Integrand size = 20, antiderivative size = 329 \[ \int \left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=2 b^2 d^2 x-\frac {8 b^2 d e x}{9 c^2}+\frac {16 b^2 e^2 x}{75 c^4}+\frac {4}{27} b^2 d e x^3-\frac {8 b^2 e^2 x^3}{225 c^2}+\frac {2}{125} b^2 e^2 x^5-\frac {2 b d^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{c}+\frac {8 b d e \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c^3}-\frac {16 b e^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{75 c^5}-\frac {4 b d e x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{9 c}+\frac {8 b e^2 x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{75 c^3}-\frac {2 b e^2 x^4 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}{25 c}+d^2 x (a+b \text {arcsinh}(c x))^2+\frac {2}{3} d e x^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{5} e^2 x^5 (a+b \text {arcsinh}(c x))^2 \]
2*b^2*d^2*x-8/9*b^2*d*e*x/c^2+16/75*b^2*e^2*x/c^4+4/27*b^2*d*e*x^3-8/225*b ^2*e^2*x^3/c^2+2/125*b^2*e^2*x^5+d^2*x*(a+b*arcsinh(c*x))^2+2/3*d*e*x^3*(a +b*arcsinh(c*x))^2+1/5*e^2*x^5*(a+b*arcsinh(c*x))^2-2*b*d^2*(a+b*arcsinh(c *x))*(c^2*x^2+1)^(1/2)/c+8/9*b*d*e*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^ 3-16/75*b*e^2*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c^5-4/9*b*d*e*x^2*(a+b* arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c+8/75*b*e^2*x^2*(a+b*arcsinh(c*x))*(c^2*x ^2+1)^(1/2)/c^3-2/25*b*e^2*x^4*(a+b*arcsinh(c*x))*(c^2*x^2+1)^(1/2)/c
Time = 0.24 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.88 \[ \int \left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {225 a^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )-30 a b \sqrt {1+c^2 x^2} \left (24 e^2-4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )+2 b^2 c x \left (360 e^2-60 c^2 e \left (25 d+e x^2\right )+c^4 \left (3375 d^2+250 d e x^2+27 e^2 x^4\right )\right )-30 b \left (-15 a c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right )+b \sqrt {1+c^2 x^2} \left (24 e^2-4 c^2 e \left (25 d+3 e x^2\right )+c^4 \left (225 d^2+50 d e x^2+9 e^2 x^4\right )\right )\right ) \text {arcsinh}(c x)+225 b^2 c^5 x \left (15 d^2+10 d e x^2+3 e^2 x^4\right ) \text {arcsinh}(c x)^2}{3375 c^5} \]
(225*a^2*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^4) - 30*a*b*Sqrt[1 + c^2*x^2 ]*(24*e^2 - 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x ^4)) + 2*b^2*c*x*(360*e^2 - 60*c^2*e*(25*d + e*x^2) + c^4*(3375*d^2 + 250* d*e*x^2 + 27*e^2*x^4)) - 30*b*(-15*a*c^5*x*(15*d^2 + 10*d*e*x^2 + 3*e^2*x^ 4) + b*Sqrt[1 + c^2*x^2]*(24*e^2 - 4*c^2*e*(25*d + 3*e*x^2) + c^4*(225*d^2 + 50*d*e*x^2 + 9*e^2*x^4)))*ArcSinh[c*x] + 225*b^2*c^5*x*(15*d^2 + 10*d*e *x^2 + 3*e^2*x^4)*ArcSinh[c*x]^2)/(3375*c^5)
Time = 0.83 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6208, 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.
| \(\displaystyle \int \left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx\) |
| \(\Big \downarrow \) 6208 |
| \(\displaystyle \int \left (d^2 (a+b \text {arcsinh}(c x))^2+2 d e x^2 (a+b \text {arcsinh}(c x))^2+e^2 x^4 (a+b \text {arcsinh}(c x))^2\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b d^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{c}-\frac {4 b d e x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c}-\frac {2 b e^2 x^4 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{25 c}-\frac {16 b e^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{75 c^5}+\frac {8 b d e \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{9 c^3}+\frac {8 b e^2 x^2 \sqrt {c^2 x^2+1} (a+b \text {arcsinh}(c x))}{75 c^3}+d^2 x (a+b \text {arcsinh}(c x))^2+\frac {2}{3} d e x^3 (a+b \text {arcsinh}(c x))^2+\frac {1}{5} e^2 x^5 (a+b \text {arcsinh}(c x))^2+\frac {16 b^2 e^2 x}{75 c^4}-\frac {8 b^2 d e x}{9 c^2}-\frac {8 b^2 e^2 x^3}{225 c^2}+2 b^2 d^2 x+\frac {4}{27} b^2 d e x^3+\frac {2}{125} b^2 e^2 x^5\) |
2*b^2*d^2*x - (8*b^2*d*e*x)/(9*c^2) + (16*b^2*e^2*x)/(75*c^4) + (4*b^2*d*e *x^3)/27 - (8*b^2*e^2*x^3)/(225*c^2) + (2*b^2*e^2*x^5)/125 - (2*b*d^2*Sqrt [1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/c + (8*b*d*e*Sqrt[1 + c^2*x^2]*(a + b* ArcSinh[c*x]))/(9*c^3) - (16*b*e^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x])) /(75*c^5) - (4*b*d*e*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(9*c) + ( 8*b*e^2*x^2*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(75*c^3) - (2*b*e^2*x^ 4*Sqrt[1 + c^2*x^2]*(a + b*ArcSinh[c*x]))/(25*c) + d^2*x*(a + b*ArcSinh[c* x])^2 + (2*d*e*x^3*(a + b*ArcSinh[c*x])^2)/3 + (e^2*x^5*(a + b*ArcSinh[c*x ])^2)/5
3.7.14.3.1 Defintions of rubi rules used
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*ArcSinh[c*x])^n, (d + e*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, n}, x] && NeQ[e, c^2*d] && IntegerQ[p] && (p > 0 || IGtQ[n, 0])
Time = 0.44 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.31
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (c^{4} d^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {2 c^{2} d e \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {e^{2} \left (225 \operatorname {arcsinh}\left (c x \right )^{2} c^{5} x^{5}-90 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+18 c^{5} x^{5}+120 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-40 c^{3} x^{3}-240 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+240 c x \right )}{1125}\right )}{c^{4}}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arcsinh}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arcsinh}\left (c x \right ) e^{2} c^{5} x^{5}}{5}-\frac {e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}-d^{2} c^{4} \sqrt {c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{4}}}{c}\) | \(431\) |
| default | \(\frac {\frac {a^{2} \left (d^{2} c^{5} x +\frac {2}{3} d \,c^{5} e \,x^{3}+\frac {1}{5} e^{2} c^{5} x^{5}\right )}{c^{4}}+\frac {b^{2} \left (c^{4} d^{2} \left (\operatorname {arcsinh}\left (c x \right )^{2} x c -2 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+2 c x \right )+\frac {2 c^{2} d e \left (9 \operatorname {arcsinh}\left (c x \right )^{2} x^{3} c^{3}-6 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+2 c^{3} x^{3}+12 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}-12 c x \right )}{27}+\frac {e^{2} \left (225 \operatorname {arcsinh}\left (c x \right )^{2} c^{5} x^{5}-90 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+18 c^{5} x^{5}+120 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}-40 c^{3} x^{3}-240 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}+240 c x \right )}{1125}\right )}{c^{4}}+\frac {2 a b \left (\operatorname {arcsinh}\left (c x \right ) d^{2} c^{5} x +\frac {2 \,\operatorname {arcsinh}\left (c x \right ) d \,c^{5} e \,x^{3}}{3}+\frac {\operatorname {arcsinh}\left (c x \right ) e^{2} c^{5} x^{5}}{5}-\frac {e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )}{5}-d^{2} c^{4} \sqrt {c^{2} x^{2}+1}-\frac {2 d \,c^{2} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{3}\right )}{c^{4}}}{c}\) | \(431\) |
| parts | \(a^{2} \left (\frac {1}{5} e^{2} x^{5}+\frac {2}{3} d e \,x^{3}+d^{2} x \right )+\frac {b^{2} \left (675 \operatorname {arcsinh}\left (c x \right )^{2} c^{5} x^{5} e^{2}+2250 \operatorname {arcsinh}\left (c x \right )^{2} c^{5} x^{3} d e +3375 \operatorname {arcsinh}\left (c x \right )^{2} c^{5} x \,d^{2}-270 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4} e^{2}-1500 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{4} x^{2} d e -6750 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{4} d^{2}+54 e^{2} c^{5} x^{5}+500 d \,c^{5} e \,x^{3}+6750 d^{2} c^{5} x +360 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} x^{2} e^{2}+3000 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c^{2} d e -120 c^{3} x^{3} e^{2}-3000 c^{3} x d e -720 \,\operatorname {arcsinh}\left (c x \right ) \sqrt {c^{2} x^{2}+1}\, e^{2}+720 c x \,e^{2}\right )}{3375 c^{5}}+\frac {2 a b \left (\frac {c \,\operatorname {arcsinh}\left (c x \right ) e^{2} x^{5}}{5}+\frac {2 c \,\operatorname {arcsinh}\left (c x \right ) d e \,x^{3}}{3}+\operatorname {arcsinh}\left (c x \right ) c x \,d^{2}-\frac {3 e^{2} \left (\frac {c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{5}-\frac {4 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{15}+\frac {8 \sqrt {c^{2} x^{2}+1}}{15}\right )+15 d^{2} c^{4} \sqrt {c^{2} x^{2}+1}+10 d \,c^{2} e \left (\frac {c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{3}-\frac {2 \sqrt {c^{2} x^{2}+1}}{3}\right )}{15 c^{4}}\right )}{c}\) | \(447\) |
1/c*(a^2/c^4*(d^2*c^5*x+2/3*d*c^5*e*x^3+1/5*e^2*c^5*x^5)+b^2/c^4*(c^4*d^2* (arcsinh(c*x)^2*x*c-2*arcsinh(c*x)*(c^2*x^2+1)^(1/2)+2*c*x)+2/27*c^2*d*e*( 9*arcsinh(c*x)^2*x^3*c^3-6*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^2*c^2+2*c^3*x^ 3+12*arcsinh(c*x)*(c^2*x^2+1)^(1/2)-12*c*x)+1/1125*e^2*(225*arcsinh(c*x)^2 *c^5*x^5-90*arcsinh(c*x)*(c^2*x^2+1)^(1/2)*x^4*c^4+18*c^5*x^5+120*arcsinh( c*x)*(c^2*x^2+1)^(1/2)*x^2*c^2-40*c^3*x^3-240*arcsinh(c*x)*(c^2*x^2+1)^(1/ 2)+240*c*x))+2*a*b/c^4*(arcsinh(c*x)*d^2*c^5*x+2/3*arcsinh(c*x)*d*c^5*e*x^ 3+1/5*arcsinh(c*x)*e^2*c^5*x^5-1/5*e^2*(1/5*c^4*x^4*(c^2*x^2+1)^(1/2)-4/15 *c^2*x^2*(c^2*x^2+1)^(1/2)+8/15*(c^2*x^2+1)^(1/2))-d^2*c^4*(c^2*x^2+1)^(1/ 2)-2/3*d*c^2*e*(1/3*c^2*x^2*(c^2*x^2+1)^(1/2)-2/3*(c^2*x^2+1)^(1/2))))
Time = 0.26 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.16 \[ \int \left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {27 \, {\left (25 \, a^{2} + 2 \, b^{2}\right )} c^{5} e^{2} x^{5} + 10 \, {\left (25 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{5} d e - 12 \, b^{2} c^{3} e^{2}\right )} x^{3} + 225 \, {\left (3 \, b^{2} c^{5} e^{2} x^{5} + 10 \, b^{2} c^{5} d e x^{3} + 15 \, b^{2} c^{5} d^{2} x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 15 \, {\left (225 \, {\left (a^{2} + 2 \, b^{2}\right )} c^{5} d^{2} - 200 \, b^{2} c^{3} d e + 48 \, b^{2} c e^{2}\right )} x + 30 \, {\left (45 \, a b c^{5} e^{2} x^{5} + 150 \, a b c^{5} d e x^{3} + 225 \, a b c^{5} d^{2} x - {\left (9 \, b^{2} c^{4} e^{2} x^{4} + 225 \, b^{2} c^{4} d^{2} - 100 \, b^{2} c^{2} d e + 24 \, b^{2} e^{2} + 2 \, {\left (25 \, b^{2} c^{4} d e - 6 \, b^{2} c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 30 \, {\left (9 \, a b c^{4} e^{2} x^{4} + 225 \, a b c^{4} d^{2} - 100 \, a b c^{2} d e + 24 \, a b e^{2} + 2 \, {\left (25 \, a b c^{4} d e - 6 \, a b c^{2} e^{2}\right )} x^{2}\right )} \sqrt {c^{2} x^{2} + 1}}{3375 \, c^{5}} \]
1/3375*(27*(25*a^2 + 2*b^2)*c^5*e^2*x^5 + 10*(25*(9*a^2 + 2*b^2)*c^5*d*e - 12*b^2*c^3*e^2)*x^3 + 225*(3*b^2*c^5*e^2*x^5 + 10*b^2*c^5*d*e*x^3 + 15*b^ 2*c^5*d^2*x)*log(c*x + sqrt(c^2*x^2 + 1))^2 + 15*(225*(a^2 + 2*b^2)*c^5*d^ 2 - 200*b^2*c^3*d*e + 48*b^2*c*e^2)*x + 30*(45*a*b*c^5*e^2*x^5 + 150*a*b*c ^5*d*e*x^3 + 225*a*b*c^5*d^2*x - (9*b^2*c^4*e^2*x^4 + 225*b^2*c^4*d^2 - 10 0*b^2*c^2*d*e + 24*b^2*e^2 + 2*(25*b^2*c^4*d*e - 6*b^2*c^2*e^2)*x^2)*sqrt( c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) - 30*(9*a*b*c^4*e^2*x^4 + 225*a *b*c^4*d^2 - 100*a*b*c^2*d*e + 24*a*b*e^2 + 2*(25*a*b*c^4*d*e - 6*a*b*c^2* e^2)*x^2)*sqrt(c^2*x^2 + 1))/c^5
Time = 0.59 (sec) , antiderivative size = 595, normalized size of antiderivative = 1.81 \[ \int \left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\begin {cases} a^{2} d^{2} x + \frac {2 a^{2} d e x^{3}}{3} + \frac {a^{2} e^{2} x^{5}}{5} + 2 a b d^{2} x \operatorname {asinh}{\left (c x \right )} + \frac {4 a b d e x^{3} \operatorname {asinh}{\left (c x \right )}}{3} + \frac {2 a b e^{2} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {2 a b d^{2} \sqrt {c^{2} x^{2} + 1}}{c} - \frac {4 a b d e x^{2} \sqrt {c^{2} x^{2} + 1}}{9 c} - \frac {2 a b e^{2} x^{4} \sqrt {c^{2} x^{2} + 1}}{25 c} + \frac {8 a b d e \sqrt {c^{2} x^{2} + 1}}{9 c^{3}} + \frac {8 a b e^{2} x^{2} \sqrt {c^{2} x^{2} + 1}}{75 c^{3}} - \frac {16 a b e^{2} \sqrt {c^{2} x^{2} + 1}}{75 c^{5}} + b^{2} d^{2} x \operatorname {asinh}^{2}{\left (c x \right )} + 2 b^{2} d^{2} x + \frac {2 b^{2} d e x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {4 b^{2} d e x^{3}}{27} + \frac {b^{2} e^{2} x^{5} \operatorname {asinh}^{2}{\left (c x \right )}}{5} + \frac {2 b^{2} e^{2} x^{5}}{125} - \frac {2 b^{2} d^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{c} - \frac {4 b^{2} d e x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} - \frac {2 b^{2} e^{2} x^{4} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{25 c} - \frac {8 b^{2} d e x}{9 c^{2}} - \frac {8 b^{2} e^{2} x^{3}}{225 c^{2}} + \frac {8 b^{2} d e \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c^{3}} + \frac {8 b^{2} e^{2} x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{75 c^{3}} + \frac {16 b^{2} e^{2} x}{75 c^{4}} - \frac {16 b^{2} e^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{75 c^{5}} & \text {for}\: c \neq 0 \\a^{2} \left (d^{2} x + \frac {2 d e x^{3}}{3} + \frac {e^{2} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \]
Piecewise((a**2*d**2*x + 2*a**2*d*e*x**3/3 + a**2*e**2*x**5/5 + 2*a*b*d**2 *x*asinh(c*x) + 4*a*b*d*e*x**3*asinh(c*x)/3 + 2*a*b*e**2*x**5*asinh(c*x)/5 - 2*a*b*d**2*sqrt(c**2*x**2 + 1)/c - 4*a*b*d*e*x**2*sqrt(c**2*x**2 + 1)/( 9*c) - 2*a*b*e**2*x**4*sqrt(c**2*x**2 + 1)/(25*c) + 8*a*b*d*e*sqrt(c**2*x* *2 + 1)/(9*c**3) + 8*a*b*e**2*x**2*sqrt(c**2*x**2 + 1)/(75*c**3) - 16*a*b* e**2*sqrt(c**2*x**2 + 1)/(75*c**5) + b**2*d**2*x*asinh(c*x)**2 + 2*b**2*d* *2*x + 2*b**2*d*e*x**3*asinh(c*x)**2/3 + 4*b**2*d*e*x**3/27 + b**2*e**2*x* *5*asinh(c*x)**2/5 + 2*b**2*e**2*x**5/125 - 2*b**2*d**2*sqrt(c**2*x**2 + 1 )*asinh(c*x)/c - 4*b**2*d*e*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(9*c) - 2* b**2*e**2*x**4*sqrt(c**2*x**2 + 1)*asinh(c*x)/(25*c) - 8*b**2*d*e*x/(9*c** 2) - 8*b**2*e**2*x**3/(225*c**2) + 8*b**2*d*e*sqrt(c**2*x**2 + 1)*asinh(c* x)/(9*c**3) + 8*b**2*e**2*x**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(75*c**3) + 16*b**2*e**2*x/(75*c**4) - 16*b**2*e**2*sqrt(c**2*x**2 + 1)*asinh(c*x)/(75 *c**5), Ne(c, 0)), (a**2*(d**2*x + 2*d*e*x**3/3 + e**2*x**5/5), True))
Time = 0.21 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.30 \[ \int \left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\frac {1}{5} \, b^{2} e^{2} x^{5} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{5} \, a^{2} e^{2} x^{5} + \frac {2}{3} \, b^{2} d e x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {2}{3} \, a^{2} d e x^{3} + b^{2} d^{2} x \operatorname {arsinh}\left (c x\right )^{2} + \frac {4}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b d e - \frac {4}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} d e + \frac {2}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} a b e^{2} - \frac {2}{1125} \, {\left (15 \, {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c \operatorname {arsinh}\left (c x\right ) - \frac {9 \, c^{4} x^{5} - 20 \, c^{2} x^{3} + 120 \, x}{c^{4}}\right )} b^{2} e^{2} + 2 \, b^{2} d^{2} {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d^{2} x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d^{2}}{c} \]
1/5*b^2*e^2*x^5*arcsinh(c*x)^2 + 1/5*a^2*e^2*x^5 + 2/3*b^2*d*e*x^3*arcsinh (c*x)^2 + 2/3*a^2*d*e*x^3 + b^2*d^2*x*arcsinh(c*x)^2 + 4/9*(3*x^3*arcsinh( c*x) - c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4))*a*b*d*e - 4/27*(3*c*(sqrt(c^2*x^2 + 1)*x^2/c^2 - 2*sqrt(c^2*x^2 + 1)/c^4)*arcsinh(c* x) - (c^2*x^3 - 6*x)/c^2)*b^2*d*e + 2/75*(15*x^5*arcsinh(c*x) - (3*sqrt(c^ 2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6 )*c)*a*b*e^2 - 2/1125*(15*(3*sqrt(c^2*x^2 + 1)*x^4/c^2 - 4*sqrt(c^2*x^2 + 1)*x^2/c^4 + 8*sqrt(c^2*x^2 + 1)/c^6)*c*arcsinh(c*x) - (9*c^4*x^5 - 20*c^2 *x^3 + 120*x)/c^4)*b^2*e^2 + 2*b^2*d^2*(x - sqrt(c^2*x^2 + 1)*arcsinh(c*x) /c) + a^2*d^2*x + 2*(c*x*arcsinh(c*x) - sqrt(c^2*x^2 + 1))*a*b*d^2/c
Exception generated. \[ \int \left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\text {Exception raised: RuntimeError} \]
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. \[ \int \left (d+e x^2\right )^2 (a+b \text {arcsinh}(c x))^2 \, dx=\int {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (e\,x^2+d\right )}^2 \,d x \]