Integrand size = 33, antiderivative size = 158 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=-\frac {2 (b d-a e)^5 (d+e x)^{7/2}}{7 e^6}+\frac {10 b (b d-a e)^4 (d+e x)^{9/2}}{9 e^6}-\frac {20 b^2 (b d-a e)^3 (d+e x)^{11/2}}{11 e^6}+\frac {20 b^3 (b d-a e)^2 (d+e x)^{13/2}}{13 e^6}-\frac {2 b^4 (b d-a e) (d+e x)^{15/2}}{3 e^6}+\frac {2 b^5 (d+e x)^{17/2}}{17 e^6} \] Output:
-2/7*(-a*e+b*d)^5*(e*x+d)^(7/2)/e^6+10/9*b*(-a*e+b*d)^4*(e*x+d)^(9/2)/e^6- 20/11*b^2*(-a*e+b*d)^3*(e*x+d)^(11/2)/e^6+20/13*b^3*(-a*e+b*d)^2*(e*x+d)^( 13/2)/e^6-2/3*b^4*(-a*e+b*d)*(e*x+d)^(15/2)/e^6+2/17*b^5*(e*x+d)^(17/2)/e^ 6
Time = 0.23 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.37 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 (d+e x)^{7/2} \left (21879 a^5 e^5+12155 a^4 b e^4 (-2 d+7 e x)+2210 a^3 b^2 e^3 \left (8 d^2-28 d e x+63 e^2 x^2\right )+510 a^2 b^3 e^2 \left (-16 d^3+56 d^2 e x-126 d e^2 x^2+231 e^3 x^3\right )+17 a b^4 e \left (128 d^4-448 d^3 e x+1008 d^2 e^2 x^2-1848 d e^3 x^3+3003 e^4 x^4\right )+b^5 \left (-256 d^5+896 d^4 e x-2016 d^3 e^2 x^2+3696 d^2 e^3 x^3-6006 d e^4 x^4+9009 e^5 x^5\right )\right )}{153153 e^6} \] Input:
Integrate[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
(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 + 5 6*d^2*e*x - 126*d*e^2*x^2 + 231*e^3*x^3) + 17*a*b^4*e*(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) + b^5*(-256*d^5 + 8 96*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)
Time = 0.50 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {1184, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right )^2 (d+e x)^{5/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^4 (a+b x)^5 (d+e x)^{5/2}dx}{b^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^5 (d+e x)^{5/2}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {5 b^4 (d+e x)^{13/2} (b d-a e)}{e^5}+\frac {10 b^3 (d+e x)^{11/2} (b d-a e)^2}{e^5}-\frac {10 b^2 (d+e x)^{9/2} (b d-a e)^3}{e^5}+\frac {5 b (d+e x)^{7/2} (b d-a e)^4}{e^5}+\frac {(d+e x)^{5/2} (a e-b d)^5}{e^5}+\frac {b^5 (d+e x)^{15/2}}{e^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 b^4 (d+e x)^{15/2} (b d-a e)}{3 e^6}+\frac {20 b^3 (d+e x)^{13/2} (b d-a e)^2}{13 e^6}-\frac {20 b^2 (d+e x)^{11/2} (b d-a e)^3}{11 e^6}+\frac {10 b (d+e x)^{9/2} (b d-a e)^4}{9 e^6}-\frac {2 (d+e x)^{7/2} (b d-a e)^5}{7 e^6}+\frac {2 b^5 (d+e x)^{17/2}}{17 e^6}\) |
Input:
Int[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^2,x]
Output:
(-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)
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[1/c^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}, x] && E qQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Time = 1.87 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.29
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {7}{17} b^{5} x^{5}+\frac {7}{3} a \,b^{4} x^{4}+\frac {70}{13} a^{2} b^{3} x^{3}+\frac {70}{11} a^{3} b^{2} x^{2}+\frac {35}{9} a^{4} b x +a^{5}\right ) e^{5}-\frac {10 \left (\frac {21}{85} b^{4} x^{4}+\frac {84}{65} a \,b^{3} x^{3}+\frac {378}{143} a^{2} b^{2} x^{2}+\frac {28}{11} a^{3} b x +a^{4}\right ) b d \,e^{4}}{9}+\frac {80 b^{2} \left (\frac {231}{1105} b^{3} x^{3}+\frac {63}{65} a \,b^{2} x^{2}+\frac {21}{13} a^{2} b x +a^{3}\right ) d^{2} e^{3}}{99}-\frac {160 b^{3} d^{3} \left (\frac {21}{85} b^{2} x^{2}+\frac {14}{15} a b x +a^{2}\right ) e^{2}}{429}+\frac {128 \left (\frac {7 b x}{17}+a \right ) b^{4} d^{4} e}{1287}-\frac {256 b^{5} d^{5}}{21879}\right ) \left (e x +d \right )^{\frac {7}{2}}}{7 e^{6}}\) | \(204\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (9009 x^{5} e^{5} b^{5}+51051 x^{4} a \,b^{4} e^{5}-6006 x^{4} b^{5} d \,e^{4}+117810 x^{3} a^{2} b^{3} e^{5}-31416 x^{3} a \,b^{4} d \,e^{4}+3696 x^{3} b^{5} d^{2} e^{3}+139230 x^{2} a^{3} b^{2} e^{5}-64260 x^{2} a^{2} b^{3} d \,e^{4}+17136 x^{2} a \,b^{4} d^{2} e^{3}-2016 x^{2} b^{5} d^{3} e^{2}+85085 a^{4} b \,e^{5} x -61880 a^{3} b^{2} d \,e^{4} x +28560 x \,a^{2} b^{3} d^{2} e^{3}-7616 x a \,b^{4} d^{3} e^{2}+896 b^{5} d^{4} e x +21879 e^{5} a^{5}-24310 a^{4} b d \,e^{4}+17680 a^{3} b^{2} d^{2} e^{3}-8160 a^{2} b^{3} d^{3} e^{2}+2176 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right )}{153153 e^{6}}\) | \(273\) |
| orering | \(\frac {2 \left (9009 x^{5} e^{5} b^{5}+51051 x^{4} a \,b^{4} e^{5}-6006 x^{4} b^{5} d \,e^{4}+117810 x^{3} a^{2} b^{3} e^{5}-31416 x^{3} a \,b^{4} d \,e^{4}+3696 x^{3} b^{5} d^{2} e^{3}+139230 x^{2} a^{3} b^{2} e^{5}-64260 x^{2} a^{2} b^{3} d \,e^{4}+17136 x^{2} a \,b^{4} d^{2} e^{3}-2016 x^{2} b^{5} d^{3} e^{2}+85085 a^{4} b \,e^{5} x -61880 a^{3} b^{2} d \,e^{4} x +28560 x \,a^{2} b^{3} d^{2} e^{3}-7616 x a \,b^{4} d^{3} e^{2}+896 b^{5} d^{4} e x +21879 e^{5} a^{5}-24310 a^{4} b d \,e^{4}+17680 a^{3} b^{2} d^{2} e^{3}-8160 a^{2} b^{3} d^{3} e^{2}+2176 a \,b^{4} d^{4} e -256 b^{5} d^{5}\right ) \left (e x +d \right )^{\frac {7}{2}} \left (b^{2} x^{2}+2 a b x +a^{2}\right )^{2}}{153153 e^{6} \left (b x +a \right )^{4}}\) | \(298\) |
| derivativedivides | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a e -b d \right ) b^{4}+2 b^{3} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (2 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+b \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{6}}\) | \(350\) |
| default | \(\frac {\frac {2 b^{5} \left (e x +d \right )^{\frac {17}{2}}}{17}+\frac {2 \left (\left (a e -b d \right ) b^{4}+2 b^{3} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (2 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+b \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (e x +d \right )^{\frac {7}{2}}}{7}}{e^{6}}\) | \(350\) |
| trager | \(\frac {2 \left (9009 b^{5} e^{8} x^{8}+51051 a \,b^{4} e^{8} x^{7}+21021 b^{5} d \,e^{7} x^{7}+117810 a^{2} b^{3} e^{8} x^{6}+121737 a \,b^{4} d \,e^{7} x^{6}+12705 b^{5} d^{2} e^{6} x^{6}+139230 a^{3} b^{2} e^{8} x^{5}+289170 a^{2} b^{3} d \,e^{7} x^{5}+76041 a \,b^{4} d^{2} e^{6} x^{5}+63 b^{5} d^{3} e^{5} x^{5}+85085 a^{4} b \,e^{8} x^{4}+355810 a^{3} b^{2} d \,e^{7} x^{4}+189210 a^{2} b^{3} d^{2} e^{6} x^{4}+595 a \,b^{4} d^{3} e^{5} x^{4}-70 b^{5} d^{4} e^{4} x^{4}+21879 a^{5} e^{8} x^{3}+230945 a^{4} b d \,e^{7} x^{3}+249730 a^{3} b^{2} d^{2} e^{6} x^{3}+2550 a^{2} b^{3} d^{3} e^{5} x^{3}-680 a \,b^{4} d^{4} e^{4} x^{3}+80 b^{5} d^{5} e^{3} x^{3}+65637 a^{5} d \,e^{7} x^{2}+182325 a^{4} b \,d^{2} e^{6} x^{2}+6630 a^{3} b^{2} d^{3} e^{5} x^{2}-3060 a^{2} b^{3} d^{4} e^{4} x^{2}+816 a \,b^{4} d^{5} e^{3} x^{2}-96 b^{5} d^{6} e^{2} x^{2}+65637 a^{5} d^{2} e^{6} x +12155 a^{4} b \,d^{3} e^{5} x -8840 a^{3} b^{2} d^{4} e^{4} x +4080 a^{2} b^{3} d^{5} e^{3} x -1088 a \,b^{4} d^{6} e^{2} x +128 b^{5} d^{7} e x +21879 a^{5} d^{3} e^{5}-24310 a^{4} b \,d^{4} e^{4}+17680 a^{3} b^{2} d^{5} e^{3}-8160 a^{2} b^{3} d^{6} e^{2}+2176 a \,b^{4} d^{7} e -256 b^{5} d^{8}\right ) \sqrt {e x +d}}{153153 e^{6}}\) | \(545\) |
| risch | \(\frac {2 \left (9009 b^{5} e^{8} x^{8}+51051 a \,b^{4} e^{8} x^{7}+21021 b^{5} d \,e^{7} x^{7}+117810 a^{2} b^{3} e^{8} x^{6}+121737 a \,b^{4} d \,e^{7} x^{6}+12705 b^{5} d^{2} e^{6} x^{6}+139230 a^{3} b^{2} e^{8} x^{5}+289170 a^{2} b^{3} d \,e^{7} x^{5}+76041 a \,b^{4} d^{2} e^{6} x^{5}+63 b^{5} d^{3} e^{5} x^{5}+85085 a^{4} b \,e^{8} x^{4}+355810 a^{3} b^{2} d \,e^{7} x^{4}+189210 a^{2} b^{3} d^{2} e^{6} x^{4}+595 a \,b^{4} d^{3} e^{5} x^{4}-70 b^{5} d^{4} e^{4} x^{4}+21879 a^{5} e^{8} x^{3}+230945 a^{4} b d \,e^{7} x^{3}+249730 a^{3} b^{2} d^{2} e^{6} x^{3}+2550 a^{2} b^{3} d^{3} e^{5} x^{3}-680 a \,b^{4} d^{4} e^{4} x^{3}+80 b^{5} d^{5} e^{3} x^{3}+65637 a^{5} d \,e^{7} x^{2}+182325 a^{4} b \,d^{2} e^{6} x^{2}+6630 a^{3} b^{2} d^{3} e^{5} x^{2}-3060 a^{2} b^{3} d^{4} e^{4} x^{2}+816 a \,b^{4} d^{5} e^{3} x^{2}-96 b^{5} d^{6} e^{2} x^{2}+65637 a^{5} d^{2} e^{6} x +12155 a^{4} b \,d^{3} e^{5} x -8840 a^{3} b^{2} d^{4} e^{4} x +4080 a^{2} b^{3} d^{5} e^{3} x -1088 a \,b^{4} d^{6} e^{2} x +128 b^{5} d^{7} e x +21879 a^{5} d^{3} e^{5}-24310 a^{4} b \,d^{4} e^{4}+17680 a^{3} b^{2} d^{5} e^{3}-8160 a^{2} b^{3} d^{6} e^{2}+2176 a \,b^{4} d^{7} e -256 b^{5} d^{8}\right ) \sqrt {e x +d}}{153153 e^{6}}\) | \(545\) |
Input:
int((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x,method=_RETURNVERBOSE)
Output:
2/7*((7/17*b^5*x^5+7/3*a*b^4*x^4+70/13*a^2*b^3*x^3+70/11*a^3*b^2*x^2+35/9* a^4*b*x+a^5)*e^5-10/9*(21/85*b^4*x^4+84/65*a*b^3*x^3+378/143*a^2*b^2*x^2+2 8/11*a^3*b*x+a^4)*b*d*e^4+80/99*b^2*(231/1105*b^3*x^3+63/65*a*b^2*x^2+21/1 3*a^2*b*x+a^3)*d^2*e^3-160/429*b^3*d^3*(21/85*b^2*x^2+14/15*a*b*x+a^2)*e^2 +128/1287*(7/17*b*x+a)*b^4*d^4*e-256/21879*b^5*d^5)*(e*x+d)^(7/2)/e^6
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (134) = 268\).
Time = 0.08 (sec) , antiderivative size = 497, normalized size of antiderivative = 3.15 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (9009 \, b^{5} e^{8} x^{8} - 256 \, b^{5} d^{8} + 2176 \, a b^{4} d^{7} e - 8160 \, a^{2} b^{3} d^{6} e^{2} + 17680 \, a^{3} b^{2} d^{5} e^{3} - 24310 \, a^{4} b d^{4} e^{4} + 21879 \, a^{5} d^{3} e^{5} + 3003 \, {\left (7 \, b^{5} d e^{7} + 17 \, a b^{4} e^{8}\right )} x^{7} + 231 \, {\left (55 \, b^{5} d^{2} e^{6} + 527 \, a b^{4} d e^{7} + 510 \, a^{2} b^{3} e^{8}\right )} x^{6} + 63 \, {\left (b^{5} d^{3} e^{5} + 1207 \, a b^{4} d^{2} e^{6} + 4590 \, a^{2} b^{3} d e^{7} + 2210 \, a^{3} b^{2} e^{8}\right )} x^{5} - 35 \, {\left (2 \, b^{5} d^{4} e^{4} - 17 \, a b^{4} d^{3} e^{5} - 5406 \, a^{2} b^{3} d^{2} e^{6} - 10166 \, a^{3} b^{2} d e^{7} - 2431 \, a^{4} b e^{8}\right )} x^{4} + {\left (80 \, b^{5} d^{5} e^{3} - 680 \, a b^{4} d^{4} e^{4} + 2550 \, a^{2} b^{3} d^{3} e^{5} + 249730 \, a^{3} b^{2} d^{2} e^{6} + 230945 \, a^{4} b d e^{7} + 21879 \, a^{5} e^{8}\right )} x^{3} - 3 \, {\left (32 \, b^{5} d^{6} e^{2} - 272 \, a b^{4} d^{5} e^{3} + 1020 \, a^{2} b^{3} d^{4} e^{4} - 2210 \, a^{3} b^{2} d^{3} e^{5} - 60775 \, a^{4} b d^{2} e^{6} - 21879 \, a^{5} d e^{7}\right )} x^{2} + {\left (128 \, b^{5} d^{7} e - 1088 \, a b^{4} d^{6} e^{2} + 4080 \, a^{2} b^{3} d^{5} e^{3} - 8840 \, a^{3} b^{2} d^{4} e^{4} + 12155 \, a^{4} b d^{3} e^{5} + 65637 \, a^{5} d^{2} e^{6}\right )} x\right )} \sqrt {e x + d}}{153153 \, e^{6}} \] Input:
integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="fric as")
Output:
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. 347 vs. \(2 (146) = 292\).
Time = 2.45 (sec) , antiderivative size = 347, normalized size of antiderivative = 2.20 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\begin {cases} \frac {2 \left (\frac {b^{5} \left (d + e x\right )^{\frac {17}{2}}}{17 e^{5}} + \frac {\left (d + e x\right )^{\frac {15}{2}} \cdot \left (5 a b^{4} e - 5 b^{5} d\right )}{15 e^{5}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (10 a^{2} b^{3} e^{2} - 20 a b^{4} d e + 10 b^{5} d^{2}\right )}{13 e^{5}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (10 a^{3} b^{2} e^{3} - 30 a^{2} b^{3} d e^{2} + 30 a b^{4} d^{2} e - 10 b^{5} d^{3}\right )}{11 e^{5}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (5 a^{4} b e^{4} - 20 a^{3} b^{2} d e^{3} + 30 a^{2} b^{3} d^{2} e^{2} - 20 a b^{4} d^{3} e + 5 b^{5} d^{4}\right )}{9 e^{5}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (a^{5} e^{5} - 5 a^{4} b d e^{4} + 10 a^{3} b^{2} d^{2} e^{3} - 10 a^{2} b^{3} d^{3} e^{2} + 5 a b^{4} d^{4} e - b^{5} d^{5}\right )}{7 e^{5}}\right )}{e} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (\begin {cases} a^{5} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{3}}{6 b} & \text {otherwise} \end {cases}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((b*x+a)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**2,x)
Output:
Piecewise((2*(b**5*(d + e*x)**(17/2)/(17*e**5) + (d + e*x)**(15/2)*(5*a*b* *4*e - 5*b**5*d)/(15*e**5) + (d + e*x)**(13/2)*(10*a**2*b**3*e**2 - 20*a*b **4*d*e + 10*b**5*d**2)/(13*e**5) + (d + e*x)**(11/2)*(10*a**3*b**2*e**3 - 30*a**2*b**3*d*e**2 + 30*a*b**4*d**2*e - 10*b**5*d**3)/(11*e**5) + (d + e *x)**(9/2)*(5*a**4*b*e**4 - 20*a**3*b**2*d*e**3 + 30*a**2*b**3*d**2*e**2 - 20*a*b**4*d**3*e + 5*b**5*d**4)/(9*e**5) + (d + e*x)**(7/2)*(a**5*e**5 - 5*a**4*b*d*e**4 + 10*a**3*b**2*d**2*e**3 - 10*a**2*b**3*d**3*e**2 + 5*a*b* *4*d**4*e - b**5*d**5)/(7*e**5))/e, Ne(e, 0)), (d**(5/2)*Piecewise((a**5*x , Eq(b, 0)), ((a**2 + 2*a*b*x + b**2*x**2)**3/(6*b), True)), True))
Time = 0.03 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.64 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \, {\left (9009 \, {\left (e x + d\right )}^{\frac {17}{2}} b^{5} - 51051 \, {\left (b^{5} d - a b^{4} e\right )} {\left (e x + d\right )}^{\frac {15}{2}} + 117810 \, {\left (b^{5} d^{2} - 2 \, a b^{4} d e + a^{2} b^{3} e^{2}\right )} {\left (e x + d\right )}^{\frac {13}{2}} - 139230 \, {\left (b^{5} d^{3} - 3 \, a b^{4} d^{2} e + 3 \, a^{2} b^{3} d e^{2} - a^{3} b^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 85085 \, {\left (b^{5} d^{4} - 4 \, a b^{4} d^{3} e + 6 \, a^{2} b^{3} d^{2} e^{2} - 4 \, a^{3} b^{2} d e^{3} + a^{4} b e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 21879 \, {\left (b^{5} d^{5} - 5 \, a b^{4} d^{4} e + 10 \, a^{2} b^{3} d^{3} e^{2} - 10 \, a^{3} b^{2} d^{2} e^{3} + 5 \, a^{4} b d e^{4} - a^{5} e^{5}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{153153 \, e^{6}} \] Input:
integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="maxi ma")
Output:
2/153153*(9009*(e*x + d)^(17/2)*b^5 - 51051*(b^5*d - a*b^4*e)*(e*x + d)^(1 5/2) + 117810*(b^5*d^2 - 2*a*b^4*d*e + a^2*b^3*e^2)*(e*x + d)^(13/2) - 139 230*(b^5*d^3 - 3*a*b^4*d^2*e + 3*a^2*b^3*d*e^2 - a^3*b^2*e^3)*(e*x + d)^(1 1/2) + 85085*(b^5*d^4 - 4*a*b^4*d^3*e + 6*a^2*b^3*d^2*e^2 - 4*a^3*b^2*d*e^ 3 + a^4*b*e^4)*(e*x + d)^(9/2) - 21879*(b^5*d^5 - 5*a*b^4*d^4*e + 10*a^2*b ^3*d^3*e^2 - 10*a^3*b^2*d^2*e^3 + 5*a^4*b*d*e^4 - a^5*e^5)*(e*x + d)^(7/2) )/e^6
Leaf count of result is larger than twice the leaf count of optimal. 1599 vs. \(2 (134) = 268\).
Time = 0.21 (sec) , antiderivative size = 1599, normalized size of antiderivative = 10.12 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x, algorithm="giac ")
Output:
2/765765*(765765*sqrt(e*x + d)*a^5*d^3 + 765765*((e*x + d)^(3/2) - 3*sqrt( e*x + d)*d)*a^5*d^2 + 1276275*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^4*b* d^3/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 + 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/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/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 + 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/e^3 + 656370*(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/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/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/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/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*sq rt(e*x + d)*d^4)*a^3*b^2*d/e^2 + 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^4*b/e + 1105*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d...
Time = 0.05 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.87 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2\,b^5\,{\left (d+e\,x\right )}^{17/2}}{17\,e^6}-\frac {\left (10\,b^5\,d-10\,a\,b^4\,e\right )\,{\left (d+e\,x\right )}^{15/2}}{15\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{7/2}}{7\,e^6}+\frac {20\,b^2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^6}+\frac {20\,b^3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^6}+\frac {10\,b\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6} \] Input:
int((a + b*x)*(d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^2,x)
Output:
(2*b^5*(d + e*x)^(17/2))/(17*e^6) - ((10*b^5*d - 10*a*b^4*e)*(d + e*x)^(15 /2))/(15*e^6) + (2*(a*e - b*d)^5*(d + e*x)^(7/2))/(7*e^6) + (20*b^2*(a*e - b*d)^3*(d + e*x)^(11/2))/(11*e^6) + (20*b^3*(a*e - b*d)^2*(d + e*x)^(13/2 ))/(13*e^6) + (10*b*(a*e - b*d)^4*(d + e*x)^(9/2))/(9*e^6)
Time = 0.20 (sec) , antiderivative size = 543, normalized size of antiderivative = 3.44 \[ \int (a+b x) (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx=\frac {2 \sqrt {e x +d}\, \left (9009 b^{5} e^{8} x^{8}+51051 a \,b^{4} e^{8} x^{7}+21021 b^{5} d \,e^{7} x^{7}+117810 a^{2} b^{3} e^{8} x^{6}+121737 a \,b^{4} d \,e^{7} x^{6}+12705 b^{5} d^{2} e^{6} x^{6}+139230 a^{3} b^{2} e^{8} x^{5}+289170 a^{2} b^{3} d \,e^{7} x^{5}+76041 a \,b^{4} d^{2} e^{6} x^{5}+63 b^{5} d^{3} e^{5} x^{5}+85085 a^{4} b \,e^{8} x^{4}+355810 a^{3} b^{2} d \,e^{7} x^{4}+189210 a^{2} b^{3} d^{2} e^{6} x^{4}+595 a \,b^{4} d^{3} e^{5} x^{4}-70 b^{5} d^{4} e^{4} x^{4}+21879 a^{5} e^{8} x^{3}+230945 a^{4} b d \,e^{7} x^{3}+249730 a^{3} b^{2} d^{2} e^{6} x^{3}+2550 a^{2} b^{3} d^{3} e^{5} x^{3}-680 a \,b^{4} d^{4} e^{4} x^{3}+80 b^{5} d^{5} e^{3} x^{3}+65637 a^{5} d \,e^{7} x^{2}+182325 a^{4} b \,d^{2} e^{6} x^{2}+6630 a^{3} b^{2} d^{3} e^{5} x^{2}-3060 a^{2} b^{3} d^{4} e^{4} x^{2}+816 a \,b^{4} d^{5} e^{3} x^{2}-96 b^{5} d^{6} e^{2} x^{2}+65637 a^{5} d^{2} e^{6} x +12155 a^{4} b \,d^{3} e^{5} x -8840 a^{3} b^{2} d^{4} e^{4} x +4080 a^{2} b^{3} d^{5} e^{3} x -1088 a \,b^{4} d^{6} e^{2} x +128 b^{5} d^{7} e x +21879 a^{5} d^{3} e^{5}-24310 a^{4} b \,d^{4} e^{4}+17680 a^{3} b^{2} d^{5} e^{3}-8160 a^{2} b^{3} d^{6} e^{2}+2176 a \,b^{4} d^{7} e -256 b^{5} d^{8}\right )}{153153 e^{6}} \] Input:
int((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^2,x)
Output:
(2*sqrt(d + e*x)*(21879*a**5*d**3*e**5 + 65637*a**5*d**2*e**6*x + 65637*a* *5*d*e**7*x**2 + 21879*a**5*e**8*x**3 - 24310*a**4*b*d**4*e**4 + 12155*a** 4*b*d**3*e**5*x + 182325*a**4*b*d**2*e**6*x**2 + 230945*a**4*b*d*e**7*x**3 + 85085*a**4*b*e**8*x**4 + 17680*a**3*b**2*d**5*e**3 - 8840*a**3*b**2*d** 4*e**4*x + 6630*a**3*b**2*d**3*e**5*x**2 + 249730*a**3*b**2*d**2*e**6*x**3 + 355810*a**3*b**2*d*e**7*x**4 + 139230*a**3*b**2*e**8*x**5 - 8160*a**2*b **3*d**6*e**2 + 4080*a**2*b**3*d**5*e**3*x - 3060*a**2*b**3*d**4*e**4*x**2 + 2550*a**2*b**3*d**3*e**5*x**3 + 189210*a**2*b**3*d**2*e**6*x**4 + 28917 0*a**2*b**3*d*e**7*x**5 + 117810*a**2*b**3*e**8*x**6 + 2176*a*b**4*d**7*e - 1088*a*b**4*d**6*e**2*x + 816*a*b**4*d**5*e**3*x**2 - 680*a*b**4*d**4*e* *4*x**3 + 595*a*b**4*d**3*e**5*x**4 + 76041*a*b**4*d**2*e**6*x**5 + 121737 *a*b**4*d*e**7*x**6 + 51051*a*b**4*e**8*x**7 - 256*b**5*d**8 + 128*b**5*d* *7*e*x - 96*b**5*d**6*e**2*x**2 + 80*b**5*d**5*e**3*x**3 - 70*b**5*d**4*e* *4*x**4 + 63*b**5*d**3*e**5*x**5 + 12705*b**5*d**2*e**6*x**6 + 21021*b**5* d*e**7*x**7 + 9009*b**5*e**8*x**8))/(153153*e**6)