Integrand size = 33, antiderivative size = 154 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 (b d-a e)^5}{5 e^6 (d+e x)^{5/2}}-\frac {10 b (b d-a e)^4}{3 e^6 (d+e x)^{3/2}}+\frac {20 b^2 (b d-a e)^3}{e^6 \sqrt {d+e x}}+\frac {20 b^3 (b d-a e)^2 \sqrt {d+e x}}{e^6}-\frac {10 b^4 (b d-a e) (d+e x)^{3/2}}{3 e^6}+\frac {2 b^5 (d+e x)^{5/2}}{5 e^6} \]
2/5*(-a*e+b*d)^5/e^6/(e*x+d)^(5/2)-10/3*b*(-a*e+b*d)^4/e^6/(e*x+d)^(3/2)-1 0/3*b^4*(-a*e+b*d)*(e*x+d)^(3/2)/e^6+2/5*b^5*(e*x+d)^(5/2)/e^6+20*b^2*(-a* e+b*d)^3/e^6/(e*x+d)^(1/2)+20*b^3*(-a*e+b*d)^2*(e*x+d)^(1/2)/e^6
Time = 0.04 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.42 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=-\frac {2 \left (3 a^5 e^5+5 a^4 b e^4 (2 d+5 e x)+10 a^3 b^2 e^3 \left (8 d^2+20 d e x+15 e^2 x^2\right )-30 a^2 b^3 e^2 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+5 a b^4 e \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )-b^5 \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )\right )}{15 e^6 (d+e x)^{5/2}} \]
(-2*(3*a^5*e^5 + 5*a^4*b*e^4*(2*d + 5*e*x) + 10*a^3*b^2*e^3*(8*d^2 + 20*d* e*x + 15*e^2*x^2) - 30*a^2*b^3*e^2*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5 *e^3*x^3) + 5*a*b^4*e*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3* x^3 - 5*e^4*x^4) - b^5*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e ^3*x^3 - 10*d*e^4*x^4 + 3*e^5*x^5)))/(15*e^6*(d + e*x)^(5/2))
Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1184, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^5}{(d+e x)^{7/2}}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^5}{(d+e x)^{7/2}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 \sqrt {d+e x} (b d-a e)}{e^5}+\frac {10 b^3 (b d-a e)^2}{e^5 \sqrt {d+e x}}-\frac {10 b^2 (b d-a e)^3}{e^5 (d+e x)^{3/2}}+\frac {5 b (b d-a e)^4}{e^5 (d+e x)^{5/2}}+\frac {(a e-b d)^5}{e^5 (d+e x)^{7/2}}+\frac {b^5 (d+e x)^{3/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {10 b^4 (d+e x)^{3/2} (b d-a e)}{3 e^6}+\frac {20 b^3 \sqrt {d+e x} (b d-a e)^2}{e^6}+\frac {20 b^2 (b d-a e)^3}{e^6 \sqrt {d+e x}}-\frac {10 b (b d-a e)^4}{3 e^6 (d+e x)^{3/2}}+\frac {2 (b d-a e)^5}{5 e^6 (d+e x)^{5/2}}+\frac {2 b^5 (d+e x)^{5/2}}{5 e^6}\) |
(2*(b*d - a*e)^5)/(5*e^6*(d + e*x)^(5/2)) - (10*b*(b*d - a*e)^4)/(3*e^6*(d + e*x)^(3/2)) + (20*b^2*(b*d - a*e)^3)/(e^6*Sqrt[d + e*x]) + (20*b^3*(b*d - a*e)^2*Sqrt[d + e*x])/e^6 - (10*b^4*(b*d - a*e)*(d + e*x)^(3/2))/(3*e^6 ) + (2*b^5*(d + e*x)^(5/2))/(5*e^6)
3.21.57.3.1 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[1/c^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}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 0.33 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\left (x^{4} b^{4}+\frac {28}{3} a \,b^{3} x^{3}+\frac {178}{3} x^{2} b^{2} a^{2}+\frac {28}{3} b \,a^{3} x +a^{4}\right ) e^{4}+\frac {16 b d \left (-b x +a \right ) \left (b^{2} x^{2}+18 a b x +a^{2}\right ) e^{3}}{3}+\frac {112 b^{2} \left (b^{2} x^{2}-\frac {34}{7} a b x +a^{2}\right ) d^{2} e^{2}}{3}-\frac {256 b^{3} d^{3} \left (-b x +a \right ) e}{3}+\frac {128 b^{4} d^{4}}{3}\right ) \left (\left (-b x +a \right ) e -2 b d \right )}{5 \left (e x +d \right )^{\frac {5}{2}} e^{6}}\) | \(148\) |
| risch | \(\frac {2 b^{3} \left (3 b^{2} e^{2} x^{2}+25 a b \,e^{2} x -19 b^{2} d e x +150 e^{2} a^{2}-275 a b d e +128 b^{2} d^{2}\right ) \sqrt {e x +d}}{15 e^{6}}-\frac {2 \left (150 b^{2} e^{2} x^{2}+25 a b \,e^{2} x +275 b^{2} d e x +3 e^{2} a^{2}+19 a b d e +128 b^{2} d^{2}\right ) \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{15 e^{6} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d e x +d^{2}\right )}\) | \(183\) |
| gosper | \(-\frac {2 \left (-3 x^{5} b^{5} e^{5}-25 x^{4} a \,b^{4} e^{5}+10 x^{4} b^{5} d \,e^{4}-150 x^{3} a^{2} b^{3} e^{5}+200 x^{3} a \,b^{4} d \,e^{4}-80 x^{3} b^{5} d^{2} e^{3}+150 x^{2} a^{3} b^{2} e^{5}-900 x^{2} a^{2} b^{3} d \,e^{4}+1200 x^{2} a \,b^{4} d^{2} e^{3}-480 x^{2} b^{5} d^{3} e^{2}+25 x \,a^{4} b \,e^{5}+200 x \,a^{3} b^{2} d \,e^{4}-1200 x \,a^{2} b^{3} d^{2} e^{3}+1600 x a \,b^{4} d^{3} e^{2}-640 x \,b^{5} d^{4} e +3 e^{5} a^{5}+10 b d \,e^{4} a^{4}+80 b^{2} d^{2} e^{3} a^{3}-480 b^{3} d^{3} e^{2} a^{2}+640 b^{4} d^{4} e a -256 b^{5} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}}\) | \(273\) |
| trager | \(-\frac {2 \left (-3 x^{5} b^{5} e^{5}-25 x^{4} a \,b^{4} e^{5}+10 x^{4} b^{5} d \,e^{4}-150 x^{3} a^{2} b^{3} e^{5}+200 x^{3} a \,b^{4} d \,e^{4}-80 x^{3} b^{5} d^{2} e^{3}+150 x^{2} a^{3} b^{2} e^{5}-900 x^{2} a^{2} b^{3} d \,e^{4}+1200 x^{2} a \,b^{4} d^{2} e^{3}-480 x^{2} b^{5} d^{3} e^{2}+25 x \,a^{4} b \,e^{5}+200 x \,a^{3} b^{2} d \,e^{4}-1200 x \,a^{2} b^{3} d^{2} e^{3}+1600 x a \,b^{4} d^{3} e^{2}-640 x \,b^{5} d^{4} e +3 e^{5} a^{5}+10 b d \,e^{4} a^{4}+80 b^{2} d^{2} e^{3} a^{3}-480 b^{3} d^{3} e^{2} a^{2}+640 b^{4} d^{4} e a -256 b^{5} d^{5}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{6}}\) | \(273\) |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {10 a \,b^{4} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {10 b^{5} d \left (e x +d \right )^{\frac {3}{2}}}{3}+20 a^{2} b^{3} e^{2} \sqrt {e x +d}-40 a \,b^{4} d e \sqrt {e x +d}+20 b^{5} d^{2} \sqrt {e x +d}-\frac {2 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {10 b \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {20 b^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(274\) |
| default | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {10 a \,b^{4} e \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {10 b^{5} d \left (e x +d \right )^{\frac {3}{2}}}{3}+20 a^{2} b^{3} e^{2} \sqrt {e x +d}-40 a \,b^{4} d e \sqrt {e x +d}+20 b^{5} d^{2} \sqrt {e x +d}-\frac {2 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {10 b \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {20 b^{2} \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(274\) |
-2/5*((x^4*b^4+28/3*a*b^3*x^3+178/3*x^2*b^2*a^2+28/3*b*a^3*x+a^4)*e^4+16/3 *b*d*(-b*x+a)*(b^2*x^2+18*a*b*x+a^2)*e^3+112/3*b^2*(b^2*x^2-34/7*a*b*x+a^2 )*d^2*e^2-256/3*b^3*d^3*(-b*x+a)*e+128/3*b^4*d^4)/(e*x+d)^(5/2)*((-b*x+a)* e-2*b*d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 294 vs. \(2 (134) = 268\).
Time = 0.29 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.91 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} - 640 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} - 80 \, a^{3} b^{2} d^{2} e^{3} - 10 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 5 \, {\left (2 \, b^{5} d e^{4} - 5 \, a b^{4} e^{5}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{2} e^{3} - 20 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 30 \, {\left (16 \, b^{5} d^{3} e^{2} - 40 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} - 5 \, a^{3} b^{2} e^{5}\right )} x^{2} + 5 \, {\left (128 \, b^{5} d^{4} e - 320 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} - 40 \, a^{3} b^{2} d e^{4} - 5 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )}} \]
2/15*(3*b^5*e^5*x^5 + 256*b^5*d^5 - 640*a*b^4*d^4*e + 480*a^2*b^3*d^3*e^2 - 80*a^3*b^2*d^2*e^3 - 10*a^4*b*d*e^4 - 3*a^5*e^5 - 5*(2*b^5*d*e^4 - 5*a*b ^4*e^5)*x^4 + 10*(8*b^5*d^2*e^3 - 20*a*b^4*d*e^4 + 15*a^2*b^3*e^5)*x^3 + 3 0*(16*b^5*d^3*e^2 - 40*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 - 5*a^3*b^2*e^5)*x ^2 + 5*(128*b^5*d^4*e - 320*a*b^4*d^3*e^2 + 240*a^2*b^3*d^2*e^3 - 40*a^3*b ^2*d*e^4 - 5*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^ 7*x + d^3*e^6)
Leaf count of result is larger than twice the leaf count of optimal. 1428 vs. \(2 (143) = 286\).
Time = 0.55 (sec) , antiderivative size = 1428, normalized size of antiderivative = 9.27 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=\begin {cases} - \frac {6 a^{5} e^{5}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {20 a^{4} b d e^{4}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {50 a^{4} b e^{5} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {160 a^{3} b^{2} d^{2} e^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {400 a^{3} b^{2} d e^{4} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {300 a^{3} b^{2} e^{5} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {960 a^{2} b^{3} d^{3} e^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {2400 a^{2} b^{3} d^{2} e^{3} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {1800 a^{2} b^{3} d e^{4} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {300 a^{2} b^{3} e^{5} x^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {1280 a b^{4} d^{4} e}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {3200 a b^{4} d^{3} e^{2} x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {2400 a b^{4} d^{2} e^{3} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {400 a b^{4} d e^{4} x^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {50 a b^{4} e^{5} x^{4}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {512 b^{5} d^{5}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {1280 b^{5} d^{4} e x}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {960 b^{5} d^{3} e^{2} x^{2}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {160 b^{5} d^{2} e^{3} x^{3}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} - \frac {20 b^{5} d e^{4} x^{4}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} + \frac {6 b^{5} e^{5} x^{5}}{15 d^{2} e^{6} \sqrt {d + e x} + 30 d e^{7} x \sqrt {d + e x} + 15 e^{8} x^{2} \sqrt {d + e x}} & \text {for}\: e \neq 0 \\\frac {a^{5} x + \frac {5 a^{4} b x^{2}}{2} + \frac {10 a^{3} b^{2} x^{3}}{3} + \frac {5 a^{2} b^{3} x^{4}}{2} + a b^{4} x^{5} + \frac {b^{5} x^{6}}{6}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((-6*a**5*e**5/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 20*a**4*b*d*e**4/(15*d**2*e**6*sqrt( d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 50*a* *4*b*e**5*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e **8*x**2*sqrt(d + e*x)) - 160*a**3*b**2*d**2*e**3/(15*d**2*e**6*sqrt(d + e *x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 400*a**3*b **2*d*e**4*x/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15* e**8*x**2*sqrt(d + e*x)) - 300*a**3*b**2*e**5*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 960*a**2* b**3*d**3*e**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 1 5*e**8*x**2*sqrt(d + e*x)) + 2400*a**2*b**3*d**2*e**3*x/(15*d**2*e**6*sqrt (d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 1800 *a**2*b**3*d*e**4*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) + 300*a**2*b**3*e**5*x**3/(15*d**2*e**6 *sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 1280*a*b**4*d**4*e/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x ) + 15*e**8*x**2*sqrt(d + e*x)) - 3200*a*b**4*d**3*e**2*x/(15*d**2*e**6*sq rt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 24 00*a*b**4*d**2*e**3*x**2/(15*d**2*e**6*sqrt(d + e*x) + 30*d*e**7*x*sqrt(d + e*x) + 15*e**8*x**2*sqrt(d + e*x)) - 400*a*b**4*d*e**4*x**3/(15*d**2*...
Time = 0.20 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.72 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} - 25 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 150 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} \sqrt {e x + d}}{e^{5}} + \frac {3 \, b^{5} d^{5} - 15 \, a b^{4} d^{4} e + 30 \, a^{2} b^{3} d^{3} e^{2} - 30 \, a^{3} b^{2} d^{2} e^{3} + 15 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} + 150 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{2} - 25 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{5}}\right )}}{15 \, e} \]
2/15*((3*(e*x + d)^(5/2)*b^5 - 25*(b^5*d - a*b^4*e)*(e*x + d)^(3/2) + 150* (b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*sqrt(e*x + d))/e^5 + (3*b^5*d^5 - 15 *a*b^4*d^4*e + 30*a^2*b^3*d^3*e^2 - 30*a^3*b^2*d^2*e^3 + 15*a^4*b*d*e^4 - 3*a^5*e^5 + 150*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)* (e*x + d)^2 - 25*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2* d*e^3 + a^4*b*e^4)*(e*x + d))/((e*x + d)^(5/2)*e^5))/e
Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (134) = 268\).
Time = 0.28 (sec) , antiderivative size = 331, normalized size of antiderivative = 2.15 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (150 \, {\left (e x + d\right )}^{2} b^{5} d^{3} - 25 \, {\left (e x + d\right )} b^{5} d^{4} + 3 \, b^{5} d^{5} - 450 \, {\left (e x + d\right )}^{2} a b^{4} d^{2} e + 100 \, {\left (e x + d\right )} a b^{4} d^{3} e - 15 \, a b^{4} d^{4} e + 450 \, {\left (e x + d\right )}^{2} a^{2} b^{3} d e^{2} - 150 \, {\left (e x + d\right )} a^{2} b^{3} d^{2} e^{2} + 30 \, a^{2} b^{3} d^{3} e^{2} - 150 \, {\left (e x + d\right )}^{2} a^{3} b^{2} e^{3} + 100 \, {\left (e x + d\right )} a^{3} b^{2} d e^{3} - 30 \, a^{3} b^{2} d^{2} e^{3} - 25 \, {\left (e x + d\right )} a^{4} b e^{4} + 15 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{6}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} e^{24} - 25 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d e^{24} + 150 \, \sqrt {e x + d} b^{5} d^{2} e^{24} + 25 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} e^{25} - 300 \, \sqrt {e x + d} a b^{4} d e^{25} + 150 \, \sqrt {e x + d} a^{2} b^{3} e^{26}\right )}}{15 \, e^{30}} \]
2/15*(150*(e*x + d)^2*b^5*d^3 - 25*(e*x + d)*b^5*d^4 + 3*b^5*d^5 - 450*(e* x + d)^2*a*b^4*d^2*e + 100*(e*x + d)*a*b^4*d^3*e - 15*a*b^4*d^4*e + 450*(e *x + d)^2*a^2*b^3*d*e^2 - 150*(e*x + d)*a^2*b^3*d^2*e^2 + 30*a^2*b^3*d^3*e ^2 - 150*(e*x + d)^2*a^3*b^2*e^3 + 100*(e*x + d)*a^3*b^2*d*e^3 - 30*a^3*b^ 2*d^2*e^3 - 25*(e*x + d)*a^4*b*e^4 + 15*a^4*b*d*e^4 - 3*a^5*e^5)/((e*x + d )^(5/2)*e^6) + 2/15*(3*(e*x + d)^(5/2)*b^5*e^24 - 25*(e*x + d)^(3/2)*b^5*d *e^24 + 150*sqrt(e*x + d)*b^5*d^2*e^24 + 25*(e*x + d)^(3/2)*a*b^4*e^25 - 3 00*sqrt(e*x + d)*a*b^4*d*e^25 + 150*sqrt(e*x + d)*a^2*b^3*e^26)/e^30
Time = 0.09 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.66 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{7/2}} \, dx=\frac {2\,b^5\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}-\frac {\left (d+e\,x\right )\,\left (\frac {10\,a^4\,b\,e^4}{3}-\frac {40\,a^3\,b^2\,d\,e^3}{3}+20\,a^2\,b^3\,d^2\,e^2-\frac {40\,a\,b^4\,d^3\,e}{3}+\frac {10\,b^5\,d^4}{3}\right )-{\left (d+e\,x\right )}^2\,\left (-20\,a^3\,b^2\,e^3+60\,a^2\,b^3\,d\,e^2-60\,a\,b^4\,d^2\,e+20\,b^5\,d^3\right )+\frac {2\,a^5\,e^5}{5}-\frac {2\,b^5\,d^5}{5}-4\,a^2\,b^3\,d^3\,e^2+4\,a^3\,b^2\,d^2\,e^3+2\,a\,b^4\,d^4\,e-2\,a^4\,b\,d\,e^4}{e^6\,{\left (d+e\,x\right )}^{5/2}}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,\sqrt {d+e\,x}}{e^6} \]
(2*b^5*(d + e*x)^(5/2))/(5*e^6) - ((d + e*x)*((10*b^5*d^4)/3 + (10*a^4*b*e ^4)/3 - (40*a^3*b^2*d*e^3)/3 + 20*a^2*b^3*d^2*e^2 - (40*a*b^4*d^3*e)/3) - (d + e*x)^2*(20*b^5*d^3 - 20*a^3*b^2*e^3 + 60*a^2*b^3*d*e^2 - 60*a*b^4*d^2 *e) + (2*a^5*e^5)/5 - (2*b^5*d^5)/5 - 4*a^2*b^3*d^3*e^2 + 4*a^3*b^2*d^2*e^ 3 + 2*a*b^4*d^4*e - 2*a^4*b*d*e^4)/(e^6*(d + e*x)^(5/2)) - ((10*b^5*d - 10 *a*b^4*e)*(d + e*x)^(3/2))/(3*e^6) + (20*b^3*(a*e - b*d)^2*(d + e*x)^(1/2) )/e^6