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

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

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Rubi [A]  time = 0.76, antiderivative size = 354, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {912, 96, 93, 208} \begin {gather*} \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}} \end {gather*}

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*
x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[m, 1])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 912

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegr
and[(d + e*x)^m*(f + g*x)^n, 1/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[c*d^2 + a*e^2,
0] &&  !IntegerQ[m] &&  !IntegerQ[n]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {d+e x} (f+g x)^{3/2} \left (a+c x^2\right )} \, dx &=\int \left (\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} (f+g x)^{3/2}}+\frac {\sqrt {-a}}{2 a \left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} (f+g x)^{3/2}}\right ) \, dx\\ &=-\frac {\int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} (f+g x)^{3/2}} \, dx}{2 \sqrt {-a}}-\frac {\int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} (f+g x)^{3/2}} \, dx}{2 \sqrt {-a}}\\ &=\frac {g \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g) \sqrt {f+g x}}-\frac {g \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g) \sqrt {f+g x}}-\frac {\sqrt {c} \int \frac {1}{\left (\sqrt {-a}-\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \left (\sqrt {-a} \sqrt {c} f-a g\right )}-\frac {\sqrt {c} \int \frac {1}{\left (\sqrt {-a}+\sqrt {c} x\right ) \sqrt {d+e x} \sqrt {f+g x}} \, dx}{2 \left (\sqrt {-a} \sqrt {c} f+a g\right )}\\ &=\frac {g \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g) \sqrt {f+g x}}-\frac {g \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g) \sqrt {f+g x}}-\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {1}{\sqrt {c} d+\sqrt {-a} e-\left (\sqrt {c} f+\sqrt {-a} g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} f-a g}-\frac {\sqrt {c} \operatorname {Subst}\left (\int \frac {1}{-\sqrt {c} d+\sqrt {-a} e-\left (-\sqrt {c} f+\sqrt {-a} g\right ) x^2} \, dx,x,\frac {\sqrt {d+e x}}{\sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {c} f+a g}\\ &=\frac {g \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} f-\sqrt {-a} g\right ) (e f-d g) \sqrt {f+g x}}-\frac {g \sqrt {d+e x}}{\sqrt {-a} \left (\sqrt {c} f+\sqrt {-a} g\right ) (e f-d g) \sqrt {f+g x}}+\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {\sqrt {c} f-\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {-a} e} \sqrt {f+g x}}\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 {\sqrt {c} f+\sqrt {-a} g} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {-a} e} \sqrt {f+g x}}\right )}{\sqrt {-a} \sqrt {\sqrt {c} d+\sqrt {-a} e} \left (\sqrt {c} f+\sqrt {-a} g\right )^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.78, size = 287, normalized size = 0.81 \begin {gather*} \frac {\frac {2 \sqrt {-a} g^2 \sqrt {d+e x}}{\sqrt {f+g x} \left (a g^2+c f^2\right ) (e f-d g)}+\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 {\sqrt {-a} e-\sqrt {c} d} \left (\sqrt {-a} g-\sqrt {c} f\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 {\sqrt {-a} e+\sqrt {c} d} \left (\sqrt {-a} g+\sqrt {c} f\right )^{3/2}}}{\sqrt {-a}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [C]  time = 1.05, size = 393, normalized size = 1.11 \begin {gather*} \frac {2 g^2 \sqrt {d+e x}}{\sqrt {f+g x} \left (a g^2+c f^2\right ) (e f-d g)}+\frac {i \sqrt {c} \left (\sqrt {c} f+i \sqrt {a} g\right )^2 \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {a g^2+c f^2}}{\sqrt {f+g x} \sqrt {-i \sqrt {a} \sqrt {c} d g+i \sqrt {a} \sqrt {c} e f-a e g-c d f}}\right )}{\sqrt {a} \left (a g^2+c f^2\right )^{3/2} \sqrt {-\left (\left (\sqrt {c} d-i \sqrt {a} e\right ) \left (\sqrt {c} f+i \sqrt {a} g\right )\right )}}-\frac {i \sqrt {c} \left (\sqrt {c} f-i \sqrt {a} g\right )^2 \tan ^{-1}\left (\frac {\sqrt {d+e x} \sqrt {a g^2+c f^2}}{\sqrt {f+g x} \sqrt {i \sqrt {a} \sqrt {c} d g-i \sqrt {a} \sqrt {c} e f-a e g-c d f}}\right )}{\sqrt {a} \left (a g^2+c f^2\right )^{3/2} \sqrt {-\left (\left (\sqrt {c} d+i \sqrt {a} e\right ) \left (\sqrt {c} f-i \sqrt {a} g\right )\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*g^2*Sqrt[d + e*x])/((e*f - d*g)*(c*f^2 + a*g^2)*Sqrt[f + g*x]) + (I*Sqrt[c]*(Sqrt[c]*f + I*Sqrt[a]*g)^2*Arc
Tan[(Sqrt[c*f^2 + a*g^2]*Sqrt[d + e*x])/(Sqrt[-(c*d*f) + I*Sqrt[a]*Sqrt[c]*e*f - I*Sqrt[a]*Sqrt[c]*d*g - a*e*g
]*Sqrt[f + g*x])])/(Sqrt[a]*Sqrt[-((Sqrt[c]*d - I*Sqrt[a]*e)*(Sqrt[c]*f + I*Sqrt[a]*g))]*(c*f^2 + a*g^2)^(3/2)
) - (I*Sqrt[c]*(Sqrt[c]*f - I*Sqrt[a]*g)^2*ArcTan[(Sqrt[c*f^2 + a*g^2]*Sqrt[d + e*x])/(Sqrt[-(c*d*f) - I*Sqrt[
a]*Sqrt[c]*e*f + I*Sqrt[a]*Sqrt[c]*d*g - a*e*g]*Sqrt[f + g*x])])/(Sqrt[a]*Sqrt[-((Sqrt[c]*d + I*Sqrt[a]*e)*(Sq
rt[c]*f - I*Sqrt[a]*g))]*(c*f^2 + a*g^2)^(3/2))

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fricas [B]  time = 126.49, size = 12028, normalized size = 33.98

result too large to display

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, 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^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)*sq
rt(-(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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.09, size = 10977, normalized size = 31.01 \begin {gather*} \text {output too large to display} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (c x^{2} + a\right )} \sqrt {e x + d} {\left (g x + f\right )}^{\frac {3}{2}}}\,{d x} \end {gather*}

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, algorithm="maxima")

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (f+g\,x\right )}^{3/2}\,\left (c\,x^2+a\right )\,\sqrt {d+e\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + c x^{2}\right ) \sqrt {d + e x} \left (f + g x\right )^{\frac {3}{2}}}\, dx \end {gather*}

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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