3.1727 \(\int (a+b x)^2 (A+B x) (d+e x)^{5/2} \, dx\)

Optimal. Leaf size=128 \[ -\frac {2 b (d+e x)^{11/2} (-2 a B e-A b e+3 b B d)}{11 e^4}+\frac {2 (d+e x)^{9/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{9 e^4}-\frac {2 (d+e x)^{7/2} (b d-a e)^2 (B d-A e)}{7 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 e^4} \]

[Out]

-2/7*(-a*e+b*d)^2*(-A*e+B*d)*(e*x+d)^(7/2)/e^4+2/9*(-a*e+b*d)*(-2*A*b*e-B*a*e+3*B*b*d)*(e*x+d)^(9/2)/e^4-2/11*
b*(-A*b*e-2*B*a*e+3*B*b*d)*(e*x+d)^(11/2)/e^4+2/13*b^2*B*(e*x+d)^(13/2)/e^4

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Rubi [A]  time = 0.05, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {77} \[ -\frac {2 b (d+e x)^{11/2} (-2 a B e-A b e+3 b B d)}{11 e^4}+\frac {2 (d+e x)^{9/2} (b d-a e) (-a B e-2 A b e+3 b B d)}{9 e^4}-\frac {2 (d+e x)^{7/2} (b d-a e)^2 (B d-A e)}{7 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 e^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^2*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(-2*(b*d - a*e)^2*(B*d - A*e)*(d + e*x)^(7/2))/(7*e^4) + (2*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x)^
(9/2))/(9*e^4) - (2*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^(11/2))/(11*e^4) + (2*b^2*B*(d + e*x)^(13/2))/(13*
e^4)

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)^2 (A+B x) (d+e x)^{5/2} \, dx &=\int \left (\frac {(-b d+a e)^2 (-B d+A e) (d+e x)^{5/2}}{e^3}+\frac {(-b d+a e) (-3 b B d+2 A b e+a B e) (d+e x)^{7/2}}{e^3}+\frac {b (-3 b B d+A b e+2 a B e) (d+e x)^{9/2}}{e^3}+\frac {b^2 B (d+e x)^{11/2}}{e^3}\right ) \, dx\\ &=-\frac {2 (b d-a e)^2 (B d-A e) (d+e x)^{7/2}}{7 e^4}+\frac {2 (b d-a e) (3 b B d-2 A b e-a B e) (d+e x)^{9/2}}{9 e^4}-\frac {2 b (3 b B d-A b e-2 a B e) (d+e x)^{11/2}}{11 e^4}+\frac {2 b^2 B (d+e x)^{13/2}}{13 e^4}\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 107, normalized size = 0.84 \[ \frac {2 (d+e x)^{7/2} \left (-819 b (d+e x)^2 (-2 a B e-A b e+3 b B d)+1001 (d+e x) (b d-a e) (-a B e-2 A b e+3 b B d)-1287 (b d-a e)^2 (B d-A e)+693 b^2 B (d+e x)^3\right )}{9009 e^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^2*(A + B*x)*(d + e*x)^(5/2),x]

[Out]

(2*(d + e*x)^(7/2)*(-1287*(b*d - a*e)^2*(B*d - A*e) + 1001*(b*d - a*e)*(3*b*B*d - 2*A*b*e - a*B*e)*(d + e*x) -
819*b*(3*b*B*d - A*b*e - 2*a*B*e)*(d + e*x)^2 + 693*b^2*B*(d + e*x)^3))/(9009*e^4)

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fricas [B]  time = 0.98, size = 356, normalized size = 2.78 \[ \frac {2 \, {\left (693 \, B b^{2} e^{6} x^{6} - 48 \, B b^{2} d^{6} + 1287 \, A a^{2} d^{3} e^{3} + 104 \, {\left (2 \, B a b + A b^{2}\right )} d^{5} e - 286 \, {\left (B a^{2} + 2 \, A a b\right )} d^{4} e^{2} + 63 \, {\left (27 \, B b^{2} d e^{5} + 13 \, {\left (2 \, B a b + A b^{2}\right )} e^{6}\right )} x^{5} + 7 \, {\left (159 \, B b^{2} d^{2} e^{4} + 299 \, {\left (2 \, B a b + A b^{2}\right )} d e^{5} + 143 \, {\left (B a^{2} + 2 \, A a b\right )} e^{6}\right )} x^{4} + {\left (15 \, B b^{2} d^{3} e^{3} + 1287 \, A a^{2} e^{6} + 1469 \, {\left (2 \, B a b + A b^{2}\right )} d^{2} e^{4} + 2717 \, {\left (B a^{2} + 2 \, A a b\right )} d e^{5}\right )} x^{3} - 3 \, {\left (6 \, B b^{2} d^{4} e^{2} - 1287 \, A a^{2} d e^{5} - 13 \, {\left (2 \, B a b + A b^{2}\right )} d^{3} e^{3} - 715 \, {\left (B a^{2} + 2 \, A a b\right )} d^{2} e^{4}\right )} x^{2} + {\left (24 \, B b^{2} d^{5} e + 3861 \, A a^{2} d^{2} e^{4} - 52 \, {\left (2 \, B a b + A b^{2}\right )} d^{4} e^{2} + 143 \, {\left (B a^{2} + 2 \, A a b\right )} d^{3} e^{3}\right )} x\right )} \sqrt {e x + d}}{9009 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(5/2),x, algorithm="fricas")

[Out]

2/9009*(693*B*b^2*e^6*x^6 - 48*B*b^2*d^6 + 1287*A*a^2*d^3*e^3 + 104*(2*B*a*b + A*b^2)*d^5*e - 286*(B*a^2 + 2*A
*a*b)*d^4*e^2 + 63*(27*B*b^2*d*e^5 + 13*(2*B*a*b + A*b^2)*e^6)*x^5 + 7*(159*B*b^2*d^2*e^4 + 299*(2*B*a*b + A*b
^2)*d*e^5 + 143*(B*a^2 + 2*A*a*b)*e^6)*x^4 + (15*B*b^2*d^3*e^3 + 1287*A*a^2*e^6 + 1469*(2*B*a*b + A*b^2)*d^2*e
^4 + 2717*(B*a^2 + 2*A*a*b)*d*e^5)*x^3 - 3*(6*B*b^2*d^4*e^2 - 1287*A*a^2*d*e^5 - 13*(2*B*a*b + A*b^2)*d^3*e^3
- 715*(B*a^2 + 2*A*a*b)*d^2*e^4)*x^2 + (24*B*b^2*d^5*e + 3861*A*a^2*d^2*e^4 - 52*(2*B*a*b + A*b^2)*d^4*e^2 + 1
43*(B*a^2 + 2*A*a*b)*d^3*e^3)*x)*sqrt(e*x + d)/e^4

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giac [B]  time = 1.43, size = 1368, normalized size = 10.69 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(5/2),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^2*d^3*e^(-1) + 30030*((x*e + d)^(3/2) - 3*sqrt(x*e +
d)*d)*A*a*b*d^3*e^(-1) + 6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a*b*d^3*e^(-
2) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*b^2*d^3*e^(-2) + 1287*(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*b^2*d^3*e^(-3) + 9009*(3*(x
*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^2*d^2*e^(-1) + 18018*(3*(x*e + d)^(5/2) - 10*
(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a*b*d^2*e^(-1) + 7722*(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*b*d^2*e^(-2) + 3861*(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*b^2*d^2*e^(-2) + 429*(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*b^2*d^2*e^(-3) + 45045
*sqrt(x*e + d)*A*a^2*d^3 + 45045*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^2*d^2 + 3861*(5*(x*e + d)^(7/2) - 2
1*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^2*d*e^(-1) + 7722*(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*d*e^(-1) + 858*(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*b*
d*e^(-2) + 429*(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^2*d*e^(-2) + 195*(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^2*d*e^(-3) + 9009*(
3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*d + 143*(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*e^(-1) + 286*(
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*e^(-1) + 130*(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*e^(-2) + 65*(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*sq
rt(x*e + d)*d^5)*A*b^2*e^(-2) + 15*(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
^2*e^(-3) + 1287*(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)*e^(-1)

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maple [A]  time = 0.01, size = 169, normalized size = 1.32 \[ \frac {2 \left (e x +d \right )^{\frac {7}{2}} \left (693 b^{2} B \,x^{3} e^{3}+819 A \,b^{2} e^{3} x^{2}+1638 B a b \,e^{3} x^{2}-378 B \,b^{2} d \,e^{2} x^{2}+2002 A a b \,e^{3} x -364 A \,b^{2} d \,e^{2} x +1001 B \,a^{2} e^{3} x -728 B a b d \,e^{2} x +168 B \,b^{2} d^{2} e x +1287 a^{2} A \,e^{3}-572 A a b d \,e^{2}+104 A \,b^{2} d^{2} e -286 B \,a^{2} d \,e^{2}+208 B a b \,d^{2} e -48 B \,b^{2} d^{3}\right )}{9009 e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^2*(B*x+A)*(e*x+d)^(5/2),x)

[Out]

2/9009*(e*x+d)^(7/2)*(693*B*b^2*e^3*x^3+819*A*b^2*e^3*x^2+1638*B*a*b*e^3*x^2-378*B*b^2*d*e^2*x^2+2002*A*a*b*e^
3*x-364*A*b^2*d*e^2*x+1001*B*a^2*e^3*x-728*B*a*b*d*e^2*x+168*B*b^2*d^2*e*x+1287*A*a^2*e^3-572*A*a*b*d*e^2+104*
A*b^2*d^2*e-286*B*a^2*d*e^2+208*B*a*b*d^2*e-48*B*b^2*d^3)/e^4

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maxima [A]  time = 0.52, size = 159, normalized size = 1.24 \[ \frac {2 \, {\left (693 \, {\left (e x + d\right )}^{\frac {13}{2}} B b^{2} - 819 \, {\left (3 \, B b^{2} d - {\left (2 \, B a b + A b^{2}\right )} e\right )} {\left (e x + d\right )}^{\frac {11}{2}} + 1001 \, {\left (3 \, B b^{2} d^{2} - 2 \, {\left (2 \, B a b + A b^{2}\right )} d e + {\left (B a^{2} + 2 \, A a b\right )} e^{2}\right )} {\left (e x + d\right )}^{\frac {9}{2}} - 1287 \, {\left (B b^{2} d^{3} - A a^{2} e^{3} - {\left (2 \, B a b + A b^{2}\right )} d^{2} e + {\left (B a^{2} + 2 \, A a b\right )} d e^{2}\right )} {\left (e x + d\right )}^{\frac {7}{2}}\right )}}{9009 \, e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^2*(B*x+A)*(e*x+d)^(5/2),x, algorithm="maxima")

[Out]

2/9009*(693*(e*x + d)^(13/2)*B*b^2 - 819*(3*B*b^2*d - (2*B*a*b + A*b^2)*e)*(e*x + d)^(11/2) + 1001*(3*B*b^2*d^
2 - 2*(2*B*a*b + A*b^2)*d*e + (B*a^2 + 2*A*a*b)*e^2)*(e*x + d)^(9/2) - 1287*(B*b^2*d^3 - A*a^2*e^3 - (2*B*a*b
+ A*b^2)*d^2*e + (B*a^2 + 2*A*a*b)*d*e^2)*(e*x + d)^(7/2))/e^4

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mupad [B]  time = 1.21, size = 115, normalized size = 0.90 \[ \frac {{\left (d+e\,x\right )}^{11/2}\,\left (2\,A\,b^2\,e-6\,B\,b^2\,d+4\,B\,a\,b\,e\right )}{11\,e^4}+\frac {2\,B\,b^2\,{\left (d+e\,x\right )}^{13/2}}{13\,e^4}+\frac {2\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b\,e+B\,a\,e-3\,B\,b\,d\right )}{9\,e^4}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{7/2}}{7\,e^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^2*(d + e*x)^(5/2),x)

[Out]

((d + e*x)^(11/2)*(2*A*b^2*e - 6*B*b^2*d + 4*B*a*b*e))/(11*e^4) + (2*B*b^2*(d + e*x)^(13/2))/(13*e^4) + (2*(a*
e - b*d)*(d + e*x)^(9/2)*(2*A*b*e + B*a*e - 3*B*b*d))/(9*e^4) + (2*(A*e - B*d)*(a*e - b*d)^2*(d + e*x)^(7/2))/
(7*e^4)

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sympy [A]  time = 4.92, size = 857, normalized size = 6.70 \[ \begin {cases} \frac {2 A a^{2} d^{3} \sqrt {d + e x}}{7 e} + \frac {6 A a^{2} d^{2} x \sqrt {d + e x}}{7} + \frac {6 A a^{2} d e x^{2} \sqrt {d + e x}}{7} + \frac {2 A a^{2} e^{2} x^{3} \sqrt {d + e x}}{7} - \frac {8 A a b d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {4 A a b d^{3} x \sqrt {d + e x}}{63 e} + \frac {20 A a b d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {76 A a b d e x^{3} \sqrt {d + e x}}{63} + \frac {4 A a b e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {16 A b^{2} d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {8 A b^{2} d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {2 A b^{2} d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {226 A b^{2} d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {46 A b^{2} d e x^{4} \sqrt {d + e x}}{99} + \frac {2 A b^{2} e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {4 B a^{2} d^{4} \sqrt {d + e x}}{63 e^{2}} + \frac {2 B a^{2} d^{3} x \sqrt {d + e x}}{63 e} + \frac {10 B a^{2} d^{2} x^{2} \sqrt {d + e x}}{21} + \frac {38 B a^{2} d e x^{3} \sqrt {d + e x}}{63} + \frac {2 B a^{2} e^{2} x^{4} \sqrt {d + e x}}{9} + \frac {32 B a b d^{5} \sqrt {d + e x}}{693 e^{3}} - \frac {16 B a b d^{4} x \sqrt {d + e x}}{693 e^{2}} + \frac {4 B a b d^{3} x^{2} \sqrt {d + e x}}{231 e} + \frac {452 B a b d^{2} x^{3} \sqrt {d + e x}}{693} + \frac {92 B a b d e x^{4} \sqrt {d + e x}}{99} + \frac {4 B a b e^{2} x^{5} \sqrt {d + e x}}{11} - \frac {32 B b^{2} d^{6} \sqrt {d + e x}}{3003 e^{4}} + \frac {16 B b^{2} d^{5} x \sqrt {d + e x}}{3003 e^{3}} - \frac {4 B b^{2} d^{4} x^{2} \sqrt {d + e x}}{1001 e^{2}} + \frac {10 B b^{2} d^{3} x^{3} \sqrt {d + e x}}{3003 e} + \frac {106 B b^{2} d^{2} x^{4} \sqrt {d + e x}}{429} + \frac {54 B b^{2} d e x^{5} \sqrt {d + e x}}{143} + \frac {2 B b^{2} e^{2} x^{6} \sqrt {d + e x}}{13} & \text {for}\: e \neq 0 \\d^{\frac {5}{2}} \left (A a^{2} x + A a b x^{2} + \frac {A b^{2} x^{3}}{3} + \frac {B a^{2} x^{2}}{2} + \frac {2 B a b x^{3}}{3} + \frac {B b^{2} x^{4}}{4}\right ) & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**2*(B*x+A)*(e*x+d)**(5/2),x)

[Out]

Piecewise((2*A*a**2*d**3*sqrt(d + e*x)/(7*e) + 6*A*a**2*d**2*x*sqrt(d + e*x)/7 + 6*A*a**2*d*e*x**2*sqrt(d + e*
x)/7 + 2*A*a**2*e**2*x**3*sqrt(d + e*x)/7 - 8*A*a*b*d**4*sqrt(d + e*x)/(63*e**2) + 4*A*a*b*d**3*x*sqrt(d + e*x
)/(63*e) + 20*A*a*b*d**2*x**2*sqrt(d + e*x)/21 + 76*A*a*b*d*e*x**3*sqrt(d + e*x)/63 + 4*A*a*b*e**2*x**4*sqrt(d
+ e*x)/9 + 16*A*b**2*d**5*sqrt(d + e*x)/(693*e**3) - 8*A*b**2*d**4*x*sqrt(d + e*x)/(693*e**2) + 2*A*b**2*d**3
*x**2*sqrt(d + e*x)/(231*e) + 226*A*b**2*d**2*x**3*sqrt(d + e*x)/693 + 46*A*b**2*d*e*x**4*sqrt(d + e*x)/99 + 2
*A*b**2*e**2*x**5*sqrt(d + e*x)/11 - 4*B*a**2*d**4*sqrt(d + e*x)/(63*e**2) + 2*B*a**2*d**3*x*sqrt(d + e*x)/(63
*e) + 10*B*a**2*d**2*x**2*sqrt(d + e*x)/21 + 38*B*a**2*d*e*x**3*sqrt(d + e*x)/63 + 2*B*a**2*e**2*x**4*sqrt(d +
e*x)/9 + 32*B*a*b*d**5*sqrt(d + e*x)/(693*e**3) - 16*B*a*b*d**4*x*sqrt(d + e*x)/(693*e**2) + 4*B*a*b*d**3*x**
2*sqrt(d + e*x)/(231*e) + 452*B*a*b*d**2*x**3*sqrt(d + e*x)/693 + 92*B*a*b*d*e*x**4*sqrt(d + e*x)/99 + 4*B*a*b
*e**2*x**5*sqrt(d + e*x)/11 - 32*B*b**2*d**6*sqrt(d + e*x)/(3003*e**4) + 16*B*b**2*d**5*x*sqrt(d + e*x)/(3003*
e**3) - 4*B*b**2*d**4*x**2*sqrt(d + e*x)/(1001*e**2) + 10*B*b**2*d**3*x**3*sqrt(d + e*x)/(3003*e) + 106*B*b**2
*d**2*x**4*sqrt(d + e*x)/429 + 54*B*b**2*d*e*x**5*sqrt(d + e*x)/143 + 2*B*b**2*e**2*x**6*sqrt(d + e*x)/13, Ne(
e, 0)), (d**(5/2)*(A*a**2*x + A*a*b*x**2 + A*b**2*x**3/3 + B*a**2*x**2/2 + 2*B*a*b*x**3/3 + B*b**2*x**4/4), Tr
ue))

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