Integrand size = 30, antiderivative size = 320 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=-\frac {2 (b d-a e)^5 (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^6 (a+b x)}+\frac {10 b (b d-a e)^4 (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^6 (a+b x)}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 e^6 (a+b x)}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^6 (a+b x)}-\frac {10 b^4 (b d-a e) (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^6 (a+b x)}+\frac {2 b^5 (d+e x)^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 e^6 (a+b x)} \] Output:
-2/5*(-a*e+b*d)^5*(e*x+d)^(5/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+10/7*b*(-a*e +b*d)^4*(e*x+d)^(7/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-20/9*b^2*(-a*e+b*d)^3* (e*x+d)^(9/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)+20/11*b^3*(-a*e+b*d)^2*(e*x+d) ^(11/2)*((b*x+a)^2)^(1/2)/e^6/(b*x+a)-10/13*b^4*(-a*e+b*d)*(e*x+d)^(13/2)* ((b*x+a)^2)^(1/2)/e^6/(b*x+a)+2/15*b^5*(e*x+d)^(15/2)*((b*x+a)^2)^(1/2)/e^ 6/(b*x+a)
Time = 0.15 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.73 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {(a+b x)^2} (d+e x)^{5/2} \left (9009 a^5 e^5+6435 a^4 b e^4 (-2 d+5 e x)+1430 a^3 b^2 e^3 \left (8 d^2-20 d e x+35 e^2 x^2\right )+390 a^2 b^3 e^2 \left (-16 d^3+40 d^2 e x-70 d e^2 x^2+105 e^3 x^3\right )+15 a b^4 e \left (128 d^4-320 d^3 e x+560 d^2 e^2 x^2-840 d e^3 x^3+1155 e^4 x^4\right )+b^5 \left (-256 d^5+640 d^4 e x-1120 d^3 e^2 x^2+1680 d^2 e^3 x^3-2310 d e^4 x^4+3003 e^5 x^5\right )\right )}{45045 e^6 (a+b x)} \] Input:
Integrate[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
Output:
(2*Sqrt[(a + b*x)^2]*(d + e*x)^(5/2)*(9009*a^5*e^5 + 6435*a^4*b*e^4*(-2*d + 5*e*x) + 1430*a^3*b^2*e^3*(8*d^2 - 20*d*e*x + 35*e^2*x^2) + 390*a^2*b^3* e^2*(-16*d^3 + 40*d^2*e*x - 70*d*e^2*x^2 + 105*e^3*x^3) + 15*a*b^4*e*(128* d^4 - 320*d^3*e*x + 560*d^2*e^2*x^2 - 840*d*e^3*x^3 + 1155*e^4*x^4) + b^5* (-256*d^5 + 640*d^4*e*x - 1120*d^3*e^2*x^2 + 1680*d^2*e^3*x^3 - 2310*d*e^4 *x^4 + 3003*e^5*x^5)))/(45045*e^6*(a + b*x))
Time = 0.52 (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.
| \(\displaystyle \int \left (a^2+2 a b x+b^2 x^2\right )^{5/2} (d+e x)^{3/2} \, dx\) |
| \(\Big \downarrow \) 1102 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int b^5 (a+b x)^5 (d+e x)^{3/2}dx}{b^5 (a+b x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int (a+b x)^5 (d+e x)^{3/2}dx}{a+b x}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {b^5 (d+e x)^{13/2}}{e^5}-\frac {5 b^4 (b d-a e) (d+e x)^{11/2}}{e^5}+\frac {10 b^3 (b d-a e)^2 (d+e x)^{9/2}}{e^5}-\frac {10 b^2 (b d-a e)^3 (d+e x)^{7/2}}{e^5}+\frac {5 b (b d-a e)^4 (d+e x)^{5/2}}{e^5}+\frac {(a e-b d)^5 (d+e x)^{3/2}}{e^5}\right )dx}{a+b x}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a^2+2 a b x+b^2 x^2} \left (-\frac {10 b^4 (d+e x)^{13/2} (b d-a e)}{13 e^6}+\frac {20 b^3 (d+e x)^{11/2} (b d-a e)^2}{11 e^6}-\frac {20 b^2 (d+e x)^{9/2} (b d-a e)^3}{9 e^6}+\frac {10 b (d+e x)^{7/2} (b d-a e)^4}{7 e^6}-\frac {2 (d+e x)^{5/2} (b d-a e)^5}{5 e^6}+\frac {2 b^5 (d+e x)^{15/2}}{15 e^6}\right )}{a+b x}\) |
Input:
Int[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(5/2),x]
Output:
(Sqrt[a^2 + 2*a*b*x + b^2*x^2]*((-2*(b*d - a*e)^5*(d + e*x)^(5/2))/(5*e^6) + (10*b*(b*d - a*e)^4*(d + e*x)^(7/2))/(7*e^6) - (20*b^2*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^6) + (20*b^3*(b*d - a*e)^2*(d + e*x)^(11/2))/(11*e^6) - (10*b^4*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^6) + (2*b^5*(d + e*x)^(15/2) )/(15*e^6)))/(a + b*x)
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 = 1.47 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.90
| method | result | size |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 x^{5} e^{5} b^{5}+17325 x^{4} a \,b^{4} e^{5}-2310 x^{4} b^{5} d \,e^{4}+40950 x^{3} a^{2} b^{3} e^{5}-12600 x^{3} a \,b^{4} d \,e^{4}+1680 x^{3} b^{5} d^{2} e^{3}+50050 x^{2} a^{3} b^{2} e^{5}-27300 x^{2} a^{2} b^{3} d \,e^{4}+8400 x^{2} a \,b^{4} d^{2} e^{3}-1120 x^{2} b^{5} d^{3} e^{2}+32175 a^{4} b \,e^{5} x -28600 a^{3} b^{2} d \,e^{4} x +15600 x \,a^{2} b^{3} d^{2} e^{3}-4800 x a \,b^{4} d^{3} e^{2}+640 b^{5} d^{4} e x +9009 e^{5} a^{5}-12870 a^{4} b d \,e^{4}+11440 a^{3} b^{2} d^{2} e^{3}-6240 a^{2} b^{3} d^{3} e^{2}+1920 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
| default | \(\frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (3003 x^{5} e^{5} b^{5}+17325 x^{4} a \,b^{4} e^{5}-2310 x^{4} b^{5} d \,e^{4}+40950 x^{3} a^{2} b^{3} e^{5}-12600 x^{3} a \,b^{4} d \,e^{4}+1680 x^{3} b^{5} d^{2} e^{3}+50050 x^{2} a^{3} b^{2} e^{5}-27300 x^{2} a^{2} b^{3} d \,e^{4}+8400 x^{2} a \,b^{4} d^{2} e^{3}-1120 x^{2} b^{5} d^{3} e^{2}+32175 a^{4} b \,e^{5} x -28600 a^{3} b^{2} d \,e^{4} x +15600 x \,a^{2} b^{3} d^{2} e^{3}-4800 x a \,b^{4} d^{3} e^{2}+640 b^{5} d^{4} e x +9009 e^{5} a^{5}-12870 a^{4} b d \,e^{4}+11440 a^{3} b^{2} d^{2} e^{3}-6240 a^{2} b^{3} d^{3} e^{2}+1920 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}}}{45045 e^{6} \left (b x +a \right )^{5}}\) | \(289\) |
| orering | \(\frac {2 \left (3003 x^{5} e^{5} b^{5}+17325 x^{4} a \,b^{4} e^{5}-2310 x^{4} b^{5} d \,e^{4}+40950 x^{3} a^{2} b^{3} e^{5}-12600 x^{3} a \,b^{4} d \,e^{4}+1680 x^{3} b^{5} d^{2} e^{3}+50050 x^{2} a^{3} b^{2} e^{5}-27300 x^{2} a^{2} b^{3} d \,e^{4}+8400 x^{2} a \,b^{4} d^{2} e^{3}-1120 x^{2} b^{5} d^{3} e^{2}+32175 a^{4} b \,e^{5} x -28600 a^{3} b^{2} d \,e^{4} x +15600 x \,a^{2} b^{3} d^{2} e^{3}-4800 x a \,b^{4} d^{3} e^{2}+640 b^{5} d^{4} e x +9009 e^{5} a^{5}-12870 a^{4} b d \,e^{4}+11440 a^{3} b^{2} d^{2} e^{3}-6240 a^{2} b^{3} d^{3} e^{2}+1920 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (e x +d \right )^{\frac {5}{2}} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{\frac {5}{2}}}{45045 e^{6} \left (b x +a \right )^{5}}\) | \(298\) |
| risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (3003 b^{5} x^{7} e^{7}+17325 a \,b^{4} e^{7} x^{6}+3696 b^{5} d \,e^{6} x^{6}+40950 a^{2} b^{3} e^{7} x^{5}+22050 a \,b^{4} d \,e^{6} x^{5}+63 b^{5} d^{2} e^{5} x^{5}+50050 a^{3} b^{2} e^{7} x^{4}+54600 a^{2} b^{3} d \,e^{6} x^{4}+525 a \,b^{4} d^{2} e^{5} x^{4}-70 b^{5} d^{3} e^{4} x^{4}+32175 a^{4} b \,e^{7} x^{3}+71500 a^{3} b^{2} d \,e^{6} x^{3}+1950 a^{2} b^{3} d^{2} e^{5} x^{3}-600 a \,b^{4} d^{3} e^{4} x^{3}+80 b^{5} d^{4} e^{3} x^{3}+9009 a^{5} e^{7} x^{2}+51480 a^{4} b d \,e^{6} x^{2}+4290 a^{3} b^{2} d^{2} e^{5} x^{2}-2340 a^{2} b^{3} d^{3} e^{4} x^{2}+720 a \,b^{4} d^{4} e^{3} x^{2}-96 b^{5} d^{5} e^{2} x^{2}+18018 a^{5} d \,e^{6} x +6435 a^{4} b \,d^{2} e^{5} x -5720 a^{3} b^{2} d^{3} e^{4} x +3120 a^{2} b^{3} d^{4} e^{3} x -960 a \,b^{4} d^{5} e^{2} x +128 b^{5} d^{6} e x +9009 a^{5} d^{2} e^{5}-12870 a^{4} b \,d^{3} e^{4}+11440 a^{3} b^{2} d^{4} e^{3}-6240 a^{2} b^{3} d^{5} e^{2}+1920 a \,b^{4} d^{6} e -256 b^{5} d^{7}\right ) \sqrt {e x +d}}{45045 \left (b x +a \right ) e^{6}}\) | \(469\) |
Input:
int((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/45045*(e*x+d)^(5/2)*(3003*b^5*e^5*x^5+17325*a*b^4*e^5*x^4-2310*b^5*d*e^4 *x^4+40950*a^2*b^3*e^5*x^3-12600*a*b^4*d*e^4*x^3+1680*b^5*d^2*e^3*x^3+5005 0*a^3*b^2*e^5*x^2-27300*a^2*b^3*d*e^4*x^2+8400*a*b^4*d^2*e^3*x^2-1120*b^5* d^3*e^2*x^2+32175*a^4*b*e^5*x-28600*a^3*b^2*d*e^4*x+15600*a^2*b^3*d^2*e^3* x-4800*a*b^4*d^3*e^2*x+640*b^5*d^4*e*x+9009*a^5*e^5-12870*a^4*b*d*e^4+1144 0*a^3*b^2*d^2*e^3-6240*a^2*b^3*d^3*e^2+1920*a*b^4*d^4*e-256*b^5*d^5)*((b*x +a)^2)^(5/2)/e^6/(b*x+a)^5
Time = 0.09 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.31 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \] Input:
integrate((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="fricas")
Output:
2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3* d^5*e^2 + 11440*a^3*b^2*d^4*e^3 - 12870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e^6 + 650*a^2*b^3*e^7)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2* b^3*d*e^6 - 1430*a^3*b^2*e^7)*x^4 + 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*(32* b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 780*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e ^5 - 17160*a^4*b*d*e^6 - 3003*a^5*e^7)*x^2 + (128*b^5*d^6*e - 960*a*b^4*d^ 5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*e^4 + 6435*a^4*b*d^2*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)/e^6
\[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int \left (d + e x\right )^{\frac {3}{2}} \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \] Input:
integrate((e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**(5/2),x)
Output:
Integral((d + e*x)**(3/2)*((a + b*x)**2)**(5/2), x)
Time = 0.04 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.31 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \, {\left (3003 \, b^{5} e^{7} x^{7} - 256 \, b^{5} d^{7} + 1920 \, a b^{4} d^{6} e - 6240 \, a^{2} b^{3} d^{5} e^{2} + 11440 \, a^{3} b^{2} d^{4} e^{3} - 12870 \, a^{4} b d^{3} e^{4} + 9009 \, a^{5} d^{2} e^{5} + 231 \, {\left (16 \, b^{5} d e^{6} + 75 \, a b^{4} e^{7}\right )} x^{6} + 63 \, {\left (b^{5} d^{2} e^{5} + 350 \, a b^{4} d e^{6} + 650 \, a^{2} b^{3} e^{7}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{3} e^{4} - 15 \, a b^{4} d^{2} e^{5} - 1560 \, a^{2} b^{3} d e^{6} - 1430 \, a^{3} b^{2} e^{7}\right )} x^{4} + 5 \, {\left (16 \, b^{5} d^{4} e^{3} - 120 \, a b^{4} d^{3} e^{4} + 390 \, a^{2} b^{3} d^{2} e^{5} + 14300 \, a^{3} b^{2} d e^{6} + 6435 \, a^{4} b e^{7}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{5} e^{2} - 240 \, a b^{4} d^{4} e^{3} + 780 \, a^{2} b^{3} d^{3} e^{4} - 1430 \, a^{3} b^{2} d^{2} e^{5} - 17160 \, a^{4} b d e^{6} - 3003 \, a^{5} e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{6} e - 960 \, a b^{4} d^{5} e^{2} + 3120 \, a^{2} b^{3} d^{4} e^{3} - 5720 \, a^{3} b^{2} d^{3} e^{4} + 6435 \, a^{4} b d^{2} e^{5} + 18018 \, a^{5} d e^{6}\right )} x\right )} \sqrt {e x + d}}{45045 \, e^{6}} \] Input:
integrate((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="maxima")
Output:
2/45045*(3003*b^5*e^7*x^7 - 256*b^5*d^7 + 1920*a*b^4*d^6*e - 6240*a^2*b^3* d^5*e^2 + 11440*a^3*b^2*d^4*e^3 - 12870*a^4*b*d^3*e^4 + 9009*a^5*d^2*e^5 + 231*(16*b^5*d*e^6 + 75*a*b^4*e^7)*x^6 + 63*(b^5*d^2*e^5 + 350*a*b^4*d*e^6 + 650*a^2*b^3*e^7)*x^5 - 35*(2*b^5*d^3*e^4 - 15*a*b^4*d^2*e^5 - 1560*a^2* b^3*d*e^6 - 1430*a^3*b^2*e^7)*x^4 + 5*(16*b^5*d^4*e^3 - 120*a*b^4*d^3*e^4 + 390*a^2*b^3*d^2*e^5 + 14300*a^3*b^2*d*e^6 + 6435*a^4*b*e^7)*x^3 - 3*(32* b^5*d^5*e^2 - 240*a*b^4*d^4*e^3 + 780*a^2*b^3*d^3*e^4 - 1430*a^3*b^2*d^2*e ^5 - 17160*a^4*b*d*e^6 - 3003*a^5*e^7)*x^2 + (128*b^5*d^6*e - 960*a*b^4*d^ 5*e^2 + 3120*a^2*b^3*d^4*e^3 - 5720*a^3*b^2*d^3*e^4 + 6435*a^4*b*d^2*e^5 + 18018*a^5*d*e^6)*x)*sqrt(e*x + d)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 1192 vs. \(2 (230) = 460\).
Time = 0.17 (sec) , antiderivative size = 1192, normalized size of antiderivative = 3.72 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x, algorithm="giac")
Output:
2/45045*(45045*sqrt(e*x + d)*a^5*d^2*sgn(b*x + a) + 30030*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^5*d*sgn(b*x + a) + 75075*((e*x + d)^(3/2) - 3*sqrt (e*x + d)*d)*a^4*b*d^2*sgn(b*x + a)/e + 3003*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^5*sgn(b*x + a) + 30030*(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^2*sgn(b*x + a)/e^2 + 30030*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^4*b*d*sgn(b*x + a)/e + 12870*(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^2*sgn(b *x + a)/e^3 + 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^2*d*sgn(b*x + a)/e^2 + 6435*(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*sgn(b*x + a)/e + 715*(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^2*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^2*b^3*d*sgn(b*x + a)/e^3 + 1430*(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*sgn(b*x + a)/e^2 + 65*(63*(e*x + d)^(1 1/2) - 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)*b^5*d^2*sgn...
Timed out. \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\int {\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \] Input:
int((d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2),x)
Output:
int((d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(5/2), x)
Time = 0.27 (sec) , antiderivative size = 451, normalized size of antiderivative = 1.41 \[ \int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx=\frac {2 \sqrt {e x +d}\, \left (3003 b^{5} e^{7} x^{7}+17325 a \,b^{4} e^{7} x^{6}+3696 b^{5} d \,e^{6} x^{6}+40950 a^{2} b^{3} e^{7} x^{5}+22050 a \,b^{4} d \,e^{6} x^{5}+63 b^{5} d^{2} e^{5} x^{5}+50050 a^{3} b^{2} e^{7} x^{4}+54600 a^{2} b^{3} d \,e^{6} x^{4}+525 a \,b^{4} d^{2} e^{5} x^{4}-70 b^{5} d^{3} e^{4} x^{4}+32175 a^{4} b \,e^{7} x^{3}+71500 a^{3} b^{2} d \,e^{6} x^{3}+1950 a^{2} b^{3} d^{2} e^{5} x^{3}-600 a \,b^{4} d^{3} e^{4} x^{3}+80 b^{5} d^{4} e^{3} x^{3}+9009 a^{5} e^{7} x^{2}+51480 a^{4} b d \,e^{6} x^{2}+4290 a^{3} b^{2} d^{2} e^{5} x^{2}-2340 a^{2} b^{3} d^{3} e^{4} x^{2}+720 a \,b^{4} d^{4} e^{3} x^{2}-96 b^{5} d^{5} e^{2} x^{2}+18018 a^{5} d \,e^{6} x +6435 a^{4} b \,d^{2} e^{5} x -5720 a^{3} b^{2} d^{3} e^{4} x +3120 a^{2} b^{3} d^{4} e^{3} x -960 a \,b^{4} d^{5} e^{2} x +128 b^{5} d^{6} e x +9009 a^{5} d^{2} e^{5}-12870 a^{4} b \,d^{3} e^{4}+11440 a^{3} b^{2} d^{4} e^{3}-6240 a^{2} b^{3} d^{5} e^{2}+1920 a \,b^{4} d^{6} e -256 b^{5} d^{7}\right )}{45045 e^{6}} \] Input:
int((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(5/2),x)
Output:
(2*sqrt(d + e*x)*(9009*a**5*d**2*e**5 + 18018*a**5*d*e**6*x + 9009*a**5*e* *7*x**2 - 12870*a**4*b*d**3*e**4 + 6435*a**4*b*d**2*e**5*x + 51480*a**4*b* d*e**6*x**2 + 32175*a**4*b*e**7*x**3 + 11440*a**3*b**2*d**4*e**3 - 5720*a* *3*b**2*d**3*e**4*x + 4290*a**3*b**2*d**2*e**5*x**2 + 71500*a**3*b**2*d*e* *6*x**3 + 50050*a**3*b**2*e**7*x**4 - 6240*a**2*b**3*d**5*e**2 + 3120*a**2 *b**3*d**4*e**3*x - 2340*a**2*b**3*d**3*e**4*x**2 + 1950*a**2*b**3*d**2*e* *5*x**3 + 54600*a**2*b**3*d*e**6*x**4 + 40950*a**2*b**3*e**7*x**5 + 1920*a *b**4*d**6*e - 960*a*b**4*d**5*e**2*x + 720*a*b**4*d**4*e**3*x**2 - 600*a* b**4*d**3*e**4*x**3 + 525*a*b**4*d**2*e**5*x**4 + 22050*a*b**4*d*e**6*x**5 + 17325*a*b**4*e**7*x**6 - 256*b**5*d**7 + 128*b**5*d**6*e*x - 96*b**5*d* *5*e**2*x**2 + 80*b**5*d**4*e**3*x**3 - 70*b**5*d**3*e**4*x**4 + 63*b**5*d **2*e**5*x**5 + 3696*b**5*d*e**6*x**6 + 3003*b**5*e**7*x**7))/(45045*e**6)