Integrand size = 27, antiderivative size = 294 \[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\frac {2 \left (20 a d+3 a^2 e+9 \left (b^2+c^2\right ) e\right ) E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {a+b \cos (x)+c \sin (x)}}{15 \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 \left (a^2-b^2-c^2\right ) (5 d+3 a e) \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right ) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}{15 \sqrt {a+b \cos (x)+c \sin (x)}}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))-\frac {2}{15} \sqrt {a+b \cos (x)+c \sin (x)} (c (5 d+3 a e) \cos (x)-b (5 d+3 a e) \sin (x)) \]
-2/5*(a+b*cos(x)+c*sin(x))^(3/2)*(c*e*cos(x)-b*e*sin(x))-2/15*(c*(3*a*e+5* d)*cos(x)-b*(3*a*e+5*d)*sin(x))*(a+b*cos(x)+c*sin(x))^(1/2)+2/15*(20*a*d+3 *a^2*e+9*(b^2+c^2)*e)*(cos(1/2*x-1/2*arctan(b,c))^2)^(1/2)/cos(1/2*x-1/2*a rctan(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)-2/15*(a^2-b^2-c^2)*(3*a*e+5*d)*(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*sin(x))/(a+(b^2+c^2)^(1/2)))^(1/2)/(a+b*cos(x)+c*sin(x))^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.59 (sec) , antiderivative size = 5218, normalized size of antiderivative = 17.75 \[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Result too large to show} \]
Time = 1.48 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.07, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.556, Rules used = {3042, 3625, 27, 3042, 3625, 27, 3042, 3628, 3042, 3598, 3042, 3132, 3606, 3042, 3140}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int (a+b \cos (x)+c \sin (x))^{3/2} (b e \cos (x)+c e \sin (x)+d) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+b \cos (x)+c \sin (x))^{3/2} (b e \cos (x)+c e \sin (x)+d)dx\) |
| \(\Big \downarrow \) 3625 |
| \(\displaystyle \frac {2 \int \frac {1}{2} \sqrt {a+b \cos (x)+c \sin (x)} \left (a \left (5 a d+3 \left (b^2+c^2\right ) e\right )+a b (5 d+3 a e) \cos (x)+a c (5 d+3 a e) \sin (x)\right )dx}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \sqrt {a+b \cos (x)+c \sin (x)} \left (a \left (5 a d+3 \left (b^2+c^2\right ) e\right )+a b (5 d+3 a e) \cos (x)+a c (5 d+3 a e) \sin (x)\right )dx}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \sqrt {a+b \cos (x)+c \sin (x)} \left (a \left (5 a d+3 \left (b^2+c^2\right ) e\right )+a b (5 d+3 a e) \cos (x)+a c (5 d+3 a e) \sin (x)\right )dx}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3625 |
| \(\displaystyle \frac {\frac {2 \int \frac {\left (15 d a^2+12 \left (b^2+c^2\right ) e a+5 \left (b^2+c^2\right ) d\right ) a^2+b \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \cos (x) a^2+c \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \sin (x) a^2}{2 \sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {\left (15 d a^2+12 \left (b^2+c^2\right ) e a+5 \left (b^2+c^2\right ) d\right ) a^2+b \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \cos (x) a^2+c \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \sin (x) a^2}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {\left (15 d a^2+12 \left (b^2+c^2\right ) e a+5 \left (b^2+c^2\right ) d\right ) a^2+b \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \cos (x) a^2+c \left (3 e a^2+20 d a+9 \left (b^2+c^2\right ) e\right ) \sin (x) a^2}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3628 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (x)+c \sin (x)}dx-a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \int \sqrt {a+b \cos (x)+c \sin (x)}dx-a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3598 |
| \(\displaystyle \frac {\frac {\frac {a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} \int \sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}dx}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {\frac {\frac {2 a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \int \frac {1}{\sqrt {a+b \cos (x)+c \sin (x)}}dx}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3606 |
| \(\displaystyle \frac {\frac {\frac {2 a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \cos \left (x-\tan ^{-1}(b,c)\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {2 a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \int \frac {1}{\sqrt {\frac {a}{a+\sqrt {b^2+c^2}}+\frac {\sqrt {b^2+c^2} \sin \left (x-\tan ^{-1}(b,c)+\frac {\pi }{2}\right )}{a+\sqrt {b^2+c^2}}}}dx}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {\frac {\frac {2 a^2 \left (3 a^2 e+20 a d+9 e \left (b^2+c^2\right )\right ) \sqrt {a+b \cos (x)+c \sin (x)} E\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right )|\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}}}-\frac {2 a^2 \left (a^2-b^2-c^2\right ) (3 a e+5 d) \sqrt {\frac {a+b \cos (x)+c \sin (x)}{a+\sqrt {b^2+c^2}}} \operatorname {EllipticF}\left (\frac {1}{2} \left (x-\tan ^{-1}(b,c)\right ),\frac {2 \sqrt {b^2+c^2}}{a+\sqrt {b^2+c^2}}\right )}{\sqrt {a+b \cos (x)+c \sin (x)}}}{3 a}-\frac {2}{3} \sqrt {a+b \cos (x)+c \sin (x)} (a c \cos (x) (3 a e+5 d)-a b \sin (x) (3 a e+5 d))}{5 a}-\frac {2}{5} (a+b \cos (x)+c \sin (x))^{3/2} (c e \cos (x)-b e \sin (x))\) |
(-2*(a + b*Cos[x] + c*Sin[x])^(3/2)*(c*e*Cos[x] - b*e*Sin[x]))/5 + ((-2*Sq rt[a + b*Cos[x] + c*Sin[x]]*(a*c*(5*d + 3*a*e)*Cos[x] - a*b*(5*d + 3*a*e)* Sin[x]))/3 + ((2*a^2*(20*a*d + 3*a^2*e + 9*(b^2 + c^2)*e)*EllipticE[(x - A rcTan[b, c])/2, (2*Sqrt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[a + b*Cos[ x] + c*Sin[x]])/Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])] - (2 *a^2*(a^2 - b^2 - c^2)*(5*d + 3*a*e)*EllipticF[(x - ArcTan[b, c])/2, (2*Sq rt[b^2 + c^2])/(a + Sqrt[b^2 + c^2])]*Sqrt[(a + b*Cos[x] + c*Sin[x])/(a + Sqrt[b^2 + c^2])])/Sqrt[a + b*Cos[x] + c*Sin[x]])/(3*a))/(5*a)
3.6.57.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[(2/(d*S qrt[a + b]))*EllipticF[(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]
Int[Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*(x_ )]], x_Symbol] :> Simp[Sqrt[a + b*Cos[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 + S qrt[b^2 + c^2]) + (Sqrt[b^2 + c^2]/(a + Sqrt[b^2 + c^2]))*Cos[d + e*x - Arc Tan[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]
Int[1/Sqrt[cos[(d_.) + (e_.)*(x_)]*(b_.) + (a_) + (c_.)*sin[(d_.) + (e_.)*( x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Cos[d + e*x] + c*Sin[d + e*x])/(a + Sq rt[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] /; 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]
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] + Simp[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]
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] :> Simp[B/b Int[Sqrt[a + b*Cos[d + e*x] + c*Sin[d + e*x]], x] , x] + Simp[(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]
Leaf count of result is larger than twice the leaf count of optimal. \(2204\) vs. \(2(322)=644\).
Time = 8.22 (sec) , antiderivative size = 2205, normalized size of antiderivative = 7.50
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(2205\) |
| parts | \(\text {Expression too large to display}\) | \(13403\) |
(-(-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)^(1/2)*(2*a^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*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^4*e+2*b^2*c^2*e+c^4*e)*(-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-arc tan(-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)*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)*Elli pticF((((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...
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.16 (sec) , antiderivative size = 1692, normalized size of antiderivative = 5.76 \[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\text {Too large to display} \]
1/45*(sqrt(2)*(5*I*(a^2*b + 3*b^3 + 3*b*c^2)*d + 5*(3*c^3 + (a^2 + 3*b^2)* c)*d - 6*I*(a^3*b - 3*a*b^3 - 3*a*b*c^2)*e + 6*(3*a*c^3 - (a^3 - 3*a*b^2)* 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)*(-5*I*(a^2*b + 3*b^3 + 3*b *c^2)*d + 5*(3*c^3 + (a^2 + 3*b^2)*c)*d + 6*I*(a^3*b - 3*a*b^3 - 3*a*b*c^2 )*e + 6*(3*a*c^3 - (a^3 - 3*a*b^2)*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)*(20*I*(a*b^2 + a*c^2)*d + 3*I*(a^2*b^2 + 3*b^4 + 3*c^4 + (a^2 + 6*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)/...
\[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\int \left (a + b \cos {\left (x \right )} + c \sin {\left (x \right )}\right )^{\frac {3}{2}} \left (b e \cos {\left (x \right )} + c e \sin {\left (x \right )} + d\right )\, dx \]
\[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\int { {\left (b e \cos \left (x\right ) + c e \sin \left (x\right ) + d\right )} {\left (b \cos \left (x\right ) + c \sin \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
\[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\int { {\left (b e \cos \left (x\right ) + c e \sin \left (x\right ) + d\right )} {\left (b \cos \left (x\right ) + c \sin \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
Timed out. \[ \int (a+b \cos (x)+c \sin (x))^{3/2} (d+b e \cos (x)+c e \sin (x)) \, dx=\int {\left (a+b\,\cos \left (x\right )+c\,\sin \left (x\right )\right )}^{3/2}\,\left (d+b\,e\,\cos \left (x\right )+c\,e\,\sin \left (x\right )\right ) \,d x \]