Integrand size = 31, antiderivative size = 100 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=-\frac {2 (b d-a e)^3 (d+e x)^{9/2}}{9 e^4}+\frac {6 b (b d-a e)^2 (d+e x)^{11/2}}{11 e^4}-\frac {6 b^2 (b d-a e) (d+e x)^{13/2}}{13 e^4}+\frac {2 b^3 (d+e x)^{15/2}}{15 e^4} \]
-2/9*(-a*e+b*d)^3*(e*x+d)^(9/2)/e^4+6/11*b*(-a*e+b*d)^2*(e*x+d)^(11/2)/e^4 -6/13*b^2*(-a*e+b*d)*(e*x+d)^(13/2)/e^4+2/15*b^3*(e*x+d)^(15/2)/e^4
Time = 0.08 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.02 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 (d+e x)^{9/2} \left (715 a^3 e^3+195 a^2 b e^2 (-2 d+9 e x)+15 a b^2 e \left (8 d^2-36 d e x+99 e^2 x^2\right )+b^3 \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )\right )}{6435 e^4} \]
(2*(d + e*x)^(9/2)*(715*a^3*e^3 + 195*a^2*b*e^2*(-2*d + 9*e*x) + 15*a*b^2* e*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + b^3*(-16*d^3 + 72*d^2*e*x - 198*d*e^2* x^2 + 429*e^3*x^3)))/(6435*e^4)
Time = 0.25 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.129, 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.
| \(\displaystyle \int (a+b x) \left (a^2+2 a b x+b^2 x^2\right ) (d+e x)^{7/2} \, dx\) |
| \(\Big \downarrow \) 1184 |
| \(\displaystyle \frac {\int b^2 (a+b x)^3 (d+e x)^{7/2}dx}{b^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \int (a+b x)^3 (d+e x)^{7/2}dx\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \int \left (-\frac {3 b^2 (d+e x)^{11/2} (b d-a e)}{e^3}+\frac {3 b (d+e x)^{9/2} (b d-a e)^2}{e^3}+\frac {(d+e x)^{7/2} (a e-b d)^3}{e^3}+\frac {b^3 (d+e x)^{13/2}}{e^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {6 b^2 (d+e x)^{13/2} (b d-a e)}{13 e^4}+\frac {6 b (d+e x)^{11/2} (b d-a e)^2}{11 e^4}-\frac {2 (d+e x)^{9/2} (b d-a e)^3}{9 e^4}+\frac {2 b^3 (d+e x)^{15/2}}{15 e^4}\) |
(-2*(b*d - a*e)^3*(d + e*x)^(9/2))/(9*e^4) + (6*b*(b*d - a*e)^2*(d + e*x)^ (11/2))/(11*e^4) - (6*b^2*(b*d - a*e)*(d + e*x)^(13/2))/(13*e^4) + (2*b^3* (d + e*x)^(15/2))/(15*e^4)
3.21.42.3.1 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[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 = 0.30 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.94
| method | result | size |
| pseudoelliptic | \(\frac {2 \left (\left (\frac {3}{5} e^{3} x^{3}-\frac {18}{65} d \,e^{2} x^{2}+\frac {72}{715} d^{2} e x -\frac {16}{715} d^{3}\right ) b^{3}+\frac {24 e \left (\frac {99}{8} e^{2} x^{2}-\frac {9}{2} d e x +d^{2}\right ) a \,b^{2}}{143}-\frac {6 e^{2} \left (-\frac {9 e x}{2}+d \right ) a^{2} b}{11}+a^{3} e^{3}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9 e^{4}}\) | \(94\) |
| gosper | \(\frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (429 b^{3} x^{3} e^{3}+1485 x^{2} a \,b^{2} e^{3}-198 x^{2} b^{3} d \,e^{2}+1755 x \,a^{2} b \,e^{3}-540 x a \,b^{2} d \,e^{2}+72 x \,b^{3} d^{2} e +715 a^{3} e^{3}-390 a^{2} b d \,e^{2}+120 a \,b^{2} d^{2} e -16 b^{3} d^{3}\right )}{6435 e^{4}}\) | \(116\) |
| derivativedivides | \(\frac {\frac {2 b^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) b^{2}+b \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\) | \(147\) |
| default | \(\frac {\frac {2 b^{3} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) b^{2}+b \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (\left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}}{e^{4}}\) | \(147\) |
| trager | \(\frac {2 \left (429 b^{3} e^{7} x^{7}+1485 a \,b^{2} e^{7} x^{6}+1518 b^{3} d \,e^{6} x^{6}+1755 a^{2} b \,e^{7} x^{5}+5400 a \,b^{2} d \,e^{6} x^{5}+1854 b^{3} d^{2} e^{5} x^{5}+715 a^{3} e^{7} x^{4}+6630 a^{2} b d \,e^{6} x^{4}+6870 a \,b^{2} d^{2} e^{5} x^{4}+800 b^{3} d^{3} e^{4} x^{4}+2860 a^{3} d \,e^{6} x^{3}+8970 a^{2} b \,d^{2} e^{5} x^{3}+3180 a \,b^{2} d^{3} e^{4} x^{3}+5 b^{3} d^{4} e^{3} x^{3}+4290 a^{3} d^{2} e^{5} x^{2}+4680 a^{2} b \,d^{3} e^{4} x^{2}+45 a \,b^{2} d^{4} e^{3} x^{2}-6 b^{3} d^{5} e^{2} x^{2}+2860 a^{3} d^{3} e^{4} x +195 a^{2} b \,d^{4} e^{3} x -60 a \,b^{2} d^{5} e^{2} x +8 b^{3} d^{6} e x +715 a^{3} d^{4} e^{3}-390 a^{2} b \,d^{5} e^{2}+120 a \,b^{2} d^{6} e -16 b^{3} d^{7}\right ) \sqrt {e x +d}}{6435 e^{4}}\) | \(344\) |
| risch | \(\frac {2 \left (429 b^{3} e^{7} x^{7}+1485 a \,b^{2} e^{7} x^{6}+1518 b^{3} d \,e^{6} x^{6}+1755 a^{2} b \,e^{7} x^{5}+5400 a \,b^{2} d \,e^{6} x^{5}+1854 b^{3} d^{2} e^{5} x^{5}+715 a^{3} e^{7} x^{4}+6630 a^{2} b d \,e^{6} x^{4}+6870 a \,b^{2} d^{2} e^{5} x^{4}+800 b^{3} d^{3} e^{4} x^{4}+2860 a^{3} d \,e^{6} x^{3}+8970 a^{2} b \,d^{2} e^{5} x^{3}+3180 a \,b^{2} d^{3} e^{4} x^{3}+5 b^{3} d^{4} e^{3} x^{3}+4290 a^{3} d^{2} e^{5} x^{2}+4680 a^{2} b \,d^{3} e^{4} x^{2}+45 a \,b^{2} d^{4} e^{3} x^{2}-6 b^{3} d^{5} e^{2} x^{2}+2860 a^{3} d^{3} e^{4} x +195 a^{2} b \,d^{4} e^{3} x -60 a \,b^{2} d^{5} e^{2} x +8 b^{3} d^{6} e x +715 a^{3} d^{4} e^{3}-390 a^{2} b \,d^{5} e^{2}+120 a \,b^{2} d^{6} e -16 b^{3} d^{7}\right ) \sqrt {e x +d}}{6435 e^{4}}\) | \(344\) |
2/9*((3/5*e^3*x^3-18/65*d*e^2*x^2+72/715*d^2*e*x-16/715*d^3)*b^3+24/143*e* (99/8*e^2*x^2-9/2*d*e*x+d^2)*a*b^2-6/11*e^2*(-9/2*e*x+d)*a^2*b+a^3*e^3)*(e *x+d)^(9/2)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 320 vs. \(2 (84) = 168\).
Time = 0.31 (sec) , antiderivative size = 320, normalized size of antiderivative = 3.20 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (429 \, b^{3} e^{7} x^{7} - 16 \, b^{3} d^{7} + 120 \, a b^{2} d^{6} e - 390 \, a^{2} b d^{5} e^{2} + 715 \, a^{3} d^{4} e^{3} + 33 \, {\left (46 \, b^{3} d e^{6} + 45 \, a b^{2} e^{7}\right )} x^{6} + 9 \, {\left (206 \, b^{3} d^{2} e^{5} + 600 \, a b^{2} d e^{6} + 195 \, a^{2} b e^{7}\right )} x^{5} + 5 \, {\left (160 \, b^{3} d^{3} e^{4} + 1374 \, a b^{2} d^{2} e^{5} + 1326 \, a^{2} b d e^{6} + 143 \, a^{3} e^{7}\right )} x^{4} + 5 \, {\left (b^{3} d^{4} e^{3} + 636 \, a b^{2} d^{3} e^{4} + 1794 \, a^{2} b d^{2} e^{5} + 572 \, a^{3} d e^{6}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{5} e^{2} - 15 \, a b^{2} d^{4} e^{3} - 1560 \, a^{2} b d^{3} e^{4} - 1430 \, a^{3} d^{2} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{6} e - 60 \, a b^{2} d^{5} e^{2} + 195 \, a^{2} b d^{4} e^{3} + 2860 \, a^{3} d^{3} e^{4}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{4}} \]
2/6435*(429*b^3*e^7*x^7 - 16*b^3*d^7 + 120*a*b^2*d^6*e - 390*a^2*b*d^5*e^2 + 715*a^3*d^4*e^3 + 33*(46*b^3*d*e^6 + 45*a*b^2*e^7)*x^6 + 9*(206*b^3*d^2 *e^5 + 600*a*b^2*d*e^6 + 195*a^2*b*e^7)*x^5 + 5*(160*b^3*d^3*e^4 + 1374*a* b^2*d^2*e^5 + 1326*a^2*b*d*e^6 + 143*a^3*e^7)*x^4 + 5*(b^3*d^4*e^3 + 636*a *b^2*d^3*e^4 + 1794*a^2*b*d^2*e^5 + 572*a^3*d*e^6)*x^3 - 3*(2*b^3*d^5*e^2 - 15*a*b^2*d^4*e^3 - 1560*a^2*b*d^3*e^4 - 1430*a^3*d^2*e^5)*x^2 + (8*b^3*d ^6*e - 60*a*b^2*d^5*e^2 + 195*a^2*b*d^4*e^3 + 2860*a^3*d^3*e^4)*x)*sqrt(e* x + d)/e^4
Leaf count of result is larger than twice the leaf count of optimal. 654 vs. \(2 (92) = 184\).
Time = 0.58 (sec) , antiderivative size = 654, normalized size of antiderivative = 6.54 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\begin {cases} \frac {2 a^{3} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 a^{3} d^{3} x \sqrt {d + e x}}{9} + \frac {4 a^{3} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 a^{3} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 a^{3} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {4 a^{2} b d^{5} \sqrt {d + e x}}{33 e^{2}} + \frac {2 a^{2} b d^{4} x \sqrt {d + e x}}{33 e} + \frac {16 a^{2} b d^{3} x^{2} \sqrt {d + e x}}{11} + \frac {92 a^{2} b d^{2} e x^{3} \sqrt {d + e x}}{33} + \frac {68 a^{2} b d e^{2} x^{4} \sqrt {d + e x}}{33} + \frac {6 a^{2} b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 a b^{2} d^{6} \sqrt {d + e x}}{429 e^{3}} - \frac {8 a b^{2} d^{5} x \sqrt {d + e x}}{429 e^{2}} + \frac {2 a b^{2} d^{4} x^{2} \sqrt {d + e x}}{143 e} + \frac {424 a b^{2} d^{3} x^{3} \sqrt {d + e x}}{429} + \frac {916 a b^{2} d^{2} e x^{4} \sqrt {d + e x}}{429} + \frac {240 a b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {6 a b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {32 b^{3} d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {16 b^{3} d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {4 b^{3} d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {2 b^{3} d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {320 b^{3} d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {412 b^{3} d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 b^{3} d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {2 b^{3} e^{3} x^{7} \sqrt {d + e x}}{15} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (a^{3} x + \frac {3 a^{2} b x^{2}}{2} + a b^{2} x^{3} + \frac {b^{3} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]
Piecewise((2*a**3*d**4*sqrt(d + e*x)/(9*e) + 8*a**3*d**3*x*sqrt(d + e*x)/9 + 4*a**3*d**2*e*x**2*sqrt(d + e*x)/3 + 8*a**3*d*e**2*x**3*sqrt(d + e*x)/9 + 2*a**3*e**3*x**4*sqrt(d + e*x)/9 - 4*a**2*b*d**5*sqrt(d + e*x)/(33*e**2 ) + 2*a**2*b*d**4*x*sqrt(d + e*x)/(33*e) + 16*a**2*b*d**3*x**2*sqrt(d + e* x)/11 + 92*a**2*b*d**2*e*x**3*sqrt(d + e*x)/33 + 68*a**2*b*d*e**2*x**4*sqr t(d + e*x)/33 + 6*a**2*b*e**3*x**5*sqrt(d + e*x)/11 + 16*a*b**2*d**6*sqrt( d + e*x)/(429*e**3) - 8*a*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 2*a*b**2* d**4*x**2*sqrt(d + e*x)/(143*e) + 424*a*b**2*d**3*x**3*sqrt(d + e*x)/429 + 916*a*b**2*d**2*e*x**4*sqrt(d + e*x)/429 + 240*a*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 6*a*b**2*e**3*x**6*sqrt(d + e*x)/13 - 32*b**3*d**7*sqrt(d + e *x)/(6435*e**4) + 16*b**3*d**6*x*sqrt(d + e*x)/(6435*e**3) - 4*b**3*d**5*x **2*sqrt(d + e*x)/(2145*e**2) + 2*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 320*b**3*d**3*x**4*sqrt(d + e*x)/1287 + 412*b**3*d**2*e*x**5*sqrt(d + e*x) /715 + 92*b**3*d*e**2*x**6*sqrt(d + e*x)/195 + 2*b**3*e**3*x**7*sqrt(d + e *x)/15, Ne(e, 0)), (d**(7/2)*(a**3*x + 3*a**2*b*x**2/2 + a*b**2*x**3 + b** 3*x**4/4), True))
Time = 0.21 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.18 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{3} - 1485 \, {\left (b^{3} d - a b^{2} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 1755 \, {\left (b^{3} d^{2} - 2 \, a b^{2} d e + a^{2} b e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 715 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{6435 \, e^{4}} \]
2/6435*(429*(e*x + d)^(15/2)*b^3 - 1485*(b^3*d - a*b^2*e)*(e*x + d)^(13/2) + 1755*(b^3*d^2 - 2*a*b^2*d*e + a^2*b*e^2)*(e*x + d)^(11/2) - 715*(b^3*d^ 3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*(e*x + d)^(9/2))/e^4
Leaf count of result is larger than twice the leaf count of optimal. 1196 vs. \(2 (84) = 168\).
Time = 0.28 (sec) , antiderivative size = 1196, normalized size of antiderivative = 11.96 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\text {Too large to display} \]
2/45045*(45045*sqrt(e*x + d)*a^3*d^4 + 60060*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^3*d^3 + 45045*((e*x + d)^(3/2) - 3*sqrt(e*x + d)*d)*a^2*b*d^4/e + 18018*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2) *a^3*d^2 + 9009*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a*b^2*d^4/e^2 + 36036*(3*(e*x + d)^(5/2) - 10*(e*x + d)^(3/2)*d + 15*sqrt(e*x + d)*d^2)*a^2*b*d^3/e + 5148*(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*d + 1287*(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)*b^3*d^4/e^3 + 15444*(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*b^2*d^3/e^2 + 23166*( 5*(e*x + d)^(7/2) - 21*(e*x + d)^(5/2)*d + 35*(e*x + d)^(3/2)*d^2 - 35*sqr t(e*x + d)*d^3)*a^2*b*d^2/e + 143*(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 + 572*(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)*b^3*d^3/e^ 3 + 2574*(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^2*d^2/e^2 + 17 16*(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*d/e + 390*(63*(e*x + d)^(11/2) - 385*(e*x + d)^(9/2)*d + 990*(e*x + d)^(7/2)*d^2 - 1386*(...
Time = 0.08 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.87 \[ \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right ) \, dx=\frac {2\,b^3\,{\left (d+e\,x\right )}^{15/2}}{15\,e^4}-\frac {\left (6\,b^3\,d-6\,a\,b^2\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^4}+\frac {6\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^4} \]