Integrand size = 30, antiderivative size = 1536 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\frac {a d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2}}{g^5}+\frac {2 b c d^2 x \sqrt {d+c^2 d x^2}}{15 g \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2}}{g^5 \sqrt {1+c^2 x^2}}-\frac {b c d^2 \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2}}{3 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f x^2 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 f \left (c^2 f^2+2 g^2\right ) x^2 \sqrt {d+c^2 d x^2}}{4 g^4 \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 x^3 \sqrt {d+c^2 d x^2}}{45 g \sqrt {1+c^2 x^2}}-\frac {b c^3 d^2 \left (c^2 f^2+2 g^2\right ) x^3 \sqrt {d+c^2 d x^2}}{9 g^3 \sqrt {1+c^2 x^2}}+\frac {b c^5 d^2 f x^4 \sqrt {d+c^2 d x^2}}{16 g^2 \sqrt {1+c^2 x^2}}-\frac {b c^5 d^2 x^5 \sqrt {d+c^2 d x^2}}{25 g \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{g^5}-\frac {c^2 d^2 f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{8 g^2}-\frac {c^2 d^2 f \left (c^2 f^2+2 g^2\right ) x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{2 g^4}-\frac {c^4 d^2 f x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{4 g^2}-\frac {d^2 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g}+\frac {d^2 \left (c^2 f^2+2 g^2\right ) \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 g^3}+\frac {d^2 \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 g}+\frac {c d^2 f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b g^2 \sqrt {1+c^2 x^2}}-\frac {c d^2 f \left (c^2 f^2+2 g^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{4 b g^4 \sqrt {1+c^2 x^2}}-\frac {c d^2 \left (c^2 f^2+g^2\right )^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b g^5 \sqrt {1+c^2 x^2}}-\frac {d^2 \left (c^2 f^2+g^2\right )^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c g^6 (f+g x) \sqrt {1+c^2 x^2}}+\frac {d^2 \left (c^2 f^2+g^2\right )^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{2 b c g^4 (f+g x)}-\frac {a d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \text {arctanh}\left (\frac {g-c^2 f x}{\sqrt {c^2 f^2+g^2} \sqrt {1+c^2 x^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \text {arcsinh}(c x) \log \left (1+\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}+\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f-\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}}-\frac {b d^2 \left (c^2 f^2+g^2\right )^{5/2} \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arcsinh}(c x)} g}{c f+\sqrt {c^2 f^2+g^2}}\right )}{g^6 \sqrt {1+c^2 x^2}} \]
-1/8*c^2*d^2*f*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/g^2-1/4*c^4*d^2*f* x^3*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/g^2+2/15*b*c*d^2*x*(c^2*d*x^2+d )^(1/2)/g/(c^2*x^2+1)^(1/2)-1/45*b*c^3*d^2*x^3*(c^2*d*x^2+d)^(1/2)/g/(c^2* x^2+1)^(1/2)-1/25*b*c^5*d^2*x^5*(c^2*d*x^2+d)^(1/2)/g/(c^2*x^2+1)^(1/2)+1/ 3*d^2*(c^2*f^2+2*g^2)*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/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^2*x^2+1) ^(1/2)-1/4*c*d^2*f*(c^2*f^2+2*g^2)*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2 )/b/g^4/(c^2*x^2+1)^(1/2)-1/2*c*d^2*(c^2*f^2+g^2)^2*x*(a+b*arcsinh(c*x))^2 *(c^2*d*x^2+d)^(1/2)/b/g^5/(c^2*x^2+1)^(1/2)-1/2*d^2*(c^2*f^2+g^2)^3*(a+b* arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/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*(a+b*arcsinh(c*x))^2*(c^2*x^2+1)^(1/2)*(c^2*d*x^2+d)^( 1/2)/b/c/g^4/(g*x+f)+a*d^2*(c^2*f^2+g^2)^2*(c^2*d*x^2+d)^(1/2)/g^5-a*d^2*( c^2*f^2+g^2)^(5/2)*arctanh((-c^2*f*x+g)/(c^2*f^2+g^2)^(1/2)/(c^2*x^2+1)^(1 /2))*(c^2*d*x^2+d)^(1/2)/g^6/(c^2*x^2+1)^(1/2)+b*d^2*(c^2*f^2+g^2)^2*arcsi nh(c*x)*(c^2*d*x^2+d)^(1/2)/g^5+b*d^2*(c^2*f^2+g^2)^(5/2)*polylog(2,-(c*x+ (c^2*x^2+1)^(1/2))*g/(c*f-(c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^6/(c ^2*x^2+1)^(1/2)-b*d^2*(c^2*f^2+g^2)^(5/2)*polylog(2,-(c*x+(c^2*x^2+1)^(1/2 ))*g/(c*f+(c^2*f^2+g^2)^(1/2)))*(c^2*d*x^2+d)^(1/2)/g^6/(c^2*x^2+1)^(1/2)- 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)*arcsinh(c*x)*ln(1+(c*x+(c^2*x^2+1)^(1/2))*g/(c*f-(c...
Result contains complex when optimal does not.
Time = 23.15 (sec) , antiderivative size = 7168, normalized size of antiderivative = 4.67 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Result too large to show} \]
Time = 2.73 (sec) , antiderivative size = 945, normalized size of antiderivative = 0.62, 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}}\) |
(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^...
3.1.46.3.1 Defintions of rubi rules used
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(3498\) vs. \(2(1420)=2840\).
Time = 0.94 (sec) , antiderivative size = 3499, normalized size of antiderivative = 2.28
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(3499\) |
| parts | \(\text {Expression too large to display}\) | \(3499\) |
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*dilo g((-(c*x+(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)))-1/25*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)^(1/2)/g*x^5*c^5+33/1 28*b*(d*(c^2*x^2+1))^(1/2)*f*d^2*c/(c^2*x^2+1)^(1/2)/g^2+1/8*b*(d*(c^2*x^2 +1))^(1/2)*f^3*d^2*c^3/(c^2*x^2+1)^(1/2)/g^4-11/45*b*(d*(c^2*x^2+1))^(1/2) *d^2/(c^2*x^2+1)^(1/2)/g*x^3*c^3-23/15*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^ 2+1)^(1/2)/g*c*x-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*dilog(((c*x+(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)))-1/2*b*(d*(c^2*x^2+1))^(1/2)*f^3*d^2*c^6/(c^2*x^2+1 )/g^4*arcsinh(c*x)*x^3-1/4*b*(d*(c^2*x^2+1))^(1/2)*f*d^2*c^6/(c^2*x^2+1)/g ^2*arcsinh(c*x)*x^5-2*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^4*dilog(((c*x+(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)))*c^2*f^2-11/8*b*(d*(c^2*x^2+1))^(1/2)*f*d^2*c^ 4/(c^2*x^2+1)/g^2*arcsinh(c*x)*x^3-9/8*b*(d*(c^2*x^2+1))^(1/2)*f*d^2*c^2/( c^2*x^2+1)/g^2*arcsinh(c*x)*x-1/2*b*(d*(c^2*x^2+1))^(1/2)*f^3*d^2*c^4/(c^2 *x^2+1)/g^4*arcsinh(c*x)*x+1/3*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g^3 *arcsinh(c*x)*x^4*c^6*f^2+8/3*b*(d*(c^2*x^2+1))^(1/2)*d^2/(c^2*x^2+1)/g^3* arcsinh(c*x)*x^2*c^4*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^6*dilog((-(c*x+(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)))*c^4*f^4-b*d^2*(d*(c^2*x^2+1))^(1/2)*(...
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}}{g x + f} \,d x } \]
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)
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}{f + g x}\, dx \]
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\text {Exception raised: TypeError} \]
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
Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))}{f+g x} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{f+g\,x} \,d x \]