Integrand size = 35, antiderivative size = 264 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 (b d-a e)^4 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {8 b (b d-a e)^3 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {4 b^2 (b d-a e)^2 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac {8 b^3 (b d-a e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac {2 b^4 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)} \]
2/5*(-a*e+b*d)^4*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-8/7*b*(-a*e+b *d)^3*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+4/3*b^2*(-a*e+b*d)^2*(e* x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^5/(b*x+a)-8/11*b^3*(-a*e+b*d)*(e*x+d)^(11/2 )*((b*x+a)^2)^(1/2)/e^5/(b*x+a)+2/13*b^4*(e*x+d)^(13/2)*((b*x+a)^2)^(1/2)/ e^5/(b*x+a)
Time = 0.04 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.65 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} \left (3003 a^4 e^4+1716 a^3 b e^3 (-2 d+5 e x)+286 a^2 b^2 e^2 \left (8 d^2-20 d e x+35 e^2 x^2\right )+52 a b^3 e \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+b^4 \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )\right )}{15015 e^5 (a+b x)} \]
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(3003*a^4*e^4 + 1716*a^3*b*e^3*(-2*d + 5*e*x) + 286*a^2*b^2*e^2*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 52*a*b^3*e*(- 16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + b^4*(128*d^4 - 320*d^3 *e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4)))/(15015*e^5*(a + b *x))
Time = 0.30 (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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^3 (a+b x)^4 (d+e x)^{3/2}dx}{b^3 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^4 (d+e x)^{3/2}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^4 (d+e x)^{11/2}}{e^4}-\frac {4 b^3 (b d-a e) (d+e x)^{9/2}}{e^4}+\frac {6 b^2 (b d-a e)^2 (d+e x)^{7/2}}{e^4}-\frac {4 b (b d-a e)^3 (d+e x)^{5/2}}{e^4}+\frac {(a e-b d)^4 (d+e x)^{3/2}}{e^4}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {8 b^3 (d+e x)^{11/2} (b d-a e)}{11 e^5}+\frac {4 b^2 (d+e x)^{9/2} (b d-a e)^2}{3 e^5}-\frac {8 b (d+e x)^{7/2} (b d-a e)^3}{7 e^5}+\frac {2 (d+e x)^{5/2} (b d-a e)^4}{5 e^5}+\frac {2 b^4 (d+e x)^{13/2}}{13 e^5}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((2*(b*d - a*e)^4*(d + e*x)^(5/2))/(5*e^5) - (8*b*(b*d - a*e)^3*(d + e*x)^(7/2))/(7*e^5) + (4*b^2*(b*d - a*e)^2*(d + e*x)^(9/2))/(3*e^5) - (8*b^3*(b*d - a*e)*(d + e*x)^(11/2))/(11*e^5) + (2*b ^4*(d + e*x)^(13/2))/(13*e^5)))/(a + b*x)
3.22.1.3.1 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 = 0.29 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.77
| method | result | size |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (1155 e^{4} x^{4} b^{4}+5460 x^{3} a \,b^{3} e^{4}-840 x^{3} b^{4} d \,e^{3}+10010 x^{2} a^{2} b^{2} e^{4}-3640 x^{2} a \,b^{3} d \,e^{3}+560 x^{2} b^{4} d^{2} e^{2}+8580 x \,a^{3} b \,e^{4}-5720 x \,a^{2} b^{2} d \,e^{3}+2080 x a \,b^{3} d^{2} e^{2}-320 x \,b^{4} d^{3} e +3003 e^{4} a^{4}-3432 b d \,e^{3} a^{3}+2288 b^{2} d^{2} e^{2} a^{2}-832 b^{3} d^{3} e a +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15015 e^{5} \left (b x +a \right )^{3}}\) | \(202\) |
| default | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (1155 e^{4} x^{4} b^{4}+5460 x^{3} a \,b^{3} e^{4}-840 x^{3} b^{4} d \,e^{3}+10010 x^{2} a^{2} b^{2} e^{4}-3640 x^{2} a \,b^{3} d \,e^{3}+560 x^{2} b^{4} d^{2} e^{2}+8580 x \,a^{3} b \,e^{4}-5720 x \,a^{2} b^{2} d \,e^{3}+2080 x a \,b^{3} d^{2} e^{2}-320 x \,b^{4} d^{3} e +3003 e^{4} a^{4}-3432 b d \,e^{3} a^{3}+2288 b^{2} d^{2} e^{2} a^{2}-832 b^{3} d^{3} e a +128 b^{4} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15015 e^{5} \left (b x +a \right )^{3}}\) | \(202\) |
| risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (1155 b^{4} x^{6} e^{6}+5460 a \,b^{3} e^{6} x^{5}+1470 b^{4} d \,e^{5} x^{5}+10010 a^{2} b^{2} e^{6} x^{4}+7280 a \,b^{3} d \,e^{5} x^{4}+35 b^{4} d^{2} e^{4} x^{4}+8580 a^{3} b \,e^{6} x^{3}+14300 a^{2} b^{2} d \,e^{5} x^{3}+260 a \,b^{3} d^{2} e^{4} x^{3}-40 b^{4} d^{3} e^{3} x^{3}+3003 a^{4} e^{6} x^{2}+13728 a^{3} b d \,e^{5} x^{2}+858 a^{2} b^{2} d^{2} e^{4} x^{2}-312 a \,b^{3} d^{3} e^{3} x^{2}+48 b^{4} d^{4} e^{2} x^{2}+6006 a^{4} d \,e^{5} x +1716 a^{3} b \,d^{2} e^{4} x -1144 a^{2} b^{2} d^{3} e^{3} x +416 a \,b^{3} d^{4} e^{2} x -64 b^{4} d^{5} e x +3003 a^{4} d^{2} e^{4}-3432 a^{3} b \,d^{3} e^{3}+2288 a^{2} b^{2} d^{4} e^{2}-832 a \,b^{3} d^{5} e +128 b^{4} d^{6}\right ) \sqrt {e x +d}}{15015 \left (b x +a \right ) e^{5}}\) | \(348\) |
2/15015*(e*x+d)^(5/2)*(1155*b^4*e^4*x^4+5460*a*b^3*e^4*x^3-840*b^4*d*e^3*x ^3+10010*a^2*b^2*e^4*x^2-3640*a*b^3*d*e^3*x^2+560*b^4*d^2*e^2*x^2+8580*a^3 *b*e^4*x-5720*a^2*b^2*d*e^3*x+2080*a*b^3*d^2*e^2*x-320*b^4*d^3*e*x+3003*a^ 4*e^4-3432*a^3*b*d*e^3+2288*a^2*b^2*d^2*e^2-832*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 = 311, normalized size of antiderivative = 1.18 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (1155 \, b^{4} e^{6} x^{6} + 128 \, b^{4} d^{6} - 832 \, a b^{3} d^{5} e + 2288 \, a^{2} b^{2} d^{4} e^{2} - 3432 \, a^{3} b d^{3} e^{3} + 3003 \, a^{4} d^{2} e^{4} + 210 \, {\left (7 \, b^{4} d e^{5} + 26 \, a b^{3} e^{6}\right )} x^{5} + 35 \, {\left (b^{4} d^{2} e^{4} + 208 \, a b^{3} d e^{5} + 286 \, a^{2} b^{2} e^{6}\right )} x^{4} - 20 \, {\left (2 \, b^{4} d^{3} e^{3} - 13 \, a b^{3} d^{2} e^{4} - 715 \, a^{2} b^{2} d e^{5} - 429 \, a^{3} b e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{4} d^{4} e^{2} - 104 \, a b^{3} d^{3} e^{3} + 286 \, a^{2} b^{2} d^{2} e^{4} + 4576 \, a^{3} b d e^{5} + 1001 \, a^{4} e^{6}\right )} x^{2} - 2 \, {\left (32 \, b^{4} d^{5} e - 208 \, a b^{3} d^{4} e^{2} + 572 \, a^{2} b^{2} d^{3} e^{3} - 858 \, a^{3} b d^{2} e^{4} - 3003 \, a^{4} d e^{5}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{5}} \]
2/15015*(1155*b^4*e^6*x^6 + 128*b^4*d^6 - 832*a*b^3*d^5*e + 2288*a^2*b^2*d ^4*e^2 - 3432*a^3*b*d^3*e^3 + 3003*a^4*d^2*e^4 + 210*(7*b^4*d*e^5 + 26*a*b ^3*e^6)*x^5 + 35*(b^4*d^2*e^4 + 208*a*b^3*d*e^5 + 286*a^2*b^2*e^6)*x^4 - 2 0*(2*b^4*d^3*e^3 - 13*a*b^3*d^2*e^4 - 715*a^2*b^2*d*e^5 - 429*a^3*b*e^6)*x ^3 + 3*(16*b^4*d^4*e^2 - 104*a*b^3*d^3*e^3 + 286*a^2*b^2*d^2*e^4 + 4576*a^ 3*b*d*e^5 + 1001*a^4*e^6)*x^2 - 2*(32*b^4*d^5*e - 208*a*b^3*d^4*e^2 + 572* a^2*b^2*d^3*e^3 - 858*a^3*b*d^2*e^4 - 3003*a^4*d*e^5)*x)*sqrt(e*x + d)/e^5
\[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (a + b x\right ) \left (d + e x\right )^{\frac {3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 488 vs. \(2 (189) = 378\).
Time = 0.22 (sec) , antiderivative size = 488, normalized size of antiderivative = 1.85 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 \, {\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \, {\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \, {\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d} a}{1155 \, e^{4}} + \frac {2 \, {\left (1155 \, b^{3} e^{6} x^{6} + 128 \, b^{3} d^{6} - 624 \, a b^{2} d^{5} e + 1144 \, a^{2} b d^{4} e^{2} - 858 \, a^{3} d^{3} e^{3} + 105 \, {\left (14 \, b^{3} d e^{5} + 39 \, a b^{2} e^{6}\right )} x^{5} + 35 \, {\left (b^{3} d^{2} e^{4} + 156 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{3} e^{3} - 39 \, a b^{2} d^{2} e^{4} - 1430 \, a^{2} b d e^{5} - 429 \, a^{3} e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{4} e^{2} - 78 \, a b^{2} d^{3} e^{3} + 143 \, a^{2} b d^{2} e^{4} + 1144 \, a^{3} d e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{5} e - 312 \, a b^{2} d^{4} e^{2} + 572 \, a^{2} b d^{3} e^{3} - 429 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d} b}{15015 \, e^{5}} \]
2/1155*(105*b^3*e^5*x^5 - 16*b^3*d^5 + 88*a*b^2*d^4*e - 198*a^2*b*d^3*e^2 + 231*a^3*d^2*e^3 + 35*(4*b^3*d*e^4 + 11*a*b^2*e^5)*x^4 + 5*(b^3*d^2*e^3 + 110*a*b^2*d*e^4 + 99*a^2*b*e^5)*x^3 - 3*(2*b^3*d^3*e^2 - 11*a*b^2*d^2*e^3 - 264*a^2*b*d*e^4 - 77*a^3*e^5)*x^2 + (8*b^3*d^4*e - 44*a*b^2*d^3*e^2 + 9 9*a^2*b*d^2*e^3 + 462*a^3*d*e^4)*x)*sqrt(e*x + d)*a/e^4 + 2/15015*(1155*b^ 3*e^6*x^6 + 128*b^3*d^6 - 624*a*b^2*d^5*e + 1144*a^2*b*d^4*e^2 - 858*a^3*d ^3*e^3 + 105*(14*b^3*d*e^5 + 39*a*b^2*e^6)*x^5 + 35*(b^3*d^2*e^4 + 156*a*b ^2*d*e^5 + 143*a^2*b*e^6)*x^4 - 5*(8*b^3*d^3*e^3 - 39*a*b^2*d^2*e^4 - 1430 *a^2*b*d*e^5 - 429*a^3*e^6)*x^3 + 3*(16*b^3*d^4*e^2 - 78*a*b^2*d^3*e^3 + 1 43*a^2*b*d^2*e^4 + 1144*a^3*d*e^5)*x^2 - (64*b^3*d^5*e - 312*a*b^2*d^4*e^2 + 572*a^2*b*d^3*e^3 - 429*a^3*d^2*e^4)*x)*sqrt(e*x + d)*b/e^5
Leaf count of result is larger than twice the leaf count of optimal. 897 vs. \(2 (189) = 378\).
Time = 0.29 (sec) , antiderivative size = 897, normalized size of antiderivative = 3.40 \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(e*x + d)*a^4*d^2*sgn(b*x + a) + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*d*sgn(b*x + a) + 60060*((e*x + d)^(3/2) - 3*sqrt (e*x + d)*d)*a^3*b*d^2*sgn(b*x + a)/e + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^4*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^2*sgn(b*x + a)/e^2 + 24024*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*b*d*sgn(b*x + a)/e + 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^2*sgn(b*x + a)/e^3 + 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^2*b^2*d*sgn(b*x + a)/e^2 + 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^3*b*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^2*sgn(b*x + a)/e^4 + 1144*(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*sqr t(e*x + d)*d^4)*a*b^3*d*sgn(b*x + a)/e^3 + 858*(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*sgn(b*x + a)/e^2 + 130*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(e*x + d)^(5/2)*d^3 + 1155*(e*x + d)^(3/2)*d^4 - 693*sqrt(e*x + d)*d^5)*b^4*d*sgn(b*x + a)...
Timed out. \[ \int (a+b x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \left (a+b\,x\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]