Integrand size = 30, antiderivative size = 651 \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=-\frac {2 b d f g x \sqrt {d+c^2 d x^2}}{5 c \sqrt {1+c^2 x^2}}-\frac {5 b c d f^2 x^2 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {b d g^2 x^2 \sqrt {d+c^2 d x^2}}{32 c \sqrt {1+c^2 x^2}}-\frac {4 b c d f g x^3 \sqrt {d+c^2 d x^2}}{15 \sqrt {1+c^2 x^2}}-\frac {b c^3 d f^2 x^4 \sqrt {d+c^2 d x^2}}{16 \sqrt {1+c^2 x^2}}-\frac {7 b c d g^2 x^4 \sqrt {d+c^2 d x^2}}{96 \sqrt {1+c^2 x^2}}-\frac {2 b c^3 d f g x^5 \sqrt {d+c^2 d x^2}}{25 \sqrt {1+c^2 x^2}}-\frac {b c^3 d g^2 x^6 \sqrt {d+c^2 d x^2}}{36 \sqrt {1+c^2 x^2}}+\frac {3}{8} d f^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {d g^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{8} d g^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{4} d f^2 x \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {1}{6} d g^2 x^3 \left (1+c^2 x^2\right ) \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))+\frac {2 d f g \left (1+c^2 x^2\right )^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {3 d f^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{16 b c \sqrt {1+c^2 x^2}}-\frac {d g^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2}{32 b c^3 \sqrt {1+c^2 x^2}} \]
3/8*d*f^2*x*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+1/16*d*g^2*x*(a+b*arcsi nh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2+1/8*d*g^2*x^3*(a+b*arcsinh(c*x))*(c^2*d*x ^2+d)^(1/2)+1/4*d*f^2*x*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2) +1/6*d*g^2*x^3*(c^2*x^2+1)*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)+2/5*d*f* g*(c^2*x^2+1)^2*(a+b*arcsinh(c*x))*(c^2*d*x^2+d)^(1/2)/c^2-2/5*b*d*f*g*x*( c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^(1/2)-5/16*b*c*d*f^2*x^2*(c^2*d*x^2+d)^(1 /2)/(c^2*x^2+1)^(1/2)-1/32*b*d*g^2*x^2*(c^2*d*x^2+d)^(1/2)/c/(c^2*x^2+1)^( 1/2)-4/15*b*c*d*f*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^2*x^4*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-7/96*b*c*d*g^2*x^4*(c^2*d*x ^2+d)^(1/2)/(c^2*x^2+1)^(1/2)-2/25*b*c^3*d*f*g*x^5*(c^2*d*x^2+d)^(1/2)/(c^ 2*x^2+1)^(1/2)-1/36*b*c^3*d*g^2*x^6*(c^2*d*x^2+d)^(1/2)/(c^2*x^2+1)^(1/2)+ 3/16*d*f^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c/(c^2*x^2+1)^(1/2)- 1/32*d*g^2*(a+b*arcsinh(c*x))^2*(c^2*d*x^2+d)^(1/2)/b/c^3/(c^2*x^2+1)^(1/2 )
Time = 1.16 (sec) , antiderivative size = 416, normalized size of antiderivative = 0.64 \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\frac {-d^2 \left (1+c^2 x^2\right ) \left (b \left (-175 g^2+90 c^2 \left (85 f^2+256 f g x+20 g^2 x^2\right )+120 c^4 x^2 \left (150 f^2+128 f g x+35 g^2 x^2\right )+16 c^6 x^4 \left (225 f^2+288 f g x+100 g^2 x^2\right )\right )-240 a c \sqrt {1+c^2 x^2} \left (96 f g \left (1+c^2 x^2\right )^2+30 c^2 f^2 x \left (5+2 c^2 x^2\right )+5 g^2 x \left (3+14 c^2 x^2+8 c^4 x^4\right )\right )\right )+1800 b d^2 \left (6 c^2 f^2-g^2\right ) \left (1+c^2 x^2\right ) \text {arcsinh}(c x)^2+3600 a d^{3/2} \left (6 c^2 f^2-g^2\right ) \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2} \log \left (c d x+\sqrt {d} \sqrt {d+c^2 d x^2}\right )+60 b d^2 \left (1+c^2 x^2\right ) \text {arcsinh}(c x) \left (15 \left (16 c^2 f^2-g^2\right ) \sinh (2 \text {arcsinh}(c x))+15 \left (2 c^2 f^2+g^2\right ) \sinh (4 \text {arcsinh}(c x))+g \left (384 c f \left (1+c^2 x^2\right )^{5/2}+5 g \sinh (6 \text {arcsinh}(c x))\right )\right )}{57600 c^3 \sqrt {1+c^2 x^2} \sqrt {d+c^2 d x^2}} \]
(-(d^2*(1 + c^2*x^2)*(b*(-175*g^2 + 90*c^2*(85*f^2 + 256*f*g*x + 20*g^2*x^ 2) + 120*c^4*x^2*(150*f^2 + 128*f*g*x + 35*g^2*x^2) + 16*c^6*x^4*(225*f^2 + 288*f*g*x + 100*g^2*x^2)) - 240*a*c*Sqrt[1 + c^2*x^2]*(96*f*g*(1 + c^2*x ^2)^2 + 30*c^2*f^2*x*(5 + 2*c^2*x^2) + 5*g^2*x*(3 + 14*c^2*x^2 + 8*c^4*x^4 )))) + 1800*b*d^2*(6*c^2*f^2 - g^2)*(1 + c^2*x^2)*ArcSinh[c*x]^2 + 3600*a* d^(3/2)*(6*c^2*f^2 - g^2)*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]*(15*( 16*c^2*f^2 - g^2)*Sinh[2*ArcSinh[c*x]] + 15*(2*c^2*f^2 + g^2)*Sinh[4*ArcSi nh[c*x]] + g*(384*c*f*(1 + c^2*x^2)^(5/2) + 5*g*Sinh[6*ArcSinh[c*x]])))/(5 7600*c^3*Sqrt[1 + c^2*x^2]*Sqrt[d + c^2*d*x^2])
Time = 1.01 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.55, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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.
| \(\displaystyle \int \left (c^2 d x^2+d\right )^{3/2} (f+g x)^2 (a+b \text {arcsinh}(c x)) \, dx\) |
| \(\Big \downarrow \) 6260 |
| \(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int (f+g x)^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 6253 |
| \(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \int \left (\left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) f^2+2 g x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x)) f+g^2 x^2 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))\right )dx}{\sqrt {c^2 x^2+1}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {d \sqrt {c^2 d x^2+d} \left (-\frac {g^2 (a+b \text {arcsinh}(c x))^2}{32 b c^3}+\frac {1}{4} f^2 x \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {3}{8} f^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {2 f g \left (c^2 x^2+1\right )^{5/2} (a+b \text {arcsinh}(c x))}{5 c^2}+\frac {g^2 x \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{16 c^2}+\frac {1}{6} g^2 x^3 \left (c^2 x^2+1\right )^{3/2} (a+b \text {arcsinh}(c x))+\frac {1}{8} g^2 x^3 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))+\frac {3 f^2 (a+b \text {arcsinh}(c x))^2}{16 b c}-\frac {1}{16} b c^3 f^2 x^4-\frac {2}{25} b c^3 f g x^5-\frac {1}{36} b c^3 g^2 x^6-\frac {5}{16} b c f^2 x^2-\frac {4}{15} b c f g x^3-\frac {2 b f g x}{5 c}-\frac {7}{96} b c g^2 x^4-\frac {b g^2 x^2}{32 c}\right )}{\sqrt {c^2 x^2+1}}\) |
(d*Sqrt[d + c^2*d*x^2]*((-2*b*f*g*x)/(5*c) - (5*b*c*f^2*x^2)/16 - (b*g^2*x ^2)/(32*c) - (4*b*c*f*g*x^3)/15 - (b*c^3*f^2*x^4)/16 - (7*b*c*g^2*x^4)/96 - (2*b*c^3*f*g*x^5)/25 - (b*c^3*g^2*x^6)/36 + (3*f^2*x*Sqrt[1 + c^2*x^2]*( a + b*ArcSinh[c*x]))/8 + (g^2*x*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(1 6*c^2) + (g^2*x^3*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/8 + (f^2*x*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/4 + (g^2*x^3*(1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x]))/6 + (2*f*g*(1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x]))/(5* c^2) + (3*f^2*(a + b*ArcSinh[c*x])^2)/(16*b*c) - (g^2*(a + b*ArcSinh[c*x]) ^2)/(32*b*c^3)))/Sqrt[1 + c^2*x^2]
3.1.40.3.1 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. \(1567\) vs. \(2(567)=1134\).
Time = 0.82 (sec) , antiderivative size = 1568, normalized size of antiderivative = 2.41
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1568\) |
| parts | \(\text {Expression too large to display}\) | \(1568\) |
a*(f^2*(1/4*x*(c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(c^2*d*x^2+d)^(1/2)+1/2*d*l n(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2*d)^(1/2)))+g^2*(1/6*x*(c ^2*d*x^2+d)^(5/2)/c^2/d-1/6/c^2*(1/4*x*(c^2*d*x^2+d)^(3/2)+3/4*d*(1/2*x*(c ^2*d*x^2+d)^(1/2)+1/2*d*ln(c^2*d*x/(c^2*d)^(1/2)+(c^2*d*x^2+d)^(1/2))/(c^2 *d)^(1/2))))+2/5*f*g/c^2/d*(c^2*d*x^2+d)^(5/2))+b*(1/32*(d*(c^2*x^2+1))^(1 /2)*arcsinh(c*x)^2*(6*c^2*f^2-g^2)*d/(c^2*x^2+1)^(1/2)/c^3+1/2304*(d*(c^2* x^2+1))^(1/2)*(32*c^7*x^7+32*c^6*x^6*(c^2*x^2+1)^(1/2)+64*c^5*x^5+48*c^4*x ^4*(c^2*x^2+1)^(1/2)+38*c^3*x^3+18*c^2*x^2*(c^2*x^2+1)^(1/2)+6*c*x+(c^2*x^ 2+1)^(1/2))*g^2*(-1+6*arcsinh(c*x))*d/c^3/(c^2*x^2+1)+1/400*(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)*f*g*(-1+5*arcsinh(c*x ))*d/c^2/(c^2*x^2+1)+1/512*(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))*(8*arcsinh(c*x)*c^2*f^2-2*c^2*f^2+4*arcsinh(c*x)*g^2-g^2)*d/c^3/(c^2*x ^2+1)+1/48*(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)*f*g*(-1+3*arcsinh(c*x))*d/c^2/(c^2*x^2+ 1)+1/256*(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))*(32*arcsinh(c*x)*c^2*f^2-16*c^2*f^2-2*arcsinh(c*x)*g^ 2+g^2)*d/c^3/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^(1/2)*(c^2*x^2+c*x*(c^2*x^2+1 )^(1/2)+1)*f*g*(-1+arcsinh(c*x))*d/c^2/(c^2*x^2+1)+1/8*(d*(c^2*x^2+1))^...
\[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int { {\left (c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}^{2} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} \,d x } \]
integral((a*c^2*d*g^2*x^4 + 2*a*c^2*d*f*g*x^3 + 2*a*d*f*g*x + a*d*f^2 + (a *c^2*d*f^2 + a*d*g^2)*x^2 + (b*c^2*d*g^2*x^4 + 2*b*c^2*d*f*g*x^3 + 2*b*d*f *g*x + b*d*f^2 + (b*c^2*d*f^2 + b*d*g^2)*x^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d), x)
\[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int \left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right ) \left (f + g x\right )^{2}\, dx \]
Exception generated. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
Exception generated. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int (f+g x)^2 \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x)) \, dx=\int {\left (f+g\,x\right )}^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^{3/2} \,d x \]