3.16.98 \(\int (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx\)

Optimal. Leaf size=173 \[ -\frac {2 b^2 (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac {6 b (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac {2 (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5}+\frac {2 (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5}+\frac {2 b^3 B (d+e x)^{11/2}}{11 e^5} \]

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Rubi [A]  time = 0.08, antiderivative size = 173, 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 (d+e x)^{9/2} (-3 a B e-A b e+4 b B d)}{9 e^5}+\frac {6 b (d+e x)^{7/2} (b d-a e) (-a B e-A b e+2 b B d)}{7 e^5}-\frac {2 (d+e x)^{5/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{5 e^5}+\frac {2 (d+e x)^{3/2} (b d-a e)^3 (B d-A e)}{3 e^5}+\frac {2 b^3 B (d+e x)^{11/2}}{11 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(3/2))/(3*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
^(5/2))/(5*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(7/2))/(7*e^5) - (2*b^2*(4*b*B*d - A*b*
e - 3*a*B*e)*(d + e*x)^(9/2))/(9*e^5) + (2*b^3*B*(d + e*x)^(11/2))/(11*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 (a+b x)^3 (A+B x) \sqrt {d+e x} \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e) \sqrt {d+e x}}{e^4}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{3/2}}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{5/2}}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{7/2}}{e^4}+\frac {b^3 B (d+e x)^{9/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{3/2}}{3 e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{5/2}}{5 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{7/2}}{7 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{9/2}}{9 e^5}+\frac {2 b^3 B (d+e x)^{11/2}}{11 e^5}\\ \end {align*}

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Mathematica [A]  time = 0.21, size = 145, normalized size = 0.84 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (-385 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+1485 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-693 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+1155 (b d-a e)^3 (B d-A e)+315 b^3 B (d+e x)^4\right )}{3465 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(1155*(b*d - a*e)^3*(B*d - A*e) - 693*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x) +
1485*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 385*b^2*(4*b*B*d - A*b*e - 3*a*B*e)*(d + e*x)^3 +
315*b^3*B*(d + e*x)^4))/(3465*e^5)

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IntegrateAlgebraic [A]  time = 0.14, size = 346, normalized size = 2.00 \begin {gather*} \frac {2 (d+e x)^{3/2} \left (1155 a^3 A e^4+693 a^3 B e^3 (d+e x)-1155 a^3 B d e^3+2079 a^2 A b e^3 (d+e x)-3465 a^2 A b d e^3+3465 a^2 b B d^2 e^2-4158 a^2 b B d e^2 (d+e x)+1485 a^2 b B e^2 (d+e x)^2+3465 a A b^2 d^2 e^2-4158 a A b^2 d e^2 (d+e x)+1485 a A b^2 e^2 (d+e x)^2-3465 a b^2 B d^3 e+6237 a b^2 B d^2 e (d+e x)-4455 a b^2 B d e (d+e x)^2+1155 a b^2 B e (d+e x)^3-1155 A b^3 d^3 e+2079 A b^3 d^2 e (d+e x)-1485 A b^3 d e (d+e x)^2+385 A b^3 e (d+e x)^3+1155 b^3 B d^4-2772 b^3 B d^3 (d+e x)+2970 b^3 B d^2 (d+e x)^2-1540 b^3 B d (d+e x)^3+315 b^3 B (d+e x)^4\right )}{3465 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(d + e*x)^(3/2)*(1155*b^3*B*d^4 - 1155*A*b^3*d^3*e - 3465*a*b^2*B*d^3*e + 3465*a*A*b^2*d^2*e^2 + 3465*a^2*b
*B*d^2*e^2 - 3465*a^2*A*b*d*e^3 - 1155*a^3*B*d*e^3 + 1155*a^3*A*e^4 - 2772*b^3*B*d^3*(d + e*x) + 2079*A*b^3*d^
2*e*(d + e*x) + 6237*a*b^2*B*d^2*e*(d + e*x) - 4158*a*A*b^2*d*e^2*(d + e*x) - 4158*a^2*b*B*d*e^2*(d + e*x) + 2
079*a^2*A*b*e^3*(d + e*x) + 693*a^3*B*e^3*(d + e*x) + 2970*b^3*B*d^2*(d + e*x)^2 - 1485*A*b^3*d*e*(d + e*x)^2
- 4455*a*b^2*B*d*e*(d + e*x)^2 + 1485*a*A*b^2*e^2*(d + e*x)^2 + 1485*a^2*b*B*e^2*(d + e*x)^2 - 1540*b^3*B*d*(d
+ e*x)^3 + 385*A*b^3*e*(d + e*x)^3 + 1155*a*b^2*B*e*(d + e*x)^3 + 315*b^3*B*(d + e*x)^4))/(3465*e^5)

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fricas [B]  time = 1.56, size = 353, normalized size = 2.04 \begin {gather*} \frac {2 \, {\left (315 \, B b^{3} e^{5} x^{5} + 128 \, B b^{3} d^{5} + 1155 \, A a^{3} d e^{4} - 176 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e + 792 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{2} - 462 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{3} + 35 \, {\left (B b^{3} d e^{4} + 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{5}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{2} e^{3} - 11 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{4} - 297 \, {\left (B a^{2} b + A a b^{2}\right )} e^{5}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{3} e^{2} - 22 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{3} + 99 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{4} + 231 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{4} e - 1155 \, A a^{3} e^{5} - 88 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{2} + 396 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{3} - 231 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{4}\right )} x\right )} \sqrt {e x + d}}{3465 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x, algorithm="fricas")

[Out]

2/3465*(315*B*b^3*e^5*x^5 + 128*B*b^3*d^5 + 1155*A*a^3*d*e^4 - 176*(3*B*a*b^2 + A*b^3)*d^4*e + 792*(B*a^2*b +
A*a*b^2)*d^3*e^2 - 462*(B*a^3 + 3*A*a^2*b)*d^2*e^3 + 35*(B*b^3*d*e^4 + 11*(3*B*a*b^2 + A*b^3)*e^5)*x^4 - 5*(8*
B*b^3*d^2*e^3 - 11*(3*B*a*b^2 + A*b^3)*d*e^4 - 297*(B*a^2*b + A*a*b^2)*e^5)*x^3 + 3*(16*B*b^3*d^3*e^2 - 22*(3*
B*a*b^2 + A*b^3)*d^2*e^3 + 99*(B*a^2*b + A*a*b^2)*d*e^4 + 231*(B*a^3 + 3*A*a^2*b)*e^5)*x^2 - (64*B*b^3*d^4*e -
1155*A*a^3*e^5 - 88*(3*B*a*b^2 + A*b^3)*d^3*e^2 + 396*(B*a^2*b + A*a*b^2)*d^2*e^3 - 231*(B*a^3 + 3*A*a^2*b)*d
*e^4)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 1.34, size = 802, normalized size = 4.64

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)^(1/2),x, algorithm="giac")

[Out]

2/3465*(1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*d*e^(-1) + 3465*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)
*A*a^2*b*d*e^(-1) + 693*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^2*b*d*e^(-2) + 6
93*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a*b^2*d*e^(-2) + 297*(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^2*d*e^(-3) + 99*(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^3*d*e^(-3) + 11*(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
^3*d*e^(-4) + 231*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^3*e^(-1) + 693*(3*(x*e
+ d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*b*e^(-1) + 297*(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*e^(-2) + 297*(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^2*e^(-2) + 33*(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^2*e^(-3) +
11*(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^3*e^(-3) + 5*(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^3*e^(-4) + 3465*sqrt(x*e + d)*A*a^
3*d + 1155*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^3)*e^(-1)

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maple [A]  time = 0.01, size = 301, normalized size = 1.74 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {3}{2}} \left (315 B \,b^{3} x^{4} e^{4}+385 A \,b^{3} e^{4} x^{3}+1155 B a \,b^{2} e^{4} x^{3}-280 B \,b^{3} d \,e^{3} x^{3}+1485 A a \,b^{2} e^{4} x^{2}-330 A \,b^{3} d \,e^{3} x^{2}+1485 B \,a^{2} b \,e^{4} x^{2}-990 B a \,b^{2} d \,e^{3} x^{2}+240 B \,b^{3} d^{2} e^{2} x^{2}+2079 A \,a^{2} b \,e^{4} x -1188 A a \,b^{2} d \,e^{3} x +264 A \,b^{3} d^{2} e^{2} x +693 B \,a^{3} e^{4} x -1188 B \,a^{2} b d \,e^{3} x +792 B a \,b^{2} d^{2} e^{2} x -192 B \,b^{3} d^{3} e x +1155 a^{3} A \,e^{4}-1386 A \,a^{2} b d \,e^{3}+792 A a \,b^{2} d^{2} e^{2}-176 A \,b^{3} d^{3} e -462 B \,a^{3} d \,e^{3}+792 B \,a^{2} b \,d^{2} e^{2}-528 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right )}{3465 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)^(1/2),x)

[Out]

2/3465*(e*x+d)^(3/2)*(315*B*b^3*e^4*x^4+385*A*b^3*e^4*x^3+1155*B*a*b^2*e^4*x^3-280*B*b^3*d*e^3*x^3+1485*A*a*b^
2*e^4*x^2-330*A*b^3*d*e^3*x^2+1485*B*a^2*b*e^4*x^2-990*B*a*b^2*d*e^3*x^2+240*B*b^3*d^2*e^2*x^2+2079*A*a^2*b*e^
4*x-1188*A*a*b^2*d*e^3*x+264*A*b^3*d^2*e^2*x+693*B*a^3*e^4*x-1188*B*a^2*b*d*e^3*x+792*B*a*b^2*d^2*e^2*x-192*B*
b^3*d^3*e*x+1155*A*a^3*e^4-1386*A*a^2*b*d*e^3+792*A*a*b^2*d^2*e^2-176*A*b^3*d^3*e-462*B*a^3*d*e^3+792*B*a^2*b*
d^2*e^2-528*B*a*b^2*d^3*e+128*B*b^3*d^4)/e^5

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maxima [A]  time = 0.49, size = 265, normalized size = 1.53 \begin {gather*} \frac {2 \, {\left (315 \, {\left (e x + d\right )}^{\frac {11}{2}} B b^{3} - 385 \, {\left (4 \, B b^{3} d - {\left (3 \, B a b^{2} + A b^{3}\right )} e\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 1485 \, {\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 )}^{\frac {7}{2}} - 693 \, {\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 )}^{\frac {5}{2}} + 1155 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 3 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} - {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}}\right )}}{3465 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(1/2),x, algorithm="maxima")

[Out]

2/3465*(315*(e*x + d)^(11/2)*B*b^3 - 385*(4*B*b^3*d - (3*B*a*b^2 + A*b^3)*e)*(e*x + d)^(9/2) + 1485*(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)^(7/2) - 693*(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)^(5/2) + 1155*(B*b^3*d^4 + A*a^3*
e^4 - (3*B*a*b^2 + A*b^3)*d^3*e + 3*(B*a^2*b + A*a*b^2)*d^2*e^2 - (B*a^3 + 3*A*a^2*b)*d*e^3)*(e*x + d)^(3/2))/
e^5

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mupad [B]  time = 1.24, size = 154, normalized size = 0.89 \begin {gather*} \frac {{\left (d+e\,x\right )}^{9/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{9\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{5/2}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{5\,e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{11/2}}{11\,e^5}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{3/2}}{3\,e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{7/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{7\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^3*(d + e*x)^(1/2),x)

[Out]

((d + e*x)^(9/2)*(2*A*b^3*e - 8*B*b^3*d + 6*B*a*b^2*e))/(9*e^5) + (2*(a*e - b*d)^2*(d + e*x)^(5/2)*(3*A*b*e +
B*a*e - 4*B*b*d))/(5*e^5) + (2*B*b^3*(d + e*x)^(11/2))/(11*e^5) + (2*(A*e - B*d)*(a*e - b*d)^3*(d + e*x)^(3/2)
)/(3*e^5) + (6*b*(a*e - b*d)*(d + e*x)^(7/2)*(A*b*e + B*a*e - 2*B*b*d))/(7*e^5)

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sympy [B]  time = 6.17, size = 342, normalized size = 1.98 \begin {gather*} \frac {2 \left (\frac {B b^{3} \left (d + e x\right )^{\frac {11}{2}}}{11 e^{4}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \left (A b^{3} e + 3 B a b^{2} e - 4 B b^{3} d\right )}{9 e^{4}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (3 A a b^{2} e^{2} - 3 A b^{3} d e + 3 B a^{2} b e^{2} - 9 B a b^{2} d e + 6 B b^{3} d^{2}\right )}{7 e^{4}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (3 A a^{2} b e^{3} - 6 A a b^{2} d e^{2} + 3 A b^{3} d^{2} e + B a^{3} e^{3} - 6 B a^{2} b d e^{2} + 9 B a b^{2} d^{2} e - 4 B b^{3} d^{3}\right )}{5 e^{4}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (A a^{3} e^{4} - 3 A a^{2} b d e^{3} + 3 A a b^{2} d^{2} e^{2} - A b^{3} d^{3} e - B a^{3} d e^{3} + 3 B a^{2} b d^{2} e^{2} - 3 B a b^{2} d^{3} e + B b^{3} d^{4}\right )}{3 e^{4}}\right )}{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)**(1/2),x)

[Out]

2*(B*b**3*(d + e*x)**(11/2)/(11*e**4) + (d + e*x)**(9/2)*(A*b**3*e + 3*B*a*b**2*e - 4*B*b**3*d)/(9*e**4) + (d
+ e*x)**(7/2)*(3*A*a*b**2*e**2 - 3*A*b**3*d*e + 3*B*a**2*b*e**2 - 9*B*a*b**2*d*e + 6*B*b**3*d**2)/(7*e**4) + (
d + e*x)**(5/2)*(3*A*a**2*b*e**3 - 6*A*a*b**2*d*e**2 + 3*A*b**3*d**2*e + B*a**3*e**3 - 6*B*a**2*b*d*e**2 + 9*B
*a*b**2*d**2*e - 4*B*b**3*d**3)/(5*e**4) + (d + e*x)**(3/2)*(A*a**3*e**4 - 3*A*a**2*b*d*e**3 + 3*A*a*b**2*d**2
*e**2 - A*b**3*d**3*e - B*a**3*d*e**3 + 3*B*a**2*b*d**2*e**2 - 3*B*a*b**2*d**3*e + B*b**3*d**4)/(3*e**4))/e

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