\(\int \frac {(d+c^2 d x^2)^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx\) [46]

Optimal result

Integrand size = 30, antiderivative size = 1500 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx =\text {Too large to display} \] Output:

-1/4*c*d^2*f*(c^2*f^2+2*g^2)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b/g^ 
4/(c^2*x^2+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*a 
rcsinh(c*x))^2/b/g^5/(c^2*x^2+1)^(1/2)-1/2*d^2*(c^2*f^2+g^2)^3*(c^2*d*x^2+ 
d)^(1/2)*(a+b*arcsinh(c*x))^2/b/c/g^6/(g*x+f)/(c^2*x^2+1)^(1/2)+1/2*d^2*(c 
^2*f^2+g^2)^2*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b 
/c/g^4/(g*x+f)+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^2*x^2+1)^(1/2)+1/5*(c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/g-b*c*d^2*(c 
^2*f^2+g^2)^2*x*(c^2*d*x^2+d)^(1/2)/g^5/(c^2*x^2+1)^(1/2)-b*d^2*(c^2*f^2+g 
^2)^(5/2)*(c^2*d*x^2+d)^(1/2)*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/ 
(c*f+(c^2*f^2+g^2)^(1/2)))/g^6/(c^2*x^2+1)^(1/2)+b*d^2*(c^2*f^2+g^2)^(5/2) 
*(c^2*d*x^2+d)^(1/2)*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c^2 
*f^2+g^2)^(1/2)))/g^6/(c^2*x^2+1)^(1/2)-1/3*b*c*d^2*(c^2*f^2+2*g^2)*x*(c^2 
*d*x^2+d)^(1/2)/g^3/(c^2*x^2+1)^(1/2)+1/16*b*c^3*d^2*f*x^2*(c^2*d*x^2+d)^( 
1/2)/g^2/(c^2*x^2+1)^(1/2)-1/9*b*c^3*d^2*(c^2*f^2+2*g^2)*x^3*(c^2*d*x^2+d) 
^(1/2)/g^3/(c^2*x^2+1)^(1/2)+1/16*b*c^5*d^2*f*x^4*(c^2*d*x^2+d)^(1/2)/g^2/ 
(c^2*x^2+1)^(1/2)-1/2*c^2*d^2*f*(c^2*f^2+2*g^2)*x*(c^2*d*x^2+d)^(1/2)*(a+b 
*arcsinh(c*x))/g^4+1/16*c*d^2*f*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))^2/b 
/g^2/(c^2*x^2+1)^(1/2)-b*d^2*(c^2*f^2+g^2)^(5/2)*(c^2*d*x^2+d)^(1/2)*polyl 
og(2,-(c*x+(c^2*x^2+1)^(1/2))*g/(c*f+(c^2*f^2+g^2)^(1/2)))/g^6/(c^2*x^2+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...
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 22.88 (sec) , antiderivative size = 7168, normalized size of antiderivative = 4.78 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 2.88 (sec) , antiderivative size = 945, normalized size of antiderivative = 0.63, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6260, 6255, 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.

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

\(\Big \downarrow \) 6260

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

\(\Big \downarrow \) 6255

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

\(\Big \downarrow \) 2009

\(\displaystyle \frac {d^2 \sqrt {c^2 d x^2+d} \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^2 x^2+1} (a+b \text {arcsinh}(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 {f \left (c^2 f^2+2 g^2\right ) x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) c^2}{2 g^4}-\frac {f x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x)) c^2}{8 g^2}-\frac {f \left (c^2 f^2+2 g^2\right ) (a+b \text {arcsinh}(c x))^2 c}{4 b g^4}-\frac {\left (c^2 f^2+g^2\right )^2 x (a+b \text {arcsinh}(c x))^2 c}{2 b g^5}+\frac {f (a+b \text {arcsinh}(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 {b \left (c^2 f^2+g^2\right )^2 \sqrt {c^2 x^2+1} \text {arcsinh}(c x)}{g^5}+\frac {\left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 g}+\frac {\left (c^2 f^2+2 g^2\right ) \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 g^3}-\frac {\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))}{3 g}-\frac {a \left (c^2 f^2+g^2\right )^{5/2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {c^2 x^2+1}}\right )}{g^6}+\frac {b \left (c^2 f^2+g^2\right )^{5/2} \text {arcsinh}(c x) \log \left (\frac {e^{\text {arcsinh}(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 {arcsinh}(c x) \log \left (\frac {e^{\text {arcsinh}(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 {arcsinh}(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 {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^6}+\frac {a \left (c^2 f^2+g^2\right )^2 \sqrt {c^2 x^2+1}}{g^5}+\frac {\left (c^2 f^2+g^2\right )^2 \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))^2}{2 b g^4 (f+g x) c}-\frac {\left (c^2 f^2+g^2\right )^3 (a+b \text {arcsinh}(c x))^2}{2 b g^6 (f+g x) c}\right )}{\sqrt {c^2 x^2+1}}\)

Input:

Int[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[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 + g^2)^2*x)/g^5 
 - (b*c*(c^2*f^2 + 2*g^2)*x)/(3*g^3) + (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*Sqrt[1 + c^2*x^2])/g^5 + (b*(c^2*f^2 + g^2)^2*Sqrt[1 + c^2*x^2 
]*ArcSinh[c*x])/g^5 - (c^2*f*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(8* 
g^2) - (c^2*f*(c^2*f^2 + 2*g^2)*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/ 
(2*g^4) - (c^4*f*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(4*g^2) - ((1 
 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*g) + ((c^2*f^2 + 2*g^2)*(1 + c^ 
2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/(3*g^3) + ((1 + c^2*x^2)^(5/2)*(a + b*A 
rcSinh[c*x]))/(5*g) + (c*f*(a + b*ArcSinh[c*x])^2)/(16*b*g^2) - (c*f*(c^2* 
f^2 + 2*g^2)*(a + b*ArcSinh[c*x])^2)/(4*b*g^4) - (c*(c^2*f^2 + g^2)^2*x*(a 
 + b*ArcSinh[c*x])^2)/(2*b*g^5) - ((c^2*f^2 + g^2)^3*(a + b*ArcSinh[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*ArcSinh 
[c*x])^2)/(2*b*c*g^4*(f + g*x)) - (a*(c^2*f^2 + g^2)^(5/2)*ArcTanh[(g - c^ 
2*f*x)/(Sqrt[c^2*f^2 + g^2]*Sqrt[1 + c^2*x^2])])/g^6 + (b*(c^2*f^2 + g^2)^ 
(5/2)*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2])] 
)/g^6 - (b*(c^2*f^2 + g^2)^(5/2)*ArcSinh[c*x]*Log[1 + (E^ArcSinh[c*x]*g)/( 
c*f + Sqrt[c^2*f^2 + g^2])])/g^6 + (b*(c^2*f^2 + g^2)^(5/2)*PolyLog[2, -(( 
E^ArcSinh[c*x]*g)/(c*f - Sqrt[c^2*f^2 + g^2]))])/g^6 - (b*(c^2*f^2 + g^...
 

Defintions of rubi rules used

rule 2009

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

rule 6255

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

rule 6260

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3498\) vs. \(2(1384)=2768\).

Time = 1.64 (sec) , antiderivative size = 3499, normalized size of antiderivative = 2.33

method	result	size
default	\(\text {Expression too large to display}\)	\(3499\)
parts	\(\text {Expression too large to display}\)	\(3499\)

Input:

int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))/(g*x+f),x,method=_RETURNVERBOSE 
)
 

Output:

14/15*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g*arcsinh(x*c)*x^4*c^4+34/15 
*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g*arcsinh(x*c)*x^2*c^2+b*(d*(c^2* 
x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g^5*arcsinh(x*c)*c^4*f^4-1/2*b*(d*(c^2*x^2+1 
))^(1/2)/(c^2*x^2+1)^(1/2)*f^5*arcsinh(x*c)^2*d^2*c^5/g^6-5/4*b*(d*(c^2*x^ 
2+1))^(1/2)/(c^2*x^2+1)^(1/2)*f^3*arcsinh(x*c)^2*d^2*c^3/g^4+1/16*b*(d*(c^ 
2*x^2+1))^(1/2)*f*d^2*c^5/(c^2*x^2+1)^(1/2)/g^2*x^4+9/16*b*(d*(c^2*x^2+1)) 
^(1/2)*f*d^2*c^3/(c^2*x^2+1)^(1/2)/g^2*x^2+1/4*b*(d*(c^2*x^2+1))^(1/2)*f^3 
*d^2*c^5/(c^2*x^2+1)^(1/2)/g^4*x^2-b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1) 
^(1/2)/g^5*x*c^5*f^4-7/3*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)^(1/2)/g^3 
*x*c^3*f^2-1/9*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)^(1/2)/g^3*x^3*c^5*f 
^2+7/3*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g^3*arcsinh(x*c)*c^2*f^2+b* 
d^2*(d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1/2)/g^2*arcsin 
h(x*c)*ln((-(x*c+(c^2*x^2+1)^(1/2))*g-c*f+(c^2*f^2+g^2)^(1/2))/(-c*f+(c^2* 
f^2+g^2)^(1/2)))-15/16*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)*f*arcsinh 
(x*c)^2*d^2*c/g^2-b*d^2*(d*(c^2*x^2+1))^(1/2)*(c^2*f^2+g^2)^(1/2)/(c^2*x^2 
+1)^(1/2)/g^2*arcsinh(x*c)*ln(((x*c+(c^2*x^2+1)^(1/2))*g+c*f+(c^2*f^2+g^2) 
^(1/2))/(c*f+(c^2*f^2+g^2)^(1/2)))+1/5*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^ 
2+1)/g*arcsinh(x*c)*x^6*c^6+23/15*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/ 
g*arcsinh(x*c)+33/128*b*(d*(c^2*x^2+1))^(1/2)*f*d^2*c/(c^2*x^2+1)^(1/2)/g^ 
2-1/4*b*(d*(c^2*x^2+1))^(1/2)*f*d^2*c^6/(c^2*x^2+1)/g^2*arcsinh(x*c)*x^...
 

Fricas [F]

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

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/(g*x+f),x, algorithm="fri 
cas")
 

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)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d)/(g*x + f), x)
 

Sympy [F]

\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{f + g x}\, dx \] Input:

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

Output:

Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))/(f + g*x), x)
 

Maxima [F(-2)]

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

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/(g*x+f),x, algorithm="max 
ima")
 

Output:

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati 
ve exponent.
 

Giac [F(-2)]

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

integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))/(g*x+f),x, algorithm="gia 
c")
 

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

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

Output:

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

Reduce [F]

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

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

Output:

(sqrt(d)*d**2*(240*sqrt(c**2*f**2 + g**2)*atan((sqrt(c**2*x**2 + 1)*g*i + 
c*f*i + c*g*i*x)/sqrt(c**2*f**2 + g**2))*a*c**4*f**4*i + 480*sqrt(c**2*f** 
2 + g**2)*atan((sqrt(c**2*x**2 + 1)*g*i + c*f*i + c*g*i*x)/sqrt(c**2*f**2 
+ g**2))*a*c**2*f**2*g**2*i + 240*sqrt(c**2*f**2 + g**2)*atan((sqrt(c**2*x 
**2 + 1)*g*i + c*f*i + c*g*i*x)/sqrt(c**2*f**2 + g**2))*a*g**4*i + 120*sqr 
t(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 + 8 
8*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)*asinh(c*x)*x**4)/(f + g*x),x)*b*c**4*g**6 + 2 
40*int((sqrt(c**2*x**2 + 1)*asinh(c*x)*x**2)/(f + g*x),x)*b*c**2*g**6 + 12 
0*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/(f + g*x),x)*b*g**6 - 120*log(sqrt( 
c**2*x**2 + 1) + c*x)*a*c**5*f**5 - 300*log(sqrt(c**2*x**2 + 1) + c*x)*a*c 
**3*f**3*g**2 - 225*log(sqrt(c**2*x**2 + 1) + c*x)*a*c*f*g**4))/(120*g**6)