3.1810 \(\int \frac{(A+B x) (d+e x)^{3/2}}{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=174 \[ -\frac{\sqrt{b d-a e} (-5 a B e+3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{\sqrt{d+e x} (-5 a B e+3 A b e+2 b B d)}{b^3}+\frac{(d+e x)^{3/2} (-5 a B e+3 A b e+2 b B d)}{3 b^2 (b d-a e)}-\frac{(d+e x)^{5/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

[Out]

((2*b*B*d + 3*A*b*e - 5*a*B*e)*Sqrt[d + e*x])/b^3 + ((2*b*B*d + 3*A*b*e - 5*a*B*
e)*(d + e*x)^(3/2))/(3*b^2*(b*d - a*e)) - ((A*b - a*B)*(d + e*x)^(5/2))/(b*(b*d
- a*e)*(a + b*x)) - (Sqrt[b*d - a*e]*(2*b*B*d + 3*A*b*e - 5*a*B*e)*ArcTanh[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(7/2)

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Rubi [A]  time = 0.344829, antiderivative size = 174, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152 \[ -\frac{\sqrt{b d-a e} (-5 a B e+3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}}+\frac{\sqrt{d+e x} (-5 a B e+3 A b e+2 b B d)}{b^3}+\frac{(d+e x)^{3/2} (-5 a B e+3 A b e+2 b B d)}{3 b^2 (b d-a e)}-\frac{(d+e x)^{5/2} (A b-a B)}{b (a+b x) (b d-a e)} \]

Antiderivative was successfully verified.

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

[Out]

((2*b*B*d + 3*A*b*e - 5*a*B*e)*Sqrt[d + e*x])/b^3 + ((2*b*B*d + 3*A*b*e - 5*a*B*
e)*(d + e*x)^(3/2))/(3*b^2*(b*d - a*e)) - ((A*b - a*B)*(d + e*x)^(5/2))/(b*(b*d
- a*e)*(a + b*x)) - (Sqrt[b*d - a*e]*(2*b*B*d + 3*A*b*e - 5*a*B*e)*ArcTanh[(Sqrt
[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(7/2)

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Rubi in Sympy [A]  time = 65.8388, size = 162, normalized size = 0.93 \[ \frac{\left (d + e x\right )^{\frac{5}{2}} \left (A b - B a\right )}{b \left (a + b x\right ) \left (a e - b d\right )} - \frac{\left (d + e x\right )^{\frac{3}{2}} \left (3 A b e - 5 B a e + 2 B b d\right )}{3 b^{2} \left (a e - b d\right )} + \frac{\sqrt{d + e x} \left (3 A b e - 5 B a e + 2 B b d\right )}{b^{3}} - \frac{\sqrt{a e - b d} \left (3 A b e - 5 B a e + 2 B b d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d + e x}}{\sqrt{a e - b d}} \right )}}{b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(e*x+d)**(3/2)/(b**2*x**2+2*a*b*x+a**2),x)

[Out]

(d + e*x)**(5/2)*(A*b - B*a)/(b*(a + b*x)*(a*e - b*d)) - (d + e*x)**(3/2)*(3*A*b
*e - 5*B*a*e + 2*B*b*d)/(3*b**2*(a*e - b*d)) + sqrt(d + e*x)*(3*A*b*e - 5*B*a*e
+ 2*B*b*d)/b**3 - sqrt(a*e - b*d)*(3*A*b*e - 5*B*a*e + 2*B*b*d)*atan(sqrt(b)*sqr
t(d + e*x)/sqrt(a*e - b*d))/b**(7/2)

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Mathematica [A]  time = 0.306084, size = 127, normalized size = 0.73 \[ \frac{\sqrt{d+e x} \left (-\frac{3 (A b-a B) (b d-a e)}{a+b x}-12 a B e+6 A b e+8 b B d+2 b B e x\right )}{3 b^3}-\frac{\sqrt{b d-a e} (-5 a B e+3 A b e+2 b B d) \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{d+e x}}{\sqrt{b d-a e}}\right )}{b^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*(8*b*B*d + 6*A*b*e - 12*a*B*e + 2*b*B*e*x - (3*(A*b - a*B)*(b*d -
a*e))/(a + b*x)))/(3*b^3) - (Sqrt[b*d - a*e]*(2*b*B*d + 3*A*b*e - 5*a*B*e)*ArcT
anh[(Sqrt[b]*Sqrt[d + e*x])/Sqrt[b*d - a*e]])/b^(7/2)

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Maple [B]  time = 0.031, size = 381, normalized size = 2.2 \[{\frac{2\,B}{3\,{b}^{2}} \left ( ex+d \right ) ^{{\frac{3}{2}}}}+2\,{\frac{Ae\sqrt{ex+d}}{{b}^{2}}}-4\,{\frac{aBe\sqrt{ex+d}}{{b}^{3}}}+2\,{\frac{Bd\sqrt{ex+d}}{{b}^{2}}}+{\frac{A{e}^{2}a}{{b}^{2} \left ( bex+ae \right ) }\sqrt{ex+d}}-{\frac{Ade}{b \left ( bex+ae \right ) }\sqrt{ex+d}}-{\frac{{a}^{2}B{e}^{2}}{{b}^{3} \left ( bex+ae \right ) }\sqrt{ex+d}}+{\frac{aBde}{{b}^{2} \left ( bex+ae \right ) }\sqrt{ex+d}}-3\,{\frac{A{e}^{2}a}{{b}^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+3\,{\frac{Ade}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+5\,{\frac{{a}^{2}B{e}^{2}}{{b}^{3}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }-7\,{\frac{aBde}{{b}^{2}\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) }+2\,{\frac{B{d}^{2}}{b\sqrt{b \left ( ae-bd \right ) }}\arctan \left ({\frac{\sqrt{ex+d}b}{\sqrt{b \left ( ae-bd \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3/b^2*B*(e*x+d)^(3/2)+2/b^2*A*e*(e*x+d)^(1/2)-4/b^3*a*B*e*(e*x+d)^(1/2)+2/b^2*
B*d*(e*x+d)^(1/2)+1/b^2*(e*x+d)^(1/2)/(b*e*x+a*e)*A*a*e^2-1/b*(e*x+d)^(1/2)/(b*e
*x+a*e)*A*d*e-1/b^3*(e*x+d)^(1/2)/(b*e*x+a*e)*a^2*B*e^2+1/b^2*(e*x+d)^(1/2)/(b*e
*x+a*e)*B*d*a*e-3/b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(
1/2))*A*a*e^2+3/b/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2)
)*A*d*e+5/b^3/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*a^
2*B*e^2-7/b^2/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*
d*a*e+2/b/(b*(a*e-b*d))^(1/2)*arctan((e*x+d)^(1/2)*b/(b*(a*e-b*d))^(1/2))*B*d^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.297225, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (2 \, B a b d -{\left (5 \, B a^{2} - 3 \, A a b\right )} e +{\left (2 \, B b^{2} d -{\left (5 \, B a b - 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt{\frac{b d - a e}{b}} \log \left (\frac{b e x + 2 \, b d - a e - 2 \, \sqrt{e x + d} b \sqrt{\frac{b d - a e}{b}}}{b x + a}\right ) + 2 \,{\left (2 \, B b^{2} e x^{2} +{\left (11 \, B a b - 3 \, A b^{2}\right )} d - 3 \,{\left (5 \, B a^{2} - 3 \, A a b\right )} e + 2 \,{\left (4 \, B b^{2} d -{\left (5 \, B a b - 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt{e x + d}}{6 \,{\left (b^{4} x + a b^{3}\right )}}, -\frac{3 \,{\left (2 \, B a b d -{\left (5 \, B a^{2} - 3 \, A a b\right )} e +{\left (2 \, B b^{2} d -{\left (5 \, B a b - 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt{-\frac{b d - a e}{b}} \arctan \left (\frac{\sqrt{e x + d}}{\sqrt{-\frac{b d - a e}{b}}}\right ) -{\left (2 \, B b^{2} e x^{2} +{\left (11 \, B a b - 3 \, A b^{2}\right )} d - 3 \,{\left (5 \, B a^{2} - 3 \, A a b\right )} e + 2 \,{\left (4 \, B b^{2} d -{\left (5 \, B a b - 3 \, A b^{2}\right )} e\right )} x\right )} \sqrt{e x + d}}{3 \,{\left (b^{4} x + a b^{3}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(3*(2*B*a*b*d - (5*B*a^2 - 3*A*a*b)*e + (2*B*b^2*d - (5*B*a*b - 3*A*b^2)*e)
*x)*sqrt((b*d - a*e)/b)*log((b*e*x + 2*b*d - a*e - 2*sqrt(e*x + d)*b*sqrt((b*d -
a*e)/b))/(b*x + a)) + 2*(2*B*b^2*e*x^2 + (11*B*a*b - 3*A*b^2)*d - 3*(5*B*a^2 -
3*A*a*b)*e + 2*(4*B*b^2*d - (5*B*a*b - 3*A*b^2)*e)*x)*sqrt(e*x + d))/(b^4*x + a*
b^3), -1/3*(3*(2*B*a*b*d - (5*B*a^2 - 3*A*a*b)*e + (2*B*b^2*d - (5*B*a*b - 3*A*b
^2)*e)*x)*sqrt(-(b*d - a*e)/b)*arctan(sqrt(e*x + d)/sqrt(-(b*d - a*e)/b)) - (2*B
*b^2*e*x^2 + (11*B*a*b - 3*A*b^2)*d - 3*(5*B*a^2 - 3*A*a*b)*e + 2*(4*B*b^2*d - (
5*B*a*b - 3*A*b^2)*e)*x)*sqrt(e*x + d))/(b^4*x + a*b^3)]

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Sympy [A]  time = 156.254, size = 2508, normalized size = 14.41 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*a**2*e**3*sqrt(d + e*x)/(2*a**2*b**2*e**2 - 2*a*b**3*d*e + 2*a*b**3*e**2*x -
2*b**4*d*e*x) - A*a**2*e**3*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/
(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/
(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**2) + A*a**2*e**3*sqrt(-1/(b*(a*e - b*
d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e -
b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**2) - 4*
A*a*d*e**2*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**2*e**2*x - 2*b**
3*d*e*x) + A*a*d*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2*sqrt(-1/(b*(a*e
- b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e
- b*d)**3)) + sqrt(d + e*x))/b - A*a*d*e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a**
2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**
2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/b - 4*A*a*e**2*Piecewise((at
an(sqrt(d + e*x)/sqrt(a*e/b - d))/(b*sqrt(a*e/b - d)), a*e/b - d > 0), (-acoth(s
qrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x
> -a*e/b + d)), (-atanh(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a
*e/b - d < 0) & (d + e*x < -a*e/b + d)))/b**2 - A*d**2*e*sqrt(-1/(b*(a*e - b*d)*
*3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*
d)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/2 + A*d**2*e*sq
rt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*
sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e
*x))/2 + 2*A*d**2*e*sqrt(d + e*x)/(2*a**2*e**2 - 2*a*b*d*e + 2*a*b*e**2*x - 2*b*
*2*d*e*x) + 4*A*d*e*Piecewise((atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b*sqrt(a*e/b
- d)), a*e/b - d > 0), (-acoth(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b +
d)), (a*e/b - d < 0) & (d + e*x > -a*e/b + d)), (-atanh(sqrt(d + e*x)/sqrt(-a*e
/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x < -a*e/b + d)))/b + 2*
A*e*sqrt(d + e*x)/b**2 - 2*B*a**3*e**3*sqrt(d + e*x)/(2*a**2*b**3*e**2 - 2*a*b**
4*d*e + 2*a*b**4*e**2*x - 2*b**5*d*e*x) + B*a**3*e**3*sqrt(-1/(b*(a*e - b*d)**3)
)*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)*
*3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b**3) - B*a**3*
e**3*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a
*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqr
t(d + e*x))/(2*b**3) + 4*B*a**2*d*e**2*sqrt(d + e*x)/(2*a**2*b**2*e**2 - 2*a*b**
3*d*e + 2*a*b**3*e**2*x - 2*b**4*d*e*x) - B*a**2*d*e**2*sqrt(-1/(b*(a*e - b*d)**
3))*log(-a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d
)**3)) - b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/b**2 + B*a**2*d*
e**2*sqrt(-1/(b*(a*e - b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a
*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqr
t(d + e*x))/b**2 + 6*B*a**2*e**2*Piecewise((atan(sqrt(d + e*x)/sqrt(a*e/b - d))/
(b*sqrt(a*e/b - d)), a*e/b - d > 0), (-acoth(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*
sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x > -a*e/b + d)), (-atanh(sqrt(d + e
*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x < -a*e/b
+ d)))/b**3 - 2*B*a*d**2*e*sqrt(d + e*x)/(2*a**2*b*e**2 - 2*a*b**2*d*e + 2*a*b**
2*e**2*x - 2*b**3*d*e*x) + B*a*d**2*e*sqrt(-1/(b*(a*e - b*d)**3))*log(-a**2*e**2
*sqrt(-1/(b*(a*e - b*d)**3)) + 2*a*b*d*e*sqrt(-1/(b*(a*e - b*d)**3)) - b**2*d**2
*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) - B*a*d**2*e*sqrt(-1/(b*(a*e
- b*d)**3))*log(a**2*e**2*sqrt(-1/(b*(a*e - b*d)**3)) - 2*a*b*d*e*sqrt(-1/(b*(a
*e - b*d)**3)) + b**2*d**2*sqrt(-1/(b*(a*e - b*d)**3)) + sqrt(d + e*x))/(2*b) -
8*B*a*d*e*Piecewise((atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b*sqrt(a*e/b - d)), a*
e/b - d > 0), (-acoth(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e
/b - d < 0) & (d + e*x > -a*e/b + d)), (-atanh(sqrt(d + e*x)/sqrt(-a*e/b + d))/(
b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x < -a*e/b + d)))/b**2 - 4*B*a*e*s
qrt(d + e*x)/b**3 + 2*B*d**2*Piecewise((atan(sqrt(d + e*x)/sqrt(a*e/b - d))/(b*s
qrt(a*e/b - d)), a*e/b - d > 0), (-acoth(sqrt(d + e*x)/sqrt(-a*e/b + d))/(b*sqrt
(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x > -a*e/b + d)), (-atanh(sqrt(d + e*x)/
sqrt(-a*e/b + d))/(b*sqrt(-a*e/b + d)), (a*e/b - d < 0) & (d + e*x < -a*e/b + d)
))/b + 2*B*d*sqrt(d + e*x)/b**2 + 2*B*(d + e*x)**(3/2)/(3*b**2)

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GIAC/XCAS [A]  time = 0.288108, size = 323, normalized size = 1.86 \[ \frac{{\left (2 \, B b^{2} d^{2} - 7 \, B a b d e + 3 \, A b^{2} d e + 5 \, B a^{2} e^{2} - 3 \, A a b e^{2}\right )} \arctan \left (\frac{\sqrt{x e + d} b}{\sqrt{-b^{2} d + a b e}}\right )}{\sqrt{-b^{2} d + a b e} b^{3}} + \frac{\sqrt{x e + d} B a b d e - \sqrt{x e + d} A b^{2} d e - \sqrt{x e + d} B a^{2} e^{2} + \sqrt{x e + d} A a b e^{2}}{{\left ({\left (x e + d\right )} b - b d + a e\right )} b^{3}} + \frac{2 \,{\left ({\left (x e + d\right )}^{\frac{3}{2}} B b^{4} + 3 \, \sqrt{x e + d} B b^{4} d - 6 \, \sqrt{x e + d} B a b^{3} e + 3 \, \sqrt{x e + d} A b^{4} e\right )}}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*B*b^2*d^2 - 7*B*a*b*d*e + 3*A*b^2*d*e + 5*B*a^2*e^2 - 3*A*a*b*e^2)*arctan(sqr
t(x*e + d)*b/sqrt(-b^2*d + a*b*e))/(sqrt(-b^2*d + a*b*e)*b^3) + (sqrt(x*e + d)*B
*a*b*d*e - sqrt(x*e + d)*A*b^2*d*e - sqrt(x*e + d)*B*a^2*e^2 + sqrt(x*e + d)*A*a
*b*e^2)/(((x*e + d)*b - b*d + a*e)*b^3) + 2/3*((x*e + d)^(3/2)*B*b^4 + 3*sqrt(x*
e + d)*B*b^4*d - 6*sqrt(x*e + d)*B*a*b^3*e + 3*sqrt(x*e + d)*A*b^4*e)/b^6