\(\int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx\) [556]

Optimal result

Integrand size = 25, antiderivative size = 197 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e} \]

[Out]

-2/231*b*(43*a^2+12*b^2)*(e*cos(d*x+c))^(7/2)/d/e+2/15*a*(3*a^2+2*b^2)*e*(e*cos(d*x+c))^(3/2)*sin(d*x+c)/d-10/
33*a*b*(e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))/d/e-2/11*b*(e*cos(d*x+c))^(7/2)*(a+b*sin(d*x+c))^2/d/e+2/5*a*(3*a
^2+2*b^2)*e^2*(cos(1/2*d*x+1/2*c)^2)^(1/2)/cos(1/2*d*x+1/2*c)*EllipticE(sin(1/2*d*x+1/2*c),2^(1/2))*(e*cos(d*x
+c))^(1/2)/d/cos(d*x+c)^(1/2)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2771, 2941, 2748, 2715, 2721, 2719} \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=\frac {2 a e^2 \left (3 a^2+2 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}-\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a e \left (3 a^2+2 b^2\right ) \sin (c+d x) (e \cos (c+d x))^{3/2}}{15 d}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e} \]

[In]

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

[Out]

(-2*b*(43*a^2 + 12*b^2)*(e*Cos[c + d*x])^(7/2))/(231*d*e) + (2*a*(3*a^2 + 2*b^2)*e^2*Sqrt[e*Cos[c + d*x]]*Elli
pticE[(c + d*x)/2, 2])/(5*d*Sqrt[Cos[c + d*x]]) + (2*a*(3*a^2 + 2*b^2)*e*(e*Cos[c + d*x])^(3/2)*Sin[c + d*x])/
(15*d) - (10*a*b*(e*Cos[c + d*x])^(7/2)*(a + b*Sin[c + d*x]))/(33*d*e) - (2*b*(e*Cos[c + d*x])^(7/2)*(a + b*Si
n[c + d*x])^2)/(11*d*e)

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2719

Int[Sqrt[sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/d)*EllipticE[(1/2)*(c - Pi/2 + d*x), 2], x] /; FreeQ[{
c, d}, x]

Rule 2721

Int[((b_)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Dist[(b*Sin[c + d*x])^n/Sin[c + d*x]^n, Int[Sin[c + d*x]
^n, x], x] /; FreeQ[{b, c, d}, x] && LtQ[-1, n, 1] && IntegerQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x
] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

Rule 2771

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-b)*(
g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Dist[1/(m + p), Int[(g*Cos[e + f*x]
)^p*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1)*Sin[e + f*x]), x], x] /; FreeQ
[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ
[m])

Rule 2941

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {2}{11} \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x)) \left (\frac {11 a^2}{2}+2 b^2+\frac {15}{2} a b \sin (c+d x)\right ) \, dx \\ & = -\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {4}{99} \int (e \cos (c+d x))^{5/2} \left (\frac {33}{4} a \left (3 a^2+2 b^2\right )+\frac {3}{4} b \left (43 a^2+12 b^2\right ) \sin (c+d x)\right ) \, dx \\ & = -\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {1}{3} \left (a \left (3 a^2+2 b^2\right )\right ) \int (e \cos (c+d x))^{5/2} \, dx \\ & = -\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {1}{5} \left (a \left (3 a^2+2 b^2\right ) e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx \\ & = -\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e}+\frac {\left (a \left (3 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}} \\ & = -\frac {2 b \left (43 a^2+12 b^2\right ) (e \cos (c+d x))^{7/2}}{231 d e}+\frac {2 a \left (3 a^2+2 b^2\right ) e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {2 a \left (3 a^2+2 b^2\right ) e (e \cos (c+d x))^{3/2} \sin (c+d x)}{15 d}-\frac {10 a b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))}{33 d e}-\frac {2 b (e \cos (c+d x))^{7/2} (a+b \sin (c+d x))^2}{11 d e} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.40 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.76 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=\frac {(e \cos (c+d x))^{5/2} \left (1848 \left (3 a^3+2 a b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^{\frac {3}{2}}(c+d x) \left (-1980 a^2 b-345 b^3-60 \left (33 a^2 b+4 b^3\right ) \cos (2 (c+d x))+105 b^3 \cos (4 (c+d x))+1848 a^3 \sin (c+d x)+462 a b^2 \sin (c+d x)-770 a b^2 \sin (3 (c+d x))\right )\right )}{4620 d \cos ^{\frac {5}{2}}(c+d x)} \]

[In]

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

[Out]

((e*Cos[c + d*x])^(5/2)*(1848*(3*a^3 + 2*a*b^2)*EllipticE[(c + d*x)/2, 2] + Cos[c + d*x]^(3/2)*(-1980*a^2*b -
345*b^3 - 60*(33*a^2*b + 4*b^3)*Cos[2*(c + d*x)] + 105*b^3*Cos[4*(c + d*x)] + 1848*a^3*Sin[c + d*x] + 462*a*b^
2*Sin[c + d*x] - 770*a*b^2*Sin[3*(c + d*x)])))/(4620*d*Cos[c + d*x]^(5/2))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(503\) vs. \(2(201)=402\).

Time = 38.95 (sec) , antiderivative size = 504, normalized size of antiderivative = 2.56

method	result	size
parts	\(-\frac {2 a^{3} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{3} \left (-8 \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+8 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{5 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}+\frac {2 b^{3} \left (\frac {\left (e \cos \left (d x +c \right )\right )^{\frac {11}{2}}}{11}-\frac {e^{2} \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7}\right )}{d \,e^{3}}-\frac {6 a^{2} b \left (e \cos \left (d x +c \right )\right )^{\frac {7}{2}}}{7 e d}+\frac {4 a \,b^{2} \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, e^{3} \left (80 \left (\cos ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-240 \left (\cos ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+272 \left (\cos ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-144 \left (\cos ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+35 \left (\cos ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-3 \cos \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{15 \sqrt {-e \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {e \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}\)	\(504\)
default	\(-\frac {2 e^{3} \left (-6720 \left (\sin ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+12320 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+20160 \left (\sin ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-24640 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+7920 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -22560 \left (\sin ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-1848 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+17248 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-15840 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b +11520 \left (\sin ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+1848 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}-4928 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}+11880 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -2340 \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}-462 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}+462 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}-693 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a^{3}-462 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, E\left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a \,b^{2}-3960 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b -60 \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}+495 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2} b +60 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) b^{3}\right )}{1155 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d}\)	\(534\)

[In]

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

[Out]

-2/5*a^3*(e*(2*cos(1/2*d*x+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e^3*(-8*sin(1/2*d*x+1/2*c)^6*cos(1/2*d*x+1/
2*c)+8*sin(1/2*d*x+1/2*c)^4*cos(1/2*d*x+1/2*c)-2*sin(1/2*d*x+1/2*c)^2*cos(1/2*d*x+1/2*c)-3*(sin(1/2*d*x+1/2*c)
^2)^(1/2)*(2*sin(1/2*d*x+1/2*c)^2-1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2)))/(-e*(2*sin(1/2*d*x+1/2*c)^4-
sin(1/2*d*x+1/2*c)^2))^(1/2)/sin(1/2*d*x+1/2*c)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d+2*b^3/d/e^3*(1/11*(e*co
s(d*x+c))^(11/2)-1/7*e^2*(e*cos(d*x+c))^(7/2))-6/7*a^2*b*(e*cos(d*x+c))^(7/2)/e/d+4/15*a*b^2*(e*(2*cos(1/2*d*x
+1/2*c)^2-1)*sin(1/2*d*x+1/2*c)^2)^(1/2)*e^3*(80*cos(1/2*d*x+1/2*c)^11-240*cos(1/2*d*x+1/2*c)^9+272*cos(1/2*d*
x+1/2*c)^7-144*cos(1/2*d*x+1/2*c)^5+35*cos(1/2*d*x+1/2*c)^3+3*(sin(1/2*d*x+1/2*c)^2)^(1/2)*(-2*cos(1/2*d*x+1/2
*c)^2+1)^(1/2)*EllipticE(cos(1/2*d*x+1/2*c),2^(1/2))-3*cos(1/2*d*x+1/2*c))/(-e*(2*sin(1/2*d*x+1/2*c)^4-sin(1/2
*d*x+1/2*c)^2))^(1/2)/sin(1/2*d*x+1/2*c)/(e*(2*cos(1/2*d*x+1/2*c)^2-1))^(1/2)/d

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.96 \[ \int (e \cos (c+d x))^{5/2} (a+b \sin (c+d x))^3 \, dx=\frac {231 i \, \sqrt {2} {\left (3 \, a^{3} + 2 \, a b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 231 i \, \sqrt {2} {\left (3 \, a^{3} + 2 \, a b^{2}\right )} e^{\frac {5}{2}} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (105 \, b^{3} e^{2} \cos \left (d x + c\right )^{5} - 165 \, {\left (3 \, a^{2} b + b^{3}\right )} e^{2} \cos \left (d x + c\right )^{3} - 77 \, {\left (5 \, a b^{2} e^{2} \cos \left (d x + c\right )^{3} - {\left (3 \, a^{3} + 2 \, a b^{2}\right )} e^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )\right )} \sqrt {e \cos \left (d x + c\right )}}{1155 \, d} \]

[In]

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

[Out]

1/1155*(231*I*sqrt(2)*(3*a^3 + 2*a*b^2)*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0, cos(d*x + c)
+ I*sin(d*x + c))) - 231*I*sqrt(2)*(3*a^3 + 2*a*b^2)*e^(5/2)*weierstrassZeta(-4, 0, weierstrassPInverse(-4, 0
, cos(d*x + c) - I*sin(d*x + c))) + 2*(105*b^3*e^2*cos(d*x + c)^5 - 165*(3*a^2*b + b^3)*e^2*cos(d*x + c)^3 - 7
7*(5*a*b^2*e^2*cos(d*x + c)^3 - (3*a^3 + 2*a*b^2)*e^2*cos(d*x + c))*sin(d*x + c))*sqrt(e*cos(d*x + c)))/d

Sympy [F(-1)]

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

[In]

integrate((e*cos(d*x+c))**(5/2)*(a+b*sin(d*x+c))**3,x)

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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