3.616 \(\int \frac{1}{\sqrt{d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx\)

Optimal. Leaf size=354 \[ \frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)}-\frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{-a} g+\sqrt{c} f\right ) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \left (\sqrt{-a} g+\sqrt{c} f\right )^{3/2}} \]

[Out]

(g*Sqrt[d + e*x])/(Sqrt[-a]*(Sqrt[c]*f - Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x])
- (g*Sqrt[d + e*x])/(Sqrt[-a]*(Sqrt[c]*f + Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x]
) + (Sqrt[c]*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*
d - Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(Sqrt[c]
*f - Sqrt[-a]*g)^(3/2)) - (Sqrt[c]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d
+ e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[Sqrt[c]*d
+ Sqrt[-a]*e]*(Sqrt[c]*f + Sqrt[-a]*g)^(3/2))

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Rubi [A]  time = 1.71511, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{c} f-\sqrt{-a} g\right ) (e f-d g)}-\frac{g \sqrt{d+e x}}{\sqrt{-a} \sqrt{f+g x} \left (\sqrt{-a} g+\sqrt{c} f\right ) (e f-d g)}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{c} f-\sqrt{-a} g}}{\sqrt{f+g x} \sqrt{\sqrt{c} d-\sqrt{-a} e}}\right )}{\sqrt{-a} \sqrt{\sqrt{c} d-\sqrt{-a} e} \left (\sqrt{c} f-\sqrt{-a} g\right )^{3/2}}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{d+e x} \sqrt{\sqrt{-a} g+\sqrt{c} f}}{\sqrt{f+g x} \sqrt{\sqrt{-a} e+\sqrt{c} d}}\right )}{\sqrt{-a} \sqrt{\sqrt{-a} e+\sqrt{c} d} \left (\sqrt{-a} g+\sqrt{c} f\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(g*Sqrt[d + e*x])/(Sqrt[-a]*(Sqrt[c]*f - Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x])
- (g*Sqrt[d + e*x])/(Sqrt[-a]*(Sqrt[c]*f + Sqrt[-a]*g)*(e*f - d*g)*Sqrt[f + g*x]
) + (Sqrt[c]*ArcTanh[(Sqrt[Sqrt[c]*f - Sqrt[-a]*g]*Sqrt[d + e*x])/(Sqrt[Sqrt[c]*
d - Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[Sqrt[c]*d - Sqrt[-a]*e]*(Sqrt[c]
*f - Sqrt[-a]*g)^(3/2)) - (Sqrt[c]*ArcTanh[(Sqrt[Sqrt[c]*f + Sqrt[-a]*g]*Sqrt[d
+ e*x])/(Sqrt[Sqrt[c]*d + Sqrt[-a]*e]*Sqrt[f + g*x])])/(Sqrt[-a]*Sqrt[Sqrt[c]*d
+ Sqrt[-a]*e]*(Sqrt[c]*f + Sqrt[-a]*g)^(3/2))

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Rubi in Sympy [A]  time = 142.261, size = 304, normalized size = 0.86 \[ - \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{d + e x} \sqrt{\sqrt{c} f + g \sqrt{- a}}}{\sqrt{f + g x} \sqrt{\sqrt{c} d + e \sqrt{- a}}} \right )}}{\sqrt{- a} \sqrt{\sqrt{c} d + e \sqrt{- a}} \left (\sqrt{c} f + g \sqrt{- a}\right )^{\frac{3}{2}}} + \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{d + e x} \sqrt{\sqrt{c} f - g \sqrt{- a}}}{\sqrt{f + g x} \sqrt{\sqrt{c} d - e \sqrt{- a}}} \right )}}{\sqrt{- a} \sqrt{\sqrt{c} d - e \sqrt{- a}} \left (\sqrt{c} f - g \sqrt{- a}\right )^{\frac{3}{2}}} + \frac{g \sqrt{d + e x}}{\sqrt{- a} \sqrt{f + g x} \left (\sqrt{c} f + g \sqrt{- a}\right ) \left (d g - e f\right )} - \frac{g \sqrt{d + e x}}{\sqrt{- a} \sqrt{f + g x} \left (\sqrt{c} f - g \sqrt{- a}\right ) \left (d g - e f\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c)*atanh(sqrt(d + e*x)*sqrt(sqrt(c)*f + g*sqrt(-a))/(sqrt(f + g*x)*sqrt(sq
rt(c)*d + e*sqrt(-a))))/(sqrt(-a)*sqrt(sqrt(c)*d + e*sqrt(-a))*(sqrt(c)*f + g*sq
rt(-a))**(3/2)) + sqrt(c)*atanh(sqrt(d + e*x)*sqrt(sqrt(c)*f - g*sqrt(-a))/(sqrt
(f + g*x)*sqrt(sqrt(c)*d - e*sqrt(-a))))/(sqrt(-a)*sqrt(sqrt(c)*d - e*sqrt(-a))*
(sqrt(c)*f - g*sqrt(-a))**(3/2)) + g*sqrt(d + e*x)/(sqrt(-a)*sqrt(f + g*x)*(sqrt
(c)*f + g*sqrt(-a))*(d*g - e*f)) - g*sqrt(d + e*x)/(sqrt(-a)*sqrt(f + g*x)*(sqrt
(c)*f - g*sqrt(-a))*(d*g - e*f))

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Mathematica [C]  time = 4.36252, size = 555, normalized size = 1.57 \[ \frac{\frac{\left (\sqrt{a} \sqrt{c} g+i c f\right ) \log \left (\frac{i \sqrt{a} \sqrt{\sqrt{c} f+i \sqrt{a} g} \left (2 \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}+i \sqrt{a} (d g+e f+2 e g x)+\sqrt{c} (2 d f+d g x+e f x)\right )}{\sqrt{c} \left (\sqrt{c} x-i \sqrt{a}\right ) \sqrt{\sqrt{c} d+i \sqrt{a} e}}\right )}{\sqrt{a} \sqrt{\sqrt{c} d+i \sqrt{a} e} \sqrt{\sqrt{c} f+i \sqrt{a} g}}+\frac{\left (\sqrt{a} \sqrt{c} g-i c f\right ) \log \left (-\frac{i \sqrt{a} \sqrt{\sqrt{c} f-i \sqrt{a} g} \left (2 \sqrt{d+e x} \sqrt{f+g x} \sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g}-i \sqrt{a} (d g+e (f+2 g x))+\sqrt{c} (2 d f+d g x+e f x)\right )}{\sqrt{c} \left (\sqrt{c} x+i \sqrt{a}\right ) \sqrt{\sqrt{c} d-i \sqrt{a} e}}\right )}{\sqrt{a} \sqrt{\sqrt{c} d-i \sqrt{a} e} \sqrt{\sqrt{c} f-i \sqrt{a} g}}+\frac{4 g^2 \sqrt{d+e x}}{\sqrt{f+g x} (e f-d g)}}{2 \left (a g^2+c f^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

((4*g^2*Sqrt[d + e*x])/((e*f - d*g)*Sqrt[f + g*x]) + ((I*c*f + Sqrt[a]*Sqrt[c]*g
)*Log[(I*Sqrt[a]*Sqrt[Sqrt[c]*f + I*Sqrt[a]*g]*(2*Sqrt[Sqrt[c]*d + I*Sqrt[a]*e]*
Sqrt[Sqrt[c]*f + I*Sqrt[a]*g]*Sqrt[d + e*x]*Sqrt[f + g*x] + Sqrt[c]*(2*d*f + e*f
*x + d*g*x) + I*Sqrt[a]*(e*f + d*g + 2*e*g*x)))/(Sqrt[c]*Sqrt[Sqrt[c]*d + I*Sqrt
[a]*e]*((-I)*Sqrt[a] + Sqrt[c]*x))])/(Sqrt[a]*Sqrt[Sqrt[c]*d + I*Sqrt[a]*e]*Sqrt
[Sqrt[c]*f + I*Sqrt[a]*g]) + (((-I)*c*f + Sqrt[a]*Sqrt[c]*g)*Log[((-I)*Sqrt[a]*S
qrt[Sqrt[c]*f - I*Sqrt[a]*g]*(2*Sqrt[Sqrt[c]*d - I*Sqrt[a]*e]*Sqrt[Sqrt[c]*f - I
*Sqrt[a]*g]*Sqrt[d + e*x]*Sqrt[f + g*x] + Sqrt[c]*(2*d*f + e*f*x + d*g*x) - I*Sq
rt[a]*(d*g + e*(f + 2*g*x))))/(Sqrt[c]*Sqrt[Sqrt[c]*d - I*Sqrt[a]*e]*(I*Sqrt[a]
+ Sqrt[c]*x))])/(Sqrt[a]*Sqrt[Sqrt[c]*d - I*Sqrt[a]*e]*Sqrt[Sqrt[c]*f - I*Sqrt[a
]*g]))/(2*(c*f^2 + a*g^2))

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Maple [B]  time = 0.112, size = 10977, normalized size = 31. \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

result too large to display

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + a\right )} \sqrt{e x + d}{\left (g x + f\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*sqrt(e*x + d)*(g*x + f)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + a)*sqrt(e*x + d)*(g*x + f)^(3/2)), x)

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Fricas [A]  time = 147.838, size = 16238, normalized size = 45.87 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*sqrt(e*x + d)*(g*x + f)^(3/2)),x, algorithm="fricas")

[Out]

1/4*(8*sqrt(e*x + d)*sqrt(g*x + f)*g^2 - (c*e*f^4 - c*d*f^3*g + a*e*f^2*g^2 - a*
d*f*g^3 + (c*e*f^3*g - c*d*f^2*g^2 + a*e*f*g^3 - a*d*g^4)*x)*sqrt(-(c^3*d*f^3 -
3*a*c^2*e*f^2*g - 3*a*c^2*d*f*g^2 + a^2*c*e*g^3 + ((a*c^4*d^2 + a^2*c^3*e^2)*f^6
+ 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 +
(a^4*c*d^2 + a^5*e^2)*g^6)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*
f^3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^
4*g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2
+ a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^
2 + 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4
+ 2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^
2 + a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g
^10 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^9*e^4)*g^12)))/((a*c^4*d^2 + a^2*c^3*e^
2)*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2
*g^4 + (a^4*c*d^2 + a^5*e^2)*g^6))*log(-(c^3*e^2*f^4 + 4*c^3*d*e*f^3*g - 4*a*c^2
*d*e*f*g^3 - a*c^2*d^2*g^4 + 3*(c^3*d^2 - a*c^2*e^2)*f^2*g^2 + 2*(c^4*d*e*f^5 -
10*a*c^3*d*e*f^3*g^2 + 5*a^2*c^2*d*e*f*g^4 + a^2*c^2*d^2*g^5 + (3*c^4*d^2 - 2*a*
c^3*e^2)*f^4*g - 2*(2*a*c^3*d^2 - 3*a^2*c^2*e^2)*f^2*g^3 - ((a*c^5*d^2*e + a^2*c
^4*e^3)*f^8 + 2*(a*c^5*d^3 + a^2*c^4*d*e^2)*f^7*g + 2*(a^2*c^4*d^2*e + a^3*c^3*e
^3)*f^6*g^2 + 6*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f^5*g^3 + 6*(a^3*c^3*d^3 + a^4*c^2
*d*e^2)*f^3*g^5 - 2*(a^4*c^2*d^2*e + a^5*c*e^3)*f^2*g^6 + 2*(a^4*c^2*d^3 + a^5*c
*d*e^2)*f*g^7 - (a^5*c*d^2*e + a^6*e^3)*g^8)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*
g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2
- 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 +
2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4
*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4
+ 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 +
2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 +
a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^9*e^4)*g^12)))*sqrt(e*
x + d)*sqrt(g*x + f)*sqrt(-(c^3*d*f^3 - 3*a*c^2*e*f^2*g - 3*a*c^2*d*f*g^2 + a^2*
c*e*g^3 + ((a*c^4*d^2 + a^2*c^3*e^2)*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2
+ 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 + (a^4*c*d^2 + a^5*e^2)*g^6)*sqrt(-(c^5*e
^2*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*
d^2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*
f^2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 +
2*a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2
+ a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*
g^6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^
4 + 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a
^9*e^4)*g^12)))/((a*c^4*d^2 + a^2*c^3*e^2)*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f
^4*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 + (a^4*c*d^2 + a^5*e^2)*g^6)) + 2*(
c^3*e^2*f^3*g + 3*c^3*d*e*f^2*g^2 - 3*a*c^2*e^2*f*g^3 - a*c^2*d*e*g^4)*x + (2*(c
^5*d^3 + a*c^4*d*e^2)*f^7 + 6*(a*c^4*d^3 + a^2*c^3*d*e^2)*f^5*g^2 + 6*(a^2*c^3*d
^3 + a^3*c^2*d*e^2)*f^3*g^4 + 2*(a^3*c^2*d^3 + a^4*c*d*e^2)*f*g^6 + ((c^5*d^2*e
+ a*c^4*e^3)*f^7 + (c^5*d^3 + a*c^4*d*e^2)*f^6*g + 3*(a*c^4*d^2*e + a^2*c^3*e^3)
*f^5*g^2 + 3*(a*c^4*d^3 + a^2*c^3*d*e^2)*f^4*g^3 + 3*(a^2*c^3*d^2*e + a^3*c^2*e^
3)*f^3*g^4 + 3*(a^2*c^3*d^3 + a^3*c^2*d*e^2)*f^2*g^5 + (a^3*c^2*d^2*e + a^4*c*e^
3)*f*g^6 + (a^3*c^2*d^3 + a^4*c*d*e^2)*g^7)*x)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^
5*g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^
2 - 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4
+ 2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a
^4*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^
4 + 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4
+ 2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2
+ a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^9*e^4)*g^12)))/x) +
(c*e*f^4 - c*d*f^3*g + a*e*f^2*g^2 - a*d*f*g^3 + (c*e*f^3*g - c*d*f^2*g^2 + a*e*
f*g^3 - a*d*g^4)*x)*sqrt(-(c^3*d*f^3 - 3*a*c^2*e*f^2*g - 3*a*c^2*d*f*g^2 + a^2*c
*e*g^3 + ((a*c^4*d^2 + a^2*c^3*e^2)*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2
+ 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 + (a^4*c*d^2 + a^5*e^2)*g^6)*sqrt(-(c^5*e^
2*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d
^2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f
^2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2
*a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 +
a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g
^6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4
+ 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^
9*e^4)*g^12)))/((a*c^4*d^2 + a^2*c^3*e^2)*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^
4*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 + (a^4*c*d^2 + a^5*e^2)*g^6))*log(-(
c^3*e^2*f^4 + 4*c^3*d*e*f^3*g - 4*a*c^2*d*e*f*g^3 - a*c^2*d^2*g^4 + 3*(c^3*d^2 -
a*c^2*e^2)*f^2*g^2 - 2*(c^4*d*e*f^5 - 10*a*c^3*d*e*f^3*g^2 + 5*a^2*c^2*d*e*f*g^
4 + a^2*c^2*d^2*g^5 + (3*c^4*d^2 - 2*a*c^3*e^2)*f^4*g - 2*(2*a*c^3*d^2 - 3*a^2*c
^2*e^2)*f^2*g^3 - ((a*c^5*d^2*e + a^2*c^4*e^3)*f^8 + 2*(a*c^5*d^3 + a^2*c^4*d*e^
2)*f^7*g + 2*(a^2*c^4*d^2*e + a^3*c^3*e^3)*f^6*g^2 + 6*(a^2*c^4*d^3 + a^3*c^3*d*
e^2)*f^5*g^3 + 6*(a^3*c^3*d^3 + a^4*c^2*d*e^2)*f^3*g^5 - 2*(a^4*c^2*d^2*e + a^5*
c*e^3)*f^2*g^6 + 2*(a^4*c^2*d^3 + a^5*c*d*e^2)*f*g^7 - (a^5*c*d^2*e + a^6*e^3)*g
^8)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*
f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 -
3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6
*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*
a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a
^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8
+ 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a
^8*c*d^2*e^2 + a^9*e^4)*g^12)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-(c^3*d*f^3 - 3
*a*c^2*e*f^2*g - 3*a*c^2*d*f*g^2 + a^2*c*e*g^3 + ((a*c^4*d^2 + a^2*c^3*e^2)*f^6
+ 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 +
(a^4*c*d^2 + a^5*e^2)*g^6)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f
^3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4
*g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2
+ a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2
+ 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4
+ 2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2
+ a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^
10 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^9*e^4)*g^12)))/((a*c^4*d^2 + a^2*c^3*e^2
)*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*
g^4 + (a^4*c*d^2 + a^5*e^2)*g^6)) + 2*(c^3*e^2*f^3*g + 3*c^3*d*e*f^2*g^2 - 3*a*c
^2*e^2*f*g^3 - a*c^2*d*e*g^4)*x + (2*(c^5*d^3 + a*c^4*d*e^2)*f^7 + 6*(a*c^4*d^3
+ a^2*c^3*d*e^2)*f^5*g^2 + 6*(a^2*c^3*d^3 + a^3*c^2*d*e^2)*f^3*g^4 + 2*(a^3*c^2*
d^3 + a^4*c*d*e^2)*f*g^6 + ((c^5*d^2*e + a*c^4*e^3)*f^7 + (c^5*d^3 + a*c^4*d*e^2
)*f^6*g + 3*(a*c^4*d^2*e + a^2*c^3*e^3)*f^5*g^2 + 3*(a*c^4*d^3 + a^2*c^3*d*e^2)*
f^4*g^3 + 3*(a^2*c^3*d^2*e + a^3*c^2*e^3)*f^3*g^4 + 3*(a^2*c^3*d^3 + a^3*c^2*d*e
^2)*f^2*g^5 + (a^3*c^2*d^2*e + a^4*c*e^3)*f*g^6 + (a^3*c^2*d^3 + a^4*c*d*e^2)*g^
7)*x)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*
e*f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2
- 3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 +
6*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 +
2*a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 +
a^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g
^8 + 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2
*a^8*c*d^2*e^2 + a^9*e^4)*g^12)))/x) - (c*e*f^4 - c*d*f^3*g + a*e*f^2*g^2 - a*d*
f*g^3 + (c*e*f^3*g - c*d*f^2*g^2 + a*e*f*g^3 - a*d*g^4)*x)*sqrt(-(c^3*d*f^3 - 3*
a*c^2*e*f^2*g - 3*a*c^2*d*f*g^2 + a^2*c*e*g^3 - ((a*c^4*d^2 + a^2*c^3*e^2)*f^6 +
3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 + (
a^4*c*d^2 + a^5*e^2)*g^6)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f^
3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4*
g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2 +
a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2
+ 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4 +
2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2
+ a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^1
0 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^9*e^4)*g^12)))/((a*c^4*d^2 + a^2*c^3*e^2)
*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g
^4 + (a^4*c*d^2 + a^5*e^2)*g^6))*log(-(c^3*e^2*f^4 + 4*c^3*d*e*f^3*g - 4*a*c^2*d
*e*f*g^3 - a*c^2*d^2*g^4 + 3*(c^3*d^2 - a*c^2*e^2)*f^2*g^2 + 2*(c^4*d*e*f^5 - 10
*a*c^3*d*e*f^3*g^2 + 5*a^2*c^2*d*e*f*g^4 + a^2*c^2*d^2*g^5 + (3*c^4*d^2 - 2*a*c^
3*e^2)*f^4*g - 2*(2*a*c^3*d^2 - 3*a^2*c^2*e^2)*f^2*g^3 + ((a*c^5*d^2*e + a^2*c^4
*e^3)*f^8 + 2*(a*c^5*d^3 + a^2*c^4*d*e^2)*f^7*g + 2*(a^2*c^4*d^2*e + a^3*c^3*e^3
)*f^6*g^2 + 6*(a^2*c^4*d^3 + a^3*c^3*d*e^2)*f^5*g^3 + 6*(a^3*c^3*d^3 + a^4*c^2*d
*e^2)*f^3*g^5 - 2*(a^4*c^2*d^2*e + a^5*c*e^3)*f^2*g^6 + 2*(a^4*c^2*d^3 + a^5*c*d
*e^2)*f*g^7 - (a^5*c*d^2*e + a^6*e^3)*g^8)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*g
- 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2 -
2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 + 2*
a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4*c
^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4 +
20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 + 2
*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 + a
^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^9*e^4)*g^12)))*sqrt(e*x
+ d)*sqrt(g*x + f)*sqrt(-(c^3*d*f^3 - 3*a*c^2*e*f^2*g - 3*a*c^2*d*f*g^2 + a^2*c*
e*g^3 - ((a*c^4*d^2 + a^2*c^3*e^2)*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2 +
3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 + (a^4*c*d^2 + a^5*e^2)*g^6)*sqrt(-(c^5*e^2
*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d^
2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f^
2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2*
a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 +
a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g^
6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4
+ 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^9
*e^4)*g^12)))/((a*c^4*d^2 + a^2*c^3*e^2)*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4
*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 + (a^4*c*d^2 + a^5*e^2)*g^6)) + 2*(c^
3*e^2*f^3*g + 3*c^3*d*e*f^2*g^2 - 3*a*c^2*e^2*f*g^3 - a*c^2*d*e*g^4)*x - (2*(c^5
*d^3 + a*c^4*d*e^2)*f^7 + 6*(a*c^4*d^3 + a^2*c^3*d*e^2)*f^5*g^2 + 6*(a^2*c^3*d^3
+ a^3*c^2*d*e^2)*f^3*g^4 + 2*(a^3*c^2*d^3 + a^4*c*d*e^2)*f*g^6 + ((c^5*d^2*e +
a*c^4*e^3)*f^7 + (c^5*d^3 + a*c^4*d*e^2)*f^6*g + 3*(a*c^4*d^2*e + a^2*c^3*e^3)*f
^5*g^2 + 3*(a*c^4*d^3 + a^2*c^3*d*e^2)*f^4*g^3 + 3*(a^2*c^3*d^2*e + a^3*c^2*e^3)
*f^3*g^4 + 3*(a^2*c^3*d^3 + a^3*c^2*d*e^2)*f^2*g^5 + (a^3*c^2*d^2*e + a^4*c*e^3)
*f*g^6 + (a^3*c^2*d^3 + a^4*c*d*e^2)*g^7)*x)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*
g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2
- 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 +
2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4
*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4
+ 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 +
2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 +
a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^9*e^4)*g^12)))/x) + (c
*e*f^4 - c*d*f^3*g + a*e*f^2*g^2 - a*d*f*g^3 + (c*e*f^3*g - c*d*f^2*g^2 + a*e*f*
g^3 - a*d*g^4)*x)*sqrt(-(c^3*d*f^3 - 3*a*c^2*e*f^2*g - 3*a*c^2*d*f*g^2 + a^2*c*e
*g^3 - ((a*c^4*d^2 + a^2*c^3*e^2)*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2 +
3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 + (a^4*c*d^2 + a^5*e^2)*g^6)*sqrt(-(c^5*e^2*
f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d^2
*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f^2
*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2*a
^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 + a
^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g^6
+ 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4 +
2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^9*
e^4)*g^12)))/((a*c^4*d^2 + a^2*c^3*e^2)*f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*
g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 + (a^4*c*d^2 + a^5*e^2)*g^6))*log(-(c^
3*e^2*f^4 + 4*c^3*d*e*f^3*g - 4*a*c^2*d*e*f*g^3 - a*c^2*d^2*g^4 + 3*(c^3*d^2 - a
*c^2*e^2)*f^2*g^2 - 2*(c^4*d*e*f^5 - 10*a*c^3*d*e*f^3*g^2 + 5*a^2*c^2*d*e*f*g^4
+ a^2*c^2*d^2*g^5 + (3*c^4*d^2 - 2*a*c^3*e^2)*f^4*g - 2*(2*a*c^3*d^2 - 3*a^2*c^2
*e^2)*f^2*g^3 + ((a*c^5*d^2*e + a^2*c^4*e^3)*f^8 + 2*(a*c^5*d^3 + a^2*c^4*d*e^2)
*f^7*g + 2*(a^2*c^4*d^2*e + a^3*c^3*e^3)*f^6*g^2 + 6*(a^2*c^4*d^3 + a^3*c^3*d*e^
2)*f^5*g^3 + 6*(a^3*c^3*d^3 + a^4*c^2*d*e^2)*f^3*g^5 - 2*(a^4*c^2*d^2*e + a^5*c*
e^3)*f^2*g^6 + 2*(a^4*c^2*d^3 + a^5*c*d*e^2)*f*g^7 - (a^5*c*d^2*e + a^6*e^3)*g^8
)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*f*
g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 - 3
*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6*(
a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*a^
4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a^6
*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8 +
6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a^8
*c*d^2*e^2 + a^9*e^4)*g^12)))*sqrt(e*x + d)*sqrt(g*x + f)*sqrt(-(c^3*d*f^3 - 3*a
*c^2*e*f^2*g - 3*a*c^2*d*f*g^2 + a^2*c*e*g^3 - ((a*c^4*d^2 + a^2*c^3*e^2)*f^6 +
3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^4 + (a
^4*c*d^2 + a^5*e^2)*g^6)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f^3
*g^3 + 6*a^2*c^3*d*e*f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4*g
^2 - 3*(2*a*c^4*d^2 - 3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2 +
a^3*c^6*e^4)*f^12 + 6*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2 +
15*(a^3*c^6*d^4 + 2*a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4 +
2*a^5*c^4*d^2*e^2 + a^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2 +
a^7*c^2*e^4)*f^4*g^8 + 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^10
+ (a^7*c^2*d^4 + 2*a^8*c*d^2*e^2 + a^9*e^4)*g^12)))/((a*c^4*d^2 + a^2*c^3*e^2)*
f^6 + 3*(a^2*c^3*d^2 + a^3*c^2*e^2)*f^4*g^2 + 3*(a^3*c^2*d^2 + a^4*c*e^2)*f^2*g^
4 + (a^4*c*d^2 + a^5*e^2)*g^6)) + 2*(c^3*e^2*f^3*g + 3*c^3*d*e*f^2*g^2 - 3*a*c^2
*e^2*f*g^3 - a*c^2*d*e*g^4)*x - (2*(c^5*d^3 + a*c^4*d*e^2)*f^7 + 6*(a*c^4*d^3 +
a^2*c^3*d*e^2)*f^5*g^2 + 6*(a^2*c^3*d^3 + a^3*c^2*d*e^2)*f^3*g^4 + 2*(a^3*c^2*d^
3 + a^4*c*d*e^2)*f*g^6 + ((c^5*d^2*e + a*c^4*e^3)*f^7 + (c^5*d^3 + a*c^4*d*e^2)*
f^6*g + 3*(a*c^4*d^2*e + a^2*c^3*e^3)*f^5*g^2 + 3*(a*c^4*d^3 + a^2*c^3*d*e^2)*f^
4*g^3 + 3*(a^2*c^3*d^2*e + a^3*c^2*e^3)*f^3*g^4 + 3*(a^2*c^3*d^3 + a^3*c^2*d*e^2
)*f^2*g^5 + (a^3*c^2*d^2*e + a^4*c*e^3)*f*g^6 + (a^3*c^2*d^3 + a^4*c*d*e^2)*g^7)
*x)*sqrt(-(c^5*e^2*f^6 + 6*c^5*d*e*f^5*g - 20*a*c^4*d*e*f^3*g^3 + 6*a^2*c^3*d*e*
f*g^5 + a^2*c^3*d^2*g^6 + 3*(3*c^5*d^2 - 2*a*c^4*e^2)*f^4*g^2 - 3*(2*a*c^4*d^2 -
3*a^2*c^3*e^2)*f^2*g^4)/((a*c^8*d^4 + 2*a^2*c^7*d^2*e^2 + a^3*c^6*e^4)*f^12 + 6
*(a^2*c^7*d^4 + 2*a^3*c^6*d^2*e^2 + a^4*c^5*e^4)*f^10*g^2 + 15*(a^3*c^6*d^4 + 2*
a^4*c^5*d^2*e^2 + a^5*c^4*e^4)*f^8*g^4 + 20*(a^4*c^5*d^4 + 2*a^5*c^4*d^2*e^2 + a
^6*c^3*e^4)*f^6*g^6 + 15*(a^5*c^4*d^4 + 2*a^6*c^3*d^2*e^2 + a^7*c^2*e^4)*f^4*g^8
+ 6*(a^6*c^3*d^4 + 2*a^7*c^2*d^2*e^2 + a^8*c*e^4)*f^2*g^10 + (a^7*c^2*d^4 + 2*a
^8*c*d^2*e^2 + a^9*e^4)*g^12)))/x))/(c*e*f^4 - c*d*f^3*g + a*e*f^2*g^2 - a*d*f*g
^3 + (c*e*f^3*g - c*d*f^2*g^2 + a*e*f*g^3 - a*d*g^4)*x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (a + c x^{2}\right ) \sqrt{d + e x} \left (f + g x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((a + c*x**2)*sqrt(d + e*x)*(f + g*x)**(3/2)), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + a)*sqrt(e*x + d)*(g*x + f)^(3/2)),x, algorithm="giac")

[Out]

Exception raised: RuntimeError