\(\int (a+b \sinh (x))^{5/2} (A+B \sinh (x)) \, dx\) [126]

Optimal result

Integrand size = 17, antiderivative size = 259 \[ \int (a+b \sinh (x))^{5/2} (A+B \sinh (x)) \, dx=\frac {2}{105} \left (56 a A b+15 a^2 B-25 b^2 B\right ) \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{35} (7 A b+5 a B) \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+b \sinh (x))^{5/2}+\frac {2 i \left (161 a^2 A b-63 A b^3+15 a^3 B-145 a b^2 B\right ) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{105 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) \left (56 a A b+15 a^2 B-25 b^2 B\right ) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{105 b \sqrt {a+b \sinh (x)}} \]

[Out]

2/35*(7*A*b+5*B*a)*cosh(x)*(a+b*sinh(x))^(3/2)+2/7*B*cosh(x)*(a+b*sinh(x))^(5/2)+2/105*(56*A*a*b+15*B*a^2-25*B
*b^2)*cosh(x)*(a+b*sinh(x))^(1/2)+2/105*I*(161*A*a^2*b-63*A*b^3+15*B*a^3-145*B*a*b^2)*(sin(1/4*Pi+1/2*I*x)^2)^
(1/2)/sin(1/4*Pi+1/2*I*x)*EllipticE(cos(1/4*Pi+1/2*I*x),2^(1/2)*(b/(I*a+b))^(1/2))*(a+b*sinh(x))^(1/2)/b/((a+b
*sinh(x))/(a-I*b))^(1/2)-2/105*I*(a^2+b^2)*(56*A*a*b+15*B*a^2-25*B*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)/b/(a+b*sinh
(x))^(1/2)

Rubi [A] (verified)

Time = 0.30 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2832, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \sinh (x))^{5/2} (A+B \sinh (x)) \, dx=\frac {2}{105} \cosh (x) \left (15 a^2 B+56 a A b-25 b^2 B\right ) \sqrt {a+b \sinh (x)}-\frac {2 i \left (a^2+b^2\right ) \left (15 a^2 B+56 a A b-25 b^2 B\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{105 b \sqrt {a+b \sinh (x)}}+\frac {2 i \left (15 a^3 B+161 a^2 A b-145 a b^2 B-63 A b^3\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{105 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2}{35} \cosh (x) (5 a B+7 A b) (a+b \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+b \sinh (x))^{5/2} \]

[In]

Int[(a + b*Sinh[x])^(5/2)*(A + B*Sinh[x]),x]

[Out]

(2*(56*a*A*b + 15*a^2*B - 25*b^2*B)*Cosh[x]*Sqrt[a + b*Sinh[x]])/105 + (2*(7*A*b + 5*a*B)*Cosh[x]*(a + b*Sinh[
x])^(3/2))/35 + (2*B*Cosh[x]*(a + b*Sinh[x])^(5/2))/7 + (((2*I)/105)*(161*a^2*A*b - 63*A*b^3 + 15*a^3*B - 145*
a*b^2*B)*EllipticE[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]*Sqrt[a + b*Sinh[x]])/(b*Sqrt[(a + b*Sinh[x])/(a - I*b)]) -
(((2*I)/105)*(a^2 + b^2)*(56*a*A*b + 15*a^2*B - 25*b^2*B)*EllipticF[Pi/4 - (I/2)*x, (2*b)/(I*a + b)]*Sqrt[(a
+ b*Sinh[x])/(a - I*b)])/(b*Sqrt[a + b*Sinh[x]])

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 2734

Int[Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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]

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 2742

Int[1/Sqrt[(a_) + (b_.)*sin[(c_.) + (d_.)*(x_)]], x_Symbol] :> Dist[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]

Rule 2831

Int[((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])/Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[(b*c
- a*d)/b, Int[1/Sqrt[a + b*Sin[e + f*x]], x], x] + Dist[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]

Rule 2832

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] + Dist[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[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]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{7} B \cosh (x) (a+b \sinh (x))^{5/2}+\frac {2}{7} \int (a+b \sinh (x))^{3/2} \left (\frac {1}{2} (7 a A-5 b B)+\frac {1}{2} (7 A b+5 a B) \sinh (x)\right ) \, dx \\ & = \frac {2}{35} (7 A b+5 a B) \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+b \sinh (x))^{5/2}+\frac {4}{35} \int \sqrt {a+b \sinh (x)} \left (\frac {1}{4} \left (35 a^2 A-21 A b^2-40 a b B\right )+\frac {1}{4} \left (56 a A b+15 a^2 B-25 b^2 B\right ) \sinh (x)\right ) \, dx \\ & = \frac {2}{105} \left (56 a A b+15 a^2 B-25 b^2 B\right ) \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{35} (7 A b+5 a B) \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+b \sinh (x))^{5/2}+\frac {8}{105} \int \frac {\frac {1}{8} \left (105 a^3 A-119 a A b^2-135 a^2 b B+25 b^3 B\right )+\frac {1}{8} \left (161 a^2 A b-63 A b^3+15 a^3 B-145 a b^2 B\right ) \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx \\ & = \frac {2}{105} \left (56 a A b+15 a^2 B-25 b^2 B\right ) \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{35} (7 A b+5 a B) \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+b \sinh (x))^{5/2}-\frac {\left (\left (a^2+b^2\right ) \left (56 a A b+15 a^2 B-25 b^2 B\right )\right ) \int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx}{105 b}+\frac {\left (161 a^2 A b-63 A b^3+15 a^3 B-145 a b^2 B\right ) \int \sqrt {a+b \sinh (x)} \, dx}{105 b} \\ & = \frac {2}{105} \left (56 a A b+15 a^2 B-25 b^2 B\right ) \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{35} (7 A b+5 a B) \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+b \sinh (x))^{5/2}+\frac {\left (\left (161 a^2 A b-63 A b^3+15 a^3 B-145 a b^2 B\right ) \sqrt {a+b \sinh (x)}\right ) \int \sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}} \, dx}{105 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (\left (a^2+b^2\right ) \left (56 a A b+15 a^2 B-25 b^2 B\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}\right ) \int \frac {1}{\sqrt {\frac {a}{a-i b}+\frac {b \sinh (x)}{a-i b}}} \, dx}{105 b \sqrt {a+b \sinh (x)}} \\ & = \frac {2}{105} \left (56 a A b+15 a^2 B-25 b^2 B\right ) \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{35} (7 A b+5 a B) \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2}{7} B \cosh (x) (a+b \sinh (x))^{5/2}+\frac {2 i \left (161 a^2 A b-63 A b^3+15 a^3 B-145 a b^2 B\right ) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{105 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) \left (56 a A b+15 a^2 B-25 b^2 B\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}}}{105 b \sqrt {a+b \sinh (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.62 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.93 \[ \int (a+b \sinh (x))^{5/2} (A+B \sinh (x)) \, dx=\frac {\frac {2 i \left (b \left (105 a^3 A-119 a A b^2-135 a^2 b B+25 b^3 B\right ) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )+\left (161 a^2 A b-63 A b^3+15 a^3 B-145 a b^2 B\right ) \left ((a-i b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )\right )\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b}+\cosh (x) (a+b \sinh (x)) \left (154 a A b+90 a^2 B-65 b^2 B+15 b^2 B \cosh (2 x)+6 b (7 A b+15 a B) \sinh (x)\right )}{105 \sqrt {a+b \sinh (x)}} \]

[In]

Integrate[(a + b*Sinh[x])^(5/2)*(A + B*Sinh[x]),x]

[Out]

(((2*I)*(b*(105*a^3*A - 119*a*A*b^2 - 135*a^2*b*B + 25*b^3*B)*EllipticF[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)
] + (161*a^2*A*b - 63*A*b^3 + 15*a^3*B - 145*a*b^2*B)*((a - I*b)*EllipticE[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I
*b)] - a*EllipticF[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)]))*Sqrt[(a + b*Sinh[x])/(a - I*b)])/b + Cosh[x]*(a +
b*Sinh[x])*(154*a*A*b + 90*a^2*B - 65*b^2*B + 15*b^2*B*Cosh[2*x] + 6*b*(7*A*b + 15*a*B)*Sinh[x]))/(105*Sqrt[a
+ b*Sinh[x]])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1877 vs. \(2 (279 ) = 558\).

Time = 5.22 (sec) , antiderivative size = 1878, normalized size of antiderivative = 7.25

method	result	size
parts	\(\text {Expression too large to display}\)	\(1878\)
default	\(\text {Expression too large to display}\)	\(1893\)

[In]

int((a+b*sinh(x))^(5/2)*(A+B*sinh(x)),x,method=_RETURNVERBOSE)

[Out]

2/15*A*(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)*Ellipti
cF((-(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)*El
lipticF((-(a+b*sinh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*a^4+6*(-(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^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-s
inh(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*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))*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)+2/21*B*(3*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^4*b-2*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*si
nh(x))/(I*b-a))^(1/2),(-(I*b-a)/(I*b+a))^(1/2))*a^2*b^3-5*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))*b^
5+3*b^5*sinh(x)^5-24*(-(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^2-24*(-(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^4-3*(-(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^5+26*(-(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^3*b^2+29*(-(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*b^4+12*a*b^4*sinh(x)^4+18*a^2*
b^3*sinh(x)^3-2*b^5*sinh(x)^3+9*a^3*b^2*sinh(x)^2+7*a*b^4*sinh(x)^2+18*a^2*b^3*sinh(x)-5*b^5*sinh(x)+9*a^3*b^2
-5*a*b^4)/b^2/cosh(x)/(a+b*sinh(x))^(1/2)

Fricas [C] (verification not implemented)

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

Time = 0.12 (sec) , antiderivative size = 1139, normalized size of antiderivative = 4.40 \[ \int (a+b \sinh (x))^{5/2} (A+B \sinh (x)) \, dx=\text {Too large to display} \]

[In]

integrate((a+b*sinh(x))^(5/2)*(A+B*sinh(x)),x, algorithm="fricas")

[Out]

-1/1260*(8*(sqrt(2)*(30*B*a^4 + 7*A*a^3*b + 115*B*a^2*b^2 + 231*A*a*b^3 - 75*B*b^4)*cosh(x)^3 + 3*sqrt(2)*(30*
B*a^4 + 7*A*a^3*b + 115*B*a^2*b^2 + 231*A*a*b^3 - 75*B*b^4)*cosh(x)^2*sinh(x) + 3*sqrt(2)*(30*B*a^4 + 7*A*a^3*
b + 115*B*a^2*b^2 + 231*A*a*b^3 - 75*B*b^4)*cosh(x)*sinh(x)^2 + sqrt(2)*(30*B*a^4 + 7*A*a^3*b + 115*B*a^2*b^2
+ 231*A*a*b^3 - 75*B*b^4)*sinh(x)^3)*sqrt(b)*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) + 24*(sqrt(2)*(15*B*a^3*b + 161*A*a^2*b^2 - 145*B*a*b^3 - 63
*A*b^4)*cosh(x)^3 + 3*sqrt(2)*(15*B*a^3*b + 161*A*a^2*b^2 - 145*B*a*b^3 - 63*A*b^4)*cosh(x)^2*sinh(x) + 3*sqrt
(2)*(15*B*a^3*b + 161*A*a^2*b^2 - 145*B*a*b^3 - 63*A*b^4)*cosh(x)*sinh(x)^2 + sqrt(2)*(15*B*a^3*b + 161*A*a^2*
b^2 - 145*B*a*b^3 - 63*A*b^4)*sinh(x)^3)*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*(15*B*b^4*cosh(x)^6 + 15*B*b^4*sinh(x)^6 + 6*(15*B*a*b^3 + 7*A*b^4)*cosh(x)^5 + 6*(15*B*b^4
*cosh(x) + 15*B*a*b^3 + 7*A*b^4)*sinh(x)^5 + 15*B*b^4 + (180*B*a^2*b^2 + 308*A*a*b^3 - 115*B*b^4)*cosh(x)^4 +
(225*B*b^4*cosh(x)^2 + 180*B*a^2*b^2 + 308*A*a*b^3 - 115*B*b^4 + 30*(15*B*a*b^3 + 7*A*b^4)*cosh(x))*sinh(x)^4
- 8*(15*B*a^3*b + 161*A*a^2*b^2 - 145*B*a*b^3 - 63*A*b^4)*cosh(x)^3 + 4*(75*B*b^4*cosh(x)^3 - 30*B*a^3*b - 322
*A*a^2*b^2 + 290*B*a*b^3 + 126*A*b^4 + 15*(15*B*a*b^3 + 7*A*b^4)*cosh(x)^2 + (180*B*a^2*b^2 + 308*A*a*b^3 - 11
5*B*b^4)*cosh(x))*sinh(x)^3 + (180*B*a^2*b^2 + 308*A*a*b^3 - 115*B*b^4)*cosh(x)^2 + (225*B*b^4*cosh(x)^4 + 180
*B*a^2*b^2 + 308*A*a*b^3 - 115*B*b^4 + 60*(15*B*a*b^3 + 7*A*b^4)*cosh(x)^3 + 6*(180*B*a^2*b^2 + 308*A*a*b^3 -
115*B*b^4)*cosh(x)^2 - 24*(15*B*a^3*b + 161*A*a^2*b^2 - 145*B*a*b^3 - 63*A*b^4)*cosh(x))*sinh(x)^2 - 6*(15*B*a
*b^3 + 7*A*b^4)*cosh(x) + 2*(45*B*b^4*cosh(x)^5 - 45*B*a*b^3 - 21*A*b^4 + 15*(15*B*a*b^3 + 7*A*b^4)*cosh(x)^4
+ 2*(180*B*a^2*b^2 + 308*A*a*b^3 - 115*B*b^4)*cosh(x)^3 - 12*(15*B*a^3*b + 161*A*a^2*b^2 - 145*B*a*b^3 - 63*A*
b^4)*cosh(x)^2 + (180*B*a^2*b^2 + 308*A*a*b^3 - 115*B*b^4)*cosh(x))*sinh(x))*sqrt(b*sinh(x) + a))/(b^2*cosh(x)
^3 + 3*b^2*cosh(x)^2*sinh(x) + 3*b^2*cosh(x)*sinh(x)^2 + b^2*sinh(x)^3)

Sympy [F(-1)]

Timed out. \[ \int (a+b \sinh (x))^{5/2} (A+B \sinh (x)) \, dx=\text {Timed out} \]

[In]

integrate((a+b*sinh(x))**(5/2)*(A+B*sinh(x)),x)

[Out]

Timed out

Maxima [F]

\[ \int (a+b \sinh (x))^{5/2} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} {\left (b \sinh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+b*sinh(x))^(5/2)*(A+B*sinh(x)),x, algorithm="maxima")

[Out]

integrate((B*sinh(x) + A)*(b*sinh(x) + a)^(5/2), x)

Giac [F]

\[ \int (a+b \sinh (x))^{5/2} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} {\left (b \sinh \left (x\right ) + a\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+b*sinh(x))^(5/2)*(A+B*sinh(x)),x, algorithm="giac")

[Out]

integrate((B*sinh(x) + A)*(b*sinh(x) + a)^(5/2), x)

Mupad [F(-1)]

Timed out. \[ \int (a+b \sinh (x))^{5/2} (A+B \sinh (x)) \, dx=\int \left (A+B\,\mathrm {sinh}\left (x\right )\right )\,{\left (a+b\,\mathrm {sinh}\left (x\right )\right )}^{5/2} \,d x \]

[In]

int((A + B*sinh(x))*(a + b*sinh(x))^(5/2),x)

[Out]

int((A + B*sinh(x))*(a + b*sinh(x))^(5/2), x)