3.19.33 \(\int (a+b x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=216 \[ -\frac {2 b^6 (d+e x)^{21/2} (b d-a e)}{3 e^8}+\frac {42 b^5 (d+e x)^{19/2} (b d-a e)^2}{19 e^8}-\frac {70 b^4 (d+e x)^{17/2} (b d-a e)^3}{17 e^8}+\frac {14 b^3 (d+e x)^{15/2} (b d-a e)^4}{3 e^8}-\frac {42 b^2 (d+e x)^{13/2} (b d-a e)^5}{13 e^8}+\frac {14 b (d+e x)^{11/2} (b d-a e)^6}{11 e^8}-\frac {2 (d+e x)^{9/2} (b d-a e)^7}{9 e^8}+\frac {2 b^7 (d+e x)^{23/2}}{23 e^8} \]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 216, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used =
{27, 43} \begin {gather*} -\frac {2 b^6 (d+e x)^{21/2} (b d-a e)}{3 e^8}+\frac {42 b^5 (d+e x)^{19/2} (b d-a e)^2}{19 e^8}-\frac {70 b^4 (d+e x)^{17/2} (b d-a e)^3}{17 e^8}+\frac {14 b^3 (d+e x)^{15/2} (b d-a e)^4}{3 e^8}-\frac {42 b^2 (d+e x)^{13/2} (b d-a e)^5}{13 e^8}+\frac {14 b (d+e x)^{11/2} (b d-a e)^6}{11 e^8}-\frac {2 (d+e x)^{9/2} (b d-a e)^7}{9 e^8}+\frac {2 b^7 (d+e x)^{23/2}}{23 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(-2*(b*d - a*e)^7*(d + e*x)^(9/2))/(9*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(11/2))/(11*e^8) - (42*b^2*(b*d - a
*e)^5*(d + e*x)^(13/2))/(13*e^8) + (14*b^3*(b*d - a*e)^4*(d + e*x)^(15/2))/(3*e^8) - (70*b^4*(b*d - a*e)^3*(d
+ e*x)^(17/2))/(17*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(19/2))/(19*e^8) - (2*b^6*(b*d - a*e)*(d + e*x)^(21/
2))/(3*e^8) + (2*b^7*(d + e*x)^(23/2))/(23*e^8)
Rule 27
Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]
Rule 43
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] && NeQ[b*c - a*d, 0] && 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])
Rubi steps
\begin {align*} \int (a+b x) (d+e x)^{7/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^{7/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^7 (d+e x)^{7/2}}{e^7}+\frac {7 b (b d-a e)^6 (d+e x)^{9/2}}{e^7}-\frac {21 b^2 (b d-a e)^5 (d+e x)^{11/2}}{e^7}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{13/2}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{15/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{17/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{19/2}}{e^7}+\frac {b^7 (d+e x)^{21/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (b d-a e)^7 (d+e x)^{9/2}}{9 e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{11/2}}{11 e^8}-\frac {42 b^2 (b d-a e)^5 (d+e x)^{13/2}}{13 e^8}+\frac {14 b^3 (b d-a e)^4 (d+e x)^{15/2}}{3 e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{17/2}}{17 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{19/2}}{19 e^8}-\frac {2 b^6 (b d-a e) (d+e x)^{21/2}}{3 e^8}+\frac {2 b^7 (d+e x)^{23/2}}{23 e^8}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 167, normalized size = 0.77 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (-3187041 b^6 (d+e x)^6 (b d-a e)+10567557 b^5 (d+e x)^5 (b d-a e)^2-19684665 b^4 (d+e x)^4 (b d-a e)^3+22309287 b^3 (d+e x)^3 (b d-a e)^4-15444891 b^2 (d+e x)^2 (b d-a e)^5+6084351 b (d+e x) (b d-a e)^6-1062347 (b d-a e)^7+415701 b^7 (d+e x)^7\right )}{9561123 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(9/2)*(-1062347*(b*d - a*e)^7 + 6084351*b*(b*d - a*e)^6*(d + e*x) - 15444891*b^2*(b*d - a*e)^5*(d
+ e*x)^2 + 22309287*b^3*(b*d - a*e)^4*(d + e*x)^3 - 19684665*b^4*(b*d - a*e)^3*(d + e*x)^4 + 10567557*b^5*(b*
d - a*e)^2*(d + e*x)^5 - 3187041*b^6*(b*d - a*e)*(d + e*x)^6 + 415701*b^7*(d + e*x)^7))/(9561123*e^8)
________________________________________________________________________________________
IntegrateAlgebraic [B] time = 0.21, size = 582, normalized size = 2.69 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (1062347 a^7 e^7+6084351 a^6 b e^6 (d+e x)-7436429 a^6 b d e^6+22309287 a^5 b^2 d^2 e^5+15444891 a^5 b^2 e^5 (d+e x)^2-36506106 a^5 b^2 d e^5 (d+e x)-37182145 a^4 b^3 d^3 e^4+91265265 a^4 b^3 d^2 e^4 (d+e x)+22309287 a^4 b^3 e^4 (d+e x)^3-77224455 a^4 b^3 d e^4 (d+e x)^2+37182145 a^3 b^4 d^4 e^3-121687020 a^3 b^4 d^3 e^3 (d+e x)+154448910 a^3 b^4 d^2 e^3 (d+e x)^2+19684665 a^3 b^4 e^3 (d+e x)^4-89237148 a^3 b^4 d e^3 (d+e x)^3-22309287 a^2 b^5 d^5 e^2+91265265 a^2 b^5 d^4 e^2 (d+e x)-154448910 a^2 b^5 d^3 e^2 (d+e x)^2+133855722 a^2 b^5 d^2 e^2 (d+e x)^3+10567557 a^2 b^5 e^2 (d+e x)^5-59053995 a^2 b^5 d e^2 (d+e x)^4+7436429 a b^6 d^6 e-36506106 a b^6 d^5 e (d+e x)+77224455 a b^6 d^4 e (d+e x)^2-89237148 a b^6 d^3 e (d+e x)^3+59053995 a b^6 d^2 e (d+e x)^4+3187041 a b^6 e (d+e x)^6-21135114 a b^6 d e (d+e x)^5-1062347 b^7 d^7+6084351 b^7 d^6 (d+e x)-15444891 b^7 d^5 (d+e x)^2+22309287 b^7 d^4 (d+e x)^3-19684665 b^7 d^3 (d+e x)^4+10567557 b^7 d^2 (d+e x)^5+415701 b^7 (d+e x)^7-3187041 b^7 d (d+e x)^6\right )}{9561123 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(a + b*x)*(d + e*x)^(7/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(9/2)*(-1062347*b^7*d^7 + 7436429*a*b^6*d^6*e - 22309287*a^2*b^5*d^5*e^2 + 37182145*a^3*b^4*d^4*e
^3 - 37182145*a^4*b^3*d^3*e^4 + 22309287*a^5*b^2*d^2*e^5 - 7436429*a^6*b*d*e^6 + 1062347*a^7*e^7 + 6084351*b^7
*d^6*(d + e*x) - 36506106*a*b^6*d^5*e*(d + e*x) + 91265265*a^2*b^5*d^4*e^2*(d + e*x) - 121687020*a^3*b^4*d^3*e
^3*(d + e*x) + 91265265*a^4*b^3*d^2*e^4*(d + e*x) - 36506106*a^5*b^2*d*e^5*(d + e*x) + 6084351*a^6*b*e^6*(d +
e*x) - 15444891*b^7*d^5*(d + e*x)^2 + 77224455*a*b^6*d^4*e*(d + e*x)^2 - 154448910*a^2*b^5*d^3*e^2*(d + e*x)^2
+ 154448910*a^3*b^4*d^2*e^3*(d + e*x)^2 - 77224455*a^4*b^3*d*e^4*(d + e*x)^2 + 15444891*a^5*b^2*e^5*(d + e*x)
^2 + 22309287*b^7*d^4*(d + e*x)^3 - 89237148*a*b^6*d^3*e*(d + e*x)^3 + 133855722*a^2*b^5*d^2*e^2*(d + e*x)^3 -
89237148*a^3*b^4*d*e^3*(d + e*x)^3 + 22309287*a^4*b^3*e^4*(d + e*x)^3 - 19684665*b^7*d^3*(d + e*x)^4 + 590539
95*a*b^6*d^2*e*(d + e*x)^4 - 59053995*a^2*b^5*d*e^2*(d + e*x)^4 + 19684665*a^3*b^4*e^3*(d + e*x)^4 + 10567557*
b^7*d^2*(d + e*x)^5 - 21135114*a*b^6*d*e*(d + e*x)^5 + 10567557*a^2*b^5*e^2*(d + e*x)^5 - 3187041*b^7*d*(d + e
*x)^6 + 3187041*a*b^6*e*(d + e*x)^6 + 415701*b^7*(d + e*x)^7))/(9561123*e^8)
________________________________________________________________________________________
fricas [B] time = 0.45, size = 891, normalized size = 4.12 \begin {gather*} \frac {2 \, {\left (415701 \, b^{7} e^{11} x^{11} - 2048 \, b^{7} d^{11} + 23552 \, a b^{6} d^{10} e - 123648 \, a^{2} b^{5} d^{9} e^{2} + 391552 \, a^{3} b^{4} d^{8} e^{3} - 832048 \, a^{4} b^{3} d^{7} e^{4} + 1248072 \, a^{5} b^{2} d^{6} e^{5} - 1352078 \, a^{6} b d^{5} e^{6} + 1062347 \, a^{7} d^{4} e^{7} + 138567 \, {\left (10 \, b^{7} d e^{10} + 23 \, a b^{6} e^{11}\right )} x^{10} + 7293 \, {\left (214 \, b^{7} d^{2} e^{9} + 1472 \, a b^{6} d e^{10} + 1449 \, a^{2} b^{5} e^{11}\right )} x^{9} + 1287 \, {\left (464 \, b^{7} d^{3} e^{8} + 9522 \, a b^{6} d^{2} e^{9} + 28014 \, a^{2} b^{5} d e^{10} + 15295 \, a^{3} b^{4} e^{11}\right )} x^{8} + 429 \, {\left (b^{7} d^{4} e^{7} + 11132 \, a b^{6} d^{3} e^{8} + 97566 \, a^{2} b^{5} d^{2} e^{9} + 159068 \, a^{3} b^{4} d e^{10} + 52003 \, a^{4} b^{3} e^{11}\right )} x^{7} - 231 \, {\left (2 \, b^{7} d^{5} e^{6} - 23 \, a b^{6} d^{4} e^{7} - 72312 \, a^{2} b^{5} d^{3} e^{8} - 350474 \, a^{3} b^{4} d^{2} e^{9} - 341734 \, a^{4} b^{3} d e^{10} - 66861 \, a^{5} b^{2} e^{11}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{6} e^{5} - 92 \, a b^{6} d^{5} e^{6} + 483 \, a^{2} b^{5} d^{4} e^{7} + 529644 \, a^{3} b^{4} d^{3} e^{8} + 1530374 \, a^{4} b^{3} d^{2} e^{9} + 891480 \, a^{5} b^{2} d e^{10} + 96577 \, a^{6} b e^{11}\right )} x^{5} - {\left (560 \, b^{7} d^{7} e^{4} - 6440 \, a b^{6} d^{6} e^{5} + 33810 \, a^{2} b^{5} d^{5} e^{6} - 107065 \, a^{3} b^{4} d^{4} e^{7} - 41602400 \, a^{4} b^{3} d^{3} e^{8} - 71452122 \, a^{5} b^{2} d^{2} e^{9} - 22985326 \, a^{6} b d e^{10} - 1062347 \, a^{7} e^{11}\right )} x^{4} + {\left (640 \, b^{7} d^{8} e^{3} - 7360 \, a b^{6} d^{7} e^{4} + 38640 \, a^{2} b^{5} d^{6} e^{5} - 122360 \, a^{3} b^{4} d^{5} e^{6} + 260015 \, a^{4} b^{3} d^{4} e^{7} + 33073908 \, a^{5} b^{2} d^{3} e^{8} + 31097794 \, a^{6} b d^{2} e^{9} + 4249388 \, a^{7} d e^{10}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{9} e^{2} - 2944 \, a b^{6} d^{8} e^{3} + 15456 \, a^{2} b^{5} d^{7} e^{4} - 48944 \, a^{3} b^{4} d^{6} e^{5} + 104006 \, a^{4} b^{3} d^{5} e^{6} - 156009 \, a^{5} b^{2} d^{4} e^{7} - 5408312 \, a^{6} b d^{3} e^{8} - 2124694 \, a^{7} d^{2} e^{9}\right )} x^{2} + {\left (1024 \, b^{7} d^{10} e - 11776 \, a b^{6} d^{9} e^{2} + 61824 \, a^{2} b^{5} d^{8} e^{3} - 195776 \, a^{3} b^{4} d^{7} e^{4} + 416024 \, a^{4} b^{3} d^{6} e^{5} - 624036 \, a^{5} b^{2} d^{5} e^{6} + 676039 \, a^{6} b d^{4} e^{7} + 4249388 \, a^{7} d^{3} e^{8}\right )} x\right )} \sqrt {e x + d}}{9561123 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
[Out]
2/9561123*(415701*b^7*e^11*x^11 - 2048*b^7*d^11 + 23552*a*b^6*d^10*e - 123648*a^2*b^5*d^9*e^2 + 391552*a^3*b^4
*d^8*e^3 - 832048*a^4*b^3*d^7*e^4 + 1248072*a^5*b^2*d^6*e^5 - 1352078*a^6*b*d^5*e^6 + 1062347*a^7*d^4*e^7 + 13
8567*(10*b^7*d*e^10 + 23*a*b^6*e^11)*x^10 + 7293*(214*b^7*d^2*e^9 + 1472*a*b^6*d*e^10 + 1449*a^2*b^5*e^11)*x^9
+ 1287*(464*b^7*d^3*e^8 + 9522*a*b^6*d^2*e^9 + 28014*a^2*b^5*d*e^10 + 15295*a^3*b^4*e^11)*x^8 + 429*(b^7*d^4*
e^7 + 11132*a*b^6*d^3*e^8 + 97566*a^2*b^5*d^2*e^9 + 159068*a^3*b^4*d*e^10 + 52003*a^4*b^3*e^11)*x^7 - 231*(2*b
^7*d^5*e^6 - 23*a*b^6*d^4*e^7 - 72312*a^2*b^5*d^3*e^8 - 350474*a^3*b^4*d^2*e^9 - 341734*a^4*b^3*d*e^10 - 66861
*a^5*b^2*e^11)*x^6 + 63*(8*b^7*d^6*e^5 - 92*a*b^6*d^5*e^6 + 483*a^2*b^5*d^4*e^7 + 529644*a^3*b^4*d^3*e^8 + 153
0374*a^4*b^3*d^2*e^9 + 891480*a^5*b^2*d*e^10 + 96577*a^6*b*e^11)*x^5 - (560*b^7*d^7*e^4 - 6440*a*b^6*d^6*e^5 +
33810*a^2*b^5*d^5*e^6 - 107065*a^3*b^4*d^4*e^7 - 41602400*a^4*b^3*d^3*e^8 - 71452122*a^5*b^2*d^2*e^9 - 229853
26*a^6*b*d*e^10 - 1062347*a^7*e^11)*x^4 + (640*b^7*d^8*e^3 - 7360*a*b^6*d^7*e^4 + 38640*a^2*b^5*d^6*e^5 - 1223
60*a^3*b^4*d^5*e^6 + 260015*a^4*b^3*d^4*e^7 + 33073908*a^5*b^2*d^3*e^8 + 31097794*a^6*b*d^2*e^9 + 4249388*a^7*
d*e^10)*x^3 - 3*(256*b^7*d^9*e^2 - 2944*a*b^6*d^8*e^3 + 15456*a^2*b^5*d^7*e^4 - 48944*a^3*b^4*d^6*e^5 + 104006
*a^4*b^3*d^5*e^6 - 156009*a^5*b^2*d^4*e^7 - 5408312*a^6*b*d^3*e^8 - 2124694*a^7*d^2*e^9)*x^2 + (1024*b^7*d^10*
e - 11776*a*b^6*d^9*e^2 + 61824*a^2*b^5*d^8*e^3 - 195776*a^3*b^4*d^7*e^4 + 416024*a^4*b^3*d^6*e^5 - 624036*a^5
*b^2*d^5*e^6 + 676039*a^6*b*d^4*e^7 + 4249388*a^7*d^3*e^8)*x)*sqrt(e*x + d)/e^8
________________________________________________________________________________________
giac [B] time = 0.41, size = 3640, normalized size = 16.85
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
[Out]
2/334639305*(780825045*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^6*b*d^4*e^(-1) + 468495027*(3*(x*e + d)^(5/2) -
10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^5*b^2*d^4*e^(-2) + 334639305*(5*(x*e + d)^(7/2) - 21*(x*e + d)
^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^4*b^3*d^4*e^(-3) + 37182145*(35*(x*e + d)^(9/2) -
180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^3*b^4*d^4
*e^(-4) + 10140585*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/
2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a^2*b^5*d^4*e^(-5) + 780045*(231*(x*e + d)^(13/2) -
1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 60
06*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a*b^6*d^4*e^(-6) + 52003*(429*(x*e + d)^(15/2) - 3465*(x*e +
d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e
+ d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*b^7*d^4*e^(-7) + 624660036*(3*(x*e + d)^(
5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^6*b*d^3*e^(-1) + 803134332*(5*(x*e + d)^(7/2) - 21*(x*e
+ d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*a^5*b^2*d^3*e^(-2) + 148728580*(35*(x*e + d)^(9/
2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^4*b^
3*d^3*e^(-3) + 67603900*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d
)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a^3*b^4*d^3*e^(-4) + 9360540*(231*(x*e + d)^(1
3/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^
4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a^2*b^5*d^3*e^(-5) + 1456084*(429*(x*e + d)^(15/2) - 34
65*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 2
7027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*a*b^6*d^3*e^(-6) + 12236*(6435*
(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 8508
50*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7
+ 109395*sqrt(x*e + d)*d^8)*b^7*d^3*e^(-7) + 334639305*sqrt(x*e + d)*a^7*d^4 + 446185740*((x*e + d)^(3/2) - 3*
sqrt(x*e + d)*d)*a^7*d^3 + 401567166*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*s
qrt(x*e + d)*d^3)*a^6*b*d^2*e^(-1) + 133855722*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/
2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^5*b^2*d^2*e^(-2) + 101405850*(63*(x*e + d)^(11/2)
- 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*
sqrt(x*e + d)*d^5)*a^4*b^3*d^2*e^(-3) + 23401350*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e +
d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e
+ d)*d^6)*a^3*b^4*d^2*e^(-4) + 6552378*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/
2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(
3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*a^2*b^5*d^2*e^(-5) + 128478*(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)
*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)
^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*a*b^6*d^2*e^(
-6) + 8694*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*d + 554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e +
d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 166
2804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230945*sqrt(x*e + d)*d^9)*b^7*d^2*e^(-7) + 133855722*(
3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^7*d^2 + 29745716*(35*(x*e + d)^(9/2) - 180*
(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*a^6*b*d*e^(-1)
+ 40562340*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 +
1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a^5*b^2*d*e^(-2) + 15600900*(231*(x*e + d)^(13/2) - 1638*(x
*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e
+ d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a^4*b^3*d*e^(-3) + 7280420*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13
/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(
5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*a^3*b^4*d*e^(-4) + 256956*(6435*(x*e + d)^(17/2
) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3 + 850850*(x*e + d)^(9
/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x
*e + d)*d^8)*a^2*b^5*d*e^(-5) + 40572*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*d + 554268*(x*e + d)^(
15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e + d)^(9/2)*d^5 + 277134
0*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230945*sqrt(x*e + d)*d^9)*a
*b^6*d*e^(-6) + 1380*(46189*(x*e + d)^(21/2) - 510510*(x*e + d)^(19/2)*d + 2567565*(x*e + d)^(17/2)*d^2 - 7759
752*(x*e + d)^(15/2)*d^3 + 15668730*(x*e + d)^(13/2)*d^4 - 22221108*(x*e + d)^(11/2)*d^5 + 22632610*(x*e + d)^
(9/2)*d^6 - 16628040*(x*e + d)^(7/2)*d^7 + 8729721*(x*e + d)^(5/2)*d^8 - 3233230*(x*e + d)^(3/2)*d^9 + 969969*
sqrt(x*e + d)*d^10)*b^7*d*e^(-7) + 38244492*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2
- 35*sqrt(x*e + d)*d^3)*a^7*d + 3380195*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^
2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*a^6*b*e^(-1) + 2340135*(231*(
x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2 - 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e +
d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*a^5*b^2*e^(-2) + 1820105*(429*(x*e + d)^(15/
2) - 3465*(x*e + d)^(13/2)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*
d^4 - 27027*(x*e + d)^(5/2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*a^4*b^3*e^(-3) + 107065*
(6435*(x*e + d)^(17/2) - 58344*(x*e + d)^(15/2)*d + 235620*(x*e + d)^(13/2)*d^2 - 556920*(x*e + d)^(11/2)*d^3
+ 850850*(x*e + d)^(9/2)*d^4 - 875160*(x*e + d)^(7/2)*d^5 + 612612*(x*e + d)^(5/2)*d^6 - 291720*(x*e + d)^(3/2
)*d^7 + 109395*sqrt(x*e + d)*d^8)*a^3*b^4*e^(-4) + 30429*(12155*(x*e + d)^(19/2) - 122265*(x*e + d)^(17/2)*d +
554268*(x*e + d)^(15/2)*d^2 - 1492260*(x*e + d)^(13/2)*d^3 + 2645370*(x*e + d)^(11/2)*d^4 - 3233230*(x*e + d)
^(9/2)*d^5 + 2771340*(x*e + d)^(7/2)*d^6 - 1662804*(x*e + d)^(5/2)*d^7 + 692835*(x*e + d)^(3/2)*d^8 - 230945*s
qrt(x*e + d)*d^9)*a^2*b^5*e^(-5) + 2415*(46189*(x*e + d)^(21/2) - 510510*(x*e + d)^(19/2)*d + 2567565*(x*e + d
)^(17/2)*d^2 - 7759752*(x*e + d)^(15/2)*d^3 + 15668730*(x*e + d)^(13/2)*d^4 - 22221108*(x*e + d)^(11/2)*d^5 +
22632610*(x*e + d)^(9/2)*d^6 - 16628040*(x*e + d)^(7/2)*d^7 + 8729721*(x*e + d)^(5/2)*d^8 - 3233230*(x*e + d)^
(3/2)*d^9 + 969969*sqrt(x*e + d)*d^10)*a*b^6*e^(-6) + 165*(88179*(x*e + d)^(23/2) - 1062347*(x*e + d)^(21/2)*d
+ 5870865*(x*e + d)^(19/2)*d^2 - 19684665*(x*e + d)^(17/2)*d^3 + 44618574*(x*e + d)^(15/2)*d^4 - 72076158*(x*
e + d)^(13/2)*d^5 + 85180914*(x*e + d)^(11/2)*d^6 - 74364290*(x*e + d)^(9/2)*d^7 + 47805615*(x*e + d)^(7/2)*d^
8 - 22309287*(x*e + d)^(5/2)*d^9 + 7436429*(x*e + d)^(3/2)*d^10 - 2028117*sqrt(x*e + d)*d^11)*b^7*e^(-7) + 106
2347*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqr
t(x*e + d)*d^4)*a^7)*e^(-1)
________________________________________________________________________________________
maple [B] time = 0.05, size = 498, normalized size = 2.31 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (415701 b^{7} x^{7} e^{7}+3187041 a \,b^{6} e^{7} x^{6}-277134 b^{7} d \,e^{6} x^{6}+10567557 a^{2} b^{5} e^{7} x^{5}-2012868 a \,b^{6} d \,e^{6} x^{5}+175032 b^{7} d^{2} e^{5} x^{5}+19684665 a^{3} b^{4} e^{7} x^{4}-6216210 a^{2} b^{5} d \,e^{6} x^{4}+1184040 a \,b^{6} d^{2} e^{5} x^{4}-102960 b^{7} d^{3} e^{4} x^{4}+22309287 a^{4} b^{3} e^{7} x^{3}-10498488 a^{3} b^{4} d \,e^{6} x^{3}+3315312 a^{2} b^{5} d^{2} e^{5} x^{3}-631488 a \,b^{6} d^{3} e^{4} x^{3}+54912 b^{7} d^{4} e^{3} x^{3}+15444891 a^{5} b^{2} e^{7} x^{2}-10296594 a^{4} b^{3} d \,e^{6} x^{2}+4845456 a^{3} b^{4} d^{2} e^{5} x^{2}-1530144 a^{2} b^{5} d^{3} e^{4} x^{2}+291456 a \,b^{6} d^{4} e^{3} x^{2}-25344 b^{7} d^{5} e^{2} x^{2}+6084351 a^{6} b \,e^{7} x -5616324 a^{5} b^{2} d \,e^{6} x +3744216 a^{4} b^{3} d^{2} e^{5} x -1761984 a^{3} b^{4} d^{3} e^{4} x +556416 a^{2} b^{5} d^{4} e^{3} x -105984 a \,b^{6} d^{5} e^{2} x +9216 b^{7} d^{6} e x +1062347 a^{7} e^{7}-1352078 a^{6} b d \,e^{6}+1248072 a^{5} b^{2} d^{2} e^{5}-832048 a^{4} b^{3} d^{3} e^{4}+391552 a^{3} b^{4} d^{4} e^{3}-123648 a^{2} b^{5} d^{5} e^{2}+23552 a \,b^{6} d^{6} e -2048 b^{7} d^{7}\right )}{9561123 e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
2/9561123*(e*x+d)^(9/2)*(415701*b^7*e^7*x^7+3187041*a*b^6*e^7*x^6-277134*b^7*d*e^6*x^6+10567557*a^2*b^5*e^7*x^
5-2012868*a*b^6*d*e^6*x^5+175032*b^7*d^2*e^5*x^5+19684665*a^3*b^4*e^7*x^4-6216210*a^2*b^5*d*e^6*x^4+1184040*a*
b^6*d^2*e^5*x^4-102960*b^7*d^3*e^4*x^4+22309287*a^4*b^3*e^7*x^3-10498488*a^3*b^4*d*e^6*x^3+3315312*a^2*b^5*d^2
*e^5*x^3-631488*a*b^6*d^3*e^4*x^3+54912*b^7*d^4*e^3*x^3+15444891*a^5*b^2*e^7*x^2-10296594*a^4*b^3*d*e^6*x^2+48
45456*a^3*b^4*d^2*e^5*x^2-1530144*a^2*b^5*d^3*e^4*x^2+291456*a*b^6*d^4*e^3*x^2-25344*b^7*d^5*e^2*x^2+6084351*a
^6*b*e^7*x-5616324*a^5*b^2*d*e^6*x+3744216*a^4*b^3*d^2*e^5*x-1761984*a^3*b^4*d^3*e^4*x+556416*a^2*b^5*d^4*e^3*
x-105984*a*b^6*d^5*e^2*x+9216*b^7*d^6*e*x+1062347*a^7*e^7-1352078*a^6*b*d*e^6+1248072*a^5*b^2*d^2*e^5-832048*a
^4*b^3*d^3*e^4+391552*a^3*b^4*d^4*e^3-123648*a^2*b^5*d^5*e^2+23552*a*b^6*d^6*e-2048*b^7*d^7)/e^8
________________________________________________________________________________________
maxima [B] time = 0.58, size = 456, normalized size = 2.11 \begin {gather*} \frac {2 \, {\left (415701 \, {\left (e x + d\right )}^{\frac {23}{2}} b^{7} - 3187041 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {21}{2}} + 10567557 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {19}{2}} - 19684665 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 22309287 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 15444891 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 6084351 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 1062347 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{9561123 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(7/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
[Out]
2/9561123*(415701*(e*x + d)^(23/2)*b^7 - 3187041*(b^7*d - a*b^6*e)*(e*x + d)^(21/2) + 10567557*(b^7*d^2 - 2*a*
b^6*d*e + a^2*b^5*e^2)*(e*x + d)^(19/2) - 19684665*(b^7*d^3 - 3*a*b^6*d^2*e + 3*a^2*b^5*d*e^2 - a^3*b^4*e^3)*(
e*x + d)^(17/2) + 22309287*(b^7*d^4 - 4*a*b^6*d^3*e + 6*a^2*b^5*d^2*e^2 - 4*a^3*b^4*d*e^3 + a^4*b^3*e^4)*(e*x
+ d)^(15/2) - 15444891*(b^7*d^5 - 5*a*b^6*d^4*e + 10*a^2*b^5*d^3*e^2 - 10*a^3*b^4*d^2*e^3 + 5*a^4*b^3*d*e^4 -
a^5*b^2*e^5)*(e*x + d)^(13/2) + 6084351*(b^7*d^6 - 6*a*b^6*d^5*e + 15*a^2*b^5*d^4*e^2 - 20*a^3*b^4*d^3*e^3 + 1
5*a^4*b^3*d^2*e^4 - 6*a^5*b^2*d*e^5 + a^6*b*e^6)*(e*x + d)^(11/2) - 1062347*(b^7*d^7 - 7*a*b^6*d^6*e + 21*a^2*
b^5*d^5*e^2 - 35*a^3*b^4*d^4*e^3 + 35*a^4*b^3*d^3*e^4 - 21*a^5*b^2*d^2*e^5 + 7*a^6*b*d*e^6 - a^7*e^7)*(e*x + d
)^(9/2))/e^8
________________________________________________________________________________________
mupad [B] time = 2.08, size = 187, normalized size = 0.87 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{23/2}}{23\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{21/2}}{21\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {14\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{15/2}}{3\,e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)*(d + e*x)^(7/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
[Out]
(2*b^7*(d + e*x)^(23/2))/(23*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(21/2))/(21*e^8) + (2*(a*e - b*d)^7*(d
+ e*x)^(9/2))/(9*e^8) + (42*b^2*(a*e - b*d)^5*(d + e*x)^(13/2))/(13*e^8) + (14*b^3*(a*e - b*d)^4*(d + e*x)^(15
/2))/(3*e^8) + (70*b^4*(a*e - b*d)^3*(d + e*x)^(17/2))/(17*e^8) + (42*b^5*(a*e - b*d)^2*(d + e*x)^(19/2))/(19*
e^8) + (14*b*(a*e - b*d)^6*(d + e*x)^(11/2))/(11*e^8)
________________________________________________________________________________________
sympy [A] time = 101.43, size = 3046, normalized size = 14.10
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)**(7/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
a**7*d**3*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 6*a**7*d**2*(-d*(d + e*x)**(3/2
)/3 + (d + e*x)**(5/2)/5)/e + 6*a**7*d*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)
/e + 2*a**7*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/
9)/e + 14*a**6*b*d**3*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 42*a**6*b*d**2*(d**2*(d + e*x)**(3/2
)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**2 + 42*a**6*b*d*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d +
e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**2 + 14*a**6*b*(d**4*(d + e*x)**(3/2)/3 - 4*d**
3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**2 + 42*a*
*5*b**2*d**3*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e**3 + 126*a**5*b**2*d**2
*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)**(9/2)/9)/e**3 + 1
26*a**5*b**2*d*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(7/2)/7 - 4*d*(d + e*x
)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**3 + 42*a**5*b**2*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5/2) - 10*
d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/13)/e**3 +
70*a**4*b**3*d**3*(-d**3*(d + e*x)**(3/2)/3 + 3*d**2*(d + e*x)**(5/2)/5 - 3*d*(d + e*x)**(7/2)/7 + (d + e*x)*
*(9/2)/9)/e**4 + 210*a**4*b**3*d**2*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(
7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**4 + 210*a**4*b**3*d*(-d**5*(d + e*x)**(3/2)/3 + d**
4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d +
e*x)**(13/2)/13)/e**4 + 70*a**4*b**3*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)
**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**
(15/2)/15)/e**4 + 70*a**3*b**4*d**3*(d**4*(d + e*x)**(3/2)/3 - 4*d**3*(d + e*x)**(5/2)/5 + 6*d**2*(d + e*x)**(
7/2)/7 - 4*d*(d + e*x)**(9/2)/9 + (d + e*x)**(11/2)/11)/e**5 + 210*a**3*b**4*d**2*(-d**5*(d + e*x)**(3/2)/3 +
d**4*(d + e*x)**(5/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (
d + e*x)**(13/2)/13)/e**5 + 210*a**3*b**4*d*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d
+ e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d +
e*x)**(15/2)/15)/e**5 + 70*a**3*b**4*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)*
*(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e
*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**5 + 42*a**2*b**5*d**3*(-d**5*(d + e*x)**(3/2)/3 + d**4*(d + e*x)**(5
/2) - 10*d**3*(d + e*x)**(7/2)/7 + 10*d**2*(d + e*x)**(9/2)/9 - 5*d*(d + e*x)**(11/2)/11 + (d + e*x)**(13/2)/1
3)/e**6 + 126*a**2*b**5*d**2*(d**6*(d + e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7
- 20*d**3*(d + e*x)**(9/2)/9 + 15*d**2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15
)/e**6 + 126*a**2*b**5*d*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*
d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/
15 + (d + e*x)**(17/2)/17)/e**6 + 42*a**2*b**5*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(
d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 2
8*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**6 + 14*a*b**6*d**3*(d**6*(d
+ e*x)**(3/2)/3 - 6*d**5*(d + e*x)**(5/2)/5 + 15*d**4*(d + e*x)**(7/2)/7 - 20*d**3*(d + e*x)**(9/2)/9 + 15*d**
2*(d + e*x)**(11/2)/11 - 6*d*(d + e*x)**(13/2)/13 + (d + e*x)**(15/2)/15)/e**7 + 42*a*b**6*d**2*(-d**7*(d + e*
x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d +
e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)**(15/2)/15 + (d + e*x)**(17/2)/17)/e**7 + 42*a*
b**6*d*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/
2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e
*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**7 + 14*a*b**6*(-d**9*(d + e*x)**(3/2)/3 + 9*d**8*(d + e*x)**(5/2)/5
- 36*d**7*(d + e*x)**(7/2)/7 + 28*d**6*(d + e*x)**(9/2)/3 - 126*d**5*(d + e*x)**(11/2)/11 + 126*d**4*(d + e*x)
**(13/2)/13 - 28*d**3*(d + e*x)**(15/2)/5 + 36*d**2*(d + e*x)**(17/2)/17 - 9*d*(d + e*x)**(19/2)/19 + (d + e*x
)**(21/2)/21)/e**7 + 2*b**7*d**3*(-d**7*(d + e*x)**(3/2)/3 + 7*d**6*(d + e*x)**(5/2)/5 - 3*d**5*(d + e*x)**(7/
2) + 35*d**4*(d + e*x)**(9/2)/9 - 35*d**3*(d + e*x)**(11/2)/11 + 21*d**2*(d + e*x)**(13/2)/13 - 7*d*(d + e*x)*
*(15/2)/15 + (d + e*x)**(17/2)/17)/e**8 + 6*b**7*d**2*(d**8*(d + e*x)**(3/2)/3 - 8*d**7*(d + e*x)**(5/2)/5 + 4
*d**6*(d + e*x)**(7/2) - 56*d**5*(d + e*x)**(9/2)/9 + 70*d**4*(d + e*x)**(11/2)/11 - 56*d**3*(d + e*x)**(13/2)
/13 + 28*d**2*(d + e*x)**(15/2)/15 - 8*d*(d + e*x)**(17/2)/17 + (d + e*x)**(19/2)/19)/e**8 + 6*b**7*d*(-d**9*(
d + e*x)**(3/2)/3 + 9*d**8*(d + e*x)**(5/2)/5 - 36*d**7*(d + e*x)**(7/2)/7 + 28*d**6*(d + e*x)**(9/2)/3 - 126*
d**5*(d + e*x)**(11/2)/11 + 126*d**4*(d + e*x)**(13/2)/13 - 28*d**3*(d + e*x)**(15/2)/5 + 36*d**2*(d + e*x)**(
17/2)/17 - 9*d*(d + e*x)**(19/2)/19 + (d + e*x)**(21/2)/21)/e**8 + 2*b**7*(d**10*(d + e*x)**(3/2)/3 - 2*d**9*(
d + e*x)**(5/2) + 45*d**8*(d + e*x)**(7/2)/7 - 40*d**7*(d + e*x)**(9/2)/3 + 210*d**6*(d + e*x)**(11/2)/11 - 25
2*d**5*(d + e*x)**(13/2)/13 + 14*d**4*(d + e*x)**(15/2) - 120*d**3*(d + e*x)**(17/2)/17 + 45*d**2*(d + e*x)**(
19/2)/19 - 10*d*(d + e*x)**(21/2)/21 + (d + e*x)**(23/2)/23)/e**8
________________________________________________________________________________________