Integrand size = 30, antiderivative size = 320 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {2 (b d-a e)^5 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}-\frac {2 b^4 (b d-a e) (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 e^6 (a+b x)} \]
-2/7*(-a*e+b*d)^5*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+10/9*b*(-a*e +b*d)^4*(e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-20/11*b^2*(-a*e+b*d)^3 *(e*x+d)^(11/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+20/13*b^3*(-a*e+b*d)^2*(e*x+ d)^(13/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-2/3*b^4*(-a*e+b*d)*(e*x+d)^(15/2)* ((b*x+a)^2)^(1/2)/e^6/(b*x+a)+2/17*b^5*(e*x+d)^(17/2)*((b*x+a)^2)^(1/2)/e^ 6/(b*x+a)
Time = 0.14 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.73 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{7/2} \left (21879 a^5 e^5+12155 a^4 b e^4 (-2 d+7 e x)+2210 a^3 b^2 e^3 \left (8 d^2-28 d e x+63 e^2 x^2\right )+510 a^2 b^3 e^2 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+17 a b^4 e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+b^5 \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )\right )}{153153 e^6 (a+b x)} \]
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(7/2)*(21879*a^5*e^5 + 12155*a^4*b*e^4*(-2* d + 7*e*x) + 2210*a^3*b^2*e^3*(8*d^2 - 28*d*e*x + 63*e^2*x^2) + 510*a^2*b^ 3*e^2*(-16*d^3 + 56*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 17*a*b^4*e*(1 28*d^4 - 448*d^3*e*x + 1008*d^2*e^2*x^2 - 1848*d*e^3*x^3 + 3003*e^4*x^4) + b^5*(-256*d^5 + 896*d^4*e*x - 2016*d^3*e^2*x^2 + 3696*d^2*e^3*x^3 - 6006* d*e^4*x^4 + 9009*e^5*x^5)))/(153153*e^6*(a + b*x))
Time = 0.32 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.58, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {1102, 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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (d+e x)^{5/2}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (d+e x)^{5/2}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^5 (d+e x)^{15/2}}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^{13/2}}{e^5}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{11/2}}{e^5}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{9/2}}{e^5}+\frac {5 b (b d-a e)^4 (d+e x)^{7/2}}{e^5}+\frac {(a e-b d)^5 (d+e x)^{5/2}}{e^5}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {2 b^4 (d+e x)^{15/2} (b d-a e)}{3 e^6}+\frac {20 b^3 (d+e x)^{13/2} (b d-a e)^2}{13 e^6}-\frac {20 b^2 (d+e x)^{11/2} (b d-a e)^3}{11 e^6}+\frac {10 b (d+e x)^{9/2} (b d-a e)^4}{9 e^6}-\frac {2 (d+e x)^{7/2} (b d-a e)^5}{7 e^6}+\frac {2 b^5 (d+e x)^{17/2}}{17 e^6}\right )}{a+b x}\) |
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((-2*(b*d - a*e)^5*(d + e*x)^(7/2))/(7*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(9/2))/(9*e^6) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(11/2))/(11*e^6) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(13/2))/(13*e^6 ) - (2*b^4*(b*d - a*e)*(d + e*x)^(15/2))/(3*e^6) + (2*b^5*(d + e*x)^(17/2) )/(17*e^6)))/(a + b*x)
3.17.92.3.1 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_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*F racPart[p])) Int[(d + e*x)^m*(b/2 + c*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && EqQ[b^2 - 4*a*c, 0]
Time = 2.30 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.90
| method | result | size |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (9009 x^{5} e^{5} b^{5}+51051 x^{4} a \,b^{4} e^{5}-6006 x^{4} b^{5} d \,e^{4}+117810 x^{3} a^{2} b^{3} e^{5}-31416 x^{3} a \,b^{4} d \,e^{4}+3696 x^{3} b^{5} d^{2} e^{3}+139230 x^{2} a^{3} b^{2} e^{5}-64260 x^{2} a^{2} b^{3} d \,e^{4}+17136 x^{2} a \,b^{4} d^{2} e^{3}-2016 x^{2} b^{5} d^{3} e^{2}+85085 a^{4} b \,e^{5} x -61880 a^{3} b^{2} d \,e^{4} x +28560 x \,a^{2} b^{3} d^{2} e^{3}-7616 x a \,b^{4} d^{3} e^{2}+896 b^{5} d^{4} e x +21879 a^{5} e^{5}-24310 a^{4} b d \,e^{4}+17680 a^{3} b^{2} d^{2} e^{3}-8160 a^{2} b^{3} d^{3} e^{2}+2176 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{153153 e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
| default | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (9009 x^{5} e^{5} b^{5}+51051 x^{4} a \,b^{4} e^{5}-6006 x^{4} b^{5} d \,e^{4}+117810 x^{3} a^{2} b^{3} e^{5}-31416 x^{3} a \,b^{4} d \,e^{4}+3696 x^{3} b^{5} d^{2} e^{3}+139230 x^{2} a^{3} b^{2} e^{5}-64260 x^{2} a^{2} b^{3} d \,e^{4}+17136 x^{2} a \,b^{4} d^{2} e^{3}-2016 x^{2} b^{5} d^{3} e^{2}+85085 a^{4} b \,e^{5} x -61880 a^{3} b^{2} d \,e^{4} x +28560 x \,a^{2} b^{3} d^{2} e^{3}-7616 x a \,b^{4} d^{3} e^{2}+896 b^{5} d^{4} e x +21879 a^{5} e^{5}-24310 a^{4} b d \,e^{4}+17680 a^{3} b^{2} d^{2} e^{3}-8160 a^{2} b^{3} d^{3} e^{2}+2176 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{153153 e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
| risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (9009 b^{5} e^{8} x^{8}+51051 a \,b^{4} e^{8} x^{7}+21021 b^{5} d \,e^{7} x^{7}+117810 a^{2} b^{3} e^{8} x^{6}+121737 a \,b^{4} d \,e^{7} x^{6}+12705 b^{5} d^{2} e^{6} x^{6}+139230 a^{3} b^{2} e^{8} x^{5}+289170 a^{2} b^{3} d \,e^{7} x^{5}+76041 a \,b^{4} d^{2} e^{6} x^{5}+63 b^{5} d^{3} e^{5} x^{5}+85085 a^{4} b \,e^{8} x^{4}+355810 a^{3} b^{2} d \,e^{7} x^{4}+189210 a^{2} b^{3} d^{2} e^{6} x^{4}+595 a \,b^{4} d^{3} e^{5} x^{4}-70 b^{5} d^{4} e^{4} x^{4}+21879 a^{5} e^{8} x^{3}+230945 a^{4} b d \,e^{7} x^{3}+249730 a^{3} b^{2} d^{2} e^{6} x^{3}+2550 a^{2} b^{3} d^{3} e^{5} x^{3}-680 a \,b^{4} d^{4} e^{4} x^{3}+80 b^{5} d^{5} e^{3} x^{3}+65637 a^{5} d \,e^{7} x^{2}+182325 a^{4} b \,d^{2} e^{6} x^{2}+6630 a^{3} b^{2} d^{3} e^{5} x^{2}-3060 a^{2} b^{3} d^{4} e^{4} x^{2}+816 a \,b^{4} d^{5} e^{3} x^{2}-96 b^{5} d^{6} e^{2} x^{2}+65637 a^{5} d^{2} e^{6} x +12155 a^{4} b \,d^{3} e^{5} x -8840 a^{3} b^{2} d^{4} e^{4} x +4080 a^{2} b^{3} d^{5} e^{3} x -1088 a \,b^{4} d^{6} e^{2} x +128 b^{5} d^{7} e x +21879 a^{5} d^{3} e^{5}-24310 a^{4} b \,d^{4} e^{4}+17680 a^{3} b^{2} d^{5} e^{3}-8160 a^{2} b^{3} d^{6} e^{2}+2176 a \,b^{4} d^{7} e -256 b^{5} d^{8}\right ) \sqrt {e x +d}}{153153 \left (b x +a \right ) e^{6}}\) | \(561\) |
2/153153*(e*x+d)^(7/2)*(9009*b^5*e^5*x^5+51051*a*b^4*e^5*x^4-6006*b^5*d*e^ 4*x^4+117810*a^2*b^3*e^5*x^3-31416*a*b^4*d*e^4*x^3+3696*b^5*d^2*e^3*x^3+13 9230*a^3*b^2*e^5*x^2-64260*a^2*b^3*d*e^4*x^2+17136*a*b^4*d^2*e^3*x^2-2016* b^5*d^3*e^2*x^2+85085*a^4*b*e^5*x-61880*a^3*b^2*d*e^4*x+28560*a^2*b^3*d^2* e^3*x-7616*a*b^4*d^3*e^2*x+896*b^5*d^4*e*x+21879*a^5*e^5-24310*a^4*b*d*e^4 +17680*a^3*b^2*d^2*e^3-8160*a^2*b^3*d^3*e^2+2176*a*b^4*d^4*e-256*b^5*d^5)* ((b*x+a)^2)^(5/2)/e^6/(b*x+a)^5
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (230) = 460\).
Time = 0.39 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.55 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{153153 \, e^{6}} \]
2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3 *d^6*e^2 + 17680*a^3*b^2*d^5*e^3 - 24310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b^4 *d*e^7 + 510*a^2*b^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 459 0*a^2*b^3*d*e^7 + 2210*a^3*b^2*e^8)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3 *e^5 - 5406*a^2*b^3*d^2*e^6 - 10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4 + 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^ 2*d^2*e^6 + 230945*a^4*b*d*e^7 + 21879*a^5*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*a^3*b^2*d^3*e^5 - 60775*a^ 4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e - 1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a^3*b^2*d^4*e^4 + 12155*a^4*b*d^3*e^5 + 65637 *a^5*d^2*e^6)*x)*sqrt(e*x + d)/e^6
\[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (d + e x\right )^{\frac {5}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (230) = 460\).
Time = 0.21 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.55 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{153153 \, e^{6}} \]
2/153153*(9009*b^5*e^8*x^8 - 256*b^5*d^8 + 2176*a*b^4*d^7*e - 8160*a^2*b^3 *d^6*e^2 + 17680*a^3*b^2*d^5*e^3 - 24310*a^4*b*d^4*e^4 + 21879*a^5*d^3*e^5 + 3003*(7*b^5*d*e^7 + 17*a*b^4*e^8)*x^7 + 231*(55*b^5*d^2*e^6 + 527*a*b^4 *d*e^7 + 510*a^2*b^3*e^8)*x^6 + 63*(b^5*d^3*e^5 + 1207*a*b^4*d^2*e^6 + 459 0*a^2*b^3*d*e^7 + 2210*a^3*b^2*e^8)*x^5 - 35*(2*b^5*d^4*e^4 - 17*a*b^4*d^3 *e^5 - 5406*a^2*b^3*d^2*e^6 - 10166*a^3*b^2*d*e^7 - 2431*a^4*b*e^8)*x^4 + (80*b^5*d^5*e^3 - 680*a*b^4*d^4*e^4 + 2550*a^2*b^3*d^3*e^5 + 249730*a^3*b^ 2*d^2*e^6 + 230945*a^4*b*d*e^7 + 21879*a^5*e^8)*x^3 - 3*(32*b^5*d^6*e^2 - 272*a*b^4*d^5*e^3 + 1020*a^2*b^3*d^4*e^4 - 2210*a^3*b^2*d^3*e^5 - 60775*a^ 4*b*d^2*e^6 - 21879*a^5*d*e^7)*x^2 + (128*b^5*d^7*e - 1088*a*b^4*d^6*e^2 + 4080*a^2*b^3*d^5*e^3 - 8840*a^3*b^2*d^4*e^4 + 12155*a^4*b*d^3*e^5 + 65637 *a^5*d^2*e^6)*x)*sqrt(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 1743 vs. \(2 (230) = 460\).
Time = 0.38 (sec) , antiderivative size = 1743, normalized size of antiderivative = 5.45 \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \]
2/765765*(765765*sqrt(e*x + d)*a^5*d^3*sgn(b*x + a) + 765765*((e*x + d)^(3 /2) - 3*sqrt(e*x + d)*d)*a^5*d^2*sgn(b*x + a) + 1276275*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*b*d^3*sgn(b*x + a)/e + 153153*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^5*d*sgn(b*x + a) + 510510* (3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^3*b^2* d^3*sgn(b*x + a)/e^2 + 765765*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^4*b*d^2*sgn(b*x + a)/e + 21879*(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^ 5*sgn(b*x + a) + 218790*(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^2*b^3*d^3*sgn(b*x + a)/e^3 + 65 6370*(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^2*d^2*sgn(b*x + a)/e^2 + 328185*(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*d*sgn(b*x + a)/e + 12155*(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*b^4*d^3*sgn(b*x + a)/e^4 + 72930*(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^3*d^2*sgn(b*x + a)/e^3 + 72930*(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^3*b^2*d*sgn(b*x + a)/e^2 + 12155*(35*(e*x + ...
Timed out. \[ \int (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^{5/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \]