Integrand size = 35, antiderivative size = 368 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {2 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{7/2}}+\frac {12 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x) (d+e x)^{5/2}}-\frac {10 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{3/2}}+\frac {40 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}}+\frac {30 b^4 (b d-a e)^2 \sqrt {d+e x} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}-\frac {4 b^5 (b d-a e) (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)} \]
-2/7*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(7/2)+12/5*b*(-a*e +b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(5/2)-10*b^2*(-a*e+b*d)^4*(( b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(3/2)-4*b^5*(-a*e+b*d)*(e*x+d)^(3/2)*( (b*x+a)^2)^(1/2)/e^7/(b*x+a)+2/5*b^6*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^7/( b*x+a)+40*b^3*(-a*e+b*d)^3*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(1/2)+30* b^4*(-a*e+b*d)^2*(e*x+d)^(1/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 0.17 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (5 a^6 e^6+6 a^5 b e^5 (2 d+7 e x)+5 a^4 b^2 e^4 \left (8 d^2+28 d e x+35 e^2 x^2\right )+20 a^3 b^3 e^3 \left (16 d^3+56 d^2 e x+70 d e^2 x^2+35 e^3 x^3\right )-15 a^2 b^4 e^2 \left (128 d^4+448 d^3 e x+560 d^2 e^2 x^2+280 d e^3 x^3+35 e^4 x^4\right )+10 a b^5 e \left (256 d^5+896 d^4 e x+1120 d^3 e^2 x^2+560 d^2 e^3 x^3+70 d e^4 x^4-7 e^5 x^5\right )-b^6 \left (1024 d^6+3584 d^5 e x+4480 d^4 e^2 x^2+2240 d^3 e^3 x^3+280 d^2 e^4 x^4-28 d e^5 x^5+7 e^6 x^6\right )\right )}{35 e^7 (a+b x) (d+e x)^{7/2}} \]
(-2*Sqrt[(a + b*x)^2]*(5*a^6*e^6 + 6*a^5*b*e^5*(2*d + 7*e*x) + 5*a^4*b^2*e ^4*(8*d^2 + 28*d*e*x + 35*e^2*x^2) + 20*a^3*b^3*e^3*(16*d^3 + 56*d^2*e*x + 70*d*e^2*x^2 + 35*e^3*x^3) - 15*a^2*b^4*e^2*(128*d^4 + 448*d^3*e*x + 560* d^2*e^2*x^2 + 280*d*e^3*x^3 + 35*e^4*x^4) + 10*a*b^5*e*(256*d^5 + 896*d^4* e*x + 1120*d^3*e^2*x^2 + 560*d^2*e^3*x^3 + 70*d*e^4*x^4 - 7*e^5*x^5) - b^6 *(1024*d^6 + 3584*d^5*e*x + 4480*d^4*e^2*x^2 + 2240*d^3*e^3*x^3 + 280*d^2* e^4*x^4 - 28*d*e^5*x^5 + 7*e^6*x^6)))/(35*e^7*(a + b*x)*(d + e*x)^(7/2))
Time = 0.35 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.56, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{9/2}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{9/2}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {(d+e x)^{3/2} b^6}{e^6}-\frac {6 (b d-a e) \sqrt {d+e x} b^5}{e^6}+\frac {15 (b d-a e)^2 b^4}{e^6 \sqrt {d+e x}}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^{3/2}}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{5/2}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{7/2}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{9/2}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {4 b^5 (d+e x)^{3/2} (b d-a e)}{e^7}+\frac {30 b^4 \sqrt {d+e x} (b d-a e)^2}{e^7}+\frac {40 b^3 (b d-a e)^3}{e^7 \sqrt {d+e x}}-\frac {10 b^2 (b d-a e)^4}{e^7 (d+e x)^{3/2}}+\frac {12 b (b d-a e)^5}{5 e^7 (d+e x)^{5/2}}-\frac {2 (b d-a e)^6}{7 e^7 (d+e x)^{7/2}}+\frac {2 b^6 (d+e x)^{5/2}}{5 e^7}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((-2*(b*d - a*e)^6)/(7*e^7*(d + e*x)^(7/2)) + (12*b*(b*d - a*e)^5)/(5*e^7*(d + e*x)^(5/2)) - (10*b^2*(b*d - a*e)^4)/( e^7*(d + e*x)^(3/2)) + (40*b^3*(b*d - a*e)^3)/(e^7*Sqrt[d + e*x]) + (30*b^ 4*(b*d - a*e)^2*Sqrt[d + e*x])/e^7 - (4*b^5*(b*d - a*e)*(d + e*x)^(3/2))/e ^7 + (2*b^6*(d + e*x)^(5/2))/(5*e^7)))/(a + b*x)
3.22.17.3.1 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 = 0.32 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.76
| method | result | size |
| risch | \(\frac {2 b^{4} \left (b^{2} e^{2} x^{2}+10 a b \,e^{2} x -8 b^{2} d e x +75 e^{2} a^{2}-140 a b d e +66 b^{2} d^{2}\right ) \sqrt {e x +d}\, \sqrt {\left (b x +a \right )^{2}}}{5 e^{7} \left (b x +a \right )}-\frac {2 \left (700 b^{3} x^{3} e^{3}+175 x^{2} a \,b^{2} e^{3}+1925 x^{2} b^{3} d \,e^{2}+42 x \,a^{2} b \,e^{3}+266 x a \,b^{2} d \,e^{2}+1792 x \,b^{3} d^{2} e +5 a^{3} e^{3}+27 a^{2} b d \,e^{2}+106 a \,b^{2} d^{2} e +562 b^{3} d^{3}\right ) \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \sqrt {\left (b x +a \right )^{2}}}{35 e^{7} \sqrt {e x +d}\, \left (e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}\right ) \left (b x +a \right )}\) | \(278\) |
| gosper | \(-\frac {2 \left (-7 b^{6} e^{6} x^{6}-70 a \,b^{5} e^{6} x^{5}+28 b^{6} d \,e^{5} x^{5}-525 a^{2} b^{4} e^{6} x^{4}+700 a \,b^{5} d \,e^{5} x^{4}-280 b^{6} d^{2} e^{4} x^{4}+700 a^{3} b^{3} e^{6} x^{3}-4200 a^{2} b^{4} d \,e^{5} x^{3}+5600 a \,b^{5} d^{2} e^{4} x^{3}-2240 b^{6} d^{3} e^{3} x^{3}+175 a^{4} b^{2} e^{6} x^{2}+1400 a^{3} b^{3} d \,e^{5} x^{2}-8400 a^{2} b^{4} d^{2} e^{4} x^{2}+11200 a \,b^{5} d^{3} e^{3} x^{2}-4480 b^{6} d^{4} e^{2} x^{2}+42 a^{5} b \,e^{6} x +140 a^{4} b^{2} d \,e^{5} x +1120 a^{3} b^{3} d^{2} e^{4} x -6720 a^{2} b^{4} d^{3} e^{3} x +8960 a \,b^{5} d^{4} e^{2} x -3584 b^{6} d^{5} e x +5 e^{6} a^{6}+12 b d \,e^{5} a^{5}+40 b^{2} d^{2} e^{4} a^{4}+320 b^{3} d^{3} e^{3} a^{3}-1920 b^{4} d^{4} e^{2} a^{2}+2560 b^{5} d^{5} e a -1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{35 \left (e x +d \right )^{\frac {7}{2}} e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
| default | \(-\frac {2 \left (-7 b^{6} e^{6} x^{6}-70 a \,b^{5} e^{6} x^{5}+28 b^{6} d \,e^{5} x^{5}-525 a^{2} b^{4} e^{6} x^{4}+700 a \,b^{5} d \,e^{5} x^{4}-280 b^{6} d^{2} e^{4} x^{4}+700 a^{3} b^{3} e^{6} x^{3}-4200 a^{2} b^{4} d \,e^{5} x^{3}+5600 a \,b^{5} d^{2} e^{4} x^{3}-2240 b^{6} d^{3} e^{3} x^{3}+175 a^{4} b^{2} e^{6} x^{2}+1400 a^{3} b^{3} d \,e^{5} x^{2}-8400 a^{2} b^{4} d^{2} e^{4} x^{2}+11200 a \,b^{5} d^{3} e^{3} x^{2}-4480 b^{6} d^{4} e^{2} x^{2}+42 a^{5} b \,e^{6} x +140 a^{4} b^{2} d \,e^{5} x +1120 a^{3} b^{3} d^{2} e^{4} x -6720 a^{2} b^{4} d^{3} e^{3} x +8960 a \,b^{5} d^{4} e^{2} x -3584 b^{6} d^{5} e x +5 e^{6} a^{6}+12 b d \,e^{5} a^{5}+40 b^{2} d^{2} e^{4} a^{4}+320 b^{3} d^{3} e^{3} a^{3}-1920 b^{4} d^{4} e^{2} a^{2}+2560 b^{5} d^{5} e a -1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{35 \left (e x +d \right )^{\frac {7}{2}} e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
2/5*b^4*(b^2*e^2*x^2+10*a*b*e^2*x-8*b^2*d*e*x+75*a^2*e^2-140*a*b*d*e+66*b^ 2*d^2)*(e*x+d)^(1/2)/e^7*((b*x+a)^2)^(1/2)/(b*x+a)-2/35*(700*b^3*e^3*x^3+1 75*a*b^2*e^3*x^2+1925*b^3*d*e^2*x^2+42*a^2*b*e^3*x+266*a*b^2*d*e^2*x+1792* b^3*d^2*e*x+5*a^3*e^3+27*a^2*b*d*e^2+106*a*b^2*d^2*e+562*b^3*d^3)*(a^3*e^3 -3*a^2*b*d*e^2+3*a*b^2*d^2*e-b^3*d^3)/e^7/(e*x+d)^(1/2)/(e^3*x^3+3*d*e^2*x ^2+3*d^2*e*x+d^3)*((b*x+a)^2)^(1/2)/(b*x+a)
Time = 0.34 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.08 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (7 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} - 2560 \, a b^{5} d^{5} e + 1920 \, a^{2} b^{4} d^{4} e^{2} - 320 \, a^{3} b^{3} d^{3} e^{3} - 40 \, a^{4} b^{2} d^{2} e^{4} - 12 \, a^{5} b d e^{5} - 5 \, a^{6} e^{6} - 14 \, {\left (2 \, b^{6} d e^{5} - 5 \, a b^{5} e^{6}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{2} e^{4} - 20 \, a b^{5} d e^{5} + 15 \, a^{2} b^{4} e^{6}\right )} x^{4} + 140 \, {\left (16 \, b^{6} d^{3} e^{3} - 40 \, a b^{5} d^{2} e^{4} + 30 \, a^{2} b^{4} d e^{5} - 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 35 \, {\left (128 \, b^{6} d^{4} e^{2} - 320 \, a b^{5} d^{3} e^{3} + 240 \, a^{2} b^{4} d^{2} e^{4} - 40 \, a^{3} b^{3} d e^{5} - 5 \, a^{4} b^{2} e^{6}\right )} x^{2} + 14 \, {\left (256 \, b^{6} d^{5} e - 640 \, a b^{5} d^{4} e^{2} + 480 \, a^{2} b^{4} d^{3} e^{3} - 80 \, a^{3} b^{3} d^{2} e^{4} - 10 \, a^{4} b^{2} d e^{5} - 3 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{35 \, {\left (e^{11} x^{4} + 4 \, d e^{10} x^{3} + 6 \, d^{2} e^{9} x^{2} + 4 \, d^{3} e^{8} x + d^{4} e^{7}\right )}} \]
2/35*(7*b^6*e^6*x^6 + 1024*b^6*d^6 - 2560*a*b^5*d^5*e + 1920*a^2*b^4*d^4*e ^2 - 320*a^3*b^3*d^3*e^3 - 40*a^4*b^2*d^2*e^4 - 12*a^5*b*d*e^5 - 5*a^6*e^6 - 14*(2*b^6*d*e^5 - 5*a*b^5*e^6)*x^5 + 35*(8*b^6*d^2*e^4 - 20*a*b^5*d*e^5 + 15*a^2*b^4*e^6)*x^4 + 140*(16*b^6*d^3*e^3 - 40*a*b^5*d^2*e^4 + 30*a^2*b ^4*d*e^5 - 5*a^3*b^3*e^6)*x^3 + 35*(128*b^6*d^4*e^2 - 320*a*b^5*d^3*e^3 + 240*a^2*b^4*d^2*e^4 - 40*a^3*b^3*d*e^5 - 5*a^4*b^2*e^6)*x^2 + 14*(256*b^6* d^5*e - 640*a*b^5*d^4*e^2 + 480*a^2*b^4*d^3*e^3 - 80*a^3*b^3*d^2*e^4 - 10* a^4*b^2*d*e^5 - 3*a^5*b*e^6)*x)*sqrt(e*x + d)/(e^11*x^4 + 4*d*e^10*x^3 + 6 *d^2*e^9*x^2 + 4*d^3*e^8*x + d^4*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 668 vs. \(2 (271) = 542\).
Time = 0.24 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.82 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (7 \, b^{5} e^{5} x^{5} - 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e - 96 \, a^{2} b^{3} d^{3} e^{2} - 16 \, a^{3} b^{2} d^{2} e^{3} - 6 \, a^{4} b d e^{4} - 3 \, a^{5} e^{5} - 35 \, {\left (2 \, b^{5} d e^{4} - 3 \, a b^{4} e^{5}\right )} x^{4} - 70 \, {\left (8 \, b^{5} d^{2} e^{3} - 12 \, a b^{4} d e^{4} + 3 \, a^{2} b^{3} e^{5}\right )} x^{3} - 70 \, {\left (16 \, b^{5} d^{3} e^{2} - 24 \, a b^{4} d^{2} e^{3} + 6 \, a^{2} b^{3} d e^{4} + a^{3} b^{2} e^{5}\right )} x^{2} - 7 \, {\left (128 \, b^{5} d^{4} e - 192 \, a b^{4} d^{3} e^{2} + 48 \, a^{2} b^{3} d^{2} e^{3} + 8 \, a^{3} b^{2} d e^{4} + 3 \, a^{4} b e^{5}\right )} x\right )} a}{21 \, {\left (e^{9} x^{3} + 3 \, d e^{8} x^{2} + 3 \, d^{2} e^{7} x + d^{3} e^{6}\right )} \sqrt {e x + d}} + \frac {2 \, {\left (21 \, b^{5} e^{6} x^{6} + 3072 \, b^{5} d^{6} - 6400 \, a b^{4} d^{5} e + 3840 \, a^{2} b^{3} d^{4} e^{2} - 480 \, a^{3} b^{2} d^{3} e^{3} - 40 \, a^{4} b d^{2} e^{4} - 6 \, a^{5} d e^{5} - 7 \, {\left (12 \, b^{5} d e^{5} - 25 \, a b^{4} e^{6}\right )} x^{5} + 70 \, {\left (12 \, b^{5} d^{2} e^{4} - 25 \, a b^{4} d e^{5} + 15 \, a^{2} b^{3} e^{6}\right )} x^{4} + 70 \, {\left (96 \, b^{5} d^{3} e^{3} - 200 \, a b^{4} d^{2} e^{4} + 120 \, a^{2} b^{3} d e^{5} - 15 \, a^{3} b^{2} e^{6}\right )} x^{3} + 35 \, {\left (384 \, b^{5} d^{4} e^{2} - 800 \, a b^{4} d^{3} e^{3} + 480 \, a^{2} b^{3} d^{2} e^{4} - 60 \, a^{3} b^{2} d e^{5} - 5 \, a^{4} b e^{6}\right )} x^{2} + 7 \, {\left (1536 \, b^{5} d^{5} e - 3200 \, a b^{4} d^{4} e^{2} + 1920 \, a^{2} b^{3} d^{3} e^{3} - 240 \, a^{3} b^{2} d^{2} e^{4} - 20 \, a^{4} b d e^{5} - 3 \, a^{5} e^{6}\right )} x\right )} b}{105 \, {\left (e^{10} x^{3} + 3 \, d e^{9} x^{2} + 3 \, d^{2} e^{8} x + d^{3} e^{7}\right )} \sqrt {e x + d}} \]
2/21*(7*b^5*e^5*x^5 - 256*b^5*d^5 + 384*a*b^4*d^4*e - 96*a^2*b^3*d^3*e^2 - 16*a^3*b^2*d^2*e^3 - 6*a^4*b*d*e^4 - 3*a^5*e^5 - 35*(2*b^5*d*e^4 - 3*a*b^ 4*e^5)*x^4 - 70*(8*b^5*d^2*e^3 - 12*a*b^4*d*e^4 + 3*a^2*b^3*e^5)*x^3 - 70* (16*b^5*d^3*e^2 - 24*a*b^4*d^2*e^3 + 6*a^2*b^3*d*e^4 + a^3*b^2*e^5)*x^2 - 7*(128*b^5*d^4*e - 192*a*b^4*d^3*e^2 + 48*a^2*b^3*d^2*e^3 + 8*a^3*b^2*d*e^ 4 + 3*a^4*b*e^5)*x)*a/((e^9*x^3 + 3*d*e^8*x^2 + 3*d^2*e^7*x + d^3*e^6)*sqr t(e*x + d)) + 2/105*(21*b^5*e^6*x^6 + 3072*b^5*d^6 - 6400*a*b^4*d^5*e + 38 40*a^2*b^3*d^4*e^2 - 480*a^3*b^2*d^3*e^3 - 40*a^4*b*d^2*e^4 - 6*a^5*d*e^5 - 7*(12*b^5*d*e^5 - 25*a*b^4*e^6)*x^5 + 70*(12*b^5*d^2*e^4 - 25*a*b^4*d*e^ 5 + 15*a^2*b^3*e^6)*x^4 + 70*(96*b^5*d^3*e^3 - 200*a*b^4*d^2*e^4 + 120*a^2 *b^3*d*e^5 - 15*a^3*b^2*e^6)*x^3 + 35*(384*b^5*d^4*e^2 - 800*a*b^4*d^3*e^3 + 480*a^2*b^3*d^2*e^4 - 60*a^3*b^2*d*e^5 - 5*a^4*b*e^6)*x^2 + 7*(1536*b^5 *d^5*e - 3200*a*b^4*d^4*e^2 + 1920*a^2*b^3*d^3*e^3 - 240*a^3*b^2*d^2*e^4 - 20*a^4*b*d*e^5 - 3*a^5*e^6)*x)*b/((e^10*x^3 + 3*d*e^9*x^2 + 3*d^2*e^8*x + d^3*e^7)*sqrt(e*x + d))
Leaf count of result is larger than twice the leaf count of optimal. 621 vs. \(2 (271) = 542\).
Time = 0.31 (sec) , antiderivative size = 621, normalized size of antiderivative = 1.69 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=\frac {2 \, {\left (700 \, {\left (e x + d\right )}^{3} b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) - 175 \, {\left (e x + d\right )}^{2} b^{6} d^{4} \mathrm {sgn}\left (b x + a\right ) + 42 \, {\left (e x + d\right )} b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) - 2100 \, {\left (e x + d\right )}^{3} a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) + 700 \, {\left (e x + d\right )}^{2} a b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) - 210 \, {\left (e x + d\right )} a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) + 30 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 2100 \, {\left (e x + d\right )}^{3} a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) - 1050 \, {\left (e x + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) + 420 \, {\left (e x + d\right )} a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) - 75 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 700 \, {\left (e x + d\right )}^{3} a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 700 \, {\left (e x + d\right )}^{2} a^{3} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) - 420 \, {\left (e x + d\right )} a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) + 100 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 175 \, {\left (e x + d\right )}^{2} a^{4} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 210 \, {\left (e x + d\right )} a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) - 75 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 42 \, {\left (e x + d\right )} a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right ) + 30 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) - 5 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )}}{35 \, {\left (e x + d\right )}^{\frac {7}{2}} e^{7}} + \frac {2 \, {\left ({\left (e x + d\right )}^{\frac {5}{2}} b^{6} e^{28} \mathrm {sgn}\left (b x + a\right ) - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} b^{6} d e^{28} \mathrm {sgn}\left (b x + a\right ) + 75 \, \sqrt {e x + d} b^{6} d^{2} e^{28} \mathrm {sgn}\left (b x + a\right ) + 10 \, {\left (e x + d\right )}^{\frac {3}{2}} a b^{5} e^{29} \mathrm {sgn}\left (b x + a\right ) - 150 \, \sqrt {e x + d} a b^{5} d e^{29} \mathrm {sgn}\left (b x + a\right ) + 75 \, \sqrt {e x + d} a^{2} b^{4} e^{30} \mathrm {sgn}\left (b x + a\right )\right )}}{5 \, e^{35}} \]
2/35*(700*(e*x + d)^3*b^6*d^3*sgn(b*x + a) - 175*(e*x + d)^2*b^6*d^4*sgn(b *x + a) + 42*(e*x + d)*b^6*d^5*sgn(b*x + a) - 5*b^6*d^6*sgn(b*x + a) - 210 0*(e*x + d)^3*a*b^5*d^2*e*sgn(b*x + a) + 700*(e*x + d)^2*a*b^5*d^3*e*sgn(b *x + a) - 210*(e*x + d)*a*b^5*d^4*e*sgn(b*x + a) + 30*a*b^5*d^5*e*sgn(b*x + a) + 2100*(e*x + d)^3*a^2*b^4*d*e^2*sgn(b*x + a) - 1050*(e*x + d)^2*a^2* b^4*d^2*e^2*sgn(b*x + a) + 420*(e*x + d)*a^2*b^4*d^3*e^2*sgn(b*x + a) - 75 *a^2*b^4*d^4*e^2*sgn(b*x + a) - 700*(e*x + d)^3*a^3*b^3*e^3*sgn(b*x + a) + 700*(e*x + d)^2*a^3*b^3*d*e^3*sgn(b*x + a) - 420*(e*x + d)*a^3*b^3*d^2*e^ 3*sgn(b*x + a) + 100*a^3*b^3*d^3*e^3*sgn(b*x + a) - 175*(e*x + d)^2*a^4*b^ 2*e^4*sgn(b*x + a) + 210*(e*x + d)*a^4*b^2*d*e^4*sgn(b*x + a) - 75*a^4*b^2 *d^2*e^4*sgn(b*x + a) - 42*(e*x + d)*a^5*b*e^5*sgn(b*x + a) + 30*a^5*b*d*e ^5*sgn(b*x + a) - 5*a^6*e^6*sgn(b*x + a))/((e*x + d)^(7/2)*e^7) + 2/5*((e* x + d)^(5/2)*b^6*e^28*sgn(b*x + a) - 10*(e*x + d)^(3/2)*b^6*d*e^28*sgn(b*x + a) + 75*sqrt(e*x + d)*b^6*d^2*e^28*sgn(b*x + a) + 10*(e*x + d)^(3/2)*a* b^5*e^29*sgn(b*x + a) - 150*sqrt(e*x + d)*a*b^5*d*e^29*sgn(b*x + a) + 75*s qrt(e*x + d)*a^2*b^4*e^30*sgn(b*x + a))/e^35
Time = 12.34 (sec) , antiderivative size = 489, normalized size of antiderivative = 1.33 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{9/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {10\,a^6\,e^6+24\,a^5\,b\,d\,e^5+80\,a^4\,b^2\,d^2\,e^4+640\,a^3\,b^3\,d^3\,e^3-3840\,a^2\,b^4\,d^4\,e^2+5120\,a\,b^5\,d^5\,e-2048\,b^6\,d^6}{35\,b\,e^{10}}-\frac {2\,b^5\,x^6}{5\,e^4}+\frac {x\,\left (84\,a^5\,b\,e^6+280\,a^4\,b^2\,d\,e^5+2240\,a^3\,b^3\,d^2\,e^4-13440\,a^2\,b^4\,d^3\,e^3+17920\,a\,b^5\,d^4\,e^2-7168\,b^6\,d^5\,e\right )}{35\,b\,e^{10}}+\frac {8\,b^2\,x^3\,\left (5\,a^3\,e^3-30\,a^2\,b\,d\,e^2+40\,a\,b^2\,d^2\,e-16\,b^3\,d^3\right )}{e^7}-\frac {4\,b^4\,x^5\,\left (5\,a\,e-2\,b\,d\right )}{5\,e^5}-\frac {2\,b^3\,x^4\,\left (15\,a^2\,e^2-20\,a\,b\,d\,e+8\,b^2\,d^2\right )}{e^6}+\frac {x^2\,\left (350\,a^4\,b^2\,e^6+2800\,a^3\,b^3\,d\,e^5-16800\,a^2\,b^4\,d^2\,e^4+22400\,a\,b^5\,d^3\,e^3-8960\,b^6\,d^4\,e^2\right )}{35\,b\,e^{10}}\right )}{x^4\,\sqrt {d+e\,x}+\frac {a\,d^3\,\sqrt {d+e\,x}}{b\,e^3}+\frac {x^3\,\left (35\,a\,e^{10}+105\,b\,d\,e^9\right )\,\sqrt {d+e\,x}}{35\,b\,e^{10}}+\frac {3\,d\,x^2\,\left (a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^2\,x\,\left (3\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}} \]
-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((10*a^6*e^6 - 2048*b^6*d^6 - 3840*a^2*b ^4*d^4*e^2 + 640*a^3*b^3*d^3*e^3 + 80*a^4*b^2*d^2*e^4 + 5120*a*b^5*d^5*e + 24*a^5*b*d*e^5)/(35*b*e^10) - (2*b^5*x^6)/(5*e^4) + (x*(84*a^5*b*e^6 - 71 68*b^6*d^5*e + 17920*a*b^5*d^4*e^2 + 280*a^4*b^2*d*e^5 - 13440*a^2*b^4*d^3 *e^3 + 2240*a^3*b^3*d^2*e^4))/(35*b*e^10) + (8*b^2*x^3*(5*a^3*e^3 - 16*b^3 *d^3 + 40*a*b^2*d^2*e - 30*a^2*b*d*e^2))/e^7 - (4*b^4*x^5*(5*a*e - 2*b*d)) /(5*e^5) - (2*b^3*x^4*(15*a^2*e^2 + 8*b^2*d^2 - 20*a*b*d*e))/e^6 + (x^2*(3 50*a^4*b^2*e^6 - 8960*b^6*d^4*e^2 + 22400*a*b^5*d^3*e^3 + 2800*a^3*b^3*d*e ^5 - 16800*a^2*b^4*d^2*e^4))/(35*b*e^10)))/(x^4*(d + e*x)^(1/2) + (a*d^3*( d + e*x)^(1/2))/(b*e^3) + (x^3*(35*a*e^10 + 105*b*d*e^9)*(d + e*x)^(1/2))/ (35*b*e^10) + (3*d*x^2*(a*e + b*d)*(d + e*x)^(1/2))/(b*e^2) + (d^2*x*(3*a* e + b*d)*(d + e*x)^(1/2))/(b*e^3))