Integrand size = 35, antiderivative size = 376 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 (b d-a e)^6 (d+e x)^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^7 (a+b x)}-\frac {12 b (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^7 (a+b x)}+\frac {30 b^2 (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^7 (a+b x)}-\frac {40 b^3 (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^7 (a+b x)}+\frac {30 b^4 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^7 (a+b x)}-\frac {12 b^5 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^7 (a+b x)}+\frac {2 b^6 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^7 (a+b x)} \]
2/3*(-a*e+b*d)^6*(e*x+d)^(3/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-12/5*b*(-a*e+ b*d)^5*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+30/7*b^2*(-a*e+b*d)^4*( e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)-40/9*b^3*(-a*e+b*d)^3*(e*x+d)^( 9/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)+30/11*b^4*(-a*e+b*d)^2*(e*x+d)^(11/2)*( (b*x+a)^2)^(1/2)/e^7/(b*x+a)-12/13*b^5*(-a*e+b*d)*(e*x+d)^(13/2)*((b*x+a)^ 2)^(1/2)/e^7/(b*x+a)+2/15*b^6*(e*x+d)^(15/2)*((b*x+a)^2)^(1/2)/e^7/(b*x+a)
Time = 0.06 (sec) , antiderivative size = 309, normalized size of antiderivative = 0.82 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{3/2} \left (15015 a^6 e^6+18018 a^5 b e^5 (-2 d+3 e x)+6435 a^4 b^2 e^4 \left (8 d^2-12 d e x+15 e^2 x^2\right )+2860 a^3 b^3 e^3 \left (-16 d^3+24 d^2 e x-30 d e^2 x^2+35 e^3 x^3\right )+195 a^2 b^4 e^2 \left (128 d^4-192 d^3 e x+240 d^2 e^2 x^2-280 d e^3 x^3+315 e^4 x^4\right )+30 a b^5 e \left (-256 d^5+384 d^4 e x-480 d^3 e^2 x^2+560 d^2 e^3 x^3-630 d e^4 x^4+693 e^5 x^5\right )+b^6 \left (1024 d^6-1536 d^5 e x+1920 d^4 e^2 x^2-2240 d^3 e^3 x^3+2520 d^2 e^4 x^4-2772 d e^5 x^5+3003 e^6 x^6\right )\right )}{45045 e^7 (a+b x)} \]
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(3/2)*(15015*a^6*e^6 + 18018*a^5*b*e^5*(-2* d + 3*e*x) + 6435*a^4*b^2*e^4*(8*d^2 - 12*d*e*x + 15*e^2*x^2) + 2860*a^3*b ^3*e^3*(-16*d^3 + 24*d^2*e*x - 30*d*e^2*x^2 + 35*e^3*x^3) + 195*a^2*b^4*e^ 2*(128*d^4 - 192*d^3*e*x + 240*d^2*e^2*x^2 - 280*d*e^3*x^3 + 315*e^4*x^4) + 30*a*b^5*e*(-256*d^5 + 384*d^4*e*x - 480*d^3*e^2*x^2 + 560*d^2*e^3*x^3 - 630*d*e^4*x^4 + 693*e^5*x^5) + b^6*(1024*d^6 - 1536*d^5*e*x + 1920*d^4*e^ 2*x^2 - 2240*d^3*e^3*x^3 + 2520*d^2*e^4*x^4 - 2772*d*e^5*x^5 + 3003*e^6*x^ 6)))/(45045*e^7*(a + b*x))
Time = 0.35 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.57, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \sqrt {d+e x} \, dx\) |
| \(\Big \downarrow \) 1187 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^6 \sqrt {d+e x}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^6 \sqrt {d+e x}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^6 (d+e x)^{13/2}}{e^6}-\frac {6 b^5 (b d-a e) (d+e x)^{11/2}}{e^6}+\frac {15 b^4 (b d-a e)^2 (d+e x)^{9/2}}{e^6}-\frac {20 b^3 (b d-a e)^3 (d+e x)^{7/2}}{e^6}+\frac {15 b^2 (b d-a e)^4 (d+e x)^{5/2}}{e^6}-\frac {6 b (b d-a e)^5 (d+e x)^{3/2}}{e^6}+\frac {(a e-b d)^6 \sqrt {d+e x}}{e^6}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {12 b^5 (d+e x)^{13/2} (b d-a e)}{13 e^7}+\frac {30 b^4 (d+e x)^{11/2} (b d-a e)^2}{11 e^7}-\frac {40 b^3 (d+e x)^{9/2} (b d-a e)^3}{9 e^7}+\frac {30 b^2 (d+e x)^{7/2} (b d-a e)^4}{7 e^7}-\frac {12 b (d+e x)^{5/2} (b d-a e)^5}{5 e^7}+\frac {2 (d+e x)^{3/2} (b d-a e)^6}{3 e^7}+\frac {2 b^6 (d+e x)^{15/2}}{15 e^7}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((2*(b*d - a*e)^6*(d + e*x)^(3/2))/(3*e^7) - (12*b*(b*d - a*e)^5*(d + e*x)^(5/2))/(5*e^7) + (30*b^2*(b*d - a*e)^4*(d + e*x)^(7/2))/(7*e^7) - (40*b^3*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^7) + ( 30*b^4*(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^7) - (12*b^5*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^7) + (2*b^6*(d + e*x)^(15/2))/(15*e^7)))/(a + b*x)
3.22.12.3.1 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 = 1.05
| method | result | size |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 b^{6} e^{6} x^{6}+20790 a \,b^{5} e^{6} x^{5}-2772 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}-18900 a \,b^{5} d \,e^{5} x^{4}+2520 b^{6} d^{2} e^{4} x^{4}+100100 a^{3} b^{3} e^{6} x^{3}-54600 a^{2} b^{4} d \,e^{5} x^{3}+16800 a \,b^{5} d^{2} e^{4} x^{3}-2240 b^{6} d^{3} e^{3} x^{3}+96525 a^{4} b^{2} e^{6} x^{2}-85800 a^{3} b^{3} d \,e^{5} x^{2}+46800 a^{2} b^{4} d^{2} e^{4} x^{2}-14400 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+54054 a^{5} b \,e^{6} x -77220 a^{4} b^{2} d \,e^{5} x +68640 a^{3} b^{3} d^{2} e^{4} x -37440 a^{2} b^{4} d^{3} e^{3} x +11520 a \,b^{5} d^{4} e^{2} x -1536 b^{6} d^{5} e x +15015 e^{6} a^{6}-36036 b d \,e^{5} a^{5}+51480 b^{2} d^{2} e^{4} a^{4}-45760 b^{3} d^{3} e^{3} a^{3}+24960 b^{4} d^{4} e^{2} a^{2}-7680 b^{5} d^{5} e a +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
| default | \(\frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (3003 b^{6} e^{6} x^{6}+20790 a \,b^{5} e^{6} x^{5}-2772 b^{6} d \,e^{5} x^{5}+61425 a^{2} b^{4} e^{6} x^{4}-18900 a \,b^{5} d \,e^{5} x^{4}+2520 b^{6} d^{2} e^{4} x^{4}+100100 a^{3} b^{3} e^{6} x^{3}-54600 a^{2} b^{4} d \,e^{5} x^{3}+16800 a \,b^{5} d^{2} e^{4} x^{3}-2240 b^{6} d^{3} e^{3} x^{3}+96525 a^{4} b^{2} e^{6} x^{2}-85800 a^{3} b^{3} d \,e^{5} x^{2}+46800 a^{2} b^{4} d^{2} e^{4} x^{2}-14400 a \,b^{5} d^{3} e^{3} x^{2}+1920 b^{6} d^{4} e^{2} x^{2}+54054 a^{5} b \,e^{6} x -77220 a^{4} b^{2} d \,e^{5} x +68640 a^{3} b^{3} d^{2} e^{4} x -37440 a^{2} b^{4} d^{3} e^{3} x +11520 a \,b^{5} d^{4} e^{2} x -1536 b^{6} d^{5} e x +15015 e^{6} a^{6}-36036 b d \,e^{5} a^{5}+51480 b^{2} d^{2} e^{4} a^{4}-45760 b^{3} d^{3} e^{3} a^{3}+24960 b^{4} d^{4} e^{2} a^{2}-7680 b^{5} d^{5} e a +1024 b^{6} d^{6}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{7} \left (b x +a \right )^{5}}\) | \(393\) |
| risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (3003 b^{6} x^{7} e^{7}+20790 a \,b^{5} e^{7} x^{6}+231 b^{6} d \,e^{6} x^{6}+61425 a^{2} b^{4} e^{7} x^{5}+1890 a \,b^{5} d \,e^{6} x^{5}-252 b^{6} d^{2} e^{5} x^{5}+100100 a^{3} b^{3} e^{7} x^{4}+6825 a^{2} b^{4} d \,e^{6} x^{4}-2100 a \,b^{5} d^{2} e^{5} x^{4}+280 b^{6} d^{3} e^{4} x^{4}+96525 a^{4} b^{2} e^{7} x^{3}+14300 a^{3} b^{3} d \,e^{6} x^{3}-7800 a^{2} b^{4} d^{2} e^{5} x^{3}+2400 a \,b^{5} d^{3} e^{4} x^{3}-320 b^{6} d^{4} e^{3} x^{3}+54054 a^{5} b \,e^{7} x^{2}+19305 a^{4} b^{2} d \,e^{6} x^{2}-17160 a^{3} b^{3} d^{2} e^{5} x^{2}+9360 a^{2} b^{4} d^{3} e^{4} x^{2}-2880 a \,b^{5} d^{4} e^{3} x^{2}+384 b^{6} d^{5} e^{2} x^{2}+15015 a^{6} e^{7} x +18018 a^{5} b d \,e^{6} x -25740 a^{4} b^{2} d^{2} e^{5} x +22880 a^{3} b^{3} d^{3} e^{4} x -12480 a^{2} b^{4} d^{4} e^{3} x +3840 a \,b^{5} d^{5} e^{2} x -512 b^{6} d^{6} e x +15015 d \,a^{6} e^{6}-36036 a^{5} b \,d^{2} e^{5}+51480 a^{4} b^{2} d^{3} e^{4}-45760 a^{3} b^{3} d^{4} e^{3}+24960 a^{2} b^{4} d^{5} e^{2}-7680 a \,b^{5} d^{6} e +1024 b^{6} d^{7}\right ) \sqrt {e x +d}}{45045 \left (b x +a \right ) e^{7}}\) | \(498\) |
2/45045*(e*x+d)^(3/2)*(3003*b^6*e^6*x^6+20790*a*b^5*e^6*x^5-2772*b^6*d*e^5 *x^5+61425*a^2*b^4*e^6*x^4-18900*a*b^5*d*e^5*x^4+2520*b^6*d^2*e^4*x^4+1001 00*a^3*b^3*e^6*x^3-54600*a^2*b^4*d*e^5*x^3+16800*a*b^5*d^2*e^4*x^3-2240*b^ 6*d^3*e^3*x^3+96525*a^4*b^2*e^6*x^2-85800*a^3*b^3*d*e^5*x^2+46800*a^2*b^4* d^2*e^4*x^2-14400*a*b^5*d^3*e^3*x^2+1920*b^6*d^4*e^2*x^2+54054*a^5*b*e^6*x -77220*a^4*b^2*d*e^5*x+68640*a^3*b^3*d^2*e^4*x-37440*a^2*b^4*d^3*e^3*x+115 20*a*b^5*d^4*e^2*x-1536*b^6*d^5*e*x+15015*a^6*e^6-36036*a^5*b*d*e^5+51480* a^4*b^2*d^2*e^4-45760*a^3*b^3*d^3*e^3+24960*a^2*b^4*d^4*e^2-7680*a*b^5*d^5 *e+1024*b^6*d^6)*((b*x+a)^2)^(5/2)/e^7/(b*x+a)^5
Time = 0.34 (sec) , antiderivative size = 447, normalized size of antiderivative = 1.19 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, b^{6} e^{7} x^{7} + 1024 \, b^{6} d^{7} - 7680 \, a b^{5} d^{6} e + 24960 \, a^{2} b^{4} d^{5} e^{2} - 45760 \, a^{3} b^{3} d^{4} e^{3} + 51480 \, a^{4} b^{2} d^{3} e^{4} - 36036 \, a^{5} b d^{2} e^{5} + 15015 \, a^{6} d e^{6} + 231 \, {\left (b^{6} d e^{6} + 90 \, a b^{5} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{6} d^{2} e^{5} - 30 \, a b^{5} d e^{6} - 975 \, a^{2} b^{4} e^{7}\right )} x^{5} + 35 \, {\left (8 \, b^{6} d^{3} e^{4} - 60 \, a b^{5} d^{2} e^{5} + 195 \, a^{2} b^{4} d e^{6} + 2860 \, a^{3} b^{3} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{6} d^{4} e^{3} - 480 \, a b^{5} d^{3} e^{4} + 1560 \, a^{2} b^{4} d^{2} e^{5} - 2860 \, a^{3} b^{3} d e^{6} - 19305 \, a^{4} b^{2} e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{6} d^{5} e^{2} - 960 \, a b^{5} d^{4} e^{3} + 3120 \, a^{2} b^{4} d^{3} e^{4} - 5720 \, a^{3} b^{3} d^{2} e^{5} + 6435 \, a^{4} b^{2} d e^{6} + 18018 \, a^{5} b e^{7}\right )} x^{2} - {\left (512 \, b^{6} d^{6} e - 3840 \, a b^{5} d^{5} e^{2} + 12480 \, a^{2} b^{4} d^{4} e^{3} - 22880 \, a^{3} b^{3} d^{3} e^{4} + 25740 \, a^{4} b^{2} d^{2} e^{5} - 18018 \, a^{5} b d e^{6} - 15015 \, a^{6} e^{7}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{7}} \]
2/45045*(3003*b^6*e^7*x^7 + 1024*b^6*d^7 - 7680*a*b^5*d^6*e + 24960*a^2*b^ 4*d^5*e^2 - 45760*a^3*b^3*d^4*e^3 + 51480*a^4*b^2*d^3*e^4 - 36036*a^5*b*d^ 2*e^5 + 15015*a^6*d*e^6 + 231*(b^6*d*e^6 + 90*a*b^5*e^7)*x^6 - 63*(4*b^6*d ^2*e^5 - 30*a*b^5*d*e^6 - 975*a^2*b^4*e^7)*x^5 + 35*(8*b^6*d^3*e^4 - 60*a* b^5*d^2*e^5 + 195*a^2*b^4*d*e^6 + 2860*a^3*b^3*e^7)*x^4 - 5*(64*b^6*d^4*e^ 3 - 480*a*b^5*d^3*e^4 + 1560*a^2*b^4*d^2*e^5 - 2860*a^3*b^3*d*e^6 - 19305* a^4*b^2*e^7)*x^3 + 3*(128*b^6*d^5*e^2 - 960*a*b^5*d^4*e^3 + 3120*a^2*b^4*d ^3*e^4 - 5720*a^3*b^3*d^2*e^5 + 6435*a^4*b^2*d*e^6 + 18018*a^5*b*e^7)*x^2 - (512*b^6*d^6*e - 3840*a*b^5*d^5*e^2 + 12480*a^2*b^4*d^4*e^3 - 22880*a^3* b^3*d^3*e^4 + 25740*a^4*b^2*d^2*e^5 - 18018*a^5*b*d*e^6 - 15015*a^6*e^7)*x )*sqrt(e*x + d)/e^7
\[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (a + b x\right ) \sqrt {d + e x} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 760 vs. \(2 (271) = 542\).
Time = 0.21 (sec) , antiderivative size = 760, normalized size of antiderivative = 2.02 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (693 \, b^{5} e^{6} x^{6} - 256 \, b^{5} d^{6} + 1664 \, a b^{4} d^{5} e - 4576 \, a^{2} b^{3} d^{4} e^{2} + 6864 \, a^{3} b^{2} d^{3} e^{3} - 6006 \, a^{4} b d^{2} e^{4} + 3003 \, a^{5} d e^{5} + 63 \, {\left (b^{5} d e^{5} + 65 \, a b^{4} e^{6}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{2} e^{4} - 13 \, a b^{4} d e^{5} - 286 \, a^{2} b^{3} e^{6}\right )} x^{4} + 10 \, {\left (8 \, b^{5} d^{3} e^{3} - 52 \, a b^{4} d^{2} e^{4} + 143 \, a^{2} b^{3} d e^{5} + 1287 \, a^{3} b^{2} e^{6}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{4} e^{2} - 208 \, a b^{4} d^{3} e^{3} + 572 \, a^{2} b^{3} d^{2} e^{4} - 858 \, a^{3} b^{2} d e^{5} - 3003 \, a^{4} b e^{6}\right )} x^{2} + {\left (128 \, b^{5} d^{5} e - 832 \, a b^{4} d^{4} e^{2} + 2288 \, a^{2} b^{3} d^{3} e^{3} - 3432 \, a^{3} b^{2} d^{2} e^{4} + 3003 \, a^{4} b d e^{5} + 3003 \, a^{5} e^{6}\right )} x\right )} \sqrt {e x + d} a}{9009 \, e^{6}} + \frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} + 1024 \, b^{5} d^{7} - 6400 \, a b^{4} d^{6} e + 16640 \, a^{2} b^{3} d^{5} e^{2} - 22880 \, a^{3} b^{2} d^{4} e^{3} + 17160 \, a^{4} b d^{3} e^{4} - 6006 \, a^{5} d^{2} e^{5} + 231 \, {\left (b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} - 63 \, {\left (4 \, b^{5} d^{2} e^{5} - 25 \, a b^{4} d e^{6} - 650 \, a^{2} b^{3} e^{7}\right )} x^{5} + 70 \, {\left (4 \, b^{5} d^{3} e^{4} - 25 \, a b^{4} d^{2} e^{5} + 65 \, a^{2} b^{3} d e^{6} + 715 \, a^{3} b^{2} e^{7}\right )} x^{4} - 5 \, {\left (64 \, b^{5} d^{4} e^{3} - 400 \, a b^{4} d^{3} e^{4} + 1040 \, a^{2} b^{3} d^{2} e^{5} - 1430 \, a^{3} b^{2} d e^{6} - 6435 \, a^{4} b e^{7}\right )} x^{3} + 3 \, {\left (128 \, b^{5} d^{5} e^{2} - 800 \, a b^{4} d^{4} e^{3} + 2080 \, a^{2} b^{3} d^{3} e^{4} - 2860 \, a^{3} b^{2} d^{2} e^{5} + 2145 \, a^{4} b d e^{6} + 3003 \, a^{5} e^{7}\right )} x^{2} - {\left (512 \, b^{5} d^{6} e - 3200 \, a b^{4} d^{5} e^{2} + 8320 \, a^{2} b^{3} d^{4} e^{3} - 11440 \, a^{3} b^{2} d^{3} e^{4} + 8580 \, a^{4} b d^{2} e^{5} - 3003 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d} b}{45045 \, e^{7}} \]
2/9009*(693*b^5*e^6*x^6 - 256*b^5*d^6 + 1664*a*b^4*d^5*e - 4576*a^2*b^3*d^ 4*e^2 + 6864*a^3*b^2*d^3*e^3 - 6006*a^4*b*d^2*e^4 + 3003*a^5*d*e^5 + 63*(b ^5*d*e^5 + 65*a*b^4*e^6)*x^5 - 35*(2*b^5*d^2*e^4 - 13*a*b^4*d*e^5 - 286*a^ 2*b^3*e^6)*x^4 + 10*(8*b^5*d^3*e^3 - 52*a*b^4*d^2*e^4 + 143*a^2*b^3*d*e^5 + 1287*a^3*b^2*e^6)*x^3 - 3*(32*b^5*d^4*e^2 - 208*a*b^4*d^3*e^3 + 572*a^2* b^3*d^2*e^4 - 858*a^3*b^2*d*e^5 - 3003*a^4*b*e^6)*x^2 + (128*b^5*d^5*e - 8 32*a*b^4*d^4*e^2 + 2288*a^2*b^3*d^3*e^3 - 3432*a^3*b^2*d^2*e^4 + 3003*a^4* b*d*e^5 + 3003*a^5*e^6)*x)*sqrt(e*x + d)*a/e^6 + 2/45045*(3003*b^5*e^7*x^7 + 1024*b^5*d^7 - 6400*a*b^4*d^6*e + 16640*a^2*b^3*d^5*e^2 - 22880*a^3*b^2 *d^4*e^3 + 17160*a^4*b*d^3*e^4 - 6006*a^5*d^2*e^5 + 231*(b^5*d*e^6 + 75*a* b^4*e^7)*x^6 - 63*(4*b^5*d^2*e^5 - 25*a*b^4*d*e^6 - 650*a^2*b^3*e^7)*x^5 + 70*(4*b^5*d^3*e^4 - 25*a*b^4*d^2*e^5 + 65*a^2*b^3*d*e^6 + 715*a^3*b^2*e^7 )*x^4 - 5*(64*b^5*d^4*e^3 - 400*a*b^4*d^3*e^4 + 1040*a^2*b^3*d^2*e^5 - 143 0*a^3*b^2*d*e^6 - 6435*a^4*b*e^7)*x^3 + 3*(128*b^5*d^5*e^2 - 800*a*b^4*d^4 *e^3 + 2080*a^2*b^3*d^3*e^4 - 2860*a^3*b^2*d^2*e^5 + 2145*a^4*b*d*e^6 + 30 03*a^5*e^7)*x^2 - (512*b^5*d^6*e - 3200*a*b^4*d^5*e^2 + 8320*a^2*b^3*d^4*e ^3 - 11440*a^3*b^2*d^3*e^4 + 8580*a^4*b*d^2*e^5 - 3003*a^5*d*e^6)*x)*sqrt( e*x + d)*b/e^7
Leaf count of result is larger than twice the leaf count of optimal. 920 vs. \(2 (271) = 542\).
Time = 0.29 (sec) , antiderivative size = 920, normalized size of antiderivative = 2.45 \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(e*x + d)*a^6*d*sgn(b*x + a) + 15015*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^6*sgn(b*x + a) + 90090*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^5*b*d*sgn(b*x + a)/e + 45045*(3*(e*x + d)^(5/2) - 10*(e*x + d)^ (3/2)*d + 15*sqrt(e*x + d)*d^2)*a^4*b^2*d*sgn(b*x + a)/e^2 + 18018*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^5*b*sgn(b*x + a)/e + 25740*(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^3*d*sgn(b*x + a)/e^3 + 19305*(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^4*b^2*sgn(b*x + a)/e^2 + 2145*(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^4*d*sgn(b*x + a)/e^4 + 2860*(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 + 31 5*sqrt(e*x + d)*d^4)*a^3*b^3*sgn(b*x + a)/e^3 + 390*(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)*a*b^5*d*sgn(b*x + a) /e^5 + 975*(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)*a^2*b^4*sgn(b*x + a)/e^4 + 15*(231*(e*x + d)^(13/2) - 1638*(e *x + d)^(11/2)*d + 5005*(e*x + d)^(9/2)*d^2 - 8580*(e*x + d)^(7/2)*d^3 + 9 009*(e*x + d)^(5/2)*d^4 - 6006*(e*x + d)^(3/2)*d^5 + 3003*sqrt(e*x + d)...
Timed out. \[ \int (a+b x) \sqrt {d+e x} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (a+b\,x\right )\,\sqrt {d+e\,x}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]