Integrand size = 12, antiderivative size = 152 \[ \int \left (a+b \sin ^2(x)\right )^{5/2} \, dx=-\frac {4}{15} b (2 a+b) \cos (x) \sin (x) \sqrt {a+b \sin ^2(x)}-\frac {1}{5} b \cos (x) \sin (x) \left (a+b \sin ^2(x)\right )^{3/2}+\frac {\left (23 a^2+23 a b+8 b^2\right ) E\left (x\left |-\frac {b}{a}\right .\right ) \sqrt {a+b \sin ^2(x)}}{15 \sqrt {\frac {a+b \sin ^2(x)}{a}}}-\frac {4 a (a+b) (2 a+b) \operatorname {EllipticF}\left (x,-\frac {b}{a}\right ) \sqrt {\frac {a+b \sin ^2(x)}{a}}}{15 \sqrt {a+b \sin ^2(x)}} \] Output:
-4/15*b*(2*a+b)*cos(x)*sin(x)*(a+b*sin(x)^2)^(1/2)-1/5*b*cos(x)*sin(x)*(a+ b*sin(x)^2)^(3/2)+1/15*(23*a^2+23*a*b+8*b^2)*EllipticE(sin(x),(-b/a)^(1/2) )*(a+b*sin(x)^2)^(1/2)/((a+b*sin(x)^2)/a)^(1/2)-4/15*a*(a+b)*(2*a+b)*Inver seJacobiAM(x,(-b/a)^(1/2))*((a+b*sin(x)^2)/a)^(1/2)/(a+b*sin(x)^2)^(1/2)
Time = 0.67 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.05 \[ \int \left (a+b \sin ^2(x)\right )^{5/2} \, dx=\frac {16 a \left (23 a^2+23 a b+8 b^2\right ) \sqrt {\frac {2 a+b-b \cos (2 x)}{a}} E\left (x\left |-\frac {b}{a}\right .\right )-64 a \left (2 a^2+3 a b+b^2\right ) \sqrt {\frac {2 a+b-b \cos (2 x)}{a}} \operatorname {EllipticF}\left (x,-\frac {b}{a}\right )-\sqrt {2} b \left (88 a^2+88 a b+25 b^2-28 b (2 a+b) \cos (2 x)+3 b^2 \cos (4 x)\right ) \sin (2 x)}{240 \sqrt {2 a+b-b \cos (2 x)}} \] Input:
Integrate[(a + b*Sin[x]^2)^(5/2),x]
Output:
(16*a*(23*a^2 + 23*a*b + 8*b^2)*Sqrt[(2*a + b - b*Cos[2*x])/a]*EllipticE[x , -(b/a)] - 64*a*(2*a^2 + 3*a*b + b^2)*Sqrt[(2*a + b - b*Cos[2*x])/a]*Elli pticF[x, -(b/a)] - Sqrt[2]*b*(88*a^2 + 88*a*b + 25*b^2 - 28*b*(2*a + b)*Co s[2*x] + 3*b^2*Cos[4*x])*Sin[2*x])/(240*Sqrt[2*a + b - b*Cos[2*x]])
Time = 0.96 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.02, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3042, 3659, 3042, 3649, 3042, 3651, 3042, 3657, 3042, 3656, 3662, 3042, 3661}
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.
| \(\displaystyle \int \left (a+b \sin ^2(x)\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a+b \sin (x)^2\right )^{5/2}dx\) |
| \(\Big \downarrow \) 3659 |
| \(\displaystyle \frac {1}{5} \int \sqrt {b \sin ^2(x)+a} \left (4 b (2 a+b) \sin ^2(x)+a (5 a+b)\right )dx-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \int \sqrt {b \sin (x)^2+a} \left (4 b (2 a+b) \sin (x)^2+a (5 a+b)\right )dx-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3649 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {b \left (23 a^2+23 b a+8 b^2\right ) \sin ^2(x)+a \left (15 a^2+11 b a+4 b^2\right )}{\sqrt {b \sin ^2(x)+a}}dx-\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\right )-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {b \left (23 a^2+23 b a+8 b^2\right ) \sin (x)^2+a \left (15 a^2+11 b a+4 b^2\right )}{\sqrt {b \sin (x)^2+a}}dx-\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\right )-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3651 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\left (23 a^2+23 a b+8 b^2\right ) \int \sqrt {b \sin ^2(x)+a}dx-4 a (a+b) (2 a+b) \int \frac {1}{\sqrt {b \sin ^2(x)+a}}dx\right )-\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\right )-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\left (23 a^2+23 a b+8 b^2\right ) \int \sqrt {b \sin (x)^2+a}dx-4 a (a+b) (2 a+b) \int \frac {1}{\sqrt {b \sin (x)^2+a}}dx\right )-\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\right )-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3657 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \sin ^2(x)} \int \sqrt {\frac {b \sin ^2(x)}{a}+1}dx}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-4 a (a+b) (2 a+b) \int \frac {1}{\sqrt {b \sin (x)^2+a}}dx\right )-\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\right )-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \sin ^2(x)} \int \sqrt {\frac {b \sin (x)^2}{a}+1}dx}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-4 a (a+b) (2 a+b) \int \frac {1}{\sqrt {b \sin (x)^2+a}}dx\right )-\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\right )-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3656 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \sin ^2(x)} E\left (x\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-4 a (a+b) (2 a+b) \int \frac {1}{\sqrt {b \sin (x)^2+a}}dx\right )-\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\right )-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3662 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \sin ^2(x)} E\left (x\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-\frac {4 a (a+b) (2 a+b) \sqrt {\frac {b \sin ^2(x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}dx}{\sqrt {a+b \sin ^2(x)}}\right )-\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\right )-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \sin ^2(x)} E\left (x\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-\frac {4 a (a+b) (2 a+b) \sqrt {\frac {b \sin ^2(x)}{a}+1} \int \frac {1}{\sqrt {\frac {b \sin (x)^2}{a}+1}}dx}{\sqrt {a+b \sin ^2(x)}}\right )-\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\right )-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
| \(\Big \downarrow \) 3661 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\frac {\left (23 a^2+23 a b+8 b^2\right ) \sqrt {a+b \sin ^2(x)} E\left (x\left |-\frac {b}{a}\right .\right )}{\sqrt {\frac {b \sin ^2(x)}{a}+1}}-\frac {4 a (a+b) (2 a+b) \sqrt {\frac {b \sin ^2(x)}{a}+1} \operatorname {EllipticF}\left (x,-\frac {b}{a}\right )}{\sqrt {a+b \sin ^2(x)}}\right )-\frac {4}{3} b (2 a+b) \sin (x) \cos (x) \sqrt {a+b \sin ^2(x)}\right )-\frac {1}{5} b \sin (x) \cos (x) \left (a+b \sin ^2(x)\right )^{3/2}\) |
Input:
Int[(a + b*Sin[x]^2)^(5/2),x]
Output:
-1/5*(b*Cos[x]*Sin[x]*(a + b*Sin[x]^2)^(3/2)) + ((-4*b*(2*a + b)*Cos[x]*Si n[x]*Sqrt[a + b*Sin[x]^2])/3 + (((23*a^2 + 23*a*b + 8*b^2)*EllipticE[x, -( b/a)]*Sqrt[a + b*Sin[x]^2])/Sqrt[1 + (b*Sin[x]^2)/a] - (4*a*(a + b)*(2*a + b)*EllipticF[x, -(b/a)]*Sqrt[1 + (b*Sin[x]^2)/a])/Sqrt[a + b*Sin[x]^2])/3 )/5
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-B)*Cos[e + f*x]*Sin[e + f*x]*((a + b* Sin[e + f*x]^2)^p/(2*f*(p + 1))), x] + Simp[1/(2*(p + 1)) Int[(a + b*Sin[ e + f*x]^2)^(p - 1)*Simp[a*B + 2*a*A*(p + 1) + (2*A*b*(p + 1) + B*(b + 2*a* p + 2*b*p))*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f, A, B}, x] && G tQ[p, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)]^2)/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[B/b Int[Sqrt[a + b*Sin[e + f*x]^2], x] , x] + Simp[(A*b - a*B)/b Int[1/Sqrt[a + b*Sin[e + f*x]^2], x], x] /; Fre eQ[{a, b, e, f, A, B}, x]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(Sqrt[a ]/f)*EllipticE[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[a + b*Sin[e + f*x]^2]/Sqrt[1 + b*(Sin[e + f*x]^2/a)] Int[Sqrt[1 + (b*Sin[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] && !GtQ[a, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2)^(p_), x_Symbol] :> Simp[(-b)*C os[e + f*x]*Sin[e + f*x]*((a + b*Sin[e + f*x]^2)^(p - 1)/(2*f*p)), x] + Sim p[1/(2*p) Int[(a + b*Sin[e + f*x]^2)^(p - 2)*Simp[a*(b + 2*a*p) + b*(2*a + b)*(2*p - 1)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, e, f}, x] && NeQ[ a + b, 0] && GtQ[p, 1]
Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[(1/(S qrt[a]*f))*EllipticF[e + f*x, -b/a], x] /; FreeQ[{a, b, e, f}, x] && GtQ[a, 0]
Int[1/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^2], x_Symbol] :> Simp[Sqrt[ 1 + b*(Sin[e + f*x]^2/a)]/Sqrt[a + b*Sin[e + f*x]^2] Int[1/Sqrt[1 + (b*Si n[e + f*x]^2)/a], x], x] /; FreeQ[{a, b, e, f}, x] && !GtQ[a, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(329\) vs. \(2(137)=274\).
Time = 4.93 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.17
| method | result | size |
| default | \(\frac {-\frac {b^{3} \cos \left (x \right )^{6} \sin \left (x \right )}{5}+\frac {\left (14 b^{2} a +10 b^{3}\right ) \cos \left (x \right )^{4} \sin \left (x \right )}{15}+\frac {\left (-11 a^{2} b -18 b^{2} a -7 b^{3}\right ) \cos \left (x \right )^{2} \sin \left (x \right )}{15}-\frac {8 a^{3} \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \sqrt {-\frac {b}{a}}\right )}{15}-\frac {4 a^{2} \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \sqrt {-\frac {b}{a}}\right ) b}{5}-\frac {4 a \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticF}\left (\sin \left (x \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}}{15}+\frac {23 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}}{15}+\frac {23 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b}{15}+\frac {8 \sqrt {\frac {\cos \left (2 x \right )}{2}+\frac {1}{2}}\, \sqrt {-\frac {b \cos \left (x \right )^{2}}{a}+\frac {a +b}{a}}\, \operatorname {EllipticE}\left (\sin \left (x \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}}{15}}{\cos \left (x \right ) \sqrt {a +b \sin \left (x \right )^{2}}}\) | \(330\) |
Input:
int((a+b*sin(x)^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
(-1/5*b^3*cos(x)^6*sin(x)+1/15*(14*a*b^2+10*b^3)*cos(x)^4*sin(x)+1/15*(-11 *a^2*b-18*a*b^2-7*b^3)*cos(x)^2*sin(x)-8/15*a^3*(cos(x)^2)^(1/2)*(-b/a*cos (x)^2+(a+b)/a)^(1/2)*EllipticF(sin(x),(-b/a)^(1/2))-4/5*a^2*(cos(x)^2)^(1/ 2)*(-b/a*cos(x)^2+(a+b)/a)^(1/2)*EllipticF(sin(x),(-b/a)^(1/2))*b-4/15*a*( cos(x)^2)^(1/2)*(-b/a*cos(x)^2+(a+b)/a)^(1/2)*EllipticF(sin(x),(-b/a)^(1/2 ))*b^2+23/15*(cos(x)^2)^(1/2)*(-b/a*cos(x)^2+(a+b)/a)^(1/2)*EllipticE(sin( x),(-b/a)^(1/2))*a^3+23/15*(cos(x)^2)^(1/2)*(-b/a*cos(x)^2+(a+b)/a)^(1/2)* EllipticE(sin(x),(-b/a)^(1/2))*a^2*b+8/15*(cos(x)^2)^(1/2)*(-b/a*cos(x)^2+ (a+b)/a)^(1/2)*EllipticE(sin(x),(-b/a)^(1/2))*a*b^2)/cos(x)/(a+b*sin(x)^2) ^(1/2)
\[ \int \left (a+b \sin ^2(x)\right )^{5/2} \, dx=\int { {\left (b \sin \left (x\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*sin(x)^2)^(5/2),x, algorithm="fricas")
Output:
integral((b^2*cos(x)^4 - 2*(a*b + b^2)*cos(x)^2 + a^2 + 2*a*b + b^2)*sqrt( -b*cos(x)^2 + a + b), x)
Timed out. \[ \int \left (a+b \sin ^2(x)\right )^{5/2} \, dx=\text {Timed out} \] Input:
integrate((a+b*sin(x)**2)**(5/2),x)
Output:
Timed out
\[ \int \left (a+b \sin ^2(x)\right )^{5/2} \, dx=\int { {\left (b \sin \left (x\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*sin(x)^2)^(5/2),x, algorithm="maxima")
Output:
integrate((b*sin(x)^2 + a)^(5/2), x)
\[ \int \left (a+b \sin ^2(x)\right )^{5/2} \, dx=\int { {\left (b \sin \left (x\right )^{2} + a\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((a+b*sin(x)^2)^(5/2),x, algorithm="giac")
Output:
integrate((b*sin(x)^2 + a)^(5/2), x)
Timed out. \[ \int \left (a+b \sin ^2(x)\right )^{5/2} \, dx=\int {\left (b\,{\sin \left (x\right )}^2+a\right )}^{5/2} \,d x \] Input:
int((a + b*sin(x)^2)^(5/2),x)
Output:
int((a + b*sin(x)^2)^(5/2), x)
\[ \int \left (a+b \sin ^2(x)\right )^{5/2} \, dx=\left (\int \sqrt {\sin \left (x \right )^{2} b +a}d x \right ) a^{2}+\left (\int \sqrt {\sin \left (x \right )^{2} b +a}\, \sin \left (x \right )^{4}d x \right ) b^{2}+2 \left (\int \sqrt {\sin \left (x \right )^{2} b +a}\, \sin \left (x \right )^{2}d x \right ) a b \] Input:
int((a+b*sin(x)^2)^(5/2),x)
Output:
int(sqrt(sin(x)**2*b + a),x)*a**2 + int(sqrt(sin(x)**2*b + a)*sin(x)**4,x) *b**2 + 2*int(sqrt(sin(x)**2*b + a)*sin(x)**2,x)*a*b