Integrand size = 28, antiderivative size = 353 \[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {b d g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {5 b c d f x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {2 b c d g x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {d g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {3 d f \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}} \]
3/8*d*f*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/4*d*f*x*(c^2*x^2+1)*(a+ b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/5*d*g*(c^2*x^2+1)^2*(a+b*arcsinh(c*x ))*(c^2*d*x^2+d)^(1/2)/c^2-1/5*b*d*g*x*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^( 1/2)-5/16*b*c*d*f*x^2*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-2/15*b*c*d*g*x ^3*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-1/16*b*c^3*d*f*x^4*(c^2*d*x^2+d)^ (1/2)/(c^2*x^2+1)^(1/2)-1/25*b*c^3*d*g*x^5*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1) ^(1/2)+3/16*d*f*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c/(c^2*x^2+1)^( 1/2)
Time = 0.63 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.82 \[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^2 \left (1+c^2 x^2\right ) \left (-240 a \sqrt {1+c^2 x^2} \left (8 g \left (1+c^2 x^2\right )^2+5 c^2 f x \left (5+2 c^2 x^2\right )\right )+b c \left (128 g x \left (15+10 c^2 x^2+3 c^4 x^4\right )+75 f \left (17+40 c^2 x^2+8 c^4 x^4\right )\right )\right )+1800 b c d^2 f \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+3600 a c d^{3/2} f \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+60 b d^2 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (32 g \left (1+c^2 x^2\right )^{5/2}+40 c f \sinh (2 \text {arcsinh}(c x))+5 c f \sinh (4 \text {arcsinh}(c x))\right )}{9600 c^2 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \]
(-(d^2*(1 + c^2*x^2)*(-240*a*Sqrt[1 + c^2*x^2]*(8*g*(1 + c^2*x^2)^2 + 5*c^ 2*f*x*(5 + 2*c^2*x^2)) + b*c*(128*g*x*(15 + 10*c^2*x^2 + 3*c^4*x^4) + 75*f *(17 + 40*c^2*x^2 + 8*c^4*x^4)))) + 1800*b*c*d^2*f*(1 + c^2*x^2)*ArcSinh[c *x]^2 + 3600*a*c*d^(3/2)*f*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2]*Log[c*d*x + Sqrt[d]*Sqrt[d + c^2*d*x^2]] + 60*b*d^2*(1 + c^2*x^2)*ArcSinh[c*x]*(32* g*(1 + c^2*x^2)^(5/2) + 40*c*f*Sinh[2*ArcSinh[c*x]] + 5*c*f*Sinh[4*ArcSinh [c*x]]))/(9600*c^2*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2])
Time = 0.60 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.53, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {6260, 6253, 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.
\(\displaystyle \int \left (c^2 d x^2+d\right )^{3/2} (f+g x) (a+b \text {arcsinh}(c x)) \, dx\) |
\(\Big \downarrow \) 6260 |
\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int (f+g x) \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 6253 |
\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int \left (f (a+b \text {arcsinh}(c x)) \left (c^2 x^2+1\right )^{3/2}+g x (a+b \text {arcsinh}(c x)) \left (c^2 x^2+1\right )^{3/2}\right )dx}{\sqrt {c^2 x^2+1}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \left (\frac {1}{4} f x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{8} f x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {g \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {3 f (a+b \text {arcsinh}(c x))^2}{16 b c}-\frac {1}{16} b c^3 f x^4-\frac {1}{25} b c^3 g x^5-\frac {5}{16} b c f x^2-\frac {2}{15} b c g x^3-\frac {b g x}{5 c}\right )}{\sqrt {c^2 x^2+1}}\) |
(d*Sqrt[d + c^2*d*x^2]*(-1/5*(b*g*x)/c - (5*b*c*f*x^2)/16 - (2*b*c*g*x^3)/ 15 - (b*c^3*f*x^4)/16 - (b*c^3*g*x^5)/25 + (3*f*x*Sqrt[1 + c^2*x^2]*(a + b *ArcSinh[c*x]))/8 + (f*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/4 + (g* (1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5*c^2) + (3*f*(a + b*ArcSinh[c* x])^2)/(16*b*c)))/Sqrt[1 + c^2*x^2]
3.1.41.3.1 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[(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[e, c^2*d] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d, 0] && IGtQ[n , 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1] || (EqQ[m, 2] && LtQ[p, -2]))
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. \(1064\) vs. \(2(305)=610\).
Time = 0.83 (sec) , antiderivative size = 1065, normalized size of antiderivative = 3.02
method | result | size |
default | \(\text {Expression too large to display}\) | \(1065\) |
parts | \(\text {Expression too large to display}\) | \(1065\) |
1/4*a*f*x*(c^2*d*x^2+d)^(3/2)+3/8*a*f*d*x*(c^2*d*x^2+d)^(1/2)+3/8*a*f*d^2* ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)+1/5*a*g/c^2/d* (c^2*d*x^2+d)^(5/2)+b*(3/16*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/c*f*ar csinh(c*x)^2*d+1/800*(d*(c^2*x^2+1))^(1/2)*(16*c^6*x^6+16*c^5*x^5*(c^2*x^2 +1)^(1/2)+28*c^4*x^4+20*c^3*x^3*(c^2*x^2+1)^(1/2)+13*c^2*x^2+5*c*x*(c^2*x^ 2+1)^(1/2)+1)*g*(-1+5*arcsinh(c*x))*d/c^2/(c^2*x^2+1)+1/256*(d*(c^2*x^2+1) )^(1/2)*(8*c^5*x^5+8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3*x^3+8*c^2*x^2*(c^2*x ^2+1)^(1/2)+4*c*x+(c^2*x^2+1)^(1/2))*f*(-1+4*arcsinh(c*x))*d/c/(c^2*x^2+1) +1/96*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4+4*c^3*x^3*(c^2*x^2+1)^(1/2)+5*c^2*x ^2+3*c*x*(c^2*x^2+1)^(1/2)+1)*g*(-1+3*arcsinh(c*x))*d/c^2/(c^2*x^2+1)+1/16 *(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3+2*c^2*x^2*(c^2*x^2+1)^(1/2)+2*c*x+(c^2*x ^2+1)^(1/2))*f*(-1+2*arcsinh(c*x))*d/c/(c^2*x^2+1)+1/16*(d*(c^2*x^2+1))^(1 /2)*(c^2*x^2+c*x*(c^2*x^2+1)^(1/2)+1)*g*(-1+arcsinh(c*x))*d/c^2/(c^2*x^2+1 )+1/16*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2-c*x*(c^2*x^2+1)^(1/2)+1)*g*(arcsinh( c*x)+1)*d/c^2/(c^2*x^2+1)+1/16*(d*(c^2*x^2+1))^(1/2)*(2*c^3*x^3-2*c^2*x^2* (c^2*x^2+1)^(1/2)+2*c*x-(c^2*x^2+1)^(1/2))*f*(1+2*arcsinh(c*x))*d/c/(c^2*x ^2+1)+1/96*(d*(c^2*x^2+1))^(1/2)*(4*c^4*x^4-4*c^3*x^3*(c^2*x^2+1)^(1/2)+5* c^2*x^2-3*c*x*(c^2*x^2+1)^(1/2)+1)*g*(3*arcsinh(c*x)+1)*d/c^2/(c^2*x^2+1)+ 1/256*(d*(c^2*x^2+1))^(1/2)*(8*c^5*x^5-8*c^4*x^4*(c^2*x^2+1)^(1/2)+12*c^3* x^3-8*c^2*x^2*(c^2*x^2+1)^(1/2)+4*c*x-(c^2*x^2+1)^(1/2))*f*(1+4*arcsinh...
\[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]
integral((a*c^2*d*g*x^3 + a*c^2*d*f*x^2 + a*d*g*x + a*d*f + (b*c^2*d*g*x^3 + b*c^2*d*f*x^2 + b*d*g*x + b*d*f)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)
\[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )\, dx \]
Exception generated. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const ve cteur & l) Error: Bad Argument Value
Timed out. \[ \int (f+g x) \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (f+g\,x\right )\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]