Integrand size = 10, antiderivative size = 179 \[ \int (a+b \sinh (x))^{5/2} \, dx=\frac {16}{15} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2 i \left (23 a^2-9 b^2\right ) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{15 \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {16 i a \left (a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{15 \sqrt {a+b \sinh (x)}} \]
2/5*b*cosh(x)*(a+b*sinh(x))^(3/2)+16/15*a*b*cosh(x)*(a+b*sinh(x))^(1/2)+2/ 15*I*(23*a^2-9*b^2)*(sin(1/4*Pi+1/2*I*x)^2)^(1/2)/sin(1/4*Pi+1/2*I*x)*Elli pticE(cos(1/4*Pi+1/2*I*x),2^(1/2)*(b/(I*a+b))^(1/2))*(a+b*sinh(x))^(1/2)/( (a+b*sinh(x))/(a-I*b))^(1/2)-16/15*I*a*(a^2+b^2)*(sin(1/4*Pi+1/2*I*x)^2)^( 1/2)/sin(1/4*Pi+1/2*I*x)*EllipticF(cos(1/4*Pi+1/2*I*x),2^(1/2)*(b/(I*a+b)) ^(1/2))*((a+b*sinh(x))/(a-I*b))^(1/2)/(a+b*sinh(x))^(1/2)
Time = 0.33 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.99 \[ \int (a+b \sinh (x))^{5/2} \, dx=\frac {2 \left (23 i a^3+23 a^2 b-9 i a b^2-9 b^3\right ) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}-16 i a \left (a^2+b^2\right ) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}+b \cosh (x) \left (22 a^2-3 b^2+3 b^2 \cosh (2 x)+28 a b \sinh (x)\right )}{15 \sqrt {a+b \sinh (x)}} \]
(2*((23*I)*a^3 + 23*a^2*b - (9*I)*a*b^2 - 9*b^3)*EllipticE[(Pi - (2*I)*x)/ 4, ((-2*I)*b)/(a - I*b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)] - (16*I)*a*(a^2 + b^2)*EllipticF[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)]*Sqrt[(a + b*Sinh[x ])/(a - I*b)] + b*Cosh[x]*(22*a^2 - 3*b^2 + 3*b^2*Cosh[2*x] + 28*a*b*Sinh[ x]))/(15*Sqrt[a + b*Sinh[x]])
Time = 0.92 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.03, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3042, 3135, 27, 3042, 3232, 27, 3042, 3231, 3042, 3134, 3042, 3132, 3142, 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.
| \(\displaystyle \int (a+b \sinh (x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a-i b \sin (i x))^{5/2}dx\) |
| \(\Big \downarrow \) 3135 |
| \(\displaystyle \frac {2}{5} \int \frac {1}{2} \sqrt {a+b \sinh (x)} \left (5 a^2+8 b \sinh (x) a-3 b^2\right )dx+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \int \sqrt {a+b \sinh (x)} \left (5 a^2+8 b \sinh (x) a-3 b^2\right )dx+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \int \sqrt {a-i b \sin (i x)} \left (5 a^2-8 i b \sin (i x) a-3 b^2\right )dx\) |
| \(\Big \downarrow \) 3232 |
| \(\displaystyle \frac {1}{5} \left (\frac {2}{3} \int \frac {a \left (15 a^2-17 b^2\right )+b \left (23 a^2-9 b^2\right ) \sinh (x)}{2 \sqrt {a+b \sinh (x)}}dx+\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}\right )+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \int \frac {a \left (15 a^2-17 b^2\right )+b \left (23 a^2-9 b^2\right ) \sinh (x)}{\sqrt {a+b \sinh (x)}}dx+\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}\right )+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \int \frac {a \left (15 a^2-17 b^2\right )-i b \left (23 a^2-9 b^2\right ) \sin (i x)}{\sqrt {a-i b \sin (i x)}}dx\right )\) |
| \(\Big \downarrow \) 3231 |
| \(\displaystyle \frac {1}{5} \left (\frac {1}{3} \left (\left (23 a^2-9 b^2\right ) \int \sqrt {a+b \sinh (x)}dx-8 a \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a+b \sinh (x)}}dx\right )+\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}\right )+\frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\left (23 a^2-9 b^2\right ) \int \sqrt {a-i b \sin (i x)}dx-8 a \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx\right )\right )\) |
| \(\Big \downarrow \) 3134 |
| \(\displaystyle \frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {\left (23 a^2-9 b^2\right ) \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}dx}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-8 a \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {\left (23 a^2-9 b^2\right ) \sqrt {a+b \sinh (x)} \int \sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}dx}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-8 a \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx\right )\right )\) |
| \(\Big \downarrow \) 3132 |
| \(\displaystyle \frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i \left (23 a^2-9 b^2\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-8 a \left (a^2+b^2\right ) \int \frac {1}{\sqrt {a-i b \sin (i x)}}dx\right )\right )\) |
| \(\Big \downarrow \) 3142 |
| \(\displaystyle \frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i \left (23 a^2-9 b^2\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {8 a \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}}dx}{\sqrt {a+b \sinh (x)}}\right )\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i \left (23 a^2-9 b^2\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {8 a \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \int \frac {1}{\sqrt {\frac {a}{a-i b}-\frac {i b \sin (i x)}{a-i b}}}dx}{\sqrt {a+b \sinh (x)}}\right )\right )\) |
| \(\Big \downarrow \) 3140 |
| \(\displaystyle \frac {2}{5} b \cosh (x) (a+b \sinh (x))^{3/2}+\frac {1}{5} \left (\frac {16}{3} a b \cosh (x) \sqrt {a+b \sinh (x)}+\frac {1}{3} \left (\frac {2 i \left (23 a^2-9 b^2\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{\sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {16 i a \left (a^2+b^2\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{\sqrt {a+b \sinh (x)}}\right )\right )\) |
(2*b*Cosh[x]*(a + b*Sinh[x])^(3/2))/5 + ((16*a*b*Cosh[x]*Sqrt[a + b*Sinh[x ]])/3 + (((2*I)*(23*a^2 - 9*b^2)*EllipticE[Pi/4 - (I/2)*x, (2*b)/(I*a + b) ]*Sqrt[a + b*Sinh[x]])/Sqrt[(a + b*Sinh[x])/(a - I*b)] - ((16*I)*a*(a^2 + b^2)*EllipticF[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]*Sqrt[(a + b*Sinh[x])/(a - I*b)])/Sqrt[a + b*Sinh[x]])/3)/5
3.2.5.3.1 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[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[a + b*Sin[c + d*x]]/Sqrt[(a + b*Sin[c + d*x])/(a + b)] Int[Sqrt[a/(a + b) + ( b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2 , 0] && !GtQ[a + b, 0]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos [c + d*x]*((a + b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[1/n Int[(a + b* Sin[c + d*x])^(n - 2)*Simp[a^2*n + b^2*(n - 1) + a*b*(2*n - 1)*Sin[c + d*x] , x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && GtQ[n, 1] && IntegerQ[2*n]
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[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Simp[Sqrt[(a + b*Sin[c + d*x])/(a + b)]/Sqrt[a + b*Sin[c + d*x]] Int[1/Sqrt[a/(a + b) + (b/(a + b))*Sin[c + d*x]], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 - b^2, 0] && !GtQ[a + b, 0]
Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + ( f_.)*(x_)]], x_Symbol] :> Simp[(b*c - a*d)/b Int[1/Sqrt[a + b*Sin[e + f*x ]], x], x] + Simp[d/b Int[Sqrt[a + b*Sin[e + f*x]], x], x] /; FreeQ[{a, b , c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/( f*(m + 1))), x] + Simp[1/(m + 1) Int[(a + b*Sin[e + f*x])^(m - 1)*Simp[b* d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ [{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 916 vs. \(2 (203 ) = 406\).
Time = 2.23 (sec) , antiderivative size = 917, normalized size of antiderivative = 5.12
| method | result | size |
| default | \(\frac {\frac {16 i \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{3} b}{15}+\frac {16 i \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a \,b^{3}}{15}+2 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{4}+\frac {4 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{2} b^{2}}{5}-\frac {6 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticF}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b^{4}}{5}-\frac {46 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{4}}{15}-\frac {28 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) a^{2} b^{2}}{15}+\frac {6 \sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}\, \sqrt {\frac {\left (i-\sinh \left (x \right )\right ) b}{i b +a}}\, \sqrt {\frac {\left (i+\sinh \left (x \right )\right ) b}{i b -a}}\, \operatorname {EllipticE}\left (\sqrt {-\frac {a +b \sinh \left (x \right )}{i b -a}}, \sqrt {-\frac {i b -a}{i b +a}}\right ) b^{4}}{5}+\frac {2 b^{4} \sinh \left (x \right )^{4}}{5}+\frac {28 a \,b^{3} \sinh \left (x \right )^{3}}{15}+\frac {22 a^{2} b^{2} \sinh \left (x \right )^{2}}{15}+\frac {2 b^{4} \sinh \left (x \right )^{2}}{5}+\frac {28 a \,b^{3} \sinh \left (x \right )}{15}+\frac {22 a^{2} b^{2}}{15}}{b \cosh \left (x \right ) \sqrt {a +b \sinh \left (x \right )}}\) | \(917\) |
2/15*(8*I*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*((I +sinh(x))*b/(I*b-a))^(1/2)*EllipticF((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b -a)/(I*b+a))^(1/2))*a^3*b+8*I*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))* b/(I*b+a))^(1/2)*((I+sinh(x))*b/(I*b-a))^(1/2)*EllipticF((-(a+b*sinh(x))/( I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*a*b^3+15*(-(a+b*sinh(x))/(I*b-a))^ (1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*((I+sinh(x))*b/(I*b-a))^(1/2)*Elliptic F((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*a^4+6*(-(a+b*si nh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*((I+sinh(x))*b/(I*b-a) )^(1/2)*EllipticF((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2)) *a^2*b^2-9*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*(( I+sinh(x))*b/(I*b-a))^(1/2)*EllipticF((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I* b-a)/(I*b+a))^(1/2))*b^4-23*(-(a+b*sinh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/ (I*b+a))^(1/2)*((I+sinh(x))*b/(I*b-a))^(1/2)*EllipticE((-(a+b*sinh(x))/(I* b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*a^4-14*(-(a+b*sinh(x))/(I*b-a))^(1/2 )*((I-sinh(x))*b/(I*b+a))^(1/2)*((I+sinh(x))*b/(I*b-a))^(1/2)*EllipticE((- (a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*a^2*b^2+9*(-(a+b*si nh(x))/(I*b-a))^(1/2)*((I-sinh(x))*b/(I*b+a))^(1/2)*((I+sinh(x))*b/(I*b-a) )^(1/2)*EllipticE((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2)) *b^4+3*b^4*sinh(x)^4+14*a*b^3*sinh(x)^3+11*a^2*b^2*sinh(x)^2+3*b^4*sinh(x) ^2+14*a*b^3*sinh(x)+11*a^2*b^2)/b/cosh(x)/(a+b*sinh(x))^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.12 (sec) , antiderivative size = 464, normalized size of antiderivative = 2.59 \[ \int (a+b \sinh (x))^{5/2} \, dx=-\frac {4 \, {\left (\sqrt {2} {\left (a^{3} + 33 \, a b^{2}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (a^{3} + 33 \, a b^{2}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (a^{3} + 33 \, a b^{2}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {b} {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right ) + 12 \, {\left (\sqrt {2} {\left (23 \, a^{2} b - 9 \, b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (23 \, a^{2} b - 9 \, b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (23 \, a^{2} b - 9 \, b^{3}\right )} \sinh \left (x\right )^{2}\right )} \sqrt {b} {\rm weierstrassZeta}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, {\rm weierstrassPInverse}\left (\frac {4 \, {\left (4 \, a^{2} + 3 \, b^{2}\right )}}{3 \, b^{2}}, -\frac {8 \, {\left (8 \, a^{3} + 9 \, a b^{2}\right )}}{27 \, b^{3}}, \frac {3 \, b \cosh \left (x\right ) + 3 \, b \sinh \left (x\right ) + 2 \, a}{3 \, b}\right )\right ) - 3 \, {\left (3 \, b^{3} \cosh \left (x\right )^{4} + 3 \, b^{3} \sinh \left (x\right )^{4} + 22 \, a b^{2} \cosh \left (x\right )^{3} + 22 \, a b^{2} \cosh \left (x\right ) + 2 \, {\left (6 \, b^{3} \cosh \left (x\right ) + 11 \, a b^{2}\right )} \sinh \left (x\right )^{3} - 3 \, b^{3} - 4 \, {\left (23 \, a^{2} b - 9 \, b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (9 \, b^{3} \cosh \left (x\right )^{2} + 33 \, a b^{2} \cosh \left (x\right ) - 46 \, a^{2} b + 18 \, b^{3}\right )} \sinh \left (x\right )^{2} + 2 \, {\left (6 \, b^{3} \cosh \left (x\right )^{3} + 33 \, a b^{2} \cosh \left (x\right )^{2} + 11 \, a b^{2} - 4 \, {\left (23 \, a^{2} b - 9 \, b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}}{90 \, {\left (b \cosh \left (x\right )^{2} + 2 \, b \cosh \left (x\right ) \sinh \left (x\right ) + b \sinh \left (x\right )^{2}\right )}} \]
-1/90*(4*(sqrt(2)*(a^3 + 33*a*b^2)*cosh(x)^2 + 2*sqrt(2)*(a^3 + 33*a*b^2)* cosh(x)*sinh(x) + sqrt(2)*(a^3 + 33*a*b^2)*sinh(x)^2)*sqrt(b)*weierstrassP Inverse(4/3*(4*a^2 + 3*b^2)/b^2, -8/27*(8*a^3 + 9*a*b^2)/b^3, 1/3*(3*b*cos h(x) + 3*b*sinh(x) + 2*a)/b) + 12*(sqrt(2)*(23*a^2*b - 9*b^3)*cosh(x)^2 + 2*sqrt(2)*(23*a^2*b - 9*b^3)*cosh(x)*sinh(x) + sqrt(2)*(23*a^2*b - 9*b^3)* sinh(x)^2)*sqrt(b)*weierstrassZeta(4/3*(4*a^2 + 3*b^2)/b^2, -8/27*(8*a^3 + 9*a*b^2)/b^3, weierstrassPInverse(4/3*(4*a^2 + 3*b^2)/b^2, -8/27*(8*a^3 + 9*a*b^2)/b^3, 1/3*(3*b*cosh(x) + 3*b*sinh(x) + 2*a)/b)) - 3*(3*b^3*cosh(x )^4 + 3*b^3*sinh(x)^4 + 22*a*b^2*cosh(x)^3 + 22*a*b^2*cosh(x) + 2*(6*b^3*c osh(x) + 11*a*b^2)*sinh(x)^3 - 3*b^3 - 4*(23*a^2*b - 9*b^3)*cosh(x)^2 + 2* (9*b^3*cosh(x)^2 + 33*a*b^2*cosh(x) - 46*a^2*b + 18*b^3)*sinh(x)^2 + 2*(6* b^3*cosh(x)^3 + 33*a*b^2*cosh(x)^2 + 11*a*b^2 - 4*(23*a^2*b - 9*b^3)*cosh( x))*sinh(x))*sqrt(b*sinh(x) + a))/(b*cosh(x)^2 + 2*b*cosh(x)*sinh(x) + b*s inh(x)^2)
\[ \int (a+b \sinh (x))^{5/2} \, dx=\int \left (a + b \sinh {\left (x \right )}\right )^{\frac {5}{2}}\, dx \]
\[ \int (a+b \sinh (x))^{5/2} \, dx=\int { {\left (b \sinh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
\[ \int (a+b \sinh (x))^{5/2} \, dx=\int { {\left (b \sinh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]
Timed out. \[ \int (a+b \sinh (x))^{5/2} \, dx=\int {\left (a+b\,\mathrm {sinh}\left (x\right )\right )}^{5/2} \,d x \]