3.16.72 \(\int (A+B x) (d+e x)^{7/2} (a^2+2 a b x+b^2 x^2)^2 \, dx\)
Optimal. Leaf size=218 \[ -\frac {2 b^3 (d+e x)^{17/2} (-4 a B e-A b e+5 b B d)}{17 e^6}+\frac {4 b^2 (d+e x)^{15/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{15 e^6}-\frac {4 b (d+e x)^{13/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{13 e^6}+\frac {2 (d+e x)^{11/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{11 e^6}-\frac {2 (d+e x)^{9/2} (b d-a e)^4 (B d-A e)}{9 e^6}+\frac {2 b^4 B (d+e x)^{19/2}}{19 e^6} \]
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Rubi [A] time = 0.14, antiderivative size = 218, 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, 77} \begin {gather*} -\frac {2 b^3 (d+e x)^{17/2} (-4 a B e-A b e+5 b B d)}{17 e^6}+\frac {4 b^2 (d+e x)^{15/2} (b d-a e) (-3 a B e-2 A b e+5 b B d)}{15 e^6}-\frac {4 b (d+e x)^{13/2} (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)}{13 e^6}+\frac {2 (d+e x)^{11/2} (b d-a e)^3 (-a B e-4 A b e+5 b B d)}{11 e^6}-\frac {2 (d+e x)^{9/2} (b d-a e)^4 (B d-A e)}{9 e^6}+\frac {2 b^4 B (d+e x)^{19/2}}{19 e^6} \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)^2,x]
[Out]
(-2*(b*d - a*e)^4*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^6) + (2*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d + e*x
)^(11/2))/(11*e^6) - (4*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^(13/2))/(13*e^6) + (4*b^2*(b*d
- a*e)*(5*b*B*d - 2*A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(15*e^6) - (2*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x
)^(17/2))/(17*e^6) + (2*b^4*B*(d + e*x)^(19/2))/(19*e^6)
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 77
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
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 )^2 \, dx &=\int (a+b x)^4 (A+B x) (d+e x)^{7/2} \, dx\\ &=\int \left (\frac {(-b d+a e)^4 (-B d+A e) (d+e x)^{7/2}}{e^5}+\frac {(-b d+a e)^3 (-5 b B d+4 A b e+a B e) (d+e x)^{9/2}}{e^5}+\frac {2 b (b d-a e)^2 (-5 b B d+3 A b e+2 a B e) (d+e x)^{11/2}}{e^5}-\frac {2 b^2 (b d-a e) (-5 b B d+2 A b e+3 a B e) (d+e x)^{13/2}}{e^5}+\frac {b^3 (-5 b B d+A b e+4 a B e) (d+e x)^{15/2}}{e^5}+\frac {b^4 B (d+e x)^{17/2}}{e^5}\right ) \, dx\\ &=-\frac {2 (b d-a e)^4 (B d-A e) (d+e x)^{9/2}}{9 e^6}+\frac {2 (b d-a e)^3 (5 b B d-4 A b e-a B e) (d+e x)^{11/2}}{11 e^6}-\frac {4 b (b d-a e)^2 (5 b B d-3 A b e-2 a B e) (d+e x)^{13/2}}{13 e^6}+\frac {4 b^2 (b d-a e) (5 b B d-2 A b e-3 a B e) (d+e x)^{15/2}}{15 e^6}-\frac {2 b^3 (5 b B d-A b e-4 a B e) (d+e x)^{17/2}}{17 e^6}+\frac {2 b^4 B (d+e x)^{19/2}}{19 e^6}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 183, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (-122265 b^3 (d+e x)^4 (-4 a B e-A b e+5 b B d)+277134 b^2 (d+e x)^3 (b d-a e) (-3 a B e-2 A b e+5 b B d)-319770 b (d+e x)^2 (b d-a e)^2 (-2 a B e-3 A b e+5 b B d)+188955 (d+e x) (b d-a e)^3 (-a B e-4 A b e+5 b B d)-230945 (b d-a e)^4 (B d-A e)+109395 b^4 B (d+e x)^5\right )}{2078505 e^6} \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)^2,x]
[Out]
(2*(d + e*x)^(9/2)*(-230945*(b*d - a*e)^4*(B*d - A*e) + 188955*(b*d - a*e)^3*(5*b*B*d - 4*A*b*e - a*B*e)*(d +
e*x) - 319770*b*(b*d - a*e)^2*(5*b*B*d - 3*A*b*e - 2*a*B*e)*(d + e*x)^2 + 277134*b^2*(b*d - a*e)*(5*b*B*d - 2*
A*b*e - 3*a*B*e)*(d + e*x)^3 - 122265*b^3*(5*b*B*d - A*b*e - 4*a*B*e)*(d + e*x)^4 + 109395*b^4*B*(d + e*x)^5))
/(2078505*e^6)
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IntegrateAlgebraic [B] time = 0.26, size = 543, normalized size = 2.49 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (230945 a^4 A e^5+188955 a^4 B e^4 (d+e x)-230945 a^4 B d e^4+755820 a^3 A b e^4 (d+e x)-923780 a^3 A b d e^4+923780 a^3 b B d^2 e^3-1511640 a^3 b B d e^3 (d+e x)+639540 a^3 b B e^3 (d+e x)^2+1385670 a^2 A b^2 d^2 e^3-2267460 a^2 A b^2 d e^3 (d+e x)+959310 a^2 A b^2 e^3 (d+e x)^2-1385670 a^2 b^2 B d^3 e^2+3401190 a^2 b^2 B d^2 e^2 (d+e x)-2877930 a^2 b^2 B d e^2 (d+e x)^2+831402 a^2 b^2 B e^2 (d+e x)^3-923780 a A b^3 d^3 e^2+2267460 a A b^3 d^2 e^2 (d+e x)-1918620 a A b^3 d e^2 (d+e x)^2+554268 a A b^3 e^2 (d+e x)^3+923780 a b^3 B d^4 e-3023280 a b^3 B d^3 e (d+e x)+3837240 a b^3 B d^2 e (d+e x)^2-2217072 a b^3 B d e (d+e x)^3+489060 a b^3 B e (d+e x)^4+230945 A b^4 d^4 e-755820 A b^4 d^3 e (d+e x)+959310 A b^4 d^2 e (d+e x)^2-554268 A b^4 d e (d+e x)^3+122265 A b^4 e (d+e x)^4-230945 b^4 B d^5+944775 b^4 B d^4 (d+e x)-1598850 b^4 B d^3 (d+e x)^2+1385670 b^4 B d^2 (d+e x)^3-611325 b^4 B d (d+e x)^4+109395 b^4 B (d+e x)^5\right )}{2078505 e^6} \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)^2,x]
[Out]
(2*(d + e*x)^(9/2)*(-230945*b^4*B*d^5 + 230945*A*b^4*d^4*e + 923780*a*b^3*B*d^4*e - 923780*a*A*b^3*d^3*e^2 - 1
385670*a^2*b^2*B*d^3*e^2 + 1385670*a^2*A*b^2*d^2*e^3 + 923780*a^3*b*B*d^2*e^3 - 923780*a^3*A*b*d*e^4 - 230945*
a^4*B*d*e^4 + 230945*a^4*A*e^5 + 944775*b^4*B*d^4*(d + e*x) - 755820*A*b^4*d^3*e*(d + e*x) - 3023280*a*b^3*B*d
^3*e*(d + e*x) + 2267460*a*A*b^3*d^2*e^2*(d + e*x) + 3401190*a^2*b^2*B*d^2*e^2*(d + e*x) - 2267460*a^2*A*b^2*d
*e^3*(d + e*x) - 1511640*a^3*b*B*d*e^3*(d + e*x) + 755820*a^3*A*b*e^4*(d + e*x) + 188955*a^4*B*e^4*(d + e*x) -
1598850*b^4*B*d^3*(d + e*x)^2 + 959310*A*b^4*d^2*e*(d + e*x)^2 + 3837240*a*b^3*B*d^2*e*(d + e*x)^2 - 1918620*
a*A*b^3*d*e^2*(d + e*x)^2 - 2877930*a^2*b^2*B*d*e^2*(d + e*x)^2 + 959310*a^2*A*b^2*e^3*(d + e*x)^2 + 639540*a^
3*b*B*e^3*(d + e*x)^2 + 1385670*b^4*B*d^2*(d + e*x)^3 - 554268*A*b^4*d*e*(d + e*x)^3 - 2217072*a*b^3*B*d*e*(d
+ e*x)^3 + 554268*a*A*b^3*e^2*(d + e*x)^3 + 831402*a^2*b^2*B*e^2*(d + e*x)^3 - 611325*b^4*B*d*(d + e*x)^4 + 12
2265*A*b^4*e*(d + e*x)^4 + 489060*a*b^3*B*e*(d + e*x)^4 + 109395*b^4*B*(d + e*x)^5))/(2078505*e^6)
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fricas [B] time = 0.42, size = 894, normalized size = 4.10 \begin {gather*} \frac {2 \, {\left (109395 \, B b^{4} e^{9} x^{9} - 1280 \, B b^{4} d^{9} + 230945 \, A a^{4} d^{4} e^{5} + 2432 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{8} e - 10336 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{7} e^{2} + 25840 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{6} e^{3} - 41990 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{5} e^{4} + 6435 \, {\left (58 \, B b^{4} d e^{8} + 19 \, {\left (4 \, B a b^{3} + A b^{4}\right )} e^{9}\right )} x^{8} + 858 \, {\left (505 \, B b^{4} d^{2} e^{7} + 494 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e^{8} + 323 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{9}\right )} x^{7} + 66 \, {\left (2620 \, B b^{4} d^{3} e^{6} + 7619 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e^{7} + 14858 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{8} + 4845 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{9}\right )} x^{6} + 9 \, {\left (35 \, B b^{4} d^{4} e^{5} + 23028 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e^{6} + 133076 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{7} + 129200 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{8} + 20995 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{9}\right )} x^{5} - 5 \, {\left (70 \, B b^{4} d^{5} e^{4} - 46189 \, A a^{4} e^{9} - 133 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e^{5} - 103360 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{6} - 295868 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{7} - 142766 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{8}\right )} x^{4} + 10 \, {\left (40 \, B b^{4} d^{6} e^{3} + 92378 \, A a^{4} d e^{8} - 76 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{5} e^{4} + 323 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{4} e^{5} + 68476 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{3} e^{6} + 96577 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{2} e^{7}\right )} x^{3} - 6 \, {\left (80 \, B b^{4} d^{7} e^{2} - 230945 \, A a^{4} d^{2} e^{7} - 152 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{6} e^{3} + 646 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{5} e^{4} - 1615 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{4} e^{5} - 83980 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{3} e^{6}\right )} x^{2} + {\left (640 \, B b^{4} d^{8} e + 923780 \, A a^{4} d^{3} e^{6} - 1216 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{7} e^{2} + 5168 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{6} e^{3} - 12920 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{5} e^{4} + 20995 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} d^{4} e^{5}\right )} x\right )} \sqrt {e x + d}}{2078505 \, e^{6}} \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)^2,x, algorithm="fricas")
[Out]
2/2078505*(109395*B*b^4*e^9*x^9 - 1280*B*b^4*d^9 + 230945*A*a^4*d^4*e^5 + 2432*(4*B*a*b^3 + A*b^4)*d^8*e - 103
36*(3*B*a^2*b^2 + 2*A*a*b^3)*d^7*e^2 + 25840*(2*B*a^3*b + 3*A*a^2*b^2)*d^6*e^3 - 41990*(B*a^4 + 4*A*a^3*b)*d^5
*e^4 + 6435*(58*B*b^4*d*e^8 + 19*(4*B*a*b^3 + A*b^4)*e^9)*x^8 + 858*(505*B*b^4*d^2*e^7 + 494*(4*B*a*b^3 + A*b^
4)*d*e^8 + 323*(3*B*a^2*b^2 + 2*A*a*b^3)*e^9)*x^7 + 66*(2620*B*b^4*d^3*e^6 + 7619*(4*B*a*b^3 + A*b^4)*d^2*e^7
+ 14858*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^8 + 4845*(2*B*a^3*b + 3*A*a^2*b^2)*e^9)*x^6 + 9*(35*B*b^4*d^4*e^5 + 2302
8*(4*B*a*b^3 + A*b^4)*d^3*e^6 + 133076*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^7 + 129200*(2*B*a^3*b + 3*A*a^2*b^2)*d*
e^8 + 20995*(B*a^4 + 4*A*a^3*b)*e^9)*x^5 - 5*(70*B*b^4*d^5*e^4 - 46189*A*a^4*e^9 - 133*(4*B*a*b^3 + A*b^4)*d^4
*e^5 - 103360*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^6 - 295868*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^7 - 142766*(B*a^4 + 4
*A*a^3*b)*d*e^8)*x^4 + 10*(40*B*b^4*d^6*e^3 + 92378*A*a^4*d*e^8 - 76*(4*B*a*b^3 + A*b^4)*d^5*e^4 + 323*(3*B*a^
2*b^2 + 2*A*a*b^3)*d^4*e^5 + 68476*(2*B*a^3*b + 3*A*a^2*b^2)*d^3*e^6 + 96577*(B*a^4 + 4*A*a^3*b)*d^2*e^7)*x^3
- 6*(80*B*b^4*d^7*e^2 - 230945*A*a^4*d^2*e^7 - 152*(4*B*a*b^3 + A*b^4)*d^6*e^3 + 646*(3*B*a^2*b^2 + 2*A*a*b^3)
*d^5*e^4 - 1615*(2*B*a^3*b + 3*A*a^2*b^2)*d^4*e^5 - 83980*(B*a^4 + 4*A*a^3*b)*d^3*e^6)*x^2 + (640*B*b^4*d^8*e
+ 923780*A*a^4*d^3*e^6 - 1216*(4*B*a*b^3 + A*b^4)*d^7*e^2 + 5168*(3*B*a^2*b^2 + 2*A*a*b^3)*d^6*e^3 - 12920*(2*
B*a^3*b + 3*A*a^2*b^2)*d^5*e^4 + 20995*(B*a^4 + 4*A*a^3*b)*d^4*e^5)*x)*sqrt(e*x + d)/e^6
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giac [B] time = 0.46, size = 3913, normalized size = 17.95
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)^2,x, algorithm="giac")
[Out]
2/14549535*(4849845*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^4*d^4*e^(-1) + 19399380*((x*e + d)^(3/2) - 3*sqr
t(x*e + d)*d)*A*a^3*b*d^4*e^(-1) + 3879876*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B
*a^3*b*d^4*e^(-2) + 5819814*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*b^2*d^4*e^
(-2) + 2494206*(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)*B*a^
2*b^2*d^4*e^(-3) + 1662804*(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*a*b^3*d^4*e^(-3) + 184756*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 42
0*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a*b^3*d^4*e^(-4) + 46189*(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*b^4*d^4*e^(-4) + 20995*
(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)*B*b^4*d^4*e^(-5) + 3879876*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d +
15*sqrt(x*e + d)*d^2)*B*a^4*d^3*e^(-1) + 15519504*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d
)*d^2)*A*a^3*b*d^3*e^(-1) + 6651216*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sq
rt(x*e + d)*d^3)*B*a^3*b*d^3*e^(-2) + 9976824*(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*a^2*b^2*d^3*e^(-2) + 1108536*(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)*B*a^2*b^2*d^3*e^(-3) + 739024*(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*a*b^3*d^3*e^(-3) + 335920*(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)*B*a*b^3*d^3*e^(-4) + 83980*(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*b^4*d^3*e^(-4) + 19380*(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)*B*b^4*d^3*e^(-5) + 14549535*sqrt(x*e + d)*A*a^4*d^4 + 19399380*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*
d)*A*a^4*d^3 + 2494206*(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)*B*a^4*d^2*e^(-1) + 9976824*(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*a^3*b*d^2*e^(-1) + 1108536*(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)*B*a^3*b*d^2*e^(-2) + 1662804*(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*a^2*b^2*d^2*e^(-
2) + 755820*(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)*B*a^2*b^2*d^2*e^(-3) + 503880*(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*a*b^3*d^2*e^(-3) + 116280*(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
)*B*a*b^3*d^2*e^(-4) + 29070*(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^4*d^2
*e^(-4) + 13566*(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*b^4*d^2*e^(-5) + 5819814*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^4*
d^2 + 184756*(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)*B*a^4*d*e^(-1) + 739024*(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*a^3*b*d*e^(-1) + 335920*(63*(x*e + d)^(11/2) - 38
5*(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)*B*a^3*b*d*e^(-2) + 503880*(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*a^2*b^2*d*e^(-2) + 116280*(
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)*B*a^2*b^2*d*e^(-3) + 77520*(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*a*b^3*d*e^(-3) + 36176*(429*(x*e + d)^(15/2) - 3
465*(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*a*b^3*d*e^(-4) + 9044*(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^4*d*e^(
-4) + 532*(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*b^4*d*e^(-5) + 1662804*(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*a^4*d + 20995*(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)*B*a^
4*e^(-1) + 83980*(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*a^3*b*e^(-1) + 19380*(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)*B*a^3*b*e^(-2) + 29070*(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*a^2*b^2*e^(-2) + 13566*(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*a^2*b^2*e^(-3) + 9044*(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*a*b^3*e^(-3) + 532*(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*a*b^3*e^(-4) + 133*(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^4*e^(-4) + 63*(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*b^4*e^(-5) + 46189*(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*a^4)*e^(-1)
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maple [B] time = 0.05, size = 469, normalized size = 2.15 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {9}{2}} \left (109395 b^{4} B \,x^{5} e^{5}+122265 A \,b^{4} e^{5} x^{4}+489060 B a \,b^{3} e^{5} x^{4}-64350 B \,b^{4} d \,e^{4} x^{4}+554268 A a \,b^{3} e^{5} x^{3}-65208 A \,b^{4} d \,e^{4} x^{3}+831402 B \,a^{2} b^{2} e^{5} x^{3}-260832 B a \,b^{3} d \,e^{4} x^{3}+34320 B \,b^{4} d^{2} e^{3} x^{3}+959310 A \,a^{2} b^{2} e^{5} x^{2}-255816 A a \,b^{3} d \,e^{4} x^{2}+30096 A \,b^{4} d^{2} e^{3} x^{2}+639540 B \,a^{3} b \,e^{5} x^{2}-383724 B \,a^{2} b^{2} d \,e^{4} x^{2}+120384 B a \,b^{3} d^{2} e^{3} x^{2}-15840 B \,b^{4} d^{3} e^{2} x^{2}+755820 A \,a^{3} b \,e^{5} x -348840 A \,a^{2} b^{2} d \,e^{4} x +93024 A a \,b^{3} d^{2} e^{3} x -10944 A \,b^{4} d^{3} e^{2} x +188955 B \,a^{4} e^{5} x -232560 B \,a^{3} b d \,e^{4} x +139536 B \,a^{2} b^{2} d^{2} e^{3} x -43776 B a \,b^{3} d^{3} e^{2} x +5760 B \,b^{4} d^{4} e x +230945 A \,a^{4} e^{5}-167960 A \,a^{3} b d \,e^{4}+77520 A \,a^{2} b^{2} d^{2} e^{3}-20672 A a \,b^{3} d^{3} e^{2}+2432 A \,b^{4} d^{4} e -41990 B \,a^{4} d \,e^{4}+51680 B \,d^{2} a^{3} b \,e^{3}-31008 B \,d^{3} a^{2} b^{2} e^{2}+9728 B a \,b^{3} d^{4} e -1280 B \,b^{4} d^{5}\right )}{2078505 e^{6}} \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)^2,x)
[Out]
2/2078505*(e*x+d)^(9/2)*(109395*B*b^4*e^5*x^5+122265*A*b^4*e^5*x^4+489060*B*a*b^3*e^5*x^4-64350*B*b^4*d*e^4*x^
4+554268*A*a*b^3*e^5*x^3-65208*A*b^4*d*e^4*x^3+831402*B*a^2*b^2*e^5*x^3-260832*B*a*b^3*d*e^4*x^3+34320*B*b^4*d
^2*e^3*x^3+959310*A*a^2*b^2*e^5*x^2-255816*A*a*b^3*d*e^4*x^2+30096*A*b^4*d^2*e^3*x^2+639540*B*a^3*b*e^5*x^2-38
3724*B*a^2*b^2*d*e^4*x^2+120384*B*a*b^3*d^2*e^3*x^2-15840*B*b^4*d^3*e^2*x^2+755820*A*a^3*b*e^5*x-348840*A*a^2*
b^2*d*e^4*x+93024*A*a*b^3*d^2*e^3*x-10944*A*b^4*d^3*e^2*x+188955*B*a^4*e^5*x-232560*B*a^3*b*d*e^4*x+139536*B*a
^2*b^2*d^2*e^3*x-43776*B*a*b^3*d^3*e^2*x+5760*B*b^4*d^4*e*x+230945*A*a^4*e^5-167960*A*a^3*b*d*e^4+77520*A*a^2*
b^2*d^2*e^3-20672*A*a*b^3*d^3*e^2+2432*A*b^4*d^4*e-41990*B*a^4*d*e^4+51680*B*a^3*b*d^2*e^3-31008*B*a^2*b^2*d^3
*e^2+9728*B*a*b^3*d^4*e-1280*B*b^4*d^5)/e^6
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maxima [B] time = 0.74, size = 409, normalized size = 1.88 \begin {gather*} \frac {2 \, {\left (109395 \, {\left (e x + d\right )}^{\frac {19}{2}} B b^{4} - 122265 \, {\left (5 \, B b^{4} d - {\left (4 \, B a b^{3} + A b^{4}\right )} e\right )} {\left (e x + d\right )}^{\frac {17}{2}} + 277134 \, {\left (5 \, B b^{4} d^{2} - 2 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d e + {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {15}{2}} - 319770 \, {\left (5 \, B b^{4} d^{3} - 3 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{2} e + 3 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d e^{2} - {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} e^{3}\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 188955 \, {\left (5 \, B b^{4} d^{4} - 4 \, {\left (4 \, B a b^{3} + A b^{4}\right )} d^{3} e + 6 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{2} e^{2} - 4 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} e^{4}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 230945 \, {\left (B b^{4} d^{5} - A a^{4} e^{5} - {\left (4 \, B a b^{3} + A b^{4}\right )} d^{4} e + 2 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} d^{3} e^{2} - 2 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} d^{2} e^{3} + {\left (B a^{4} + 4 \, A a^{3} b\right )} d e^{4}\right )} {\left (e x + d\right )}^{\frac {9}{2}}\right )}}{2078505 \, e^{6}} \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)^2,x, algorithm="maxima")
[Out]
2/2078505*(109395*(e*x + d)^(19/2)*B*b^4 - 122265*(5*B*b^4*d - (4*B*a*b^3 + A*b^4)*e)*(e*x + d)^(17/2) + 27713
4*(5*B*b^4*d^2 - 2*(4*B*a*b^3 + A*b^4)*d*e + (3*B*a^2*b^2 + 2*A*a*b^3)*e^2)*(e*x + d)^(15/2) - 319770*(5*B*b^4
*d^3 - 3*(4*B*a*b^3 + A*b^4)*d^2*e + 3*(3*B*a^2*b^2 + 2*A*a*b^3)*d*e^2 - (2*B*a^3*b + 3*A*a^2*b^2)*e^3)*(e*x +
d)^(13/2) + 188955*(5*B*b^4*d^4 - 4*(4*B*a*b^3 + A*b^4)*d^3*e + 6*(3*B*a^2*b^2 + 2*A*a*b^3)*d^2*e^2 - 4*(2*B*
a^3*b + 3*A*a^2*b^2)*d*e^3 + (B*a^4 + 4*A*a^3*b)*e^4)*(e*x + d)^(11/2) - 230945*(B*b^4*d^5 - A*a^4*e^5 - (4*B*
a*b^3 + A*b^4)*d^4*e + 2*(3*B*a^2*b^2 + 2*A*a*b^3)*d^3*e^2 - 2*(2*B*a^3*b + 3*A*a^2*b^2)*d^2*e^3 + (B*a^4 + 4*
A*a^3*b)*d*e^4)*(e*x + d)^(9/2))/e^6
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mupad [B] time = 1.98, size = 197, normalized size = 0.90 \begin {gather*} \frac {{\left (d+e\,x\right )}^{17/2}\,\left (2\,A\,b^4\,e-10\,B\,b^4\,d+8\,B\,a\,b^3\,e\right )}{17\,e^6}+\frac {2\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{11/2}\,\left (4\,A\,b\,e+B\,a\,e-5\,B\,b\,d\right )}{11\,e^6}+\frac {2\,B\,b^4\,{\left (d+e\,x\right )}^{19/2}}{19\,e^6}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{9/2}}{9\,e^6}+\frac {4\,b\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{13/2}\,\left (3\,A\,b\,e+2\,B\,a\,e-5\,B\,b\,d\right )}{13\,e^6}+\frac {4\,b^2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{15/2}\,\left (2\,A\,b\,e+3\,B\,a\,e-5\,B\,b\,d\right )}{15\,e^6} \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)^2,x)
[Out]
((d + e*x)^(17/2)*(2*A*b^4*e - 10*B*b^4*d + 8*B*a*b^3*e))/(17*e^6) + (2*(a*e - b*d)^3*(d + e*x)^(11/2)*(4*A*b*
e + B*a*e - 5*B*b*d))/(11*e^6) + (2*B*b^4*(d + e*x)^(19/2))/(19*e^6) + (2*(A*e - B*d)*(a*e - b*d)^4*(d + e*x)^
(9/2))/(9*e^6) + (4*b*(a*e - b*d)^2*(d + e*x)^(13/2)*(3*A*b*e + 2*B*a*e - 5*B*b*d))/(13*e^6) + (4*b^2*(a*e - b
*d)*(d + e*x)^(15/2)*(2*A*b*e + 3*B*a*e - 5*B*b*d))/(15*e^6)
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sympy [A] time = 16.31, size = 2091, normalized size = 9.59
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)**2,x)
[Out]
Piecewise((2*A*a**4*d**4*sqrt(d + e*x)/(9*e) + 8*A*a**4*d**3*x*sqrt(d + e*x)/9 + 4*A*a**4*d**2*e*x**2*sqrt(d +
e*x)/3 + 8*A*a**4*d*e**2*x**3*sqrt(d + e*x)/9 + 2*A*a**4*e**3*x**4*sqrt(d + e*x)/9 - 16*A*a**3*b*d**5*sqrt(d
+ e*x)/(99*e**2) + 8*A*a**3*b*d**4*x*sqrt(d + e*x)/(99*e) + 64*A*a**3*b*d**3*x**2*sqrt(d + e*x)/33 + 368*A*a**
3*b*d**2*e*x**3*sqrt(d + e*x)/99 + 272*A*a**3*b*d*e**2*x**4*sqrt(d + e*x)/99 + 8*A*a**3*b*e**3*x**5*sqrt(d + e
*x)/11 + 32*A*a**2*b**2*d**6*sqrt(d + e*x)/(429*e**3) - 16*A*a**2*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 4*A*a
**2*b**2*d**4*x**2*sqrt(d + e*x)/(143*e) + 848*A*a**2*b**2*d**3*x**3*sqrt(d + e*x)/429 + 1832*A*a**2*b**2*d**2
*e*x**4*sqrt(d + e*x)/429 + 480*A*a**2*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 12*A*a**2*b**2*e**3*x**6*sqrt(d +
e*x)/13 - 128*A*a*b**3*d**7*sqrt(d + e*x)/(6435*e**4) + 64*A*a*b**3*d**6*x*sqrt(d + e*x)/(6435*e**3) - 16*A*a*
b**3*d**5*x**2*sqrt(d + e*x)/(2145*e**2) + 8*A*a*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 1280*A*a*b**3*d**3*x*
*4*sqrt(d + e*x)/1287 + 1648*A*a*b**3*d**2*e*x**5*sqrt(d + e*x)/715 + 368*A*a*b**3*d*e**2*x**6*sqrt(d + e*x)/1
95 + 8*A*a*b**3*e**3*x**7*sqrt(d + e*x)/15 + 256*A*b**4*d**8*sqrt(d + e*x)/(109395*e**5) - 128*A*b**4*d**7*x*s
qrt(d + e*x)/(109395*e**4) + 32*A*b**4*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 16*A*b**4*d**5*x**3*sqrt(d + e*x
)/(21879*e**2) + 14*A*b**4*d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424*A*b**4*d**3*x**5*sqrt(d + e*x)/12155 + 160
4*A*b**4*d**2*e*x**6*sqrt(d + e*x)/3315 + 104*A*b**4*d*e**2*x**7*sqrt(d + e*x)/255 + 2*A*b**4*e**3*x**8*sqrt(d
+ e*x)/17 - 4*B*a**4*d**5*sqrt(d + e*x)/(99*e**2) + 2*B*a**4*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a**4*d**3*x**
2*sqrt(d + e*x)/33 + 92*B*a**4*d**2*e*x**3*sqrt(d + e*x)/99 + 68*B*a**4*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a**
4*e**3*x**5*sqrt(d + e*x)/11 + 64*B*a**3*b*d**6*sqrt(d + e*x)/(1287*e**3) - 32*B*a**3*b*d**5*x*sqrt(d + e*x)/(
1287*e**2) + 8*B*a**3*b*d**4*x**2*sqrt(d + e*x)/(429*e) + 1696*B*a**3*b*d**3*x**3*sqrt(d + e*x)/1287 + 3664*B*
a**3*b*d**2*e*x**4*sqrt(d + e*x)/1287 + 320*B*a**3*b*d*e**2*x**5*sqrt(d + e*x)/143 + 8*B*a**3*b*e**3*x**6*sqrt
(d + e*x)/13 - 64*B*a**2*b**2*d**7*sqrt(d + e*x)/(2145*e**4) + 32*B*a**2*b**2*d**6*x*sqrt(d + e*x)/(2145*e**3)
- 8*B*a**2*b**2*d**5*x**2*sqrt(d + e*x)/(715*e**2) + 4*B*a**2*b**2*d**4*x**3*sqrt(d + e*x)/(429*e) + 640*B*a*
*2*b**2*d**3*x**4*sqrt(d + e*x)/429 + 2472*B*a**2*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 184*B*a**2*b**2*d*e**2*
x**6*sqrt(d + e*x)/65 + 4*B*a**2*b**2*e**3*x**7*sqrt(d + e*x)/5 + 1024*B*a*b**3*d**8*sqrt(d + e*x)/(109395*e**
5) - 512*B*a*b**3*d**7*x*sqrt(d + e*x)/(109395*e**4) + 128*B*a*b**3*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 64*
B*a*b**3*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 56*B*a*b**3*d**4*x**4*sqrt(d + e*x)/(21879*e) + 9696*B*a*b**3*
d**3*x**5*sqrt(d + e*x)/12155 + 6416*B*a*b**3*d**2*e*x**6*sqrt(d + e*x)/3315 + 416*B*a*b**3*d*e**2*x**7*sqrt(d
+ e*x)/255 + 8*B*a*b**3*e**3*x**8*sqrt(d + e*x)/17 - 512*B*b**4*d**9*sqrt(d + e*x)/(415701*e**6) + 256*B*b**4
*d**8*x*sqrt(d + e*x)/(415701*e**5) - 64*B*b**4*d**7*x**2*sqrt(d + e*x)/(138567*e**4) + 160*B*b**4*d**6*x**3*s
qrt(d + e*x)/(415701*e**3) - 140*B*b**4*d**5*x**4*sqrt(d + e*x)/(415701*e**2) + 14*B*b**4*d**4*x**5*sqrt(d + e
*x)/(46189*e) + 2096*B*b**4*d**3*x**6*sqrt(d + e*x)/12597 + 404*B*b**4*d**2*e*x**7*sqrt(d + e*x)/969 + 116*B*b
**4*d*e**2*x**8*sqrt(d + e*x)/323 + 2*B*b**4*e**3*x**9*sqrt(d + e*x)/19, Ne(e, 0)), (d**(7/2)*(A*a**4*x + 2*A*
a**3*b*x**2 + 2*A*a**2*b**2*x**3 + A*a*b**3*x**4 + A*b**4*x**5/5 + B*a**4*x**2/2 + 4*B*a**3*b*x**3/3 + 3*B*a**
2*b**2*x**4/2 + 4*B*a*b**3*x**5/5 + B*b**4*x**6/6), True))
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