Integrand size = 35, antiderivative size = 370 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {2 (b d-a e)^6 \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x) (d+e x)^{13/2}}+\frac {12 b (b d-a e)^5 \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x) (d+e x)^{11/2}}-\frac {10 b^2 (b d-a e)^4 \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x) (d+e x)^{9/2}}+\frac {40 b^3 (b d-a e)^3 \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x) (d+e x)^{7/2}}-\frac {6 b^4 (b d-a e)^2 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{5/2}}+\frac {4 b^5 (b d-a e) \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) (d+e x)^{3/2}}-\frac {2 b^6 \sqrt {a^2+2 a b x+b^2 x^2}}{e^7 (a+b x) \sqrt {d+e x}} \]
-2/13*(-a*e+b*d)^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(13/2)+12/11*b*(- a*e+b*d)^5*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(11/2)-10/3*b^2*(-a*e+b*d )^4*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(9/2)+40/7*b^3*(-a*e+b*d)^3*((b* x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(7/2)-6*b^4*(-a*e+b*d)^2*((b*x+a)^2)^(1/ 2)/e^7/(b*x+a)/(e*x+d)^(5/2)+4*b^5*(-a*e+b*d)*((b*x+a)^2)^(1/2)/e^7/(b*x+a )/(e*x+d)^(3/2)-2*b^6*((b*x+a)^2)^(1/2)/e^7/(b*x+a)/(e*x+d)^(1/2)
Time = 0.18 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (231 a^6 e^6+126 a^5 b e^5 (2 d+13 e x)+35 a^4 b^2 e^4 \left (8 d^2+52 d e x+143 e^2 x^2\right )+20 a^3 b^3 e^3 \left (16 d^3+104 d^2 e x+286 d e^2 x^2+429 e^3 x^3\right )+3 a^2 b^4 e^2 \left (128 d^4+832 d^3 e x+2288 d^2 e^2 x^2+3432 d e^3 x^3+3003 e^4 x^4\right )+2 a b^5 e \left (256 d^5+1664 d^4 e x+4576 d^3 e^2 x^2+6864 d^2 e^3 x^3+6006 d e^4 x^4+3003 e^5 x^5\right )+b^6 \left (1024 d^6+6656 d^5 e x+18304 d^4 e^2 x^2+27456 d^3 e^3 x^3+24024 d^2 e^4 x^4+12012 d e^5 x^5+3003 e^6 x^6\right )\right )}{3003 e^7 (a+b x) (d+e x)^{13/2}} \]
(-2*Sqrt[(a + b*x)^2]*(231*a^6*e^6 + 126*a^5*b*e^5*(2*d + 13*e*x) + 35*a^4 *b^2*e^4*(8*d^2 + 52*d*e*x + 143*e^2*x^2) + 20*a^3*b^3*e^3*(16*d^3 + 104*d ^2*e*x + 286*d*e^2*x^2 + 429*e^3*x^3) + 3*a^2*b^4*e^2*(128*d^4 + 832*d^3*e *x + 2288*d^2*e^2*x^2 + 3432*d*e^3*x^3 + 3003*e^4*x^4) + 2*a*b^5*e*(256*d^ 5 + 1664*d^4*e*x + 4576*d^3*e^2*x^2 + 6864*d^2*e^3*x^3 + 6006*d*e^4*x^4 + 3003*e^5*x^5) + b^6*(1024*d^6 + 6656*d^5*e*x + 18304*d^4*e^2*x^2 + 27456*d ^3*e^3*x^3 + 24024*d^2*e^4*x^4 + 12012*d*e^5*x^5 + 3003*e^6*x^6)))/(3003*e ^7*(a + b*x)*(d + e*x)^(13/2))
Time = 0.34 (sec) , antiderivative size = 209, 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.
| \(\displaystyle \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {b^5 (a+b x)^6}{(d+e x)^{15/2}}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {(a+b x)^6}{(d+e x)^{15/2}}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6}{e^6 (d+e x)^{3/2}}-\frac {6 (b d-a e) b^5}{e^6 (d+e x)^{5/2}}+\frac {15 (b d-a e)^2 b^4}{e^6 (d+e x)^{7/2}}-\frac {20 (b d-a e)^3 b^3}{e^6 (d+e x)^{9/2}}+\frac {15 (b d-a e)^4 b^2}{e^6 (d+e x)^{11/2}}-\frac {6 (b d-a e)^5 b}{e^6 (d+e x)^{13/2}}+\frac {(a e-b d)^6}{e^6 (d+e x)^{15/2}}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (\frac {4 b^5 (b d-a e)}{e^7 (d+e x)^{3/2}}-\frac {6 b^4 (b d-a e)^2}{e^7 (d+e x)^{5/2}}+\frac {40 b^3 (b d-a e)^3}{7 e^7 (d+e x)^{7/2}}-\frac {10 b^2 (b d-a e)^4}{3 e^7 (d+e x)^{9/2}}+\frac {12 b (b d-a e)^5}{11 e^7 (d+e x)^{11/2}}-\frac {2 (b d-a e)^6}{13 e^7 (d+e x)^{13/2}}-\frac {2 b^6}{e^7 \sqrt {d+e x}}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((-2*(b*d - a*e)^6)/(13*e^7*(d + e*x)^(13/2 )) + (12*b*(b*d - a*e)^5)/(11*e^7*(d + e*x)^(11/2)) - (10*b^2*(b*d - a*e)^ 4)/(3*e^7*(d + e*x)^(9/2)) + (40*b^3*(b*d - a*e)^3)/(7*e^7*(d + e*x)^(7/2) ) - (6*b^4*(b*d - a*e)^2)/(e^7*(d + e*x)^(5/2)) + (4*b^5*(b*d - a*e))/(e^7 *(d + e*x)^(3/2)) - (2*b^6)/(e^7*Sqrt[d + e*x])))/(a + b*x)
3.22.20.3.1 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 = 393, normalized size of antiderivative = 1.06
| method | result | size |
| gosper | \(-\frac {2 \left (3003 b^{6} e^{6} x^{6}+6006 a \,b^{5} e^{6} x^{5}+12012 b^{6} d \,e^{5} x^{5}+9009 a^{2} b^{4} e^{6} x^{4}+12012 a \,b^{5} d \,e^{5} x^{4}+24024 b^{6} d^{2} e^{4} x^{4}+8580 a^{3} b^{3} e^{6} x^{3}+10296 a^{2} b^{4} d \,e^{5} x^{3}+13728 a \,b^{5} d^{2} e^{4} x^{3}+27456 b^{6} d^{3} e^{3} x^{3}+5005 a^{4} b^{2} e^{6} x^{2}+5720 a^{3} b^{3} d \,e^{5} x^{2}+6864 a^{2} b^{4} d^{2} e^{4} x^{2}+9152 a \,b^{5} d^{3} e^{3} x^{2}+18304 b^{6} d^{4} e^{2} x^{2}+1638 a^{5} b \,e^{6} x +1820 a^{4} b^{2} d \,e^{5} x +2080 a^{3} b^{3} d^{2} e^{4} x +2496 a^{2} b^{4} d^{3} e^{3} x +3328 a \,b^{5} d^{4} e^{2} x +6656 b^{6} d^{5} e x +231 e^{6} a^{6}+252 b d \,e^{5} a^{5}+280 b^{2} d^{2} e^{4} a^{4}+320 b^{3} d^{3} e^{3} a^{3}+384 b^{4} d^{4} e^{2} a^{2}+512 b^{5} d^{5} e a +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{3003 \left (e x +d \right )^{\frac {13}{2}} e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
| default | \(-\frac {2 \left (3003 b^{6} e^{6} x^{6}+6006 a \,b^{5} e^{6} x^{5}+12012 b^{6} d \,e^{5} x^{5}+9009 a^{2} b^{4} e^{6} x^{4}+12012 a \,b^{5} d \,e^{5} x^{4}+24024 b^{6} d^{2} e^{4} x^{4}+8580 a^{3} b^{3} e^{6} x^{3}+10296 a^{2} b^{4} d \,e^{5} x^{3}+13728 a \,b^{5} d^{2} e^{4} x^{3}+27456 b^{6} d^{3} e^{3} x^{3}+5005 a^{4} b^{2} e^{6} x^{2}+5720 a^{3} b^{3} d \,e^{5} x^{2}+6864 a^{2} b^{4} d^{2} e^{4} x^{2}+9152 a \,b^{5} d^{3} e^{3} x^{2}+18304 b^{6} d^{4} e^{2} x^{2}+1638 a^{5} b \,e^{6} x +1820 a^{4} b^{2} d \,e^{5} x +2080 a^{3} b^{3} d^{2} e^{4} x +2496 a^{2} b^{4} d^{3} e^{3} x +3328 a \,b^{5} d^{4} e^{2} x +6656 b^{6} d^{5} e x +231 e^{6} a^{6}+252 b d \,e^{5} a^{5}+280 b^{2} d^{2} e^{4} a^{4}+320 b^{3} d^{3} e^{3} a^{3}+384 b^{4} d^{4} e^{2} a^{2}+512 b^{5} d^{5} e a +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{3003 \left (e x +d \right )^{\frac {13}{2}} e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
-2/3003/(e*x+d)^(13/2)*(3003*b^6*e^6*x^6+6006*a*b^5*e^6*x^5+12012*b^6*d*e^ 5*x^5+9009*a^2*b^4*e^6*x^4+12012*a*b^5*d*e^5*x^4+24024*b^6*d^2*e^4*x^4+858 0*a^3*b^3*e^6*x^3+10296*a^2*b^4*d*e^5*x^3+13728*a*b^5*d^2*e^4*x^3+27456*b^ 6*d^3*e^3*x^3+5005*a^4*b^2*e^6*x^2+5720*a^3*b^3*d*e^5*x^2+6864*a^2*b^4*d^2 *e^4*x^2+9152*a*b^5*d^3*e^3*x^2+18304*b^6*d^4*e^2*x^2+1638*a^5*b*e^6*x+182 0*a^4*b^2*d*e^5*x+2080*a^3*b^3*d^2*e^4*x+2496*a^2*b^4*d^3*e^3*x+3328*a*b^5 *d^4*e^2*x+6656*b^6*d^5*e*x+231*a^6*e^6+252*a^5*b*d*e^5+280*a^4*b^2*d^2*e^ 4+320*a^3*b^3*d^3*e^3+384*a^2*b^4*d^4*e^2+512*a*b^5*d^5*e+1024*b^6*d^6)*(( b*x+a)^2)^(5/2)/e^7/(b*x+a)^5
Time = 0.37 (sec) , antiderivative size = 431, normalized size of antiderivative = 1.16 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {2 \, {\left (3003 \, b^{6} e^{6} x^{6} + 1024 \, b^{6} d^{6} + 512 \, a b^{5} d^{5} e + 384 \, a^{2} b^{4} d^{4} e^{2} + 320 \, a^{3} b^{3} d^{3} e^{3} + 280 \, a^{4} b^{2} d^{2} e^{4} + 252 \, a^{5} b d e^{5} + 231 \, a^{6} e^{6} + 6006 \, {\left (2 \, b^{6} d e^{5} + a b^{5} e^{6}\right )} x^{5} + 3003 \, {\left (8 \, b^{6} d^{2} e^{4} + 4 \, a b^{5} d e^{5} + 3 \, a^{2} b^{4} e^{6}\right )} x^{4} + 1716 \, {\left (16 \, b^{6} d^{3} e^{3} + 8 \, a b^{5} d^{2} e^{4} + 6 \, a^{2} b^{4} d e^{5} + 5 \, a^{3} b^{3} e^{6}\right )} x^{3} + 143 \, {\left (128 \, b^{6} d^{4} e^{2} + 64 \, a b^{5} d^{3} e^{3} + 48 \, a^{2} b^{4} d^{2} e^{4} + 40 \, a^{3} b^{3} d e^{5} + 35 \, a^{4} b^{2} e^{6}\right )} x^{2} + 26 \, {\left (256 \, b^{6} d^{5} e + 128 \, a b^{5} d^{4} e^{2} + 96 \, a^{2} b^{4} d^{3} e^{3} + 80 \, a^{3} b^{3} d^{2} e^{4} + 70 \, a^{4} b^{2} d e^{5} + 63 \, a^{5} b e^{6}\right )} x\right )} \sqrt {e x + d}}{3003 \, {\left (e^{14} x^{7} + 7 \, d e^{13} x^{6} + 21 \, d^{2} e^{12} x^{5} + 35 \, d^{3} e^{11} x^{4} + 35 \, d^{4} e^{10} x^{3} + 21 \, d^{5} e^{9} x^{2} + 7 \, d^{6} e^{8} x + d^{7} e^{7}\right )}} \]
-2/3003*(3003*b^6*e^6*x^6 + 1024*b^6*d^6 + 512*a*b^5*d^5*e + 384*a^2*b^4*d ^4*e^2 + 320*a^3*b^3*d^3*e^3 + 280*a^4*b^2*d^2*e^4 + 252*a^5*b*d*e^5 + 231 *a^6*e^6 + 6006*(2*b^6*d*e^5 + a*b^5*e^6)*x^5 + 3003*(8*b^6*d^2*e^4 + 4*a* b^5*d*e^5 + 3*a^2*b^4*e^6)*x^4 + 1716*(16*b^6*d^3*e^3 + 8*a*b^5*d^2*e^4 + 6*a^2*b^4*d*e^5 + 5*a^3*b^3*e^6)*x^3 + 143*(128*b^6*d^4*e^2 + 64*a*b^5*d^3 *e^3 + 48*a^2*b^4*d^2*e^4 + 40*a^3*b^3*d*e^5 + 35*a^4*b^2*e^6)*x^2 + 26*(2 56*b^6*d^5*e + 128*a*b^5*d^4*e^2 + 96*a^2*b^4*d^3*e^3 + 80*a^3*b^3*d^2*e^4 + 70*a^4*b^2*d*e^5 + 63*a^5*b*e^6)*x)*sqrt(e*x + d)/(e^14*x^7 + 7*d*e^13* x^6 + 21*d^2*e^12*x^5 + 35*d^3*e^11*x^4 + 35*d^4*e^10*x^3 + 21*d^5*e^9*x^2 + 7*d^6*e^8*x + d^7*e^7)
Timed out. \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 735 vs. \(2 (271) = 542\).
Time = 0.26 (sec) , antiderivative size = 735, normalized size of antiderivative = 1.99 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {2 \, {\left (3003 \, b^{5} e^{5} x^{5} + 256 \, b^{5} d^{5} + 384 \, a b^{4} d^{4} e + 480 \, a^{2} b^{3} d^{3} e^{2} + 560 \, a^{3} b^{2} d^{2} e^{3} + 630 \, a^{4} b d e^{4} + 693 \, a^{5} e^{5} + 3003 \, {\left (2 \, b^{5} d e^{4} + 3 \, a b^{4} e^{5}\right )} x^{4} + 858 \, {\left (8 \, b^{5} d^{2} e^{3} + 12 \, a b^{4} d e^{4} + 15 \, a^{2} b^{3} e^{5}\right )} x^{3} + 286 \, {\left (16 \, b^{5} d^{3} e^{2} + 24 \, a b^{4} d^{2} e^{3} + 30 \, a^{2} b^{3} d e^{4} + 35 \, a^{3} b^{2} e^{5}\right )} x^{2} + 13 \, {\left (128 \, b^{5} d^{4} e + 192 \, a b^{4} d^{3} e^{2} + 240 \, a^{2} b^{3} d^{2} e^{3} + 280 \, a^{3} b^{2} d e^{4} + 315 \, a^{4} b e^{5}\right )} x\right )} a}{9009 \, {\left (e^{12} x^{6} + 6 \, d e^{11} x^{5} + 15 \, d^{2} e^{10} x^{4} + 20 \, d^{3} e^{9} x^{3} + 15 \, d^{4} e^{8} x^{2} + 6 \, d^{5} e^{7} x + d^{6} e^{6}\right )} \sqrt {e x + d}} - \frac {2 \, {\left (9009 \, b^{5} e^{6} x^{6} + 3072 \, b^{5} d^{6} + 1280 \, a b^{4} d^{5} e + 768 \, a^{2} b^{3} d^{4} e^{2} + 480 \, a^{3} b^{2} d^{3} e^{3} + 280 \, a^{4} b d^{2} e^{4} + 126 \, a^{5} d e^{5} + 3003 \, {\left (12 \, b^{5} d e^{5} + 5 \, a b^{4} e^{6}\right )} x^{5} + 6006 \, {\left (12 \, b^{5} d^{2} e^{4} + 5 \, a b^{4} d e^{5} + 3 \, a^{2} b^{3} e^{6}\right )} x^{4} + 858 \, {\left (96 \, b^{5} d^{3} e^{3} + 40 \, a b^{4} d^{2} e^{4} + 24 \, a^{2} b^{3} d e^{5} + 15 \, a^{3} b^{2} e^{6}\right )} x^{3} + 143 \, {\left (384 \, b^{5} d^{4} e^{2} + 160 \, a b^{4} d^{3} e^{3} + 96 \, a^{2} b^{3} d^{2} e^{4} + 60 \, a^{3} b^{2} d e^{5} + 35 \, a^{4} b e^{6}\right )} x^{2} + 13 \, {\left (1536 \, b^{5} d^{5} e + 640 \, a b^{4} d^{4} e^{2} + 384 \, a^{2} b^{3} d^{3} e^{3} + 240 \, a^{3} b^{2} d^{2} e^{4} + 140 \, a^{4} b d e^{5} + 63 \, a^{5} e^{6}\right )} x\right )} b}{9009 \, {\left (e^{13} x^{6} + 6 \, d e^{12} x^{5} + 15 \, d^{2} e^{11} x^{4} + 20 \, d^{3} e^{10} x^{3} + 15 \, d^{4} e^{9} x^{2} + 6 \, d^{5} e^{8} x + d^{6} e^{7}\right )} \sqrt {e x + d}} \]
-2/9009*(3003*b^5*e^5*x^5 + 256*b^5*d^5 + 384*a*b^4*d^4*e + 480*a^2*b^3*d^ 3*e^2 + 560*a^3*b^2*d^2*e^3 + 630*a^4*b*d*e^4 + 693*a^5*e^5 + 3003*(2*b^5* d*e^4 + 3*a*b^4*e^5)*x^4 + 858*(8*b^5*d^2*e^3 + 12*a*b^4*d*e^4 + 15*a^2*b^ 3*e^5)*x^3 + 286*(16*b^5*d^3*e^2 + 24*a*b^4*d^2*e^3 + 30*a^2*b^3*d*e^4 + 3 5*a^3*b^2*e^5)*x^2 + 13*(128*b^5*d^4*e + 192*a*b^4*d^3*e^2 + 240*a^2*b^3*d ^2*e^3 + 280*a^3*b^2*d*e^4 + 315*a^4*b*e^5)*x)*a/((e^12*x^6 + 6*d*e^11*x^5 + 15*d^2*e^10*x^4 + 20*d^3*e^9*x^3 + 15*d^4*e^8*x^2 + 6*d^5*e^7*x + d^6*e ^6)*sqrt(e*x + d)) - 2/9009*(9009*b^5*e^6*x^6 + 3072*b^5*d^6 + 1280*a*b^4* d^5*e + 768*a^2*b^3*d^4*e^2 + 480*a^3*b^2*d^3*e^3 + 280*a^4*b*d^2*e^4 + 12 6*a^5*d*e^5 + 3003*(12*b^5*d*e^5 + 5*a*b^4*e^6)*x^5 + 6006*(12*b^5*d^2*e^4 + 5*a*b^4*d*e^5 + 3*a^2*b^3*e^6)*x^4 + 858*(96*b^5*d^3*e^3 + 40*a*b^4*d^2 *e^4 + 24*a^2*b^3*d*e^5 + 15*a^3*b^2*e^6)*x^3 + 143*(384*b^5*d^4*e^2 + 160 *a*b^4*d^3*e^3 + 96*a^2*b^3*d^2*e^4 + 60*a^3*b^2*d*e^5 + 35*a^4*b*e^6)*x^2 + 13*(1536*b^5*d^5*e + 640*a*b^4*d^4*e^2 + 384*a^2*b^3*d^3*e^3 + 240*a^3* b^2*d^2*e^4 + 140*a^4*b*d*e^5 + 63*a^5*e^6)*x)*b/((e^13*x^6 + 6*d*e^12*x^5 + 15*d^2*e^11*x^4 + 20*d^3*e^10*x^3 + 15*d^4*e^9*x^2 + 6*d^5*e^8*x + d^6* e^7)*sqrt(e*x + d))
Leaf count of result is larger than twice the leaf count of optimal. 602 vs. \(2 (271) = 542\).
Time = 0.30 (sec) , antiderivative size = 602, normalized size of antiderivative = 1.63 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {2 \, {\left (3003 \, {\left (e x + d\right )}^{6} b^{6} \mathrm {sgn}\left (b x + a\right ) - 6006 \, {\left (e x + d\right )}^{5} b^{6} d \mathrm {sgn}\left (b x + a\right ) + 9009 \, {\left (e x + d\right )}^{4} b^{6} d^{2} \mathrm {sgn}\left (b x + a\right ) - 8580 \, {\left (e x + d\right )}^{3} b^{6} d^{3} \mathrm {sgn}\left (b x + a\right ) + 5005 \, {\left (e x + d\right )}^{2} b^{6} d^{4} \mathrm {sgn}\left (b x + a\right ) - 1638 \, {\left (e x + d\right )} b^{6} d^{5} \mathrm {sgn}\left (b x + a\right ) + 231 \, b^{6} d^{6} \mathrm {sgn}\left (b x + a\right ) + 6006 \, {\left (e x + d\right )}^{5} a b^{5} e \mathrm {sgn}\left (b x + a\right ) - 18018 \, {\left (e x + d\right )}^{4} a b^{5} d e \mathrm {sgn}\left (b x + a\right ) + 25740 \, {\left (e x + d\right )}^{3} a b^{5} d^{2} e \mathrm {sgn}\left (b x + a\right ) - 20020 \, {\left (e x + d\right )}^{2} a b^{5} d^{3} e \mathrm {sgn}\left (b x + a\right ) + 8190 \, {\left (e x + d\right )} a b^{5} d^{4} e \mathrm {sgn}\left (b x + a\right ) - 1386 \, a b^{5} d^{5} e \mathrm {sgn}\left (b x + a\right ) + 9009 \, {\left (e x + d\right )}^{4} a^{2} b^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) - 25740 \, {\left (e x + d\right )}^{3} a^{2} b^{4} d e^{2} \mathrm {sgn}\left (b x + a\right ) + 30030 \, {\left (e x + d\right )}^{2} a^{2} b^{4} d^{2} e^{2} \mathrm {sgn}\left (b x + a\right ) - 16380 \, {\left (e x + d\right )} a^{2} b^{4} d^{3} e^{2} \mathrm {sgn}\left (b x + a\right ) + 3465 \, a^{2} b^{4} d^{4} e^{2} \mathrm {sgn}\left (b x + a\right ) + 8580 \, {\left (e x + d\right )}^{3} a^{3} b^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) - 20020 \, {\left (e x + d\right )}^{2} a^{3} b^{3} d e^{3} \mathrm {sgn}\left (b x + a\right ) + 16380 \, {\left (e x + d\right )} a^{3} b^{3} d^{2} e^{3} \mathrm {sgn}\left (b x + a\right ) - 4620 \, a^{3} b^{3} d^{3} e^{3} \mathrm {sgn}\left (b x + a\right ) + 5005 \, {\left (e x + d\right )}^{2} a^{4} b^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) - 8190 \, {\left (e x + d\right )} a^{4} b^{2} d e^{4} \mathrm {sgn}\left (b x + a\right ) + 3465 \, a^{4} b^{2} d^{2} e^{4} \mathrm {sgn}\left (b x + a\right ) + 1638 \, {\left (e x + d\right )} a^{5} b e^{5} \mathrm {sgn}\left (b x + a\right ) - 1386 \, a^{5} b d e^{5} \mathrm {sgn}\left (b x + a\right ) + 231 \, a^{6} e^{6} \mathrm {sgn}\left (b x + a\right )\right )}}{3003 \, {\left (e x + d\right )}^{\frac {13}{2}} e^{7}} \]
-2/3003*(3003*(e*x + d)^6*b^6*sgn(b*x + a) - 6006*(e*x + d)^5*b^6*d*sgn(b* x + a) + 9009*(e*x + d)^4*b^6*d^2*sgn(b*x + a) - 8580*(e*x + d)^3*b^6*d^3* sgn(b*x + a) + 5005*(e*x + d)^2*b^6*d^4*sgn(b*x + a) - 1638*(e*x + d)*b^6* d^5*sgn(b*x + a) + 231*b^6*d^6*sgn(b*x + a) + 6006*(e*x + d)^5*a*b^5*e*sgn (b*x + a) - 18018*(e*x + d)^4*a*b^5*d*e*sgn(b*x + a) + 25740*(e*x + d)^3*a *b^5*d^2*e*sgn(b*x + a) - 20020*(e*x + d)^2*a*b^5*d^3*e*sgn(b*x + a) + 819 0*(e*x + d)*a*b^5*d^4*e*sgn(b*x + a) - 1386*a*b^5*d^5*e*sgn(b*x + a) + 900 9*(e*x + d)^4*a^2*b^4*e^2*sgn(b*x + a) - 25740*(e*x + d)^3*a^2*b^4*d*e^2*s gn(b*x + a) + 30030*(e*x + d)^2*a^2*b^4*d^2*e^2*sgn(b*x + a) - 16380*(e*x + d)*a^2*b^4*d^3*e^2*sgn(b*x + a) + 3465*a^2*b^4*d^4*e^2*sgn(b*x + a) + 85 80*(e*x + d)^3*a^3*b^3*e^3*sgn(b*x + a) - 20020*(e*x + d)^2*a^3*b^3*d*e^3* sgn(b*x + a) + 16380*(e*x + d)*a^3*b^3*d^2*e^3*sgn(b*x + a) - 4620*a^3*b^3 *d^3*e^3*sgn(b*x + a) + 5005*(e*x + d)^2*a^4*b^2*e^4*sgn(b*x + a) - 8190*( e*x + d)*a^4*b^2*d*e^4*sgn(b*x + a) + 3465*a^4*b^2*d^2*e^4*sgn(b*x + a) + 1638*(e*x + d)*a^5*b*e^5*sgn(b*x + a) - 1386*a^5*b*d*e^5*sgn(b*x + a) + 23 1*a^6*e^6*sgn(b*x + a))/((e*x + d)^(13/2)*e^7)
Time = 12.46 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.52 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{(d+e x)^{15/2}} \, dx=-\frac {\sqrt {a^2+2\,a\,b\,x+b^2\,x^2}\,\left (\frac {462\,a^6\,e^6+504\,a^5\,b\,d\,e^5+560\,a^4\,b^2\,d^2\,e^4+640\,a^3\,b^3\,d^3\,e^3+768\,a^2\,b^4\,d^4\,e^2+1024\,a\,b^5\,d^5\,e+2048\,b^6\,d^6}{3003\,b\,e^{13}}+\frac {2\,b^5\,x^6}{e^7}+\frac {x\,\left (3276\,a^5\,b\,e^6+3640\,a^4\,b^2\,d\,e^5+4160\,a^3\,b^3\,d^2\,e^4+4992\,a^2\,b^4\,d^3\,e^3+6656\,a\,b^5\,d^4\,e^2+13312\,b^6\,d^5\,e\right )}{3003\,b\,e^{13}}+\frac {8\,b^2\,x^3\,\left (5\,a^3\,e^3+6\,a^2\,b\,d\,e^2+8\,a\,b^2\,d^2\,e+16\,b^3\,d^3\right )}{7\,e^{10}}+\frac {2\,b\,x^2\,\left (35\,a^4\,e^4+40\,a^3\,b\,d\,e^3+48\,a^2\,b^2\,d^2\,e^2+64\,a\,b^3\,d^3\,e+128\,b^4\,d^4\right )}{21\,e^{11}}+\frac {4\,b^4\,x^5\,\left (a\,e+2\,b\,d\right )}{e^8}+\frac {2\,b^3\,x^4\,\left (3\,a^2\,e^2+4\,a\,b\,d\,e+8\,b^2\,d^2\right )}{e^9}\right )}{x^7\,\sqrt {d+e\,x}+\frac {a\,d^6\,\sqrt {d+e\,x}}{b\,e^6}+\frac {x^6\,\left (a\,e+6\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e}+\frac {3\,d\,x^5\,\left (2\,a\,e+5\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^2}+\frac {d^5\,x\,\left (6\,a\,e+b\,d\right )\,\sqrt {d+e\,x}}{b\,e^6}+\frac {5\,d^2\,x^4\,\left (3\,a\,e+4\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^3}+\frac {5\,d^3\,x^3\,\left (4\,a\,e+3\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^4}+\frac {3\,d^4\,x^2\,\left (5\,a\,e+2\,b\,d\right )\,\sqrt {d+e\,x}}{b\,e^5}} \]
-((a^2 + b^2*x^2 + 2*a*b*x)^(1/2)*((462*a^6*e^6 + 2048*b^6*d^6 + 768*a^2*b ^4*d^4*e^2 + 640*a^3*b^3*d^3*e^3 + 560*a^4*b^2*d^2*e^4 + 1024*a*b^5*d^5*e + 504*a^5*b*d*e^5)/(3003*b*e^13) + (2*b^5*x^6)/e^7 + (x*(3276*a^5*b*e^6 + 13312*b^6*d^5*e + 6656*a*b^5*d^4*e^2 + 3640*a^4*b^2*d*e^5 + 4992*a^2*b^4*d ^3*e^3 + 4160*a^3*b^3*d^2*e^4))/(3003*b*e^13) + (8*b^2*x^3*(5*a^3*e^3 + 16 *b^3*d^3 + 8*a*b^2*d^2*e + 6*a^2*b*d*e^2))/(7*e^10) + (2*b*x^2*(35*a^4*e^4 + 128*b^4*d^4 + 48*a^2*b^2*d^2*e^2 + 64*a*b^3*d^3*e + 40*a^3*b*d*e^3))/(2 1*e^11) + (4*b^4*x^5*(a*e + 2*b*d))/e^8 + (2*b^3*x^4*(3*a^2*e^2 + 8*b^2*d^ 2 + 4*a*b*d*e))/e^9))/(x^7*(d + e*x)^(1/2) + (a*d^6*(d + e*x)^(1/2))/(b*e^ 6) + (x^6*(a*e + 6*b*d)*(d + e*x)^(1/2))/(b*e) + (3*d*x^5*(2*a*e + 5*b*d)* (d + e*x)^(1/2))/(b*e^2) + (d^5*x*(6*a*e + b*d)*(d + e*x)^(1/2))/(b*e^6) + (5*d^2*x^4*(3*a*e + 4*b*d)*(d + e*x)^(1/2))/(b*e^3) + (5*d^3*x^3*(4*a*e + 3*b*d)*(d + e*x)^(1/2))/(b*e^4) + (3*d^4*x^2*(5*a*e + 2*b*d)*(d + e*x)^(1 /2))/(b*e^5))