\(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx\) [53]

Optimal result

Integrand size = 31, antiderivative size = 1702 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx =\text {Too large to display} \] Output:

b*d^2*(c^2*f^2-g^2)^2*(-c^2*d*x^2+d)^(1/2)*arccosh(c*x)/g^5-2/15*d^2*(-c*x 
+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/g+1/8*c^2*d^2*f*x*(-c^ 
2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/g^2-1/4*c^4*d^2*f*x^3*(-c^2*d*x^2+d)^( 
1/2)*(a+b*arccosh(c*x))/g^2-1/3*d^2*(c^2*f^2-2*g^2)*(-c*x+1)*(c*x+1)*(-c^2 
*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))/g^3+1/4*b*c^3*d^2*f*(c^2*f^2-2*g^2)*x^2 
*(-c^2*d*x^2+d)^(1/2)/g^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)+1/2*d^2*(c^2*f^2-g^2 
)^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/ 
b/c/g^4/(g*x+f)+1/4*c*d^2*f*(c^2*f^2-2*g^2)*(-c^2*d*x^2+d)^(1/2)*(a+b*arcc 
osh(c*x))^2/b/g^4/(c*x-1)^(1/2)/(c*x+1)^(1/2)-1/2*c*d^2*(c^2*f^2-g^2)^2*x* 
(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/g^5/(c*x-1)^(1/2)/(c*x+1)^(1/2 
)-1/2*d^2*(c^2*f^2-g^2)^3*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh(c*x))^2/b/c/g^ 
6/(c*x-1)^(1/2)/(c*x+1)^(1/2)/(g*x+f)+a*d^2*(c^2*f^2-g^2)^2*(-c^2*d*x^2+d) 
^(1/2)/g^5-b*d^2*(c^2*f^2-g^2)^(5/2)*(-c^2*d*x^2+d)^(1/2)*polylog(2,-(c*x+ 
(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2)^(1/2)))/g^6/(c*x-1)^(1/2 
)/(c*x+1)^(1/2)+b*d^2*(c^2*f^2-g^2)^(5/2)*(-c^2*d*x^2+d)^(1/2)*polylog(2,- 
(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))/g^6/(c*x-1) 
^(1/2)/(c*x+1)^(1/2)-1/25*b*c^5*d^2*x^5*(-c^2*d*x^2+d)^(1/2)/g/(c*x-1)^(1/ 
2)/(c*x+1)^(1/2)+2/15*b*c*d^2*x*(-c^2*d*x^2+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+ 
1)^(1/2)+1/45*b*c^3*d^2*x^3*(-c^2*d*x^2+d)^(1/2)/g/(c*x-1)^(1/2)/(c*x+1)^( 
1/2)-1/5*c^2*d^2*x^2*(-c*x+1)*(c*x+1)*(-c^2*d*x^2+d)^(1/2)*(a+b*arccosh...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 16.00 (sec) , antiderivative size = 6244, normalized size of antiderivative = 3.67 \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Result too large to show} \] Input:

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(f + g*x),x]
 

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 5.18 (sec) , antiderivative size = 1070, normalized size of antiderivative = 0.63, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6387, 6392, 2009}

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.

\(\displaystyle \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx\)

\(\Big \downarrow \) 6387

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \frac {(c x-1)^{5/2} (c x+1)^{5/2} (a+b \text {arccosh}(c x))}{f+g x}dx}{\sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 6392

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \int \left (\frac {x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^4}{g}-\frac {f x^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^4}{g^2}-\frac {f \left (c^2 f^2-2 g^2\right ) \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^2}{g^4}+\frac {\left (c^2 f^2-2 g^2\right ) x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^2}{g^3}+\frac {\left (g^2-c^2 f^2\right )^2 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{g^4 (f+g x)}\right )dx}{\sqrt {c x-1} \sqrt {c x+1}}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {d-c^2 d x^2} \left (-\frac {b x^5 c^5}{25 g}+\frac {b f x^4 c^5}{16 g^2}-\frac {f x^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^4}{4 g^2}-\frac {b \left (c^2 f^2-2 g^2\right ) x^3 c^3}{9 g^3}+\frac {b x^3 c^3}{45 g}+\frac {b f \left (c^2 f^2-2 g^2\right ) x^2 c^3}{4 g^4}-\frac {b f x^2 c^3}{16 g^2}+\frac {x^2 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x)) c^2}{5 g}-\frac {f \left (c^2 f^2-2 g^2\right ) x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^2}{2 g^4}+\frac {f x \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x)) c^2}{8 g^2}+\frac {f \left (c^2 f^2-2 g^2\right ) (a+b \text {arccosh}(c x))^2 c}{4 b g^4}-\frac {\left (c^2 f^2-g^2\right )^2 x (a+b \text {arccosh}(c x))^2 c}{2 b g^5}+\frac {f (a+b \text {arccosh}(c x))^2 c}{16 b g^2}-\frac {b \left (c^2 f^2-g^2\right )^2 x c}{g^5}+\frac {b \left (c^2 f^2-2 g^2\right ) x c}{3 g^3}+\frac {2 b x c}{15 g}-\frac {a \left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right )}{g^5 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (c^2 f^2-g^2\right )^2 \sqrt {c x-1} \sqrt {c x+1} \text {arccosh}(c x)}{g^5}+\frac {\left (c^2 f^2-2 g^2\right ) (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{3 g^3}+\frac {2 (c x-1)^{3/2} (c x+1)^{3/2} (a+b \text {arccosh}(c x))}{15 g}+\frac {a \left (c^2 f^2-g^2\right )^{5/2} \sqrt {c^2 x^2-1} \text {arctanh}\left (\frac {f x c^2+g}{\sqrt {c^2 f^2-g^2} \sqrt {c^2 x^2-1}}\right )}{g^6 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (c^2 f^2-g^2\right )^{5/2} \text {arccosh}(c x) \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g^6}-\frac {b \left (c^2 f^2-g^2\right )^{5/2} \text {arccosh}(c x) \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )}{g^6}+\frac {b \left (c^2 f^2-g^2\right )^{5/2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g^6}-\frac {b \left (c^2 f^2-g^2\right )^{5/2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g^6}-\frac {\left (c^2 f^2-g^2\right )^2 \left (1-c^2 x^2\right ) (a+b \text {arccosh}(c x))^2}{2 b g^4 (f+g x) c}-\frac {\left (c^2 f^2-g^2\right )^3 (a+b \text {arccosh}(c x))^2}{2 b g^6 (f+g x) c}\right )}{\sqrt {c x-1} \sqrt {c x+1}}\)

Input:

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x]))/(f + g*x),x]
 

Output:

(d^2*Sqrt[d - c^2*d*x^2]*((2*b*c*x)/(15*g) + (b*c*(c^2*f^2 - 2*g^2)*x)/(3* 
g^3) - (b*c*(c^2*f^2 - g^2)^2*x)/g^5 - (b*c^3*f*x^2)/(16*g^2) + (b*c^3*f*( 
c^2*f^2 - 2*g^2)*x^2)/(4*g^4) + (b*c^3*x^3)/(45*g) - (b*c^3*(c^2*f^2 - 2*g 
^2)*x^3)/(9*g^3) + (b*c^5*f*x^4)/(16*g^2) - (b*c^5*x^5)/(25*g) - (a*(c^2*f 
^2 - g^2)^2*(1 - c^2*x^2))/(g^5*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*(c^2*f^ 
2 - g^2)^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*ArcCosh[c*x])/g^5 + (c^2*f*x*Sqrt[ 
-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(8*g^2) - (c^2*f*(c^2*f^2 - 
2*g^2)*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*g^4) - (c^4 
*f*x^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(4*g^2) + (2*(-1 
 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(15*g) + ((c^2*f^2 - 2 
*g^2)*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(3*g^3) + (c^ 
2*x^2*(-1 + c*x)^(3/2)*(1 + c*x)^(3/2)*(a + b*ArcCosh[c*x]))/(5*g) + (c*f* 
(a + b*ArcCosh[c*x])^2)/(16*b*g^2) + (c*f*(c^2*f^2 - 2*g^2)*(a + b*ArcCosh 
[c*x])^2)/(4*b*g^4) - (c*(c^2*f^2 - g^2)^2*x*(a + b*ArcCosh[c*x])^2)/(2*b* 
g^5) - ((c^2*f^2 - g^2)^3*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^6*(f + g*x)) - 
((c^2*f^2 - g^2)^2*(1 - c^2*x^2)*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^4*(f + g 
*x)) + (a*(c^2*f^2 - g^2)^(5/2)*Sqrt[-1 + c^2*x^2]*ArcTanh[(g + c^2*f*x)/( 
Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2])])/(g^6*Sqrt[-1 + c*x]*Sqrt[1 + c*x 
]) + (b*(c^2*f^2 - g^2)^(5/2)*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f 
 - Sqrt[c^2*f^2 - g^2])])/g^6 - (b*(c^2*f^2 - g^2)^(5/2)*ArcCosh[c*x]*L...
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 6387

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d 
_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra 
cPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p]))   Int[(f + g*x)^m* 
(-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, 
d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]
 

rule 6392

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( 
d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand 
Integrand[Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]*(a + b*ArcCosh[c*x])^n, (f + g*x) 
^m*(d1 + e1*x)^(p - 1/2)*(d2 + e2*x)^(p - 1/2), x], x] /; FreeQ[{a, b, c, d 
1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && Integ 
erQ[m] && IGtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3768\) vs. \(2(1558)=3116\).

Time = 0.75 (sec) , antiderivative size = 3769, normalized size of antiderivative = 2.21

method	result	size
default	\(\text {Expression too large to display}\)	\(3769\)
parts	\(\text {Expression too large to display}\)	\(3769\)

Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/(g*x+f),x,method=_RETURNVERBOS 
E)
 

Output:

b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/ 
2)/g^2*dilog((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-f*c+(c^2*f^2-g^2)^(1/2) 
)/(-f*c+(c^2*f^2-g^2)^(1/2)))-b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^( 
1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*dilog(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/ 
2))*g+f*c+(c^2*f^2-g^2)^(1/2))/(f*c+(c^2*f^2-g^2)^(1/2)))-1/25*b*(-d*(c^2* 
x^2-1))^(1/2)*d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g*x^5*c^5+11/45*b*(-d*(c^2*x 
^2-1))^(1/2)*d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g*x^3*c^3-23/15*b*(-d*(c^2*x^ 
2-1))^(1/2)*d^2/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g*x*c+33/128*b*(-d*(c^2*x^2-1) 
)^(1/2)*f*d^2*c/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2-1/8*b*(-d*(c^2*x^2-1))^(1/ 
2)*f^3*d^2*c^3/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4-23/15*b*(-d*(c^2*x^2-1))^(1 
/2)*d^2/(c*x-1)/(c*x+1)/g*arccosh(c*x)+b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f 
^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^6*arccosh(c*x)*ln((-(c*x+(c*x- 
1)^(1/2)*(c*x+1)^(1/2))*g-f*c+(c^2*f^2-g^2)^(1/2))/(-f*c+(c^2*f^2-g^2)^(1/ 
2)))*c^4*f^4-b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2 
)/(c*x+1)^(1/2)/g^6*arccosh(c*x)*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+f 
*c+(c^2*f^2-g^2)^(1/2))/(f*c+(c^2*f^2-g^2)^(1/2)))*c^4*f^4-2*b*d^2*(-d*(c^ 
2*x^2-1))^(1/2)*(c^2*f^2-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*arccos 
h(c*x)*ln((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-f*c+(c^2*f^2-g^2)^(1/2))/( 
-f*c+(c^2*f^2-g^2)^(1/2)))*c^2*f^2+2*b*d^2*(-d*(c^2*x^2-1))^(1/2)*(c^2*f^2 
-g^2)^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*arccosh(c*x)*ln(((c*x+(c*x-...
 

Fricas [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int { \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{g x + f} \,d x } \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="fr 
icas")
 

Output:

integral((a*c^4*d^2*x^4 - 2*a*c^2*d^2*x^2 + a*d^2 + (b*c^4*d^2*x^4 - 2*b*c 
^2*d^2*x^2 + b*d^2)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/(g*x + f), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Timed out} \] Input:

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))/(g*x+f),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="ma 
xima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(g-c*f>0)', see `assume?` for mor 
e details)
 

Giac [F(-2)]

Exception generated. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="gi 
ac")
 

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const 
index_m & i,const vecteur & l) Error: Bad Argument Value
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{f+g\,x} \,d x \] Input:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/(f + g*x),x)
 

Output:

int(((a + b*acosh(c*x))*(d - c^2*d*x^2)^(5/2))/(f + g*x), x)
 

Reduce [F]

\[ \int \frac {\left (d-c^2 d x^2\right )^{5/2} (a+b \text {arccosh}(c x))}{f+g x} \, dx=\frac {\sqrt {d}\, d^{2} \left (120 \mathit {asin} \left (c x \right ) a \,c^{5} f^{5}-300 \mathit {asin} \left (c x \right ) a \,c^{3} f^{3} g^{2}+225 \mathit {asin} \left (c x \right ) a c f \,g^{4}-240 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{4} f^{4}+480 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f^{2} g^{2}-240 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,g^{4}+120 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{4} g -60 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{3} g^{2} x +40 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f^{2} g^{3} x^{2}-30 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} f \,g^{4} x^{3}+24 \sqrt {-c^{2} x^{2}+1}\, a \,c^{4} g^{5} x^{4}-280 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g^{3}+135 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f \,g^{4} x -88 \sqrt {-c^{2} x^{2}+1}\, a \,c^{2} g^{5} x^{2}+184 \sqrt {-c^{2} x^{2}+1}\, a \,g^{5}+120 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{4}}{g x +f}d x \right ) b \,c^{4} g^{6}-240 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right ) x^{2}}{g x +f}d x \right ) b \,c^{2} g^{6}+120 \left (\int \frac {\sqrt {-c^{2} x^{2}+1}\, \mathit {acosh} \left (c x \right )}{g x +f}d x \right ) b \,g^{6}+72 a \,c^{4} f^{4} g -136 a \,c^{2} f^{2} g^{3}+40 a \,g^{5}\right )}{120 g^{6}} \] Input:

int((-c^2*d*x^2+d)^(5/2)*(a+b*acosh(c*x))/(g*x+f),x)
 

Output:

(sqrt(d)*d**2*(120*asin(c*x)*a*c**5*f**5 - 300*asin(c*x)*a*c**3*f**3*g**2 
+ 225*asin(c*x)*a*c*f*g**4 - 240*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x 
)/2)*c*f + g)/sqrt(c**2*f**2 - g**2))*a*c**4*f**4 + 480*sqrt(c**2*f**2 - g 
**2)*atan((tan(asin(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - g**2))*a*c**2*f**2*g 
**2 - 240*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sqrt(c**2 
*f**2 - g**2))*a*g**4 + 120*sqrt( - c**2*x**2 + 1)*a*c**4*f**4*g - 60*sqrt 
( - c**2*x**2 + 1)*a*c**4*f**3*g**2*x + 40*sqrt( - c**2*x**2 + 1)*a*c**4*f 
**2*g**3*x**2 - 30*sqrt( - c**2*x**2 + 1)*a*c**4*f*g**4*x**3 + 24*sqrt( - 
c**2*x**2 + 1)*a*c**4*g**5*x**4 - 280*sqrt( - c**2*x**2 + 1)*a*c**2*f**2*g 
**3 + 135*sqrt( - c**2*x**2 + 1)*a*c**2*f*g**4*x - 88*sqrt( - c**2*x**2 + 
1)*a*c**2*g**5*x**2 + 184*sqrt( - c**2*x**2 + 1)*a*g**5 + 120*int((sqrt( - 
 c**2*x**2 + 1)*acosh(c*x)*x**4)/(f + g*x),x)*b*c**4*g**6 - 240*int((sqrt( 
 - c**2*x**2 + 1)*acosh(c*x)*x**2)/(f + g*x),x)*b*c**2*g**6 + 120*int((sqr 
t( - c**2*x**2 + 1)*acosh(c*x))/(f + g*x),x)*b*g**6 + 72*a*c**4*f**4*g - 1 
36*a*c**2*f**2*g**3 + 40*a*g**5))/(120*g**6)