Integrand size = 33, antiderivative size = 152 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 (b d-a e)^5}{3 e^6 (d+e x)^{3/2}}-\frac {10 b (b d-a e)^4}{e^6 \sqrt {d+e x}}-\frac {20 b^2 (b d-a e)^3 \sqrt {d+e x}}{e^6}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{3/2}}{3 e^6}-\frac {2 b^4 (b d-a e) (d+e x)^{5/2}}{e^6}+\frac {2 b^5 (d+e x)^{7/2}}{7 e^6} \]
2/3*(-a*e+b*d)^5/e^6/(e*x+d)^(3/2)+20/3*b^3*(-a*e+b*d)^2*(e*x+d)^(3/2)/e^6 -2*b^4*(-a*e+b*d)*(e*x+d)^(5/2)/e^6+2/7*b^5*(e*x+d)^(7/2)/e^6-10*b*(-a*e+b *d)^4/e^6/(e*x+d)^(1/2)-20*b^2*(-a*e+b*d)^3*(e*x+d)^(1/2)/e^6
Time = 0.04 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (7 a^5 e^5+35 a^4 b e^4 (2 d+3 e x)-70 a^3 b^2 e^3 \left (8 d^2+12 d e x+3 e^2 x^2\right )+70 a^2 b^3 e^2 \left (16 d^3+24 d^2 e x+6 d e^2 x^2-e^3 x^3\right )-7 a b^4 e \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+b^5 \left (256 d^5+384 d^4 e x+96 d^3 e^2 x^2-16 d^2 e^3 x^3+6 d e^4 x^4-3 e^5 x^5\right )\right )}{21 e^6 (d+e x)^{3/2}} \]
(-2*(7*a^5*e^5 + 35*a^4*b*e^4*(2*d + 3*e*x) - 70*a^3*b^2*e^3*(8*d^2 + 12*d *e*x + 3*e^2*x^2) + 70*a^2*b^3*e^2*(16*d^3 + 24*d^2*e*x + 6*d*e^2*x^2 - e^ 3*x^3) - 7*a*b^4*e*(128*d^4 + 192*d^3*e*x + 48*d^2*e^2*x^2 - 8*d*e^3*x^3 + 3*e^4*x^4) + b^5*(256*d^5 + 384*d^4*e*x + 96*d^3*e^2*x^2 - 16*d^2*e^3*x^3 + 6*d*e^4*x^4 - 3*e^5*x^5)))/(21*e^6*(d + e*x)^(3/2))
Time = 0.31 (sec) , antiderivative size = 152, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int \frac {b^4 (a+b x)^5}{(d+e x)^{5/2}}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int \frac {(a+b x)^5}{(d+e x)^{5/2}}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (d+e x)^{3/2} (b d-a e)}{e^5}+\frac {10 b^3 \sqrt {d+e x} (b d-a e)^2}{e^5}-\frac {10 b^2 (b d-a e)^3}{e^5 \sqrt {d+e x}}+\frac {5 b (b d-a e)^4}{e^5 (d+e x)^{3/2}}+\frac {(a e-b d)^5}{e^5 (d+e x)^{5/2}}+\frac {b^5 (d+e x)^{5/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^4 (d+e x)^{5/2} (b d-a e)}{e^6}+\frac {20 b^3 (d+e x)^{3/2} (b d-a e)^2}{3 e^6}-\frac {20 b^2 \sqrt {d+e x} (b d-a e)^3}{e^6}-\frac {10 b (b d-a e)^4}{e^6 \sqrt {d+e x}}+\frac {2 (b d-a e)^5}{3 e^6 (d+e x)^{3/2}}+\frac {2 b^5 (d+e x)^{7/2}}{7 e^6}\) |
(2*(b*d - a*e)^5)/(3*e^6*(d + e*x)^(3/2)) - (10*b*(b*d - a*e)^4)/(e^6*Sqrt [d + e*x]) - (20*b^2*(b*d - a*e)^3*Sqrt[d + e*x])/e^6 + (20*b^3*(b*d - a*e )^2*(d + e*x)^(3/2))/(3*e^6) - (2*b^4*(b*d - a*e)*(d + e*x)^(5/2))/e^6 + ( 2*b^5*(d + e*x)^(7/2))/(7*e^6)
3.21.56.3.1 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 = 0.34 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.28
| method | result | size |
| risch | \(\frac {2 b^{2} \left (3 b^{3} x^{3} e^{3}+21 x^{2} a \,b^{2} e^{3}-12 x^{2} b^{3} d \,e^{2}+70 x \,a^{2} b \,e^{3}-98 x a \,b^{2} d \,e^{2}+37 x \,b^{3} d^{2} e +210 a^{3} e^{3}-560 a^{2} b d \,e^{2}+511 a \,b^{2} d^{2} e -158 b^{3} d^{3}\right ) \sqrt {e x +d}}{21 e^{6}}-\frac {2 \left (15 b e x +a e +14 b d \right ) \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}{3 e^{6} \left (e x +d \right )^{\frac {3}{2}}}\) | \(194\) |
| pseudoelliptic | \(-\frac {2 \left (\left (-\frac {3}{7} b^{5} x^{5}-3 a \,b^{4} x^{4}-10 a^{2} b^{3} x^{3}-30 a^{3} b^{2} x^{2}+15 a^{4} b x +a^{5}\right ) e^{5}+10 b \left (\frac {3}{35} x^{4} b^{4}+\frac {4}{5} a \,b^{3} x^{3}+6 x^{2} b^{2} a^{2}-12 b \,a^{3} x +a^{4}\right ) d \,e^{4}-80 b^{2} \left (\frac {1}{35} x^{3} b^{3}+\frac {3}{5} a \,b^{2} x^{2}-3 b \,a^{2} x +a^{3}\right ) d^{2} e^{3}+160 b^{3} \left (\frac {3}{35} b^{2} x^{2}-\frac {6}{5} a b x +a^{2}\right ) d^{3} e^{2}-128 b^{4} \left (-\frac {3 b x}{7}+a \right ) d^{4} e +\frac {256 b^{5} d^{5}}{7}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(204\) |
| gosper | \(-\frac {2 \left (-3 x^{5} b^{5} e^{5}-21 x^{4} a \,b^{4} e^{5}+6 x^{4} b^{5} d \,e^{4}-70 x^{3} a^{2} b^{3} e^{5}+56 x^{3} a \,b^{4} d \,e^{4}-16 x^{3} b^{5} d^{2} e^{3}-210 x^{2} a^{3} b^{2} e^{5}+420 x^{2} a^{2} b^{3} d \,e^{4}-336 x^{2} a \,b^{4} d^{2} e^{3}+96 x^{2} b^{5} d^{3} e^{2}+105 x \,a^{4} b \,e^{5}-840 x \,a^{3} b^{2} d \,e^{4}+1680 x \,a^{2} b^{3} d^{2} e^{3}-1344 x a \,b^{4} d^{3} e^{2}+384 x \,b^{5} d^{4} e +7 e^{5} a^{5}+70 b d \,e^{4} a^{4}-560 b^{2} d^{2} e^{3} a^{3}+1120 b^{3} d^{3} e^{2} a^{2}-896 b^{4} d^{4} e a +256 b^{5} d^{5}\right )}{21 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(273\) |
| trager | \(-\frac {2 \left (-3 x^{5} b^{5} e^{5}-21 x^{4} a \,b^{4} e^{5}+6 x^{4} b^{5} d \,e^{4}-70 x^{3} a^{2} b^{3} e^{5}+56 x^{3} a \,b^{4} d \,e^{4}-16 x^{3} b^{5} d^{2} e^{3}-210 x^{2} a^{3} b^{2} e^{5}+420 x^{2} a^{2} b^{3} d \,e^{4}-336 x^{2} a \,b^{4} d^{2} e^{3}+96 x^{2} b^{5} d^{3} e^{2}+105 x \,a^{4} b \,e^{5}-840 x \,a^{3} b^{2} d \,e^{4}+1680 x \,a^{2} b^{3} d^{2} e^{3}-1344 x a \,b^{4} d^{3} e^{2}+384 x \,b^{5} d^{4} e +7 e^{5} a^{5}+70 b d \,e^{4} a^{4}-560 b^{2} d^{2} e^{3} a^{3}+1120 b^{3} d^{3} e^{2} a^{2}-896 b^{4} d^{4} e a +256 b^{5} d^{5}\right )}{21 \left (e x +d \right )^{\frac {3}{2}} e^{6}}\) | \(273\) |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {7}{2}}}{7}+2 a \,b^{4} e \left (e x +d \right )^{\frac {5}{2}}-2 b^{5} d \left (e x +d \right )^{\frac {5}{2}}+\frac {20 a^{2} b^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {40 a \,b^{4} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {20 b^{5} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+20 a^{3} b^{2} e^{3} \sqrt {e x +d}-60 a^{2} b^{3} d \,e^{2} \sqrt {e x +d}+60 a \,b^{4} d^{2} e \sqrt {e x +d}-20 b^{5} d^{3} \sqrt {e x +d}-\frac {2 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {10 b \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(294\) |
| default | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {7}{2}}}{7}+2 a \,b^{4} e \left (e x +d \right )^{\frac {5}{2}}-2 b^{5} d \left (e x +d \right )^{\frac {5}{2}}+\frac {20 a^{2} b^{3} e^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}-\frac {40 a \,b^{4} d e \left (e x +d \right )^{\frac {3}{2}}}{3}+\frac {20 b^{5} d^{2} \left (e x +d \right )^{\frac {3}{2}}}{3}+20 a^{3} b^{2} e^{3} \sqrt {e x +d}-60 a^{2} b^{3} d \,e^{2} \sqrt {e x +d}+60 a \,b^{4} d^{2} e \sqrt {e x +d}-20 b^{5} d^{3} \sqrt {e x +d}-\frac {2 \left (e^{5} a^{5}-5 b d \,e^{4} a^{4}+10 b^{2} d^{2} e^{3} a^{3}-10 b^{3} d^{3} e^{2} a^{2}+5 b^{4} d^{4} e a -b^{5} d^{5}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}-\frac {10 b \left (e^{4} a^{4}-4 b d \,e^{3} a^{3}+6 b^{2} d^{2} e^{2} a^{2}-4 b^{3} d^{3} e a +b^{4} d^{4}\right )}{\sqrt {e x +d}}}{e^{6}}\) | \(294\) |
2/21*b^2*(3*b^3*e^3*x^3+21*a*b^2*e^3*x^2-12*b^3*d*e^2*x^2+70*a^2*b*e^3*x-9 8*a*b^2*d*e^2*x+37*b^3*d^2*e*x+210*a^3*e^3-560*a^2*b*d*e^2+511*a*b^2*d^2*e -158*b^3*d^3)*(e*x+d)^(1/2)/e^6-2/3*(15*b*e*x+a*e+14*b*d)*(a^4*e^4-4*a^3*b *d*e^3+6*a^2*b^2*d^2*e^2-4*a*b^3*d^3*e+b^4*d^4)/e^6/(e*x+d)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (134) = 268\).
Time = 0.30 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.86 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (3 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 896 \, a b^{4} d^{4} e - 1120 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} - 70 \, a^{4} b d e^{4} - 7 \, a^{5} e^{5} - 3 \, {\left (2 \, b^{5} d e^{4} - 7 \, a b^{4} e^{5}\right )} x^{4} + 2 \, {\left (8 \, b^{5} d^{2} e^{3} - 28 \, a b^{4} d e^{4} + 35 \, a^{2} b^{3} e^{5}\right )} x^{3} - 6 \, {\left (16 \, b^{5} d^{3} e^{2} - 56 \, a b^{4} d^{2} e^{3} + 70 \, a^{2} b^{3} d e^{4} - 35 \, a^{3} b^{2} e^{5}\right )} x^{2} - 3 \, {\left (128 \, b^{5} d^{4} e - 448 \, a b^{4} d^{3} e^{2} + 560 \, a^{2} b^{3} d^{2} e^{3} - 280 \, a^{3} b^{2} d e^{4} + 35 \, a^{4} b e^{5}\right )} x\right )} \sqrt {e x + d}}{21 \, {\left (e^{8} x^{2} + 2 \, d e^{7} x + d^{2} e^{6}\right )}} \]
2/21*(3*b^5*e^5*x^5 - 256*b^5*d^5 + 896*a*b^4*d^4*e - 1120*a^2*b^3*d^3*e^2 + 560*a^3*b^2*d^2*e^3 - 70*a^4*b*d*e^4 - 7*a^5*e^5 - 3*(2*b^5*d*e^4 - 7*a *b^4*e^5)*x^4 + 2*(8*b^5*d^2*e^3 - 28*a*b^4*d*e^4 + 35*a^2*b^3*e^5)*x^3 - 6*(16*b^5*d^3*e^2 - 56*a*b^4*d^2*e^3 + 70*a^2*b^3*d*e^4 - 35*a^3*b^2*e^5)* x^2 - 3*(128*b^5*d^4*e - 448*a*b^4*d^3*e^2 + 560*a^2*b^3*d^2*e^3 - 280*a^3 *b^2*d*e^4 + 35*a^4*b*e^5)*x)*sqrt(e*x + d)/(e^8*x^2 + 2*d*e^7*x + d^2*e^6 )
Time = 6.82 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.55 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (d + e x\right )^{\frac {7}{2}}}{7 e^{5}} - \frac {5 b \left (a e - b d\right )^{4}}{e^{5} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (5 a b^{4} e - 5 b^{5} d\right )}{5 e^{5}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (10 a^{2} b^{3} e^{2} - 20 a b^{4} d e + 10 b^{5} d^{2}\right )}{3 e^{5}} + \frac {\sqrt {d + e x} \left (10 a^{3} b^{2} e^{3} - 30 a^{2} b^{3} d e^{2} + 30 a b^{4} d^{2} e - 10 b^{5} d^{3}\right )}{e^{5}} - \frac {\left (a e - b d\right )^{5}}{3 e^{5} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{3}}{6 b} & \text {otherwise} \end {cases}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
Piecewise((2*(b**5*(d + e*x)**(7/2)/(7*e**5) - 5*b*(a*e - b*d)**4/(e**5*sq rt(d + e*x)) + (d + e*x)**(5/2)*(5*a*b**4*e - 5*b**5*d)/(5*e**5) + (d + e* x)**(3/2)*(10*a**2*b**3*e**2 - 20*a*b**4*d*e + 10*b**5*d**2)/(3*e**5) + sq rt(d + e*x)*(10*a**3*b**2*e**3 - 30*a**2*b**3*d*e**2 + 30*a*b**4*d**2*e - 10*b**5*d**3)/e**5 - (a*e - b*d)**5/(3*e**5*(d + e*x)**(3/2)))/e, Ne(e, 0) ), (Piecewise((a**5*x, Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**3/(6*b), True))/d**(5/2), True))
Time = 0.19 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.74 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} - 21 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 70 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 210 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} \sqrt {e x + d}}{e^{5}} + \frac {7 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5} - 15 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{5}}\right )}}{21 \, e} \]
2/21*((3*(e*x + d)^(7/2)*b^5 - 21*(b^5*d - a*b^4*e)*(e*x + d)^(5/2) + 70*( b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(3/2) - 210*(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)*sqrt(e*x + d))/e^5 + 7*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5 - 15*(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)^(3/2)*e^5))/e
Leaf count of result is larger than twice the leaf count of optimal. 335 vs. \(2 (134) = 268\).
Time = 0.28 (sec) , antiderivative size = 335, normalized size of antiderivative = 2.20 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=-\frac {2 \, {\left (15 \, {\left (e x + d\right )} b^{5} d^{4} - b^{5} d^{5} - 60 \, {\left (e x + d\right )} a b^{4} d^{3} e + 5 \, a b^{4} d^{4} e + 90 \, {\left (e x + d\right )} a^{2} b^{3} d^{2} e^{2} - 10 \, a^{2} b^{3} d^{3} e^{2} - 60 \, {\left (e x + d\right )} a^{3} b^{2} d e^{3} + 10 \, a^{3} b^{2} d^{2} e^{3} + 15 \, {\left (e x + d\right )} a^{4} b e^{4} - 5 \, a^{4} b d e^{4} + a^{5} e^{5}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{6}} + \frac {2 \, {\left (3 \, {\left (e x + d\right )}^{\frac {7}{2}} b^{5} e^{36} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} b^{5} d e^{36} + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{5} d^{2} e^{36} - 210 \, \sqrt {e x + d} b^{5} d^{3} e^{36} + 21 \, {\left (e x + d\right )}^{\frac {5}{2}} a b^{4} e^{37} - 140 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{4} d e^{37} + 630 \, \sqrt {e x + d} a b^{4} d^{2} e^{37} + 70 \, {\left (e x + d\right )}^{\frac {3}{2}} a^{2} b^{3} e^{38} - 630 \, \sqrt {e x + d} a^{2} b^{3} d e^{38} + 210 \, \sqrt {e x + d} a^{3} b^{2} e^{39}\right )}}{21 \, e^{42}} \]
-2/3*(15*(e*x + d)*b^5*d^4 - b^5*d^5 - 60*(e*x + d)*a*b^4*d^3*e + 5*a*b^4* d^4*e + 90*(e*x + d)*a^2*b^3*d^2*e^2 - 10*a^2*b^3*d^3*e^2 - 60*(e*x + d)*a ^3*b^2*d*e^3 + 10*a^3*b^2*d^2*e^3 + 15*(e*x + d)*a^4*b*e^4 - 5*a^4*b*d*e^4 + a^5*e^5)/((e*x + d)^(3/2)*e^6) + 2/21*(3*(e*x + d)^(7/2)*b^5*e^36 - 21* (e*x + d)^(5/2)*b^5*d*e^36 + 70*(e*x + d)^(3/2)*b^5*d^2*e^36 - 210*sqrt(e* x + d)*b^5*d^3*e^36 + 21*(e*x + d)^(5/2)*a*b^4*e^37 - 140*(e*x + d)^(3/2)* a*b^4*d*e^37 + 630*sqrt(e*x + d)*a*b^4*d^2*e^37 + 70*(e*x + d)^(3/2)*a^2*b ^3*e^38 - 630*sqrt(e*x + d)*a^2*b^3*d*e^38 + 210*sqrt(e*x + d)*a^3*b^2*e^3 9)/e^42
Time = 10.70 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.51 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2}{(d+e x)^{5/2}} \, dx=\frac {2\,b^5\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^6}-\frac {\left (d+e\,x\right )\,\left (10\,a^4\,b\,e^4-40\,a^3\,b^2\,d\,e^3+60\,a^2\,b^3\,d^2\,e^2-40\,a\,b^4\,d^3\,e+10\,b^5\,d^4\right )+\frac {2\,a^5\,e^5}{3}-\frac {2\,b^5\,d^5}{3}-\frac {20\,a^2\,b^3\,d^3\,e^2}{3}+\frac {20\,a^3\,b^2\,d^2\,e^3}{3}+\frac {10\,a\,b^4\,d^4\,e}{3}-\frac {10\,a^4\,b\,d\,e^4}{3}}{e^6\,{\left (d+e\,x\right )}^{3/2}}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,\sqrt {d+e\,x}}{e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{3/2}}{3\,e^6} \]
(2*b^5*(d + e*x)^(7/2))/(7*e^6) - ((10*b^5*d - 10*a*b^4*e)*(d + e*x)^(5/2) )/(5*e^6) - ((d + e*x)*(10*b^5*d^4 + 10*a^4*b*e^4 - 40*a^3*b^2*d*e^3 + 60* a^2*b^3*d^2*e^2 - 40*a*b^4*d^3*e) + (2*a^5*e^5)/3 - (2*b^5*d^5)/3 - (20*a^ 2*b^3*d^3*e^2)/3 + (20*a^3*b^2*d^2*e^3)/3 + (10*a*b^4*d^4*e)/3 - (10*a^4*b *d*e^4)/3)/(e^6*(d + e*x)^(3/2)) + (20*b^2*(a*e - b*d)^3*(d + e*x)^(1/2))/ e^6 + (20*b^3*(a*e - b*d)^2*(d + e*x)^(3/2))/(3*e^6)