Integrand size = 35, antiderivative size = 264 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}-\frac {8 b (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^5 (a+b x)}+\frac {12 b^2 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}-\frac {8 b^3 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}+\frac {2 b^4 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^5 (a+b x)} \]
2/7*(-a*e+b*d)^4*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-8/9*b*(-a*e+b *d)^3*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+12/11*b^2*(-a*e+b*d)^2*( e*x+d)^(11/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-8/13*b^3*(-a*e+b*d)*(e*x+d)^(1 3/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+2/15*b^4*(e*x+d)^(15/2)*((b*x+a)^2)^(1/ 2)/e^5/(b*x+a)
Time = 0.05 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.65 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (6435 a^4 e^4+2860 a^3 b e^3 (-2 d+7 e x)+390 a^2 b^2 e^2 \left (8 d^2-28 d e x+63 e^2 x^2\right )+60 a b^3 e \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+b^4 \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )\right )}{45045 e^5 (a+b x)} \]
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(6435*a^4*e^4 + 2860*a^3*b*e^3*(-2*d + 7*e*x) + 390*a^2*b^2*e^2*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 60*a*b^3*e*(- 16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + b^4*(128*d^4 - 448*d^ 3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4)))/(45045*e^5*(a + b*x))
Time = 0.31 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.59, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.114, Rules used = {1187, 27, 53, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^4 (d+e x)^{5/2}dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^4 (d+e x)^{5/2}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^4 (d+e x)^{13/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{11/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{9/2}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{7/2}}{e^4}+\frac {(a e-b d)^4 (d+e x)^{5/2}}{e^4}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {8 b^3 (d+e x)^{13/2} (b d-a e)}{13 e^5}+\frac {12 b^2 (d+e x)^{11/2} (b d-a e)^2}{11 e^5}-\frac {8 b (d+e x)^{9/2} (b d-a e)^3}{9 e^5}+\frac {2 (d+e x)^{7/2} (b d-a e)^4}{7 e^5}+\frac {2 b^4 (d+e x)^{15/2}}{15 e^5}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((2*(b*d - a*e)^4*(d + e*x)^(7/2))/(7*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^5) + (12*b^2*(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^5) + ( 2*b^4*(d + e*x)^(15/2))/(15*e^5)))/(a + b*x)
3.21.100.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (b_.)*(x_ ) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^ IntPart[p]*(b/2 + c*x)^(2*FracPart[p])) Int[(d + e*x)^m*(f + g*x)^n*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p}, x] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[p]
Time = 0.30 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.77
| method | result | size |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 e^{4} x^{4} b^{4}+13860 x^{3} a \,b^{3} e^{4}-1848 x^{3} b^{4} d \,e^{3}+24570 x^{2} a^{2} b^{2} e^{4}-7560 x^{2} a \,b^{3} d \,e^{3}+1008 x^{2} b^{4} d^{2} e^{2}+20020 x \,a^{3} b \,e^{4}-10920 x \,a^{2} b^{2} d \,e^{3}+3360 x a \,b^{3} d^{2} e^{2}-448 x \,b^{4} d^{3} e +6435 e^{4} a^{4}-5720 b d \,e^{3} a^{3}+3120 b^{2} d^{2} e^{2} a^{2}-960 b^{3} d^{3} e a +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{45045 e^{5} \left (b x +a \right )^{3}}\) | \(202\) |
| default | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (3003 e^{4} x^{4} b^{4}+13860 x^{3} a \,b^{3} e^{4}-1848 x^{3} b^{4} d \,e^{3}+24570 x^{2} a^{2} b^{2} e^{4}-7560 x^{2} a \,b^{3} d \,e^{3}+1008 x^{2} b^{4} d^{2} e^{2}+20020 x \,a^{3} b \,e^{4}-10920 x \,a^{2} b^{2} d \,e^{3}+3360 x a \,b^{3} d^{2} e^{2}-448 x \,b^{4} d^{3} e +6435 e^{4} a^{4}-5720 b d \,e^{3} a^{3}+3120 b^{2} d^{2} e^{2} a^{2}-960 b^{3} d^{3} e a +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{45045 e^{5} \left (b x +a \right )^{3}}\) | \(202\) |
| risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (3003 b^{4} e^{7} x^{7}+13860 a \,b^{3} e^{7} x^{6}+7161 b^{4} d \,e^{6} x^{6}+24570 a^{2} b^{2} e^{7} x^{5}+34020 a \,b^{3} d \,e^{6} x^{5}+4473 b^{4} d^{2} e^{5} x^{5}+20020 a^{3} b \,e^{7} x^{4}+62790 a^{2} b^{2} d \,e^{6} x^{4}+22260 a \,b^{3} d^{2} e^{5} x^{4}+35 b^{4} d^{3} e^{4} x^{4}+6435 a^{4} e^{7} x^{3}+54340 a^{3} b d \,e^{6} x^{3}+44070 a^{2} b^{2} d^{2} e^{5} x^{3}+300 a \,b^{3} d^{3} e^{4} x^{3}-40 b^{4} d^{4} e^{3} x^{3}+19305 a^{4} d \,e^{6} x^{2}+42900 a^{3} b \,d^{2} e^{5} x^{2}+1170 a^{2} b^{2} d^{3} e^{4} x^{2}-360 a \,b^{3} d^{4} e^{3} x^{2}+48 b^{4} d^{5} e^{2} x^{2}+19305 a^{4} d^{2} e^{5} x +2860 a^{3} b \,d^{3} e^{4} x -1560 a^{2} b^{2} d^{4} e^{3} x +480 a \,b^{3} d^{5} e^{2} x -64 b^{4} d^{6} e x +6435 a^{4} d^{3} e^{4}-5720 a^{3} b \,d^{4} e^{3}+3120 a^{2} b^{2} d^{5} e^{2}-960 a \,b^{3} d^{6} e +128 b^{4} d^{7}\right ) \sqrt {e x +d}}{45045 \left (b x +a \right ) e^{5}}\) | \(423\) |
2/45045*(e*x+d)^(7/2)*(3003*b^4*e^4*x^4+13860*a*b^3*e^4*x^3-1848*b^4*d*e^3 *x^3+24570*a^2*b^2*e^4*x^2-7560*a*b^3*d*e^3*x^2+1008*b^4*d^2*e^2*x^2+20020 *a^3*b*e^4*x-10920*a^2*b^2*d*e^3*x+3360*a*b^3*d^2*e^2*x-448*b^4*d^3*e*x+64 35*a^4*e^4-5720*a^3*b*d*e^3+3120*a^2*b^2*d^2*e^2-960*a*b^3*d^3*e+128*b^4*d ^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3
Time = 0.31 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.43 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (3003 \, b^{4} e^{7} x^{7} + 128 \, b^{4} d^{7} - 960 \, a b^{3} d^{6} e + 3120 \, a^{2} b^{2} d^{5} e^{2} - 5720 \, a^{3} b d^{4} e^{3} + 6435 \, a^{4} d^{3} e^{4} + 231 \, {\left (31 \, b^{4} d e^{6} + 60 \, a b^{3} e^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{4} d^{2} e^{5} + 540 \, a b^{3} d e^{6} + 390 \, a^{2} b^{2} e^{7}\right )} x^{5} + 35 \, {\left (b^{4} d^{3} e^{4} + 636 \, a b^{3} d^{2} e^{5} + 1794 \, a^{2} b^{2} d e^{6} + 572 \, a^{3} b e^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{4} d^{4} e^{3} - 60 \, a b^{3} d^{3} e^{4} - 8814 \, a^{2} b^{2} d^{2} e^{5} - 10868 \, a^{3} b d e^{6} - 1287 \, a^{4} e^{7}\right )} x^{3} + 3 \, {\left (16 \, b^{4} d^{5} e^{2} - 120 \, a b^{3} d^{4} e^{3} + 390 \, a^{2} b^{2} d^{3} e^{4} + 14300 \, a^{3} b d^{2} e^{5} + 6435 \, a^{4} d e^{6}\right )} x^{2} - {\left (64 \, b^{4} d^{6} e - 480 \, a b^{3} d^{5} e^{2} + 1560 \, a^{2} b^{2} d^{4} e^{3} - 2860 \, a^{3} b d^{3} e^{4} - 19305 \, a^{4} d^{2} e^{5}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{5}} \]
2/45045*(3003*b^4*e^7*x^7 + 128*b^4*d^7 - 960*a*b^3*d^6*e + 3120*a^2*b^2*d ^5*e^2 - 5720*a^3*b*d^4*e^3 + 6435*a^4*d^3*e^4 + 231*(31*b^4*d*e^6 + 60*a* b^3*e^7)*x^6 + 63*(71*b^4*d^2*e^5 + 540*a*b^3*d*e^6 + 390*a^2*b^2*e^7)*x^5 + 35*(b^4*d^3*e^4 + 636*a*b^3*d^2*e^5 + 1794*a^2*b^2*d*e^6 + 572*a^3*b*e^ 7)*x^4 - 5*(8*b^4*d^4*e^3 - 60*a*b^3*d^3*e^4 - 8814*a^2*b^2*d^2*e^5 - 1086 8*a^3*b*d*e^6 - 1287*a^4*e^7)*x^3 + 3*(16*b^4*d^5*e^2 - 120*a*b^3*d^4*e^3 + 390*a^2*b^2*d^3*e^4 + 14300*a^3*b*d^2*e^5 + 6435*a^4*d*e^6)*x^2 - (64*b^ 4*d^6*e - 480*a*b^3*d^5*e^2 + 1560*a^2*b^2*d^4*e^3 - 2860*a^3*b*d^3*e^4 - 19305*a^4*d^2*e^5)*x)*sqrt(e*x + d)/e^5
Timed out. \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (189) = 378\).
Time = 0.22 (sec) , antiderivative size = 592, normalized size of antiderivative = 2.24 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (231 \, b^{3} e^{6} x^{6} - 16 \, b^{3} d^{6} + 104 \, a b^{2} d^{5} e - 286 \, a^{2} b d^{4} e^{2} + 429 \, a^{3} d^{3} e^{3} + 63 \, {\left (9 \, b^{3} d e^{5} + 13 \, a b^{2} e^{6}\right )} x^{5} + 7 \, {\left (53 \, b^{3} d^{2} e^{4} + 299 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} + {\left (5 \, b^{3} d^{3} e^{3} + 1469 \, a b^{2} d^{2} e^{4} + 2717 \, a^{2} b d e^{5} + 429 \, a^{3} e^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{4} e^{2} - 13 \, a b^{2} d^{3} e^{3} - 715 \, a^{2} b d^{2} e^{4} - 429 \, a^{3} d e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{5} e - 52 \, a b^{2} d^{4} e^{2} + 143 \, a^{2} b d^{3} e^{3} + 1287 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d} a}{3003 \, e^{4}} + \frac {2 \, {\left (3003 \, b^{3} e^{7} x^{7} + 128 \, b^{3} d^{7} - 720 \, a b^{2} d^{6} e + 1560 \, a^{2} b d^{5} e^{2} - 1430 \, a^{3} d^{4} e^{3} + 231 \, {\left (31 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 63 \, {\left (71 \, b^{3} d^{2} e^{5} + 405 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 35 \, {\left (b^{3} d^{3} e^{4} + 477 \, a b^{2} d^{2} e^{5} + 897 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{4} e^{3} - 45 \, a b^{2} d^{3} e^{4} - 4407 \, a^{2} b d^{2} e^{5} - 2717 \, a^{3} d e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{5} e^{2} - 90 \, a b^{2} d^{4} e^{3} + 195 \, a^{2} b d^{3} e^{4} + 3575 \, a^{3} d^{2} e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{6} e - 360 \, a b^{2} d^{5} e^{2} + 780 \, a^{2} b d^{4} e^{3} - 715 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d} b}{45045 \, e^{5}} \]
2/3003*(231*b^3*e^6*x^6 - 16*b^3*d^6 + 104*a*b^2*d^5*e - 286*a^2*b*d^4*e^2 + 429*a^3*d^3*e^3 + 63*(9*b^3*d*e^5 + 13*a*b^2*e^6)*x^5 + 7*(53*b^3*d^2*e ^4 + 299*a*b^2*d*e^5 + 143*a^2*b*e^6)*x^4 + (5*b^3*d^3*e^3 + 1469*a*b^2*d^ 2*e^4 + 2717*a^2*b*d*e^5 + 429*a^3*e^6)*x^3 - 3*(2*b^3*d^4*e^2 - 13*a*b^2* d^3*e^3 - 715*a^2*b*d^2*e^4 - 429*a^3*d*e^5)*x^2 + (8*b^3*d^5*e - 52*a*b^2 *d^4*e^2 + 143*a^2*b*d^3*e^3 + 1287*a^3*d^2*e^4)*x)*sqrt(e*x + d)*a/e^4 + 2/45045*(3003*b^3*e^7*x^7 + 128*b^3*d^7 - 720*a*b^2*d^6*e + 1560*a^2*b*d^5 *e^2 - 1430*a^3*d^4*e^3 + 231*(31*b^3*d*e^6 + 45*a*b^2*e^7)*x^6 + 63*(71*b ^3*d^2*e^5 + 405*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 35*(b^3*d^3*e^4 + 477* a*b^2*d^2*e^5 + 897*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 - 5*(8*b^3*d^4*e^3 - 45 *a*b^2*d^3*e^4 - 4407*a^2*b*d^2*e^5 - 2717*a^3*d*e^6)*x^3 + 3*(16*b^3*d^5* e^2 - 90*a*b^2*d^4*e^3 + 195*a^2*b*d^3*e^4 + 3575*a^3*d^2*e^5)*x^2 - (64*b ^3*d^6*e - 360*a*b^2*d^5*e^2 + 780*a^2*b*d^4*e^3 - 715*a^3*d^3*e^4)*x)*sqr t(e*x + d)*b/e^5
Leaf count of result is larger than twice the leaf count of optimal. 1324 vs. \(2 (189) = 378\).
Time = 0.39 (sec) , antiderivative size = 1324, normalized size of antiderivative = 5.02 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(e*x + d)*a^4*d^3*sgn(b*x + a) + 45045*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*d^2*sgn(b*x + a) + 60060*((e*x + d)^(3/2) - 3*sq rt(e*x + d)*d)*a^3*b*d^3*sgn(b*x + a)/e + 9009*(3*(e*x + d)^(5/2) - 10*(e* x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^4*d*sgn(b*x + a) + 18018*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b^2*d^3*sgn( b*x + a)/e^2 + 36036*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e *x + d)*d^2)*a^3*b*d^2*sgn(b*x + a)/e + 1287*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^4*sgn(b*x + a) + 5148*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2) *d^2 - 35*sqrt(e*x + d)*d^3)*a*b^3*d^3*sgn(b*x + a)/e^3 + 23166*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d )*d^3)*a^2*b^2*d^2*sgn(b*x + a)/e^2 + 15444*(5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqrt(e*x + d)*d^3)*a^3*b*d*sgn(b *x + a)/e + 143*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d )^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*b^4*d^3*sgn (b*x + a)/e^4 + 1716*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e* x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4)*a*b^3* d^2*sgn(b*x + a)/e^3 + 2574*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(7/2)*d + 378*(e*x + d)^(5/2)*d^2 - 420*(e*x + d)^(3/2)*d^3 + 315*sqrt(e*x + d)*d^4) *a^2*b^2*d*sgn(b*x + a)/e^2 + 572*(35*(e*x + d)^(9/2) - 180*(e*x + d)^(...
Timed out. \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]