3.19.34 \(\int (a+b x) (d+e x)^{5/2} (a^2+2 a b x+b^2 x^2)^3 \, dx\)
Optimal. Leaf size=216 \[ -\frac {14 b^6 (d+e x)^{19/2} (b d-a e)}{19 e^8}+\frac {42 b^5 (d+e x)^{17/2} (b d-a e)^2}{17 e^8}-\frac {14 b^4 (d+e x)^{15/2} (b d-a e)^3}{3 e^8}+\frac {70 b^3 (d+e x)^{13/2} (b d-a e)^4}{13 e^8}-\frac {42 b^2 (d+e x)^{11/2} (b d-a e)^5}{11 e^8}+\frac {14 b (d+e x)^{9/2} (b d-a e)^6}{9 e^8}-\frac {2 (d+e x)^{7/2} (b d-a e)^7}{7 e^8}+\frac {2 b^7 (d+e x)^{21/2}}{21 e^8} \]
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Rubi [A] time = 0.08, 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 {14 b^6 (d+e x)^{19/2} (b d-a e)}{19 e^8}+\frac {42 b^5 (d+e x)^{17/2} (b d-a e)^2}{17 e^8}-\frac {14 b^4 (d+e x)^{15/2} (b d-a e)^3}{3 e^8}+\frac {70 b^3 (d+e x)^{13/2} (b d-a e)^4}{13 e^8}-\frac {42 b^2 (d+e x)^{11/2} (b d-a e)^5}{11 e^8}+\frac {14 b (d+e x)^{9/2} (b d-a e)^6}{9 e^8}-\frac {2 (d+e x)^{7/2} (b d-a e)^7}{7 e^8}+\frac {2 b^7 (d+e x)^{21/2}}{21 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(-2*(b*d - a*e)^7*(d + e*x)^(7/2))/(7*e^8) + (14*b*(b*d - a*e)^6*(d + e*x)^(9/2))/(9*e^8) - (42*b^2*(b*d - a*e
)^5*(d + e*x)^(11/2))/(11*e^8) + (70*b^3*(b*d - a*e)^4*(d + e*x)^(13/2))/(13*e^8) - (14*b^4*(b*d - a*e)^3*(d +
e*x)^(15/2))/(3*e^8) + (42*b^5*(b*d - a*e)^2*(d + e*x)^(17/2))/(17*e^8) - (14*b^6*(b*d - a*e)*(d + e*x)^(19/2
))/(19*e^8) + (2*b^7*(d + e*x)^(21/2))/(21*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)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^3 \, dx &=\int (a+b x)^7 (d+e x)^{5/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^7 (d+e x)^{5/2}}{e^7}+\frac {7 b (b d-a e)^6 (d+e x)^{7/2}}{e^7}-\frac {21 b^2 (b d-a e)^5 (d+e x)^{9/2}}{e^7}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{11/2}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{13/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{15/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{17/2}}{e^7}+\frac {b^7 (d+e x)^{19/2}}{e^7}\right ) \, dx\\ &=-\frac {2 (b d-a e)^7 (d+e x)^{7/2}}{7 e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{9/2}}{9 e^8}-\frac {42 b^2 (b d-a e)^5 (d+e x)^{11/2}}{11 e^8}+\frac {70 b^3 (b d-a e)^4 (d+e x)^{13/2}}{13 e^8}-\frac {14 b^4 (b d-a e)^3 (d+e x)^{15/2}}{3 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{17/2}}{17 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{19/2}}{19 e^8}+\frac {2 b^7 (d+e x)^{21/2}}{21 e^8}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 167, normalized size = 0.77 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (-1072071 b^6 (d+e x)^6 (b d-a e)+3594591 b^5 (d+e x)^5 (b d-a e)^2-6789783 b^4 (d+e x)^4 (b d-a e)^3+7834365 b^3 (d+e x)^3 (b d-a e)^4-5555277 b^2 (d+e x)^2 (b d-a e)^5+2263261 b (d+e x) (b d-a e)^6-415701 (b d-a e)^7+138567 b^7 (d+e x)^7\right )}{2909907 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(7/2)*(-415701*(b*d - a*e)^7 + 2263261*b*(b*d - a*e)^6*(d + e*x) - 5555277*b^2*(b*d - a*e)^5*(d +
e*x)^2 + 7834365*b^3*(b*d - a*e)^4*(d + e*x)^3 - 6789783*b^4*(b*d - a*e)^3*(d + e*x)^4 + 3594591*b^5*(b*d - a
*e)^2*(d + e*x)^5 - 1072071*b^6*(b*d - a*e)*(d + e*x)^6 + 138567*b^7*(d + e*x)^7))/(2909907*e^8)
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IntegrateAlgebraic [B] time = 0.20, size = 582, normalized size = 2.69 \begin {gather*} \frac {2 (d+e x)^{7/2} \left (415701 a^7 e^7+2263261 a^6 b e^6 (d+e x)-2909907 a^6 b d e^6+8729721 a^5 b^2 d^2 e^5+5555277 a^5 b^2 e^5 (d+e x)^2-13579566 a^5 b^2 d e^5 (d+e x)-14549535 a^4 b^3 d^3 e^4+33948915 a^4 b^3 d^2 e^4 (d+e x)+7834365 a^4 b^3 e^4 (d+e x)^3-27776385 a^4 b^3 d e^4 (d+e x)^2+14549535 a^3 b^4 d^4 e^3-45265220 a^3 b^4 d^3 e^3 (d+e x)+55552770 a^3 b^4 d^2 e^3 (d+e x)^2+6789783 a^3 b^4 e^3 (d+e x)^4-31337460 a^3 b^4 d e^3 (d+e x)^3-8729721 a^2 b^5 d^5 e^2+33948915 a^2 b^5 d^4 e^2 (d+e x)-55552770 a^2 b^5 d^3 e^2 (d+e x)^2+47006190 a^2 b^5 d^2 e^2 (d+e x)^3+3594591 a^2 b^5 e^2 (d+e x)^5-20369349 a^2 b^5 d e^2 (d+e x)^4+2909907 a b^6 d^6 e-13579566 a b^6 d^5 e (d+e x)+27776385 a b^6 d^4 e (d+e x)^2-31337460 a b^6 d^3 e (d+e x)^3+20369349 a b^6 d^2 e (d+e x)^4+1072071 a b^6 e (d+e x)^6-7189182 a b^6 d e (d+e x)^5-415701 b^7 d^7+2263261 b^7 d^6 (d+e x)-5555277 b^7 d^5 (d+e x)^2+7834365 b^7 d^4 (d+e x)^3-6789783 b^7 d^3 (d+e x)^4+3594591 b^7 d^2 (d+e x)^5+138567 b^7 (d+e x)^7-1072071 b^7 d (d+e x)^6\right )}{2909907 e^8} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[(a + b*x)*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^3,x]
[Out]
(2*(d + e*x)^(7/2)*(-415701*b^7*d^7 + 2909907*a*b^6*d^6*e - 8729721*a^2*b^5*d^5*e^2 + 14549535*a^3*b^4*d^4*e^3
- 14549535*a^4*b^3*d^3*e^4 + 8729721*a^5*b^2*d^2*e^5 - 2909907*a^6*b*d*e^6 + 415701*a^7*e^7 + 2263261*b^7*d^6
*(d + e*x) - 13579566*a*b^6*d^5*e*(d + e*x) + 33948915*a^2*b^5*d^4*e^2*(d + e*x) - 45265220*a^3*b^4*d^3*e^3*(d
+ e*x) + 33948915*a^4*b^3*d^2*e^4*(d + e*x) - 13579566*a^5*b^2*d*e^5*(d + e*x) + 2263261*a^6*b*e^6*(d + e*x)
- 5555277*b^7*d^5*(d + e*x)^2 + 27776385*a*b^6*d^4*e*(d + e*x)^2 - 55552770*a^2*b^5*d^3*e^2*(d + e*x)^2 + 5555
2770*a^3*b^4*d^2*e^3*(d + e*x)^2 - 27776385*a^4*b^3*d*e^4*(d + e*x)^2 + 5555277*a^5*b^2*e^5*(d + e*x)^2 + 7834
365*b^7*d^4*(d + e*x)^3 - 31337460*a*b^6*d^3*e*(d + e*x)^3 + 47006190*a^2*b^5*d^2*e^2*(d + e*x)^3 - 31337460*a
^3*b^4*d*e^3*(d + e*x)^3 + 7834365*a^4*b^3*e^4*(d + e*x)^3 - 6789783*b^7*d^3*(d + e*x)^4 + 20369349*a*b^6*d^2*
e*(d + e*x)^4 - 20369349*a^2*b^5*d*e^2*(d + e*x)^4 + 6789783*a^3*b^4*e^3*(d + e*x)^4 + 3594591*b^7*d^2*(d + e*
x)^5 - 7189182*a*b^6*d*e*(d + e*x)^5 + 3594591*a^2*b^5*e^2*(d + e*x)^5 - 1072071*b^7*d*(d + e*x)^6 + 1072071*a
*b^6*e*(d + e*x)^6 + 138567*b^7*(d + e*x)^7))/(2909907*e^8)
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fricas [B] time = 0.43, size = 783, normalized size = 3.62 \begin {gather*} \frac {2 \, {\left (138567 \, b^{7} e^{10} x^{10} - 2048 \, b^{7} d^{10} + 21504 \, a b^{6} d^{9} e - 102144 \, a^{2} b^{5} d^{8} e^{2} + 289408 \, a^{3} b^{4} d^{7} e^{3} - 542640 \, a^{4} b^{3} d^{6} e^{4} + 705432 \, a^{5} b^{2} d^{5} e^{5} - 646646 \, a^{6} b d^{4} e^{6} + 415701 \, a^{7} d^{3} e^{7} + 7293 \, {\left (43 \, b^{7} d e^{9} + 147 \, a b^{6} e^{10}\right )} x^{9} + 3861 \, {\left (47 \, b^{7} d^{2} e^{8} + 637 \, a b^{6} d e^{9} + 931 \, a^{2} b^{5} e^{10}\right )} x^{8} + 429 \, {\left (b^{7} d^{3} e^{7} + 3381 \, a b^{6} d^{2} e^{8} + 19551 \, a^{2} b^{5} d e^{9} + 15827 \, a^{3} b^{4} e^{10}\right )} x^{7} - 231 \, {\left (2 \, b^{7} d^{4} e^{6} - 21 \, a b^{6} d^{3} e^{7} - 21945 \, a^{2} b^{5} d^{2} e^{8} - 70091 \, a^{3} b^{4} d e^{9} - 33915 \, a^{4} b^{3} e^{10}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{5} e^{5} - 84 \, a b^{6} d^{4} e^{6} + 399 \, a^{2} b^{5} d^{3} e^{7} + 160531 \, a^{3} b^{4} d^{2} e^{8} + 305235 \, a^{4} b^{3} d e^{9} + 88179 \, a^{5} b^{2} e^{10}\right )} x^{5} - 7 \, {\left (80 \, b^{7} d^{6} e^{4} - 840 \, a b^{6} d^{5} e^{5} + 3990 \, a^{2} b^{5} d^{4} e^{6} - 11305 \, a^{3} b^{4} d^{3} e^{7} - 1797495 \, a^{4} b^{3} d^{2} e^{8} - 2028117 \, a^{5} b^{2} d e^{9} - 323323 \, a^{6} b e^{10}\right )} x^{4} + {\left (640 \, b^{7} d^{7} e^{3} - 6720 \, a b^{6} d^{6} e^{4} + 31920 \, a^{2} b^{5} d^{5} e^{5} - 90440 \, a^{3} b^{4} d^{4} e^{6} + 169575 \, a^{4} b^{3} d^{3} e^{7} + 9964227 \, a^{5} b^{2} d^{2} e^{8} + 6143137 \, a^{6} b d e^{9} + 415701 \, a^{7} e^{10}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{8} e^{2} - 2688 \, a b^{6} d^{7} e^{3} + 12768 \, a^{2} b^{5} d^{6} e^{4} - 36176 \, a^{3} b^{4} d^{5} e^{5} + 67830 \, a^{4} b^{3} d^{4} e^{6} - 88179 \, a^{5} b^{2} d^{3} e^{7} - 1616615 \, a^{6} b d^{2} e^{8} - 415701 \, a^{7} d e^{9}\right )} x^{2} + {\left (1024 \, b^{7} d^{9} e - 10752 \, a b^{6} d^{8} e^{2} + 51072 \, a^{2} b^{5} d^{7} e^{3} - 144704 \, a^{3} b^{4} d^{6} e^{4} + 271320 \, a^{4} b^{3} d^{5} e^{5} - 352716 \, a^{5} b^{2} d^{4} e^{6} + 323323 \, a^{6} b d^{3} e^{7} + 1247103 \, a^{7} d^{2} e^{8}\right )} x\right )} \sqrt {e x + d}}{2909907 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="fricas")
[Out]
2/2909907*(138567*b^7*e^10*x^10 - 2048*b^7*d^10 + 21504*a*b^6*d^9*e - 102144*a^2*b^5*d^8*e^2 + 289408*a^3*b^4*
d^7*e^3 - 542640*a^4*b^3*d^6*e^4 + 705432*a^5*b^2*d^5*e^5 - 646646*a^6*b*d^4*e^6 + 415701*a^7*d^3*e^7 + 7293*(
43*b^7*d*e^9 + 147*a*b^6*e^10)*x^9 + 3861*(47*b^7*d^2*e^8 + 637*a*b^6*d*e^9 + 931*a^2*b^5*e^10)*x^8 + 429*(b^7
*d^3*e^7 + 3381*a*b^6*d^2*e^8 + 19551*a^2*b^5*d*e^9 + 15827*a^3*b^4*e^10)*x^7 - 231*(2*b^7*d^4*e^6 - 21*a*b^6*
d^3*e^7 - 21945*a^2*b^5*d^2*e^8 - 70091*a^3*b^4*d*e^9 - 33915*a^4*b^3*e^10)*x^6 + 63*(8*b^7*d^5*e^5 - 84*a*b^6
*d^4*e^6 + 399*a^2*b^5*d^3*e^7 + 160531*a^3*b^4*d^2*e^8 + 305235*a^4*b^3*d*e^9 + 88179*a^5*b^2*e^10)*x^5 - 7*(
80*b^7*d^6*e^4 - 840*a*b^6*d^5*e^5 + 3990*a^2*b^5*d^4*e^6 - 11305*a^3*b^4*d^3*e^7 - 1797495*a^4*b^3*d^2*e^8 -
2028117*a^5*b^2*d*e^9 - 323323*a^6*b*e^10)*x^4 + (640*b^7*d^7*e^3 - 6720*a*b^6*d^6*e^4 + 31920*a^2*b^5*d^5*e^5
- 90440*a^3*b^4*d^4*e^6 + 169575*a^4*b^3*d^3*e^7 + 9964227*a^5*b^2*d^2*e^8 + 6143137*a^6*b*d*e^9 + 415701*a^7
*e^10)*x^3 - 3*(256*b^7*d^8*e^2 - 2688*a*b^6*d^7*e^3 + 12768*a^2*b^5*d^6*e^4 - 36176*a^3*b^4*d^5*e^5 + 67830*a
^4*b^3*d^4*e^6 - 88179*a^5*b^2*d^3*e^7 - 1616615*a^6*b*d^2*e^8 - 415701*a^7*d*e^9)*x^2 + (1024*b^7*d^9*e - 107
52*a*b^6*d^8*e^2 + 51072*a^2*b^5*d^7*e^3 - 144704*a^3*b^4*d^6*e^4 + 271320*a^4*b^3*d^5*e^5 - 352716*a^5*b^2*d^
4*e^6 + 323323*a^6*b*d^3*e^7 + 1247103*a^7*d^2*e^8)*x)*sqrt(e*x + d)/e^8
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giac [B] time = 0.43, size = 2696, normalized size = 12.48
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="giac")
[Out]
2/14549535*(33948915*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^6*b*d^3*e^(-1) + 20369349*(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^3*e^(-2) + 14549535*(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^3*e^(-3) + 1616615*(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^3*e^(-
4) + 440895*(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^3*e^(-5) + 33915*(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*b^6*d^3*e^(-6) + 2261*(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^3*e^(-7) + 20369349*(3*(x*e + d)^(5/2) - 10*
(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*a^6*b*d^2*e^(-1) + 26189163*(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^2*e^(-2) + 4849845*(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^2*e^(-3)
+ 2204475*(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^2*e^(-4) + 305235*(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^2*b^5*d^2*e^(-5) + 47481*(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*b^6*d^2*e^(-6) + 399*(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)*b^7*d^2*e^(-7) + 14549535*sqrt(x*e + d)*a^7*d^3 + 14549535*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*a^7*
d^2 + 8729721*(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^6*b
*d*e^(-1) + 2909907*(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*e^(-2) + 2204475*(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*e^
(-3) + 508725*(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*e^(-4) + 14244
3*(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 + 3
2175*(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*e^(-5) + 2793*(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*e^(-6) + 189*(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*sqrt(x*e + d)*d^9)*b^7*d*e^(-7) + 2909907*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqr
t(x*e + d)*d^2)*a^7*d + 323323*(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*e^(-1) + 440895*(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*e^(-2) + 169575*(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*e^(-3) + 791
35*(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*e^(-4) + 2793*(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 - 2
91720*(x*e + d)^(3/2)*d^7 + 109395*sqrt(x*e + d)*d^8)*a^2*b^5*e^(-5) + 441*(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*sqrt(x*e + d)*d^9)*a*b^6*e^(-6) + 15*(46189*(x*e + d)^(21/2) - 510510*(x*e + d)^(19/2)*d + 25
67565*(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 - 3233
230*(x*e + d)^(3/2)*d^9 + 969969*sqrt(x*e + d)*d^10)*b^7*e^(-7) + 415701*(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)*e^(-1)
________________________________________________________________________________________
maple [B] time = 0.05, size = 498, normalized size = 2.31 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (138567 b^{7} x^{7} e^{7}+1072071 a \,b^{6} e^{7} x^{6}-102102 b^{7} d \,e^{6} x^{6}+3594591 a^{2} b^{5} e^{7} x^{5}-756756 a \,b^{6} d \,e^{6} x^{5}+72072 b^{7} d^{2} e^{5} x^{5}+6789783 a^{3} b^{4} e^{7} x^{4}-2396394 a^{2} b^{5} d \,e^{6} x^{4}+504504 a \,b^{6} d^{2} e^{5} x^{4}-48048 b^{7} d^{3} e^{4} x^{4}+7834365 a^{4} b^{3} e^{7} x^{3}-4178328 a^{3} b^{4} d \,e^{6} x^{3}+1474704 a^{2} b^{5} d^{2} e^{5} x^{3}-310464 a \,b^{6} d^{3} e^{4} x^{3}+29568 b^{7} d^{4} e^{3} x^{3}+5555277 a^{5} b^{2} e^{7} x^{2}-4273290 a^{4} b^{3} d \,e^{6} x^{2}+2279088 a^{3} b^{4} d^{2} e^{5} x^{2}-804384 a^{2} b^{5} d^{3} e^{4} x^{2}+169344 a \,b^{6} d^{4} e^{3} x^{2}-16128 b^{7} d^{5} e^{2} x^{2}+2263261 a^{6} b \,e^{7} x -2469012 a^{5} b^{2} d \,e^{6} x +1899240 a^{4} b^{3} d^{2} e^{5} x -1012928 a^{3} b^{4} d^{3} e^{4} x +357504 a^{2} b^{5} d^{4} e^{3} x -75264 a \,b^{6} d^{5} e^{2} x +7168 b^{7} d^{6} e x +415701 a^{7} e^{7}-646646 a^{6} b d \,e^{6}+705432 a^{5} b^{2} d^{2} e^{5}-542640 a^{4} b^{3} d^{3} e^{4}+289408 a^{3} b^{4} d^{4} e^{3}-102144 a^{2} b^{5} d^{5} e^{2}+21504 a \,b^{6} d^{6} e -2048 b^{7} d^{7}\right )}{2909907 e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x)
[Out]
2/2909907*(e*x+d)^(7/2)*(138567*b^7*e^7*x^7+1072071*a*b^6*e^7*x^6-102102*b^7*d*e^6*x^6+3594591*a^2*b^5*e^7*x^5
-756756*a*b^6*d*e^6*x^5+72072*b^7*d^2*e^5*x^5+6789783*a^3*b^4*e^7*x^4-2396394*a^2*b^5*d*e^6*x^4+504504*a*b^6*d
^2*e^5*x^4-48048*b^7*d^3*e^4*x^4+7834365*a^4*b^3*e^7*x^3-4178328*a^3*b^4*d*e^6*x^3+1474704*a^2*b^5*d^2*e^5*x^3
-310464*a*b^6*d^3*e^4*x^3+29568*b^7*d^4*e^3*x^3+5555277*a^5*b^2*e^7*x^2-4273290*a^4*b^3*d*e^6*x^2+2279088*a^3*
b^4*d^2*e^5*x^2-804384*a^2*b^5*d^3*e^4*x^2+169344*a*b^6*d^4*e^3*x^2-16128*b^7*d^5*e^2*x^2+2263261*a^6*b*e^7*x-
2469012*a^5*b^2*d*e^6*x+1899240*a^4*b^3*d^2*e^5*x-1012928*a^3*b^4*d^3*e^4*x+357504*a^2*b^5*d^4*e^3*x-75264*a*b
^6*d^5*e^2*x+7168*b^7*d^6*e*x+415701*a^7*e^7-646646*a^6*b*d*e^6+705432*a^5*b^2*d^2*e^5-542640*a^4*b^3*d^3*e^4+
289408*a^3*b^4*d^4*e^3-102144*a^2*b^5*d^5*e^2+21504*a*b^6*d^6*e-2048*b^7*d^7)/e^8
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maxima [B] time = 0.65, size = 456, normalized size = 2.11 \begin {gather*} \frac {2 \, {\left (138567 \, {\left (e x + d\right )}^{\frac {21}{2}} b^{7} - 1072071 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {19}{2}} + 3594591 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {17}{2}} - 6789783 \, {\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 {15}{2}} + 7834365 \, {\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 {13}{2}} - 5555277 \, {\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 {11}{2}} + 2263261 \, {\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 {9}{2}} - 415701 \, {\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 {7}{2}}\right )}}{2909907 \, e^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)^(5/2)*(b^2*x^2+2*a*b*x+a^2)^3,x, algorithm="maxima")
[Out]
2/2909907*(138567*(e*x + d)^(21/2)*b^7 - 1072071*(b^7*d - a*b^6*e)*(e*x + d)^(19/2) + 3594591*(b^7*d^2 - 2*a*b
^6*d*e + a^2*b^5*e^2)*(e*x + d)^(17/2) - 6789783*(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)^(15/2) + 7834365*(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
)^(13/2) - 5555277*(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)^(11/2) + 2263261*(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)*(e*x + d)^(9/2) - 415701*(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)^(7/2
))/e^8
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mupad [B] time = 0.06, size = 187, normalized size = 0.87 \begin {gather*} \frac {2\,b^7\,{\left (d+e\,x\right )}^{21/2}}{21\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{19/2}}{19\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,{\left (d+e\,x\right )}^{7/2}}{7\,e^8}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {70\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {14\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{15/2}}{3\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{17/2}}{17\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((a + b*x)*(d + e*x)^(5/2)*(a^2 + b^2*x^2 + 2*a*b*x)^3,x)
[Out]
(2*b^7*(d + e*x)^(21/2))/(21*e^8) - ((14*b^7*d - 14*a*b^6*e)*(d + e*x)^(19/2))/(19*e^8) + (2*(a*e - b*d)^7*(d
+ e*x)^(7/2))/(7*e^8) + (42*b^2*(a*e - b*d)^5*(d + e*x)^(11/2))/(11*e^8) + (70*b^3*(a*e - b*d)^4*(d + e*x)^(13
/2))/(13*e^8) + (14*b^4*(a*e - b*d)^3*(d + e*x)^(15/2))/(3*e^8) + (42*b^5*(a*e - b*d)^2*(d + e*x)^(17/2))/(17*
e^8) + (14*b*(a*e - b*d)^6*(d + e*x)^(9/2))/(9*e^8)
________________________________________________________________________________________
sympy [A] time = 70.41, size = 2096, normalized size = 9.70
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((b*x+a)*(e*x+d)**(5/2)*(b**2*x**2+2*a*b*x+a**2)**3,x)
[Out]
a**7*d**2*Piecewise((sqrt(d)*x, Eq(e, 0)), (2*(d + e*x)**(3/2)/(3*e), True)) + 4*a**7*d*(-d*(d + e*x)**(3/2)/3
+ (d + e*x)**(5/2)/5)/e + 2*a**7*(d**2*(d + e*x)**(3/2)/3 - 2*d*(d + e*x)**(5/2)/5 + (d + e*x)**(7/2)/7)/e +
14*a**6*b*d**2*(-d*(d + e*x)**(3/2)/3 + (d + e*x)**(5/2)/5)/e**2 + 28*a**6*b*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 + 14*a**6*b*(-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 + 42*a**5*b**2*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**3 + 84*a**5*b**2*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**3 + 42*a**5*b**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**3 + 70*a**4*b**3*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**4
+ 140*a**4*b**3*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**4 + 70*a**4*b**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)/13)/e**
4 + 70*a**3*b**4*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**5 + 140*a**3*b**4*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**5 + 70*a**3*b**4*(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 + 42*a**2*b**5*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**6 + 84*a**2*b**5*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**6 + 42*a**2*b**5*(-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 + 14*a*b**6*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**7 + 28*a*b**6*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**7 +
14*a*b**6*(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 + 2*b**7*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**8 + 4*b**7*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**8 + 2*b**7*(-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
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