\(\int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx\) [127]

Optimal result

Integrand size = 17, antiderivative size = 207 \[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\frac {2}{15} (5 A b+3 a B) \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2 i \left (20 a A b+3 a^2 B-9 b^2 B\right ) E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right ) \sqrt {a+b \sinh (x)}}{15 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) (5 A b+3 a B) \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{15 b \sqrt {a+b \sinh (x)}} \]

[Out]

2/5*B*cosh(x)*(a+b*sinh(x))^(3/2)+2/15*(5*A*b+3*B*a)*cosh(x)*(a+b*sinh(x))^(1/2)+2/15*I*(20*A*a*b+3*B*a^2-9*B*
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)-2/15*I*(a^2+b^2)*(5*A*b+3*B*a)*(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 = 0.21 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2832, 2831, 2742, 2740, 2734, 2732} \[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=-\frac {2 i \left (a^2+b^2\right ) (3 a B+5 A b) \sqrt {\frac {a+b \sinh (x)}{a-i b}} \operatorname {EllipticF}\left (\frac {\pi }{4}-\frac {i x}{2},\frac {2 b}{i a+b}\right )}{15 b \sqrt {a+b \sinh (x)}}+\frac {2 i \left (3 a^2 B+20 a A b-9 b^2 B\right ) \sqrt {a+b \sinh (x)} E\left (\frac {\pi }{4}-\frac {i x}{2}|\frac {2 b}{i a+b}\right )}{15 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}+\frac {2}{15} \cosh (x) (3 a B+5 A b) \sqrt {a+b \sinh (x)}+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2} \]

[In]

Int[(a + b*Sinh[x])^(3/2)*(A + B*Sinh[x]),x]

[Out]

(2*(5*A*b + 3*a*B)*Cosh[x]*Sqrt[a + b*Sinh[x]])/15 + (2*B*Cosh[x]*(a + b*Sinh[x])^(3/2))/5 + (((2*I)/15)*(20*a
*A*b + 3*a^2*B - 9*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)/15)*(a^2 + b^2)*(5*A*b + 3*a*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}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {2}{5} \int \sqrt {a+b \sinh (x)} \left (\frac {1}{2} (5 a A-3 b B)+\frac {1}{2} (5 A b+3 a B) \sinh (x)\right ) \, dx \\ & = \frac {2}{15} (5 A b+3 a B) \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {4}{15} \int \frac {\frac {1}{4} \left (15 a^2 A-5 A b^2-12 a b B\right )+\frac {1}{4} \left (20 a A b+3 a^2 B-9 b^2 B\right ) \sinh (x)}{\sqrt {a+b \sinh (x)}} \, dx \\ & = \frac {2}{15} (5 A b+3 a B) \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}-\frac {\left (\left (a^2+b^2\right ) (5 A b+3 a B)\right ) \int \frac {1}{\sqrt {a+b \sinh (x)}} \, dx}{15 b}+\frac {\left (20 a A b+3 a^2 B-9 b^2 B\right ) \int \sqrt {a+b \sinh (x)} \, dx}{15 b} \\ & = \frac {2}{15} (5 A b+3 a B) \cosh (x) \sqrt {a+b \sinh (x)}+\frac {2}{5} B \cosh (x) (a+b \sinh (x))^{3/2}+\frac {\left (\left (20 a A b+3 a^2 B-9 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}{15 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {\left (\left (a^2+b^2\right ) (5 A b+3 a B) \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}{15 b \sqrt {a+b \sinh (x)}} \\ & = \frac {2}{15} (5 A b+3 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 (20 a A b+3 a^2 B-9 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)}}{15 b \sqrt {\frac {a+b \sinh (x)}{a-i b}}}-\frac {2 i \left (a^2+b^2\right ) (5 A b+3 a B) \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 b \sqrt {a+b \sinh (x)}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.44 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.95 \[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\frac {2 \left (\frac {i \left (b \left (15 a^2 A-5 A b^2-12 a b B\right ) \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )+\left (20 a A b+3 a^2 B-9 b^2 B\right ) \left ((a-i b) E\left (\frac {1}{4} (\pi -2 i x)|-\frac {2 i b}{a-i b}\right )-a \operatorname {EllipticF}\left (\frac {1}{4} (\pi -2 i x),-\frac {2 i b}{a-i b}\right )\right )\right ) \sqrt {\frac {a+b \sinh (x)}{a-i b}}}{b}+\cosh (x) (a+b \sinh (x)) (5 A b+6 a B+3 b B \sinh (x))\right )}{15 \sqrt {a+b \sinh (x)}} \]

[In]

Integrate[(a + b*Sinh[x])^(3/2)*(A + B*Sinh[x]),x]

[Out]

(2*((I*(b*(15*a^2*A - 5*A*b^2 - 12*a*b*B)*EllipticF[(Pi - (2*I)*x)/4, ((-2*I)*b)/(a - I*b)] + (20*a*A*b + 3*a^
2*B - 9*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])*(5*A*b + 6*a*B + 3*b*B*Sinh
[x])))/(15*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. 1036 vs. \(2 (231 ) = 462\).

Time = 4.32 (sec) , antiderivative size = 1037, normalized size of antiderivative = 5.01

method	result	size
default	\(\text {Expression too large to display}\)	\(1037\)
parts	\(\text {Expression too large to display}\)	\(1489\)

[In]

int((a+b*sinh(x))^(3/2)*(A+B*sinh(x)),x,method=_RETURNVERBOSE)

[Out]

(cosh(x)^2*(a+b*sinh(x)))^(1/2)*(2*a^2*A*(a/b-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)/(cosh(x)^2*(a+b*sinh(x)))^(1/2)*EllipticF(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b
)/(I*b+a))^(1/2))+B*b^2*(2/5/b*sinh(x)*(cosh(x)^2*(a+b*sinh(x)))^(1/2)-8/15*a/b^2*(cosh(x)^2*(a+b*sinh(x)))^(1
/2)-4/15*a/b*(a/b-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
)/(cosh(x)^2*(a+b*sinh(x)))^(1/2)*EllipticF(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/(I*b+a))^(1/2))+2*(-3/5+8/
15*a^2/b^2)*(a/b-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)
/(cosh(x)^2*(a+b*sinh(x)))^(1/2)*((-a/b-I)*EllipticE(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/(I*b+a))^(1/2))+I
*EllipticF(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/(I*b+a))^(1/2))))+(A*b^2+2*B*a*b)*(2/3/b*(cosh(x)^2*(a+b*si
nh(x)))^(1/2)-2/3*(a/b-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)/(cosh(x)^2*(a+b*sinh(x)))^(1/2)*EllipticF(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/(I*b+a))^(1/2))-4/3*a
/b*(a/b-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)/(cosh(x)
^2*(a+b*sinh(x)))^(1/2)*((-a/b-I)*EllipticE(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/(I*b+a))^(1/2))+I*Elliptic
F(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/(I*b+a))^(1/2))))+2*(2*A*a*b+B*a^2)*(a/b-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)/(cosh(x)^2*(a+b*sinh(x)))^(1/2)*((-a/b-I)*E
llipticE(((-a-b*sinh(x))/(I*b-a))^(1/2),((a-I*b)/(I*b+a))^(1/2))+I*EllipticF(((-a-b*sinh(x))/(I*b-a))^(1/2),((
a-I*b)/(I*b+a))^(1/2))))/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 = 635, normalized size of antiderivative = 3.07 \[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=-\frac {4 \, {\left (\sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b + 18 \, B a b^{2} + 15 \, A b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b + 18 \, B a b^{2} + 15 \, A b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (6 \, B a^{3} - 5 \, A a^{2} b + 18 \, B a b^{2} + 15 \, A b^{3}\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 (3 \, B a^{2} b + 20 \, A a b^{2} - 9 \, B b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, \sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} - 9 \, B b^{3}\right )} \cosh \left (x\right ) \sinh \left (x\right ) + \sqrt {2} {\left (3 \, B a^{2} b + 20 \, A a b^{2} - 9 \, B 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 b^{3} \cosh \left (x\right )^{4} + 3 \, B b^{3} \sinh \left (x\right )^{4} - 3 \, B b^{3} + 2 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )^{3} + 2 \, {\left (6 \, B b^{3} \cosh \left (x\right ) + 6 \, B a b^{2} + 5 \, A b^{3}\right )} \sinh \left (x\right )^{3} - 4 \, {\left (3 \, B a^{2} b + 20 \, A a b^{2} - 9 \, B b^{3}\right )} \cosh \left (x\right )^{2} + 2 \, {\left (9 \, B b^{3} \cosh \left (x\right )^{2} - 6 \, B a^{2} b - 40 \, A a b^{2} + 18 \, B b^{3} + 3 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 2 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right ) + 2 \, {\left (6 \, B b^{3} \cosh \left (x\right )^{3} + 6 \, B a b^{2} + 5 \, A b^{3} + 3 \, {\left (6 \, B a b^{2} + 5 \, A b^{3}\right )} \cosh \left (x\right )^{2} - 4 \, {\left (3 \, B a^{2} b + 20 \, A a b^{2} - 9 \, B b^{3}\right )} \cosh \left (x\right )\right )} \sinh \left (x\right )\right )} \sqrt {b \sinh \left (x\right ) + a}}{90 \, {\left (b^{2} \cosh \left (x\right )^{2} + 2 \, b^{2} \cosh \left (x\right ) \sinh \left (x\right ) + b^{2} \sinh \left (x\right )^{2}\right )}} \]

[In]

integrate((a+b*sinh(x))^(3/2)*(A+B*sinh(x)),x, algorithm="fricas")

[Out]

-1/90*(4*(sqrt(2)*(6*B*a^3 - 5*A*a^2*b + 18*B*a*b^2 + 15*A*b^3)*cosh(x)^2 + 2*sqrt(2)*(6*B*a^3 - 5*A*a^2*b + 1
8*B*a*b^2 + 15*A*b^3)*cosh(x)*sinh(x) + sqrt(2)*(6*B*a^3 - 5*A*a^2*b + 18*B*a*b^2 + 15*A*b^3)*sinh(x)^2)*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) + 12*(sqrt(2)*(3*B*a^2*b + 20*A*a*b^2 - 9*B*b^3)*cosh(x)^2 + 2*sqrt(2)*(3*B*a^2*b + 20*A*a*b^2 - 9*B*b
^3)*cosh(x)*sinh(x) + sqrt(2)*(3*B*a^2*b + 20*A*a*b^2 - 9*B*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*b^3*cosh(x)^4 + 3*B*b^3*sinh(x)^4 - 3*B*b^3 + 2*(6*
B*a*b^2 + 5*A*b^3)*cosh(x)^3 + 2*(6*B*b^3*cosh(x) + 6*B*a*b^2 + 5*A*b^3)*sinh(x)^3 - 4*(3*B*a^2*b + 20*A*a*b^2
- 9*B*b^3)*cosh(x)^2 + 2*(9*B*b^3*cosh(x)^2 - 6*B*a^2*b - 40*A*a*b^2 + 18*B*b^3 + 3*(6*B*a*b^2 + 5*A*b^3)*cos
h(x))*sinh(x)^2 + 2*(6*B*a*b^2 + 5*A*b^3)*cosh(x) + 2*(6*B*b^3*cosh(x)^3 + 6*B*a*b^2 + 5*A*b^3 + 3*(6*B*a*b^2
+ 5*A*b^3)*cosh(x)^2 - 4*(3*B*a^2*b + 20*A*a*b^2 - 9*B*b^3)*cosh(x))*sinh(x))*sqrt(b*sinh(x) + a))/(b^2*cosh(x
)^2 + 2*b^2*cosh(x)*sinh(x) + b^2*sinh(x)^2)

Sympy [F]

\[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int \left (A + B \sinh {\left (x \right )}\right ) \left (a + b \sinh {\left (x \right )}\right )^{\frac {3}{2}}\, dx \]

[In]

integrate((a+b*sinh(x))**(3/2)*(A+B*sinh(x)),x)

[Out]

Integral((A + B*sinh(x))*(a + b*sinh(x))**(3/2), x)

Maxima [F]

\[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} {\left (b \sinh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*sinh(x))^(3/2)*(A+B*sinh(x)),x, algorithm="maxima")

[Out]

integrate((B*sinh(x) + A)*(b*sinh(x) + a)^(3/2), x)

Giac [F]

\[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int { {\left (B \sinh \left (x\right ) + A\right )} {\left (b \sinh \left (x\right ) + a\right )}^{\frac {3}{2}} \,d x } \]

[In]

integrate((a+b*sinh(x))^(3/2)*(A+B*sinh(x)),x, algorithm="giac")

[Out]

integrate((B*sinh(x) + A)*(b*sinh(x) + a)^(3/2), x)

Mupad [F(-1)]

Timed out. \[ \int (a+b \sinh (x))^{3/2} (A+B \sinh (x)) \, dx=\int \left (A+B\,\mathrm {sinh}\left (x\right )\right )\,{\left (a+b\,\mathrm {sinh}\left (x\right )\right )}^{3/2} \,d x \]

[In]

int((A + B*sinh(x))*(a + b*sinh(x))^(3/2),x)

[Out]

int((A + B*sinh(x))*(a + b*sinh(x))^(3/2), x)