3.835 \(\int \frac{(d+e x)^{3/2} \left (a+b x+c x^2\right )}{\sqrt{f+g x}} \, dx\)

Optimal. Leaf size=333 \[ -\frac{\sqrt{d+e x} \sqrt{f+g x} (e f-d g) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^2 g^4}+\frac{(d+e x)^{3/2} \sqrt{f+g x} \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{96 e^2 g^3}+\frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^{5/2} g^{9/2}}-\frac{(d+e x)^{5/2} \sqrt{f+g x} (-8 b e g+9 c d g+7 c e f)}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g} \]

[Out]

-((e*f - d*g)*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e
*f + d*g)))*Sqrt[d + e*x]*Sqrt[f + g*x])/(64*e^2*g^4) + ((c*(35*e^2*f^2 + 10*d*e
*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*(d + e*x)^(3/2)*Sqrt[f +
g*x])/(96*e^2*g^3) - ((7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(5/2)*Sqrt[f + g*x
])/(24*e^2*g^2) + (c*(d + e*x)^(7/2)*Sqrt[f + g*x])/(4*e^2*g) + ((e*f - d*g)^2*(
c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*Arc
Tanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(64*e^(5/2)*g^(9/2))

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Rubi [A]  time = 0.788135, antiderivative size = 333, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{\sqrt{d+e x} \sqrt{f+g x} (e f-d g) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^2 g^4}+\frac{(d+e x)^{3/2} \sqrt{f+g x} \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{96 e^2 g^3}+\frac{(e f-d g)^2 \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )}{64 e^{5/2} g^{9/2}}-\frac{(d+e x)^{5/2} \sqrt{f+g x} (-8 b e g+9 c d g+7 c e f)}{24 e^2 g^2}+\frac{c (d+e x)^{7/2} \sqrt{f+g x}}{4 e^2 g} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^(3/2)*(a + b*x + c*x^2))/Sqrt[f + g*x],x]

[Out]

-((e*f - d*g)*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e
*f + d*g)))*Sqrt[d + e*x]*Sqrt[f + g*x])/(64*e^2*g^4) + ((c*(35*e^2*f^2 + 10*d*e
*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*(d + e*x)^(3/2)*Sqrt[f +
g*x])/(96*e^2*g^3) - ((7*c*e*f + 9*c*d*g - 8*b*e*g)*(d + e*x)^(5/2)*Sqrt[f + g*x
])/(24*e^2*g^2) + (c*(d + e*x)^(7/2)*Sqrt[f + g*x])/(4*e^2*g) + ((e*f - d*g)^2*(
c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(5*e*f + d*g)))*Arc
Tanh[(Sqrt[g]*Sqrt[d + e*x])/(Sqrt[e]*Sqrt[f + g*x])])/(64*e^(5/2)*g^(9/2))

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Rubi in Sympy [A]  time = 74.4476, size = 434, normalized size = 1.3 \[ \frac{b \left (d + e x\right )^{\frac{5}{2}} \sqrt{f + g x}}{3 e g} + \frac{c x \left (d + e x\right )^{\frac{5}{2}} \sqrt{f + g x}}{4 e g} - \frac{c \left (d + e x\right )^{\frac{5}{2}} \sqrt{f + g x} \left (3 d g + 7 e f\right )}{24 e^{2} g^{2}} + \frac{c \left (d + e x\right )^{\frac{3}{2}} \sqrt{f + g x} \left (3 d^{2} g^{2} + 10 d e f g + 35 e^{2} f^{2}\right )}{96 e^{2} g^{3}} + \frac{c \sqrt{d + e x} \sqrt{f + g x} \left (d g - e f\right ) \left (3 d^{2} g^{2} + 10 d e f g + 35 e^{2} f^{2}\right )}{64 e^{2} g^{4}} + \frac{c \left (d g - e f\right )^{2} \left (3 d^{2} g^{2} + 10 d e f g + 35 e^{2} f^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{g} \sqrt{d + e x}} \right )}}{64 e^{\frac{5}{2}} g^{\frac{9}{2}}} + \frac{\left (d + e x\right )^{\frac{3}{2}} \sqrt{f + g x} \left (6 a e g - b d g - 5 b e f\right )}{12 e g^{2}} + \frac{\sqrt{d + e x} \sqrt{f + g x} \left (d g - e f\right ) \left (6 a e g - b d g - 5 b e f\right )}{8 e g^{3}} + \frac{\left (d g - e f\right )^{2} \left (6 a e g - b d g - 5 b e f\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{g} \sqrt{d + e x}} \right )}}{8 e^{\frac{3}{2}} g^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(c*x**2+b*x+a)/(g*x+f)**(1/2),x)

[Out]

b*(d + e*x)**(5/2)*sqrt(f + g*x)/(3*e*g) + c*x*(d + e*x)**(5/2)*sqrt(f + g*x)/(4
*e*g) - c*(d + e*x)**(5/2)*sqrt(f + g*x)*(3*d*g + 7*e*f)/(24*e**2*g**2) + c*(d +
e*x)**(3/2)*sqrt(f + g*x)*(3*d**2*g**2 + 10*d*e*f*g + 35*e**2*f**2)/(96*e**2*g*
*3) + c*sqrt(d + e*x)*sqrt(f + g*x)*(d*g - e*f)*(3*d**2*g**2 + 10*d*e*f*g + 35*e
**2*f**2)/(64*e**2*g**4) + c*(d*g - e*f)**2*(3*d**2*g**2 + 10*d*e*f*g + 35*e**2*
f**2)*atanh(sqrt(e)*sqrt(f + g*x)/(sqrt(g)*sqrt(d + e*x)))/(64*e**(5/2)*g**(9/2)
) + (d + e*x)**(3/2)*sqrt(f + g*x)*(6*a*e*g - b*d*g - 5*b*e*f)/(12*e*g**2) + sqr
t(d + e*x)*sqrt(f + g*x)*(d*g - e*f)*(6*a*e*g - b*d*g - 5*b*e*f)/(8*e*g**3) + (d
*g - e*f)**2*(6*a*e*g - b*d*g - 5*b*e*f)*atanh(sqrt(e)*sqrt(f + g*x)/(sqrt(g)*sq
rt(d + e*x)))/(8*e**(3/2)*g**(7/2))

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Mathematica [A]  time = 0.573291, size = 306, normalized size = 0.92 \[ \frac{3 (e f-d g)^2 \log \left (2 \sqrt{e} \sqrt{g} \sqrt{d+e x} \sqrt{f+g x}+d g+e f+2 e g x\right ) \left (8 e g (6 a e g-b (d g+5 e f))+c \left (3 d^2 g^2+10 d e f g+35 e^2 f^2\right )\right )-2 \sqrt{e} \sqrt{g} \sqrt{d+e x} \sqrt{f+g x} \left (c \left (9 d^3 g^3+3 d^2 e g^2 (5 f-2 g x)+d e^2 g \left (-145 f^2+92 f g x-72 g^2 x^2\right )+e^3 \left (105 f^3-70 f^2 g x+56 f g^2 x^2-48 g^3 x^3\right )\right )-8 e g \left (6 a e g (5 d g-3 e f+2 e g x)+b \left (3 d^2 g^2+2 d e g (7 g x-11 f)+e^2 \left (15 f^2-10 f g x+8 g^2 x^2\right )\right )\right )\right )}{384 e^{5/2} g^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^(3/2)*(a + b*x + c*x^2))/Sqrt[f + g*x],x]

[Out]

(-2*Sqrt[e]*Sqrt[g]*Sqrt[d + e*x]*Sqrt[f + g*x]*(c*(9*d^3*g^3 + 3*d^2*e*g^2*(5*f
- 2*g*x) + d*e^2*g*(-145*f^2 + 92*f*g*x - 72*g^2*x^2) + e^3*(105*f^3 - 70*f^2*g
*x + 56*f*g^2*x^2 - 48*g^3*x^3)) - 8*e*g*(6*a*e*g*(-3*e*f + 5*d*g + 2*e*g*x) + b
*(3*d^2*g^2 + 2*d*e*g*(-11*f + 7*g*x) + e^2*(15*f^2 - 10*f*g*x + 8*g^2*x^2)))) +
3*(e*f - d*g)^2*(c*(35*e^2*f^2 + 10*d*e*f*g + 3*d^2*g^2) + 8*e*g*(6*a*e*g - b*(
5*e*f + d*g)))*Log[e*f + d*g + 2*e*g*x + 2*Sqrt[e]*Sqrt[g]*Sqrt[d + e*x]*Sqrt[f
+ g*x]])/(384*e^(5/2)*g^(9/2))

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Maple [B]  time = 0.041, size = 1207, normalized size = 3.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(c*x^2+b*x+a)/(g*x+f)^(1/2),x)

[Out]

1/384*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(-184*((e*x+d)*(g*x+f))^(1/2)*x*d*c*f*g^2*(e*g
)^(1/2)*e^2-112*x^2*c*e^3*f*g^2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+224*((e*x+d)
*(g*x+f))^(1/2)*x*d*b*g^3*(e*g)^(1/2)*e^2-160*((e*x+d)*(g*x+f))^(1/2)*x*f*b*e^3*
g^2*(e*g)^(1/2)+12*((e*x+d)*(g*x+f))^(1/2)*x*d^2*c*g^3*(e*g)^(1/2)*e+140*((e*x+d
)*(g*x+f))^(1/2)*x*f^2*c*e^3*g*(e*g)^(1/2)-30*((e*x+d)*(g*x+f))^(1/2)*d^2*c*f*g^
2*(e*g)^(1/2)*e-352*((e*x+d)*(g*x+f))^(1/2)*b*d*f*g^2*(e*g)^(1/2)*e^2+290*((e*x+
d)*(g*x+f))^(1/2)*c*d*f^2*g*(e*g)^(1/2)*e^2+9*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f)
)^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*d^4*c*g^4+105*ln(1/2*(2*e*g*x+2*((e*x+
d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*e^4*f^4*c-288*ln(1/2*(2*e*g*
x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*d*a*e^3*f*g^3+144*
ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*d^2*
a*g^4*e^2+144*x^2*c*d*e^2*g^3*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)-18*((e*x+d)*(g
*x+f))^(1/2)*d^3*c*g^3*(e*g)^(1/2)+144*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)
*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*e^4*f^2*a*g^2-120*ln(1/2*(2*e*g*x+2*((e*x+d)*
(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*e^4*f^3*b*g-210*((e*x+d)*(g*x+f
))^(1/2)*c*e^3*f^3*(e*g)^(1/2)-24*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g
)^(1/2)+d*g+e*f)/(e*g)^(1/2))*d^3*b*g^4*e+96*x^3*c*e^3*g^3*((e*x+d)*(g*x+f))^(1/
2)*(e*g)^(1/2)+128*x^2*b*e^3*g^3*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)-72*ln(1/2*(
2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*d^2*b*f*g^3*
e^2+216*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/
2))*d*b*e^3*f^2*g^2+54*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g
+e*f)/(e*g)^(1/2))*d^2*c*f^2*g^2*e^2-180*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/
2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*d*c*e^3*f^3*g+480*((e*x+d)*(g*x+f))^(1/2)*a
*d*g^3*(e*g)^(1/2)*e^2-288*((e*x+d)*(g*x+f))^(1/2)*a*e^3*f*g^2*(e*g)^(1/2)+240*(
(e*x+d)*(g*x+f))^(1/2)*b*e^3*f^2*g*(e*g)^(1/2)+192*a*e^3*((e*x+d)*(g*x+f))^(1/2)
*x*g^3*(e*g)^(1/2)+12*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+
e*f)/(e*g)^(1/2))*d^3*c*f*g^3*e+48*((e*x+d)*(g*x+f))^(1/2)*d^2*b*g^3*(e*g)^(1/2)
*e)/((e*x+d)*(g*x+f))^(1/2)/g^4/(e*g)^(1/2)/e^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((c*x^2 + b*x + a)*(e*x + d)^(3/2)/sqrt(g*x + f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.8049, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^(3/2)/sqrt(g*x + f),x, algorithm="fricas")

[Out]

[1/768*(4*(48*c*e^3*g^3*x^3 - 105*c*e^3*f^3 + 5*(29*c*d*e^2 + 24*b*e^3)*f^2*g -
(15*c*d^2*e + 176*b*d*e^2 + 144*a*e^3)*f*g^2 - 3*(3*c*d^3 - 8*b*d^2*e - 80*a*d*e
^2)*g^3 - 8*(7*c*e^3*f*g^2 - (9*c*d*e^2 + 8*b*e^3)*g^3)*x^2 + 2*(35*c*e^3*f^2*g
- 2*(23*c*d*e^2 + 20*b*e^3)*f*g^2 + (3*c*d^2*e + 56*b*d*e^2 + 48*a*e^3)*g^3)*x)*
sqrt(e*g)*sqrt(e*x + d)*sqrt(g*x + f) + 3*(35*c*e^4*f^4 - 20*(3*c*d*e^3 + 2*b*e^
4)*f^3*g + 6*(3*c*d^2*e^2 + 12*b*d*e^3 + 8*a*e^4)*f^2*g^2 + 4*(c*d^3*e - 6*b*d^2
*e^2 - 24*a*d*e^3)*f*g^3 + (3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*g^4)*log(4*(2*e^
2*g^2*x + e^2*f*g + d*e*g^2)*sqrt(e*x + d)*sqrt(g*x + f) + (8*e^2*g^2*x^2 + e^2*
f^2 + 6*d*e*f*g + d^2*g^2 + 8*(e^2*f*g + d*e*g^2)*x)*sqrt(e*g)))/(sqrt(e*g)*e^2*
g^4), 1/384*(2*(48*c*e^3*g^3*x^3 - 105*c*e^3*f^3 + 5*(29*c*d*e^2 + 24*b*e^3)*f^2
*g - (15*c*d^2*e + 176*b*d*e^2 + 144*a*e^3)*f*g^2 - 3*(3*c*d^3 - 8*b*d^2*e - 80*
a*d*e^2)*g^3 - 8*(7*c*e^3*f*g^2 - (9*c*d*e^2 + 8*b*e^3)*g^3)*x^2 + 2*(35*c*e^3*f
^2*g - 2*(23*c*d*e^2 + 20*b*e^3)*f*g^2 + (3*c*d^2*e + 56*b*d*e^2 + 48*a*e^3)*g^3
)*x)*sqrt(-e*g)*sqrt(e*x + d)*sqrt(g*x + f) + 3*(35*c*e^4*f^4 - 20*(3*c*d*e^3 +
2*b*e^4)*f^3*g + 6*(3*c*d^2*e^2 + 12*b*d*e^3 + 8*a*e^4)*f^2*g^2 + 4*(c*d^3*e - 6
*b*d^2*e^2 - 24*a*d*e^3)*f*g^3 + (3*c*d^4 - 8*b*d^3*e + 48*a*d^2*e^2)*g^4)*arcta
n(1/2*(2*e*g*x + e*f + d*g)*sqrt(-e*g)/(sqrt(e*x + d)*sqrt(g*x + f)*e*g)))/(sqrt
(-e*g)*e^2*g^4)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**(3/2)*(c*x**2+b*x+a)/(g*x+f)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.299809, size = 605, normalized size = 1.82 \[ \frac{1}{192} \, \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}}{\left (2 \,{\left (4 \,{\left (x e + d\right )}{\left (\frac{6 \,{\left (x e + d\right )} c e^{\left (-3\right )}}{g} - \frac{{\left (9 \, c d g^{6} e^{6} + 7 \, c f g^{5} e^{7} - 8 \, b g^{6} e^{7}\right )} e^{\left (-9\right )}}{g^{7}}\right )} + \frac{{\left (3 \, c d^{2} g^{6} e^{6} + 10 \, c d f g^{5} e^{7} - 8 \, b d g^{6} e^{7} + 35 \, c f^{2} g^{4} e^{8} - 40 \, b f g^{5} e^{8} + 48 \, a g^{6} e^{8}\right )} e^{\left (-9\right )}}{g^{7}}\right )}{\left (x e + d\right )} + \frac{3 \,{\left (3 \, c d^{3} g^{6} e^{6} + 7 \, c d^{2} f g^{5} e^{7} - 8 \, b d^{2} g^{6} e^{7} + 25 \, c d f^{2} g^{4} e^{8} - 32 \, b d f g^{5} e^{8} + 48 \, a d g^{6} e^{8} - 35 \, c f^{3} g^{3} e^{9} + 40 \, b f^{2} g^{4} e^{9} - 48 \, a f g^{5} e^{9}\right )} e^{\left (-9\right )}}{g^{7}}\right )} \sqrt{x e + d} - \frac{{\left (3 \, c d^{4} g^{4} + 4 \, c d^{3} f g^{3} e - 8 \, b d^{3} g^{4} e + 18 \, c d^{2} f^{2} g^{2} e^{2} - 24 \, b d^{2} f g^{3} e^{2} + 48 \, a d^{2} g^{4} e^{2} - 60 \, c d f^{3} g e^{3} + 72 \, b d f^{2} g^{2} e^{3} - 96 \, a d f g^{3} e^{3} + 35 \, c f^{4} e^{4} - 40 \, b f^{3} g e^{4} + 48 \, a f^{2} g^{2} e^{4}\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -\sqrt{x e + d} \sqrt{g} e^{\frac{1}{2}} + \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}} \right |}\right )}{64 \, g^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2 + b*x + a)*(e*x + d)^(3/2)/sqrt(g*x + f),x, algorithm="giac")

[Out]

1/192*sqrt((x*e + d)*g*e - d*g*e + f*e^2)*(2*(4*(x*e + d)*(6*(x*e + d)*c*e^(-3)/
g - (9*c*d*g^6*e^6 + 7*c*f*g^5*e^7 - 8*b*g^6*e^7)*e^(-9)/g^7) + (3*c*d^2*g^6*e^6
+ 10*c*d*f*g^5*e^7 - 8*b*d*g^6*e^7 + 35*c*f^2*g^4*e^8 - 40*b*f*g^5*e^8 + 48*a*g
^6*e^8)*e^(-9)/g^7)*(x*e + d) + 3*(3*c*d^3*g^6*e^6 + 7*c*d^2*f*g^5*e^7 - 8*b*d^2
*g^6*e^7 + 25*c*d*f^2*g^4*e^8 - 32*b*d*f*g^5*e^8 + 48*a*d*g^6*e^8 - 35*c*f^3*g^3
*e^9 + 40*b*f^2*g^4*e^9 - 48*a*f*g^5*e^9)*e^(-9)/g^7)*sqrt(x*e + d) - 1/64*(3*c*
d^4*g^4 + 4*c*d^3*f*g^3*e - 8*b*d^3*g^4*e + 18*c*d^2*f^2*g^2*e^2 - 24*b*d^2*f*g^
3*e^2 + 48*a*d^2*g^4*e^2 - 60*c*d*f^3*g*e^3 + 72*b*d*f^2*g^2*e^3 - 96*a*d*f*g^3*
e^3 + 35*c*f^4*e^4 - 40*b*f^3*g*e^4 + 48*a*f^2*g^2*e^4)*e^(-5/2)*ln(abs(-sqrt(x*
e + d)*sqrt(g)*e^(1/2) + sqrt((x*e + d)*g*e - d*g*e + f*e^2)))/g^(9/2)