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

Optimal result

Integrand size = 27, antiderivative size = 390 \[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\frac {2 \left (161 a^2 d+63 \left (b^2+c^2\right ) d+15 a^3 e+145 a \left (b^2+c^2\right ) e\right ) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{105 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \left (a^2-b^2-c^2\right ) \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{105 \sqrt {a+b \cos (x)+c \sin (x)}}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))-\frac {2}{35} (a+b \cos (x)+c \sin (x))^{3/2} (c (7 d+5 a e) \cos (x)-b (7 d+5 a e) \sin (x))-\frac {2}{105} \sqrt {a+b \cos (x)+c \sin (x)} \left (c \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \sin (x)\right ) \]

[Out]

-2/7*(a+b*cos(x)+c*sin(x))^(5/2)*(c*e*cos(x)-b*e*sin(x))-2/35*(a+b*cos(x)+c*sin(x))^(3/2)*(c*(5*a*e+7*d)*cos(x
)-b*(5*a*e+7*d)*sin(x))-2/105*(c*(56*a*d+15*a^2*e+25*(b^2+c^2)*e)*cos(x)-b*(56*a*d+15*a^2*e+25*(b^2+c^2)*e)*si
n(x))*(a+b*cos(x)+c*sin(x))^(1/2)+2/105*(161*a^2*d+63*(b^2+c^2)*d+15*a^3*e+145*a*(b^2+c^2)*e)*(cos(1/2*x-1/2*a
rctan(b,c))^2)^(1/2)/cos(1/2*x-1/2*arctan(b,c))*EllipticE(sin(1/2*x-1/2*arctan(b,c)),2^(1/2)*((b^2+c^2)^(1/2)/
(a+(b^2+c^2)^(1/2)))^(1/2))*(a+b*cos(x)+c*sin(x))^(1/2)/((a+b*cos(x)+c*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)-2/10
5*(a^2-b^2-c^2)*(56*a*d+15*a^2*e+25*(b^2+c^2)*e)*(cos(1/2*x-1/2*arctan(b,c))^2)^(1/2)/cos(1/2*x-1/2*arctan(b,c
))*EllipticF(sin(1/2*x-1/2*arctan(b,c)),2^(1/2)*((b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2))*((a+b*cos(x)+c*si
n(x))/(a+(b^2+c^2)^(1/2)))^(1/2)/(a+b*cos(x)+c*sin(x))^(1/2)

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 390, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {3225, 3228, 3198, 2732, 3206, 2740} \[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=-\frac {2 \left (a^2-b^2-c^2\right ) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{105 \sqrt {a+b \cos (x)+c \sin (x)}}+\frac {2 \left (15 a^3 e+161 a^2 d+145 a e \left (b^2+c^2\right )+63 d \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{105 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2}{105} \sqrt {a+b \cos (x)+c \sin (x)} \left (c \cos (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )-b \sin (x) \left (15 a^2 e+56 a d+25 e \left (b^2+c^2\right )\right )\right )-\frac {2}{35} (a+b \cos (x)+c \sin (x))^{3/2} (c \cos (x) (5 a e+7 d)-b \sin (x) (5 a e+7 d))-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x)) \]

[In]

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

[Out]

(2*(161*a^2*d + 63*(b^2 + c^2)*d + 15*a^3*e + 145*a*(b^2 + c^2)*e)*EllipticE[(x - ArcTan[b, c])/2, (2*Sqrt[b^2
+ c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[x] + c*Sin[x]])/(105*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b
^2 + c^2])]) - (2*(a^2 - b^2 - c^2)*(56*a*d + 15*a^2*e + 25*(b^2 + c^2)*e)*EllipticF[(x - ArcTan[b, c])/2, (2*
Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/(105*Sqrt[a + b
*Cos[x] + c*Sin[x]]) - (2*(a + b*Cos[x] + c*Sin[x])^(5/2)*(c*e*Cos[x] - b*e*Sin[x]))/7 - (2*(a + b*Cos[x] + c*
Sin[x])^(3/2)*(c*(7*d + 5*a*e)*Cos[x] - b*(7*d + 5*a*e)*Sin[x]))/35 - (2*Sqrt[a + b*Cos[x] + c*Sin[x]]*(c*(56*
a*d + 15*a^2*e + 25*(b^2 + c^2)*e)*Cos[x] - b*(56*a*d + 15*a^2*e + 25*(b^2 + c^2)*e)*Sin[x]))/105

Rule 2732

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[2*(Sqrt[a + b]/d)*EllipticE[(1/2)*(c - Pi/2
+ d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 2740

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*Sqrt[a + b]))*EllipticF[(1/2)*(c - P
i/2 + d*x), 2*(b/(a + b))], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[a + b, 0]

Rule 3198

Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[a + b*C
os[d + e*x] + c*Sin[d + e*x]]/Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])], Int[Sqrt[a/(a
+ Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; FreeQ[{a
, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rule 3206

Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[Sqrt[(a +
b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sqrt[b^2 + c^2])]/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], Int[1/Sqrt[
a/(a + Sqrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - ArcTan[b, c]]], x], x] /; Free
Q[{a, b, c, d, e}, x] && NeQ[a^2 - b^2 - c^2, 0] && NeQ[b^2 + c^2, 0] &&  !GtQ[a + Sqrt[b^2 + c^2], 0]

Rule 3225

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

Rule 3228

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.)
+ (a_) + (c_.)*sin[(d_.) + (e_.)*(x_)]], x_Symbol] :> Dist[B/b, Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]]
, x], x] + Dist[(A*b - a*B)/b, Int[1/Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x], x] /; FreeQ[{a, b, c, d, e
, A, B, C}, x] && EqQ[B*c - b*C, 0] && NeQ[A*b - a*B, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))+\frac {2 \int (a+b \cos (x)+c \sin (x))^{3/2} \left (\frac {1}{2} a \left (7 a d+5 \left (b^2+c^2\right ) e\right )+\frac {1}{2} a b (7 d+5 a e) \cos (x)+\frac {1}{2} a c (7 d+5 a e) \sin (x)\right ) \, dx}{7 a} \\ & = -\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))-\frac {2}{35} (a+b \cos (x)+c \sin (x))^{3/2} (c (7 d+5 a e) \cos (x)-b (7 d+5 a e) \sin (x))+\frac {4 \int \sqrt {a+b \cos (x)+c \sin (x)} \left (\frac {1}{4} a^2 \left (35 a^2 d+21 \left (b^2+c^2\right ) d+40 a \left (b^2+c^2\right ) e\right )+\frac {1}{4} a^2 b \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \cos (x)+\frac {1}{4} a^2 c \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \sin (x)\right ) \, dx}{35 a^2} \\ & = -\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))-\frac {2}{35} (a+b \cos (x)+c \sin (x))^{3/2} (c (7 d+5 a e) \cos (x)-b (7 d+5 a e) \sin (x))-\frac {2}{105} \sqrt {a+b \cos (x)+c \sin (x)} \left (c \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \sin (x)\right )+\frac {8 \int \frac {\frac {1}{8} a^3 \left (105 a^3 d+119 a \left (b^2+c^2\right ) d+135 a^2 \left (b^2+c^2\right ) e+25 \left (b^2+c^2\right )^2 e\right )+\frac {1}{8} a^3 b \left (161 a^2 d+63 \left (b^2+c^2\right ) d+15 a^3 e+145 a \left (b^2+c^2\right ) e\right ) \cos (x)+\frac {1}{8} a^3 c \left (161 a^2 d+63 \left (b^2+c^2\right ) d+15 a^3 e+145 a \left (b^2+c^2\right ) e\right ) \sin (x)}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx}{105 a^3} \\ & = -\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))-\frac {2}{35} (a+b \cos (x)+c \sin (x))^{3/2} (c (7 d+5 a e) \cos (x)-b (7 d+5 a e) \sin (x))-\frac {2}{105} \sqrt {a+b \cos (x)+c \sin (x)} \left (c \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \sin (x)\right )-\frac {1}{105} \left (\left (a^2-b^2-c^2\right ) \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right )\right ) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}} \, dx+\frac {1}{105} \left (161 a^2 d+63 \left (b^2+c^2\right ) d+15 a^3 e+145 a \left (b^2+c^2\right ) e\right ) \int \sqrt {a+b \cos (x)+c \sin (x)} \, dx \\ & = -\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))-\frac {2}{35} (a+b \cos (x)+c \sin (x))^{3/2} (c (7 d+5 a e) \cos (x)-b (7 d+5 a e) \sin (x))-\frac {2}{105} \sqrt {a+b \cos (x)+c \sin (x)} \left (c \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \sin (x)\right )+\frac {\left (\left (161 a^2 d+63 \left (b^2+c^2\right ) d+15 a^3 e+145 a \left (b^2+c^2\right ) e\right ) \sqrt {a+b \cos (x)+c \sin (x)}\right ) \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}} \, dx}{105 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {\left (\left (a^2-b^2-c^2\right ) \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}\right ) \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}} \, dx}{105 \sqrt {a+b \cos (x)+c \sin (x)}} \\ & = \frac {2 \left (161 a^2 d+63 \left (b^2+c^2\right ) d+15 a^3 e+145 a \left (b^2+c^2\right ) e\right ) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{105 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \left (a^2-b^2-c^2\right ) \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{105 \sqrt {a+b \cos (x)+c \sin (x)}}-\frac {2}{7} (a+b \cos (x)+c \sin (x))^{5/2} (c e \cos (x)-b e \sin (x))-\frac {2}{35} (a+b \cos (x)+c \sin (x))^{3/2} (c (7 d+5 a e) \cos (x)-b (7 d+5 a e) \sin (x))-\frac {2}{105} \sqrt {a+b \cos (x)+c \sin (x)} \left (c \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \cos (x)-b \left (56 a d+15 a^2 e+25 \left (b^2+c^2\right ) e\right ) \sin (x)\right ) \\ \end{align*}

Mathematica [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.

Time = 6.78 (sec) , antiderivative size = 7823, normalized size of antiderivative = 20.06 \[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Result too large to show} \]

[In]

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

[Out]

Result too large to show

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3451\) vs. \(2(414)=828\).

Time = 12.96 (sec) , antiderivative size = 3452, normalized size of antiderivative = 8.85

method	result	size
default	\(\text {Expression too large to display}\)	\(3452\)
parts	\(\text {Expression too large to display}\)	\(38062\)

[In]

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

[Out]

(-(-b^2*sin(x-arctan(-b,c))-c^2*sin(x-arctan(-b,c))-a*(b^2+c^2)^(1/2))*cos(x-arctan(-b,c))^2/(b^2+c^2)^(1/2))^
(1/2)/(b^2+c^2)*(2*a^3*b^2*d*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/
2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2
+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(-(-b^2*sin(x-arctan(-b,c))-c^2*sin(x-arctan(-b,c))-a*(b^2+c^2)^(1/2))*
cos(x-arctan(-b,c))^2/(b^2+c^2)^(1/2))^(1/2)*EllipticF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(
1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))+2*a^3*c^2*d*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2
)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^
2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(-(-b^2*sin(x-arctan(-b,
c))-c^2*sin(x-arctan(-b,c))-a*(b^2+c^2)^(1/2))*cos(x-arctan(-b,c))^2/(b^2+c^2)^(1/2))^(1/2)*EllipticF((((b^2+c
^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))
+(b^6*e+3*b^4*c^2*e+3*b^2*c^4*e+c^6*e)*(-2/7/(b^2+c^2)^(1/2)*sin(x-arctan(-b,c))^2*(((b^2+c^2)^(1/2)*sin(x-arc
tan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)+12/35/(b^2+c^2)*a*sin(x-arctan(-b,c))*(((b^2+c^2)^(1/2)*sin(x-arcta
n(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)-2/3*(5/7+24/35/(b^2+c^2)*a^2)/(b^2+c^2)^(1/2)*(((b^2+c^2)^(1/2)*sin(x
-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)+2*(-4/35/(b^2+c^2)*a^2+5/21)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2
)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^
2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(x-
arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)*EllipticF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^
(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))+2*(-48*a^3-44*a*b^2-44*a*c^2)/(105*(b^2+c^2)^
(1/2)*b^2+105*(b^2+c^2)^(1/2)*c^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^
2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1
)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(
1/2)*((-1/(b^2+c^2)^(1/2)*a+1)*EllipticE((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),(
(-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-EllipticF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c
^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))))+(3*a^2*b^4*e+6*a^2*b^2*c^2*e+3*a^2*c^4*
e+3*a*b^4*d+6*a*b^2*c^2*d+3*a*c^4*d)*(-2/3/(b^2+c^2)^(1/2)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arct
an(-b,c))^2)^(1/2)+2/3*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(
1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^
(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)*Ellipti
cF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/
2)))^(1/2))-4/3/(b^2+c^2)^(1/2)*a*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2
)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)
*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1
/2)*((-1/(b^2+c^2)^(1/2)*a+1)*EllipticE((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((
-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-EllipticF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^
2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))))+2*((b^2+c^2)^(1/2)*a^3*b^2*e+(b^2+c^2)^(
1/2)*a^3*c^2*e+(b^2+c^2)^(3/2)*a^2*d+2*a^2*b^2*d*(b^2+c^2)^(1/2)+2*a^2*c^2*d*(b^2+c^2)^(1/2))*(1/(b^2+c^2)^(1/
2)*a+1)*(((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)
^(1/2)/(-a+(b^2+c^2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2
+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)*((-1/(b^2+c^2)^(1/2)*a+1)*EllipticE((((b^2+c^2
)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-E
llipticF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^
2)^(1/2)))^(1/2)))+(3*(b^2+c^2)^(1/2)*a*b^4*e+6*(b^2+c^2)^(1/2)*a*b^2*c^2*e+3*(b^2+c^2)^(1/2)*a*c^4*e+(b^2+c^2
)^(1/2)*b^4*d+2*(b^2+c^2)^(1/2)*b^2*c^2*d+(b^2+c^2)^(1/2)*c^4*d)*(-2/5/(b^2+c^2)^(1/2)*sin(x-arctan(-b,c))*(((
b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)+8/15/(b^2+c^2)*a*(((b^2+c^2)^(1/2)*sin(x-ar
ctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)+4/15/(b^2+c^2)^(1/2)*a*(1/(b^2+c^2)^(1/2)*a+1)*(((b^2+c^2)^(1/2)*s
in(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+(b^2+c^2)^(1/2))
)^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)*sin(x-arctan(-b
,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)*EllipticF((((b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(
1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))+2*(3/5+8/15/(b^2+c^2)*a^2)*(1/(b^2+c^2)^(1/2)*a+1)*(((
b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2)*((sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(-a+
(b^2+c^2)^(1/2)))^(1/2)*((-sin(x-arctan(-b,c))+1)*(b^2+c^2)^(1/2)/(a+(b^2+c^2)^(1/2)))^(1/2)/(((b^2+c^2)^(1/2)
*sin(x-arctan(-b,c))+a)*cos(x-arctan(-b,c))^2)^(1/2)*((-1/(b^2+c^2)^(1/2)*a+1)*EllipticE((((b^2+c^2)^(1/2)*sin
(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^(1/2))-EllipticF(((
(b^2+c^2)^(1/2)*sin(x-arctan(-b,c))+a)/(a+(b^2+c^2)^(1/2)))^(1/2),((-a-(b^2+c^2)^(1/2))/(-a+(b^2+c^2)^(1/2)))^
(1/2)))))/cos(x-arctan(-b,c))/((b^2*sin(x-arctan(-b,c))+c^2*sin(x-arctan(-b,c))+a*(b^2+c^2)^(1/2))/(b^2+c^2)^(
1/2))^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.21 (sec) , antiderivative size = 2014, normalized size of antiderivative = 5.16 \[ \int (a+b \cos (x)+c \sin (x))^{5/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Too large to display} \]

[In]

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

[Out]

1/315*(sqrt(2)*(-7*I*(a^3*b - 33*a*b^3 - 33*a*b*c^2)*d + 7*(33*a*c^3 - (a^3 - 33*a*b^2)*c)*d - 5*I*(6*a^4*b -
23*a^2*b^3 - 15*b^5 - 15*b*c^4 - (23*a^2*b + 30*b^3)*c^2)*e + 5*(15*c^5 + (23*a^2 + 30*b^2)*c^3 - (6*a^4 - 23*
a^2*b^2 - 15*b^4)*c)*e)*sqrt(b + I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c
^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2
*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c
^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3*(b^2 + c^2)*cos(x) - 3*(I*b^2 + I*c^2)*sin(x))/(b^2 + c^2)) + sqrt(2)*(7*I
*(a^3*b - 33*a*b^3 - 33*a*b*c^2)*d + 7*(33*a*c^3 - (a^3 - 33*a*b^2)*c)*d + 5*I*(6*a^4*b - 23*a^2*b^3 - 15*b^5
- 15*b*c^4 - (23*a^2*b + 30*b^3)*c^2)*e + 5*(15*c^5 + (23*a^2 + 30*b^2)*c^3 - (6*a^4 - 23*a^2*b^2 - 15*b^4)*c)
*e)*sqrt(b - I*c)*weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3 + 3*c^4 + 2*I*(4*a^2*b -
3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 + 9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)
*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*
b + 2*I*a*c + 3*(b^2 + c^2)*cos(x) - 3*(-I*b^2 - I*c^2)*sin(x))/(b^2 + c^2)) - 3*sqrt(2)*(7*I*(23*a^2*b^2 + 9*
b^4 + 9*c^4 + (23*a^2 + 18*b^2)*c^2)*d + 5*I*(3*a^3*b^2 + 29*a*b^4 + 29*a*c^4 + (3*a^3 + 58*a*b^2)*c^2)*e)*sqr
t(b + I*c)*weierstrassZeta(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(
b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4
*a^3*b - 3*a*b^3)*c^2 - 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), weierstrassPInverse(
4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 + 6*I*b*c^3 + 3*c^4 - 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/
27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4 - 9*I*a*c^5 + 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 - 3*I
*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b - 2*I*a*c + 3*(b^2 + c^2)*cos(x) - 3
*(I*b^2 + I*c^2)*sin(x))/(b^2 + c^2))) - 3*sqrt(2)*(-7*I*(23*a^2*b^2 + 9*b^4 + 9*c^4 + (23*a^2 + 18*b^2)*c^2)*
d - 5*I*(3*a^3*b^2 + 29*a*b^4 + 29*a*c^4 + (3*a^3 + 58*a*b^2)*c^2)*e)*sqrt(b - I*c)*weierstrassZeta(4/3*(4*a^2
*b^2 - 3*b^4 - 4*a^2*c^2 - 6*I*b*c^3 + 3*c^4 + 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*
b^3 - 9*a*b^5 + 27*a*b*c^4 + 9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^
2 - 9*a*b^4)*c)/(b^6 + 3*b^4*c^2 + 3*b^2*c^4 + c^6), weierstrassPInverse(4/3*(4*a^2*b^2 - 3*b^4 - 4*a^2*c^2 -
6*I*b*c^3 + 3*c^4 + 2*I*(4*a^2*b - 3*b^3)*c)/(b^4 + 2*b^2*c^2 + c^4), -8/27*(8*a^3*b^3 - 9*a*b^5 + 27*a*b*c^4
+ 9*I*a*c^5 - 2*I*(4*a^3 + 9*a*b^2)*c^3 - 6*(4*a^3*b - 3*a*b^3)*c^2 + 3*I*(8*a^3*b^2 - 9*a*b^4)*c)/(b^6 + 3*b^
4*c^2 + 3*b^2*c^4 + c^6), 1/3*(2*a*b + 2*I*a*c + 3*(b^2 + c^2)*cos(x) - 3*(-I*b^2 - I*c^2)*sin(x))/(b^2 + c^2)
)) - 6*(15*(3*b^4*c + 2*b^2*c^3 - c^5)*e*cos(x)^3 + 6*(7*(b^3*c + b*c^3)*d + 15*(a*b^3*c + a*b*c^3)*e)*cos(x)^
2 - 21*(b^3*c + b*c^3)*d - 45*(a*b^3*c + a*b*c^3)*e + (77*(a*b^2*c + a*c^3)*d + 5*(8*c^5 + (9*a^2 + 7*b^2)*c^3
+ (9*a^2*b^2 - b^4)*c)*e)*cos(x) - (15*(b^5 - 2*b^3*c^2 - 3*b*c^4)*e*cos(x)^2 + 77*(a*b^3 + a*b*c^2)*d + 5*(9
*a^2*b^3 + 5*b^5 + 8*b*c^4 + (9*a^2*b + 13*b^3)*c^2)*e + 3*(7*(b^4 - c^4)*d + 15*(a*b^4 - a*c^4)*e)*cos(x))*si
n(x))*sqrt(b*cos(x) + c*sin(x) + a))/(b^2 + c^2)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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