3.17.2 \(\int \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=169 \[ -\frac {2 b^2 \sqrt {d+e x} (-3 a B e-A b e+4 b B d)}{e^5}-\frac {6 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 \sqrt {d+e x}}+\frac {2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (d+e x)^{3/2}}-\frac {2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac {2 b^3 B (d+e x)^{3/2}}{3 e^5} \]

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Rubi [A]  time = 0.07, antiderivative size = 169, 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} \begin {gather*} -\frac {2 b^2 \sqrt {d+e x} (-3 a B e-A b e+4 b B d)}{e^5}-\frac {6 b (b d-a e) (-a B e-A b e+2 b B d)}{e^5 \sqrt {d+e x}}+\frac {2 (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{3 e^5 (d+e x)^{3/2}}-\frac {2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac {2 b^3 B (d+e x)^{3/2}}{3 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((a + b*x)^3*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(-2*(b*d - a*e)^3*(B*d - A*e))/(5*e^5*(d + e*x)^(5/2)) + (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e))/(3*e^5*
(d + e*x)^(3/2)) - (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e))/(e^5*Sqrt[d + e*x]) - (2*b^2*(4*b*B*d - A*b*e -
3*a*B*e)*Sqrt[d + e*x])/e^5 + (2*b^3*B*(d + e*x)^(3/2))/(3*e^5)

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 \frac {(a+b x)^3 (A+B x)}{(d+e x)^{7/2}} \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e)}{e^4 (d+e x)^{7/2}}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e)}{e^4 (d+e x)^{5/2}}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e)}{e^4 (d+e x)^{3/2}}+\frac {b^2 (-4 b B d+A b e+3 a B e)}{e^4 \sqrt {d+e x}}+\frac {b^3 B \sqrt {d+e x}}{e^4}\right ) \, dx\\ &=-\frac {2 (b d-a e)^3 (B d-A e)}{5 e^5 (d+e x)^{5/2}}+\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e)}{3 e^5 (d+e x)^{3/2}}-\frac {6 b (b d-a e) (2 b B d-A b e-a B e)}{e^5 \sqrt {d+e x}}-\frac {2 b^2 (4 b B d-A b e-3 a B e) \sqrt {d+e x}}{e^5}+\frac {2 b^3 B (d+e x)^{3/2}}{3 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 145, normalized size = 0.86 \begin {gather*} \frac {2 \left (-15 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)-45 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)+5 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)-3 (b d-a e)^3 (B d-A e)+5 b^3 B (d+e x)^4\right )}{15 e^5 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(2*(-3*(b*d - a*e)^3*(B*d - A*e) + 5*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x) - 45*b*(b*d - a*e)*(2
*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 15*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^3 + 5*b^3*B*(d + e*x)^4))/(
15*e^5*(d + e*x)^(5/2))

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IntegrateAlgebraic [B]  time = 0.10, size = 346, normalized size = 2.05 \begin {gather*} \frac {2 \left (-3 a^3 A e^4-5 a^3 B e^3 (d+e x)+3 a^3 B d e^3-15 a^2 A b e^3 (d+e x)+9 a^2 A b d e^3-9 a^2 b B d^2 e^2+30 a^2 b B d e^2 (d+e x)-45 a^2 b B e^2 (d+e x)^2-9 a A b^2 d^2 e^2+30 a A b^2 d e^2 (d+e x)-45 a A b^2 e^2 (d+e x)^2+9 a b^2 B d^3 e-45 a b^2 B d^2 e (d+e x)+135 a b^2 B d e (d+e x)^2+45 a b^2 B e (d+e x)^3+3 A b^3 d^3 e-15 A b^3 d^2 e (d+e x)+45 A b^3 d e (d+e x)^2+15 A b^3 e (d+e x)^3-3 b^3 B d^4+20 b^3 B d^3 (d+e x)-90 b^3 B d^2 (d+e x)^2-60 b^3 B d (d+e x)^3+5 b^3 B (d+e x)^4\right )}{15 e^5 (d+e x)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((a + b*x)^3*(A + B*x))/(d + e*x)^(7/2),x]

[Out]

(2*(-3*b^3*B*d^4 + 3*A*b^3*d^3*e + 9*a*b^2*B*d^3*e - 9*a*A*b^2*d^2*e^2 - 9*a^2*b*B*d^2*e^2 + 9*a^2*A*b*d*e^3 +
3*a^3*B*d*e^3 - 3*a^3*A*e^4 + 20*b^3*B*d^3*(d + e*x) - 15*A*b^3*d^2*e*(d + e*x) - 45*a*b^2*B*d^2*e*(d + e*x)
+ 30*a*A*b^2*d*e^2*(d + e*x) + 30*a^2*b*B*d*e^2*(d + e*x) - 15*a^2*A*b*e^3*(d + e*x) - 5*a^3*B*e^3*(d + e*x) -
90*b^3*B*d^2*(d + e*x)^2 + 45*A*b^3*d*e*(d + e*x)^2 + 135*a*b^2*B*d*e*(d + e*x)^2 - 45*a*A*b^2*e^2*(d + e*x)^
2 - 45*a^2*b*B*e^2*(d + e*x)^2 - 60*b^3*B*d*(d + e*x)^3 + 15*A*b^3*e*(d + e*x)^3 + 45*a*b^2*B*e*(d + e*x)^3 +
5*b^3*B*(d + e*x)^4))/(15*e^5*(d + e*x)^(5/2))

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fricas [A]  time = 1.54, size = 294, normalized size = 1.74 \begin {gather*} \frac {2 \, {\left (5 \, B b^{3} e^{4} x^{4} - 128 \, B b^{3} d^{4} - 3 \, A a^{3} e^{4} + 48 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e - 24 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 2 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} - 5 \, {\left (8 \, B b^{3} d e^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} - 15 \, {\left (16 \, B b^{3} d^{2} e^{2} - 6 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 3 \, {\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} - 5 \, {\left (64 \, B b^{3} d^{3} e - 24 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 12 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{3} + {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x\right )} \sqrt {e x + d}}{15 \, {\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/15*(5*B*b^3*e^4*x^4 - 128*B*b^3*d^4 - 3*A*a^3*e^4 + 48*(3*B*a*b^2 + A*b^3)*d^3*e - 24*(B*a^2*b + A*a*b^2)*d^
2*e^2 - 2*(B*a^3 + 3*A*a^2*b)*d*e^3 - 5*(8*B*b^3*d*e^3 - 3*(3*B*a*b^2 + A*b^3)*e^4)*x^3 - 15*(16*B*b^3*d^2*e^2
- 6*(3*B*a*b^2 + A*b^3)*d*e^3 + 3*(B*a^2*b + A*a*b^2)*e^4)*x^2 - 5*(64*B*b^3*d^3*e - 24*(3*B*a*b^2 + A*b^3)*d
^2*e^2 + 12*(B*a^2*b + A*a*b^2)*d*e^3 + (B*a^3 + 3*A*a^2*b)*e^4)*x)*sqrt(e*x + d)/(e^8*x^3 + 3*d*e^7*x^2 + 3*d
^2*e^6*x + d^3*e^5)

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giac [B]  time = 1.37, size = 364, normalized size = 2.15 \begin {gather*} \frac {2}{3} \, {\left ({\left (x e + d\right )}^{\frac {3}{2}} B b^{3} e^{10} - 12 \, \sqrt {x e + d} B b^{3} d e^{10} + 9 \, \sqrt {x e + d} B a b^{2} e^{11} + 3 \, \sqrt {x e + d} A b^{3} e^{11}\right )} e^{\left (-15\right )} - \frac {2 \, {\left (90 \, {\left (x e + d\right )}^{2} B b^{3} d^{2} - 20 \, {\left (x e + d\right )} B b^{3} d^{3} + 3 \, B b^{3} d^{4} - 135 \, {\left (x e + d\right )}^{2} B a b^{2} d e - 45 \, {\left (x e + d\right )}^{2} A b^{3} d e + 45 \, {\left (x e + d\right )} B a b^{2} d^{2} e + 15 \, {\left (x e + d\right )} A b^{3} d^{2} e - 9 \, B a b^{2} d^{3} e - 3 \, A b^{3} d^{3} e + 45 \, {\left (x e + d\right )}^{2} B a^{2} b e^{2} + 45 \, {\left (x e + d\right )}^{2} A a b^{2} e^{2} - 30 \, {\left (x e + d\right )} B a^{2} b d e^{2} - 30 \, {\left (x e + d\right )} A a b^{2} d e^{2} + 9 \, B a^{2} b d^{2} e^{2} + 9 \, A a b^{2} d^{2} e^{2} + 5 \, {\left (x e + d\right )} B a^{3} e^{3} + 15 \, {\left (x e + d\right )} A a^{2} b e^{3} - 3 \, B a^{3} d e^{3} - 9 \, A a^{2} b d e^{3} + 3 \, A a^{3} e^{4}\right )} e^{\left (-5\right )}}{15 \, {\left (x e + d\right )}^{\frac {5}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/3*((x*e + d)^(3/2)*B*b^3*e^10 - 12*sqrt(x*e + d)*B*b^3*d*e^10 + 9*sqrt(x*e + d)*B*a*b^2*e^11 + 3*sqrt(x*e +
d)*A*b^3*e^11)*e^(-15) - 2/15*(90*(x*e + d)^2*B*b^3*d^2 - 20*(x*e + d)*B*b^3*d^3 + 3*B*b^3*d^4 - 135*(x*e + d)
^2*B*a*b^2*d*e - 45*(x*e + d)^2*A*b^3*d*e + 45*(x*e + d)*B*a*b^2*d^2*e + 15*(x*e + d)*A*b^3*d^2*e - 9*B*a*b^2*
d^3*e - 3*A*b^3*d^3*e + 45*(x*e + d)^2*B*a^2*b*e^2 + 45*(x*e + d)^2*A*a*b^2*e^2 - 30*(x*e + d)*B*a^2*b*d*e^2 -
30*(x*e + d)*A*a*b^2*d*e^2 + 9*B*a^2*b*d^2*e^2 + 9*A*a*b^2*d^2*e^2 + 5*(x*e + d)*B*a^3*e^3 + 15*(x*e + d)*A*a
^2*b*e^3 - 3*B*a^3*d*e^3 - 9*A*a^2*b*d*e^3 + 3*A*a^3*e^4)*e^(-5)/(x*e + d)^(5/2)

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maple [A]  time = 0.01, size = 301, normalized size = 1.78 \begin {gather*} -\frac {2 \left (-5 B \,b^{3} x^{4} e^{4}-15 A \,b^{3} e^{4} x^{3}-45 B a \,b^{2} e^{4} x^{3}+40 B \,b^{3} d \,e^{3} x^{3}+45 A a \,b^{2} e^{4} x^{2}-90 A \,b^{3} d \,e^{3} x^{2}+45 B \,a^{2} b \,e^{4} x^{2}-270 B a \,b^{2} d \,e^{3} x^{2}+240 B \,b^{3} d^{2} e^{2} x^{2}+15 A \,a^{2} b \,e^{4} x +60 A a \,b^{2} d \,e^{3} x -120 A \,b^{3} d^{2} e^{2} x +5 B \,a^{3} e^{4} x +60 B \,a^{2} b d \,e^{3} x -360 B a \,b^{2} d^{2} e^{2} x +320 B \,b^{3} d^{3} e x +3 a^{3} A \,e^{4}+6 A \,a^{2} b d \,e^{3}+24 A a \,b^{2} d^{2} e^{2}-48 A \,b^{3} d^{3} e +2 B \,a^{3} d \,e^{3}+24 B \,a^{2} b \,d^{2} e^{2}-144 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right )}{15 \left (e x +d \right )^{\frac {5}{2}} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x)

[Out]

-2/15/(e*x+d)^(5/2)*(-5*B*b^3*e^4*x^4-15*A*b^3*e^4*x^3-45*B*a*b^2*e^4*x^3+40*B*b^3*d*e^3*x^3+45*A*a*b^2*e^4*x^
2-90*A*b^3*d*e^3*x^2+45*B*a^2*b*e^4*x^2-270*B*a*b^2*d*e^3*x^2+240*B*b^3*d^2*e^2*x^2+15*A*a^2*b*e^4*x+60*A*a*b^
2*d*e^3*x-120*A*b^3*d^2*e^2*x+5*B*a^3*e^4*x+60*B*a^2*b*d*e^3*x-360*B*a*b^2*d^2*e^2*x+320*B*b^3*d^3*e*x+3*A*a^3
*e^4+6*A*a^2*b*d*e^3+24*A*a*b^2*d^2*e^2-48*A*b^3*d^3*e+2*B*a^3*d*e^3+24*B*a^2*b*d^2*e^2-144*B*a*b^2*d^3*e+128*
B*b^3*d^4)/e^5

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maxima [A]  time = 0.57, size = 273, normalized size = 1.62 \begin {gather*} \frac {2 \, {\left (\frac {5 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} B b^{3} - 3 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} \sqrt {e x + d}\right )}}{e^{4}} - \frac {3 \, B b^{3} d^{4} + 3 \, A a^{3} e^{4} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 9 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - 3 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 45 \, {\left (2 \, B b^{3} d^{2} - {\left (3 \, B a b^{2} + A b^{3}\right )} d e + {\left (B a^{2} b + A a b^{2}\right )} e^{2}\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (4 \, B b^{3} d^{3} - 3 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e + 6 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{3}\right )} {\left (e x + d\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{4}}\right )}}{15 \, e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)/(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/15*(5*((e*x + d)^(3/2)*B*b^3 - 3*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*sqrt(e*x + d))/e^4 - (3*B*b^3*d^4 + 3*A
*a^3*e^4 - 3*(3*B*a*b^2 + A*b^3)*d^3*e + 9*(B*a^2*b + A*a*b^2)*d^2*e^2 - 3*(B*a^3 + 3*A*a^2*b)*d*e^3 + 45*(2*B
*b^3*d^2 - (3*B*a*b^2 + A*b^3)*d*e + (B*a^2*b + A*a*b^2)*e^2)*(e*x + d)^2 - 5*(4*B*b^3*d^3 - 3*(3*B*a*b^2 + A*
b^3)*d^2*e + 6*(B*a^2*b + A*a*b^2)*d*e^2 - (B*a^3 + 3*A*a^2*b)*e^3)*(e*x + d))/((e*x + d)^(5/2)*e^4))/e

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mupad [B]  time = 0.13, size = 300, normalized size = 1.78 \begin {gather*} -\frac {2\,\left (2\,B\,a^3\,d\,e^3+5\,B\,a^3\,e^4\,x+3\,A\,a^3\,e^4+24\,B\,a^2\,b\,d^2\,e^2+60\,B\,a^2\,b\,d\,e^3\,x+6\,A\,a^2\,b\,d\,e^3+45\,B\,a^2\,b\,e^4\,x^2+15\,A\,a^2\,b\,e^4\,x-144\,B\,a\,b^2\,d^3\,e-360\,B\,a\,b^2\,d^2\,e^2\,x+24\,A\,a\,b^2\,d^2\,e^2-270\,B\,a\,b^2\,d\,e^3\,x^2+60\,A\,a\,b^2\,d\,e^3\,x-45\,B\,a\,b^2\,e^4\,x^3+45\,A\,a\,b^2\,e^4\,x^2+128\,B\,b^3\,d^4+320\,B\,b^3\,d^3\,e\,x-48\,A\,b^3\,d^3\,e+240\,B\,b^3\,d^2\,e^2\,x^2-120\,A\,b^3\,d^2\,e^2\,x+40\,B\,b^3\,d\,e^3\,x^3-90\,A\,b^3\,d\,e^3\,x^2-5\,B\,b^3\,e^4\,x^4-15\,A\,b^3\,e^4\,x^3\right )}{15\,e^5\,{\left (d+e\,x\right )}^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((A + B*x)*(a + b*x)^3)/(d + e*x)^(7/2),x)

[Out]

-(2*(3*A*a^3*e^4 + 128*B*b^3*d^4 - 48*A*b^3*d^3*e + 2*B*a^3*d*e^3 + 5*B*a^3*e^4*x - 15*A*b^3*e^4*x^3 - 5*B*b^3
*e^4*x^4 + 320*B*b^3*d^3*e*x + 24*A*a*b^2*d^2*e^2 + 24*B*a^2*b*d^2*e^2 + 45*A*a*b^2*e^4*x^2 + 45*B*a^2*b*e^4*x
^2 - 45*B*a*b^2*e^4*x^3 - 120*A*b^3*d^2*e^2*x - 90*A*b^3*d*e^3*x^2 + 40*B*b^3*d*e^3*x^3 + 240*B*b^3*d^2*e^2*x^
2 + 6*A*a^2*b*d*e^3 - 144*B*a*b^2*d^3*e + 15*A*a^2*b*e^4*x + 60*A*a*b^2*d*e^3*x + 60*B*a^2*b*d*e^3*x - 360*B*a
*b^2*d^2*e^2*x - 270*B*a*b^2*d*e^3*x^2))/(15*e^5*(d + e*x)^(5/2))

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sympy [A]  time = 4.33, size = 1654, normalized size = 9.79

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)/(e*x+d)**(7/2),x)

[Out]

Piecewise((-6*A*a**3*e**4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)
) - 12*A*a**2*b*d*e**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) -
30*A*a**2*b*e**4*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 48
*A*a*b**2*d**2*e**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 12
0*A*a*b**2*d*e**3*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 90
*A*a*b**2*e**4*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 96
*A*b**3*d**3*e/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 240*A*b
**3*d**2*e**2*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 180*A*
b**3*d*e**3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 30*A*
b**3*e**4*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 4*B*a**
3*d*e**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 10*B*a**3*e**
4*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 48*B*a**2*b*d**2*e
**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 120*B*a**2*b*d*e**
3*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 90*B*a**2*b*e**4*x
**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 288*B*a*b**2*d**3*
e/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 720*B*a*b**2*d**2*e*
*2*x/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 540*B*a*b**2*d*e*
*3*x**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 90*B*a*b**2*e*
*4*x**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 256*B*b**3*d**
4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 640*B*b**3*d**3*e*x/
(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 480*B*b**3*d**2*e**2*x
**2/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) - 80*B*b**3*d*e**3*x
**3/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)) + 10*B*b**3*e**4*x**
4/(15*d**2*e**5*sqrt(d + e*x) + 30*d*e**6*x*sqrt(d + e*x) + 15*e**7*x**2*sqrt(d + e*x)), Ne(e, 0)), ((A*a**3*x
+ 3*A*a**2*b*x**2/2 + A*a*b**2*x**3 + A*b**3*x**4/4 + B*a**3*x**2/2 + B*a**2*b*x**3 + 3*B*a*b**2*x**4/4 + B*b
**3*x**5/5)/d**(7/2), True))

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