[3]*(1 + Sin[e + f*x])^(5/2)*(((c + d)*(c^2 - 6*c*d + 25*d^2)*(2*ArcTan[(Sqrt[2]*Sqrt[d]*Sin[(2*e - Pi
+ 2*f*x)/4])/Sqrt[c + d*Sin[e + f*x]]] + ArcTanh[(Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4])/Sqrt[c + d*Sin[e
+ f*x]]] - Log[Sqrt[2]*Sqrt[d]*Cos[(2*e - Pi + 2*f*x)/4] + Sqrt[c + d*Sin[e + f*x]]]))/d^(5/2) + (2*(Cos[(e +
f*x)/2] - Sin[(e + f*x)/2])*Sqrt[c + d*Sin[e + f*x]]*(3*c^2 - 16*c*d - 79*d^2 + 4*d^2*Cos[2*(e + f*x)] - 2*d*(
c + 17*d)*Sin[e + f*x]))/(3*d^2)))/(16*f*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^5)

Maple [F(-1)]

Timed out.

\[\int \left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}} \sqrt {c +d \sin \left (f x +e \right )}d x\]

[In]

int((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(1/2),x)

[Out]

int((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(1/2),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 498 vs. \(2 (209) = 418\).

Time = 0.76 (sec) , antiderivative size = 1455, normalized size of antiderivative = 6.33 \[ \int (3+3 \sin (e+f x))^{5/2} \sqrt {c+d \sin (e+f x)} \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

[1/192*(3*(a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3 + (a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*
d^3)*cos(f*x + e) + (a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3)*sin(f*x + e))*sqrt(-a/d)*log((128*a*d^
4*cos(f*x + e)^5 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 + 128*(2*a*c*d^3 - a*d^4)*cos(f*x + e)^
4 - 32*(5*a*c^2*d^2 - 14*a*c*d^3 + 13*a*d^4)*cos(f*x + e)^3 - 32*(a*c^3*d - 2*a*c^2*d^2 + 9*a*c*d^3 - 4*a*d^4)
*cos(f*x + e)^2 - 8*(16*d^4*cos(f*x + e)^4 - c^3*d + 17*c^2*d^2 - 59*c*d^3 + 51*d^4 + 24*(c*d^3 - d^4)*cos(f*x
+ e)^3 - 2*(5*c^2*d^2 - 26*c*d^3 + 33*d^4)*cos(f*x + e)^2 - (c^3*d - 7*c^2*d^2 + 31*c*d^3 - 25*d^4)*cos(f*x +
e) + (16*d^4*cos(f*x + e)^3 + c^3*d - 17*c^2*d^2 + 59*c*d^3 - 51*d^4 - 8*(3*c*d^3 - 5*d^4)*cos(f*x + e)^2 - 2
*(5*c^2*d^2 - 14*c*d^3 + 13*d^4)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c)
*sqrt(-a/d) + (a*c^4 - 28*a*c^3*d + 230*a*c^2*d^2 - 476*a*c*d^3 + 289*a*d^4)*cos(f*x + e) + (128*a*d^4*cos(f*x
+ e)^4 + a*c^4 + 4*a*c^3*d + 6*a*c^2*d^2 + 4*a*c*d^3 + a*d^4 - 256*(a*c*d^3 - a*d^4)*cos(f*x + e)^3 - 32*(5*a
*c^2*d^2 - 6*a*c*d^3 + 5*a*d^4)*cos(f*x + e)^2 + 32*(a*c^3*d - 7*a*c^2*d^2 + 15*a*c*d^3 - 9*a*d^4)*cos(f*x + e
))*sin(f*x + e))/(cos(f*x + e) + sin(f*x + e) + 1)) + 8*(8*a^2*d^2*cos(f*x + e)^3 + 3*a^2*c^2 - 14*a^2*c*d - 4
9*a^2*d^2 - 2*(a^2*c*d + 13*a^2*d^2)*cos(f*x + e)^2 + (3*a^2*c^2 - 16*a^2*c*d - 83*a^2*d^2)*cos(f*x + e) - (8*
a^2*d^2*cos(f*x + e)^2 + 3*a^2*c^2 - 14*a^2*c*d - 49*a^2*d^2 + 2*(a^2*c*d + 17*a^2*d^2)*cos(f*x + e))*sin(f*x
+ e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f), 1/
96*(3*(a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3 + (a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3)
*cos(f*x + e) + (a^2*c^3 - 5*a^2*c^2*d + 19*a^2*c*d^2 + 25*a^2*d^3)*sin(f*x + e))*sqrt(a/d)*arctan(1/4*(8*d^2*
cos(f*x + e)^2 - c^2 + 6*c*d - 9*d^2 - 8*(c*d - d^2)*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*sqrt(d*sin(f*x + e
) + c)*sqrt(a/d)/(2*a*d^2*cos(f*x + e)^3 - (3*a*c*d - a*d^2)*cos(f*x + e)*sin(f*x + e) - (a*c^2 - a*c*d + 2*a*
d^2)*cos(f*x + e))) + 4*(8*a^2*d^2*cos(f*x + e)^3 + 3*a^2*c^2 - 14*a^2*c*d - 49*a^2*d^2 - 2*(a^2*c*d + 13*a^2*
d^2)*cos(f*x + e)^2 + (3*a^2*c^2 - 16*a^2*c*d - 83*a^2*d^2)*cos(f*x + e) - (8*a^2*d^2*cos(f*x + e)^2 + 3*a^2*c
^2 - 14*a^2*c*d - 49*a^2*d^2 + 2*(a^2*c*d + 17*a^2*d^2)*cos(f*x + e))*sin(f*x + e))*sqrt(a*sin(f*x + e) + a)*s
qrt(d*sin(f*x + e) + c))/(d^2*f*cos(f*x + e) + d^2*f*sin(f*x + e) + d^2*f)]

Sympy [F(-1)]

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

[In]

integrate((a+a*sin(f*x+e))**(5/2)*(c+d*sin(f*x+e))**(1/2),x)

[Out]

Timed out

Maxima [F]

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

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate((a*sin(f*x + e) + a)^(5/2)*sqrt(d*sin(f*x + e) + c), x)

Giac [F]

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

[In]

integrate((a+a*sin(f*x+e))^(5/2)*(c+d*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

integrate((a*sin(f*x + e) + a)^(5/2)*sqrt(d*sin(f*x + e) + c), x)

Mupad [F(-1)]

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

[In]

int((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(1/2),x)

[Out]

int((a + a*sin(e + f*x))^(5/2)*(c + d*sin(e + f*x))^(1/2), x)