3.23.100 \(\int \frac {\sqrt {c+d x}}{\sqrt {a+b x} (e+f x)} \, dx\) [2300]

Optimal. Leaf size=119 \[ \frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} f}-\frac {2 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {d e-c f} \sqrt {a+b x}}{\sqrt {b e-a f} \sqrt {c+d x}}\right )}{f \sqrt {b e-a f}} \]

[Out]

2*arctanh(d^(1/2)*(b*x+a)^(1/2)/b^(1/2)/(d*x+c)^(1/2))*d^(1/2)/f/b^(1/2)-2*arctanh((-c*f+d*e)^(1/2)*(b*x+a)^(1
/2)/(-a*f+b*e)^(1/2)/(d*x+c)^(1/2))*(-c*f+d*e)^(1/2)/f/(-a*f+b*e)^(1/2)

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Rubi [A]
time = 0.06, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {132, 65, 223, 212, 12, 95, 214} \begin {gather*} \frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b} f}-\frac {2 \sqrt {d e-c f} \tanh ^{-1}\left (\frac {\sqrt {a+b x} \sqrt {d e-c f}}{\sqrt {c+d x} \sqrt {b e-a f}}\right )}{f \sqrt {b e-a f}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x]/(Sqrt[a + b*x]*(e + f*x)),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 95

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 132

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 214

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

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

Rubi steps

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

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Mathematica [A]
time = 0.57, size = 163, normalized size = 1.37 \begin {gather*} \frac {2 \left (-\frac {\sqrt {b} \sqrt {-d e+c f} \tan ^{-1}\left (\frac {\sqrt {d} \left (-\sqrt {\frac {b}{d}} f \sqrt {a+b x} \sqrt {c+d x}+b (e+f x)\right )}{\sqrt {b} \sqrt {b e-a f} \sqrt {-d e+c f}}\right )}{\sqrt {b e-a f}}-\sqrt {d} \log \left (\sqrt {a+b x}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )\right )}{\sqrt {\frac {b}{d}} \sqrt {d} f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[c + d*x]/(Sqrt[a + b*x]*(e + f*x)),x]

[Out]

(2*(-((Sqrt[b]*Sqrt[-(d*e) + c*f]*ArcTan[(Sqrt[d]*(-(Sqrt[b/d]*f*Sqrt[a + b*x]*Sqrt[c + d*x]) + b*(e + f*x)))/
(Sqrt[b]*Sqrt[b*e - a*f]*Sqrt[-(d*e) + c*f])])/Sqrt[b*e - a*f]) - Sqrt[d]*Log[Sqrt[a + b*x] - Sqrt[b/d]*Sqrt[c
+ d*x]]))/(Sqrt[b/d]*Sqrt[d]*f)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(299\) vs. \(2(95)=190\).
time = 0.12, size = 300, normalized size = 2.52

method	result	size
default	\(\frac {\left (\sqrt {\frac {\left (c f -d e \right ) \left (a f -b e \right )}{f^{2}}}\, \ln \left (\frac {2 b d x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) d f -\ln \left (\frac {a d f x +b c f x -2 b d e x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {\frac {\left (c f -d e \right ) \left (a f -b e \right )}{f^{2}}}\, f +2 a c f -a d e -b c e}{f x +e}\right ) \sqrt {b d}\, c f +\ln \left (\frac {a d f x +b c f x -2 b d e x +2 \sqrt {\left (d x +c \right ) \left (b x +a \right )}\, \sqrt {\frac {\left (c f -d e \right ) \left (a f -b e \right )}{f^{2}}}\, f +2 a c f -a d e -b c e}{f x +e}\right ) \sqrt {b d}\, d e \right ) \sqrt {b x +a}\, \sqrt {d x +c}}{\sqrt {\frac {\left (c f -d e \right ) \left (a f -b e \right )}{f^{2}}}\, \sqrt {b d}\, f^{2} \sqrt {\left (d x +c \right ) \left (b x +a \right )}}\)	\(300\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x+c)^(1/2)/(f*x+e)/(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

(((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*ln(1/2*(2*b*d*x+2*((d*x+c)*(b*x+a))^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*d
*f-ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((d*x+c)*(b*x+a))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)^(1/2)*f+2*a*c*f-a*d*e-b*c
*e)/(f*x+e))*(b*d)^(1/2)*c*f+ln((a*d*f*x+b*c*f*x-2*b*d*e*x+2*((d*x+c)*(b*x+a))^(1/2)*((c*f-d*e)*(a*f-b*e)/f^2)
^(1/2)*f+2*a*c*f-a*d*e-b*c*e)/(f*x+e))*(b*d)^(1/2)*d*e)*(b*x+a)^(1/2)*(d*x+c)^(1/2)/((c*f-d*e)*(a*f-b*e)/f^2)^
(1/2)/(b*d)^(1/2)/f^2/((d*x+c)*(b*x+a))^(1/2)

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(f*x+e)/(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(((a*d)/f>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (99) = 198\).
time = 5.28, size = 1354, normalized size = 11.38 \begin {gather*} \left [\frac {\sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + \sqrt {\frac {c f - d e}{a f - b e}} \log \left (\frac {8 \, a^{2} c^{2} f^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} f^{2} x^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} f^{2} x - 4 \, {\left (2 \, a^{2} c f^{2} + {\left (a b c + a^{2} d\right )} f^{2} x + {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} e^{2} - {\left ({\left (b^{2} c + 3 \, a b d\right )} f x + {\left (3 \, a b c + a^{2} d\right )} f\right )} e\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c f - d e}{a f - b e}} + {\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} e^{2} - 2 \, {\left (4 \, {\left (b^{2} c d + a b d^{2}\right )} f x^{2} + {\left (3 \, b^{2} c^{2} + 10 \, a b c d + 3 \, a^{2} d^{2}\right )} f x + 4 \, {\left (a b c^{2} + a^{2} c d\right )} f\right )} e}{f^{2} x^{2} + 2 \, f x e + e^{2}}\right )}{2 \, f}, -\frac {2 \, \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) - \sqrt {\frac {c f - d e}{a f - b e}} \log \left (\frac {8 \, a^{2} c^{2} f^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} f^{2} x^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} f^{2} x - 4 \, {\left (2 \, a^{2} c f^{2} + {\left (a b c + a^{2} d\right )} f^{2} x + {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} e^{2} - {\left ({\left (b^{2} c + 3 \, a b d\right )} f x + {\left (3 \, a b c + a^{2} d\right )} f\right )} e\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c f - d e}{a f - b e}} + {\left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )} e^{2} - 2 \, {\left (4 \, {\left (b^{2} c d + a b d^{2}\right )} f x^{2} + {\left (3 \, b^{2} c^{2} + 10 \, a b c d + 3 \, a^{2} d^{2}\right )} f x + 4 \, {\left (a b c^{2} + a^{2} c d\right )} f\right )} e}{f^{2} x^{2} + 2 \, f x e + e^{2}}\right )}{2 \, f}, \frac {2 \, \sqrt {-\frac {c f - d e}{a f - b e}} \arctan \left (\frac {{\left (2 \, a c f + {\left (b c + a d\right )} f x - {\left (2 \, b d x + b c + a d\right )} e\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c f - d e}{a f - b e}}}{2 \, {\left (b c d f x^{2} + a c^{2} f + {\left (b c^{2} + a c d\right )} f x - {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )} e\right )}}\right ) + \sqrt {\frac {d}{b}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right )}{2 \, f}, \frac {\sqrt {-\frac {c f - d e}{a f - b e}} \arctan \left (\frac {{\left (2 \, a c f + {\left (b c + a d\right )} f x - {\left (2 \, b d x + b c + a d\right )} e\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {c f - d e}{a f - b e}}}{2 \, {\left (b c d f x^{2} + a c^{2} f + {\left (b c^{2} + a c d\right )} f x - {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )} e\right )}}\right ) - \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right )}{f}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(f*x+e)/(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

[1/2*(sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a
)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x) + sqrt((c*f - d*e)/(a*f - b*e))*log((8*a^2*c^2*f^2 + (b^2
*c^2 + 6*a*b*c*d + a^2*d^2)*f^2*x^2 + 8*(a*b*c^2 + a^2*c*d)*f^2*x - 4*(2*a^2*c*f^2 + (a*b*c + a^2*d)*f^2*x + (
2*b^2*d*x + b^2*c + a*b*d)*e^2 - ((b^2*c + 3*a*b*d)*f*x + (3*a*b*c + a^2*d)*f)*e)*sqrt(b*x + a)*sqrt(d*x + c)*
sqrt((c*f - d*e)/(a*f - b*e)) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x)*e^2
- 2*(4*(b^2*c*d + a*b*d^2)*f*x^2 + (3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*f*x + 4*(a*b*c^2 + a^2*c*d)*f)*e)/(f^2
*x^2 + 2*f*x*e + e^2)))/f, -1/2*(2*sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqr
t(-d/b)/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)) - sqrt((c*f - d*e)/(a*f - b*e))*log((8*a^2*c^2*f^2 + (b^2*c^2
+ 6*a*b*c*d + a^2*d^2)*f^2*x^2 + 8*(a*b*c^2 + a^2*c*d)*f^2*x - 4*(2*a^2*c*f^2 + (a*b*c + a^2*d)*f^2*x + (2*b^
2*d*x + b^2*c + a*b*d)*e^2 - ((b^2*c + 3*a*b*d)*f*x + (3*a*b*c + a^2*d)*f)*e)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt
((c*f - d*e)/(a*f - b*e)) + (8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x)*e^2 - 2*
(4*(b^2*c*d + a*b*d^2)*f*x^2 + (3*b^2*c^2 + 10*a*b*c*d + 3*a^2*d^2)*f*x + 4*(a*b*c^2 + a^2*c*d)*f)*e)/(f^2*x^2
+ 2*f*x*e + e^2)))/f, 1/2*(2*sqrt(-(c*f - d*e)/(a*f - b*e))*arctan(1/2*(2*a*c*f + (b*c + a*d)*f*x - (2*b*d*x
+ b*c + a*d)*e)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-(c*f - d*e)/(a*f - b*e))/(b*c*d*f*x^2 + a*c^2*f + (b*c^2 + a
*c*d)*f*x - (b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)*e)) + sqrt(d/b)*log(8*b^2*d^2*x^2 + b^2*c^2 + 6*a*b*c*d +
a^2*d^2 + 4*(2*b^2*d*x + b^2*c + a*b*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(d/b) + 8*(b^2*c*d + a*b*d^2)*x))/f, (
sqrt(-(c*f - d*e)/(a*f - b*e))*arctan(1/2*(2*a*c*f + (b*c + a*d)*f*x - (2*b*d*x + b*c + a*d)*e)*sqrt(b*x + a)*
sqrt(d*x + c)*sqrt(-(c*f - d*e)/(a*f - b*e))/(b*c*d*f*x^2 + a*c^2*f + (b*c^2 + a*c*d)*f*x - (b*d^2*x^2 + a*c*d
+ (b*c*d + a*d^2)*x)*e)) - sqrt(-d/b)*arctan(1/2*(2*b*d*x + b*c + a*d)*sqrt(b*x + a)*sqrt(d*x + c)*sqrt(-d/b)
/(b*d^2*x^2 + a*c*d + (b*c*d + a*d^2)*x)))/f]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)**(1/2)/(f*x+e)/(b*x+a)**(1/2),x)

[Out]

Integral(sqrt(c + d*x)/(sqrt(a + b*x)*(e + f*x)), x)

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/(f*x+e)/(b*x+a)^(1/2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:index.cc index_m i_lex_is_greater Error: Bad Argument Value

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Mupad [B]
time = 107.43, size = 2500, normalized size = 21.01 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (d^(1/2)*atan(((d^(1/2)*((65536*(24*a^(1/2)*b^8*c^(7/2)*d^5*e^4 - 24*a^(3/2)*b^7*c^(5/2)*d^6*e^4 - 24*a^(5/2
)*b^6*c^(3/2)*d^7*e^4 + 24*a^(7/2)*b^5*c^(1/2)*d^8*e^4 - 40*a^(3/2)*b^7*c^(13/2)*d^2*f^4 + 8*a^(5/2)*b^6*c^(11
/2)*d^3*f^4 + 24*a^(7/2)*b^5*c^(9/2)*d^4*f^4 - 32*a^(9/2)*b^4*c^(7/2)*d^5*f^4 + 16*a^(11/2)*b^3*c^(5/2)*d^6*f^
4 + 24*a^(1/2)*b^8*c^(15/2)*d*f^4 - 108*a^(1/2)*b^8*c^(9/2)*d^4*e^3*f + 176*a^(3/2)*b^7*c^(7/2)*d^5*e^3*f - 72
*a^(5/2)*b^6*c^(5/2)*d^6*e^3*f + 48*a^(7/2)*b^5*c^(3/2)*d^7*e^3*f - 44*a^(9/2)*b^4*c^(1/2)*d^8*e^3*f - 108*a^(
1/2)*b^8*c^(13/2)*d^2*e*f^3 + 208*a^(3/2)*b^7*c^(11/2)*d^3*e*f^3 - 136*a^(5/2)*b^6*c^(9/2)*d^4*e*f^3 + 48*a^(7
/2)*b^5*c^(7/2)*d^5*e*f^3 + 20*a^(9/2)*b^4*c^(5/2)*d^6*e*f^3 - 32*a^(11/2)*b^3*c^(3/2)*d^7*e*f^3 + 168*a^(1/2)
*b^8*c^(11/2)*d^3*e^2*f^2 - 320*a^(3/2)*b^7*c^(9/2)*d^4*e^2*f^2 + 224*a^(5/2)*b^6*c^(7/2)*d^5*e^2*f^2 - 144*a^
(7/2)*b^5*c^(5/2)*d^6*e^2*f^2 + 56*a^(9/2)*b^4*c^(3/2)*d^7*e^2*f^2 + 16*a^(11/2)*b^3*c^(1/2)*d^8*e^2*f^2))/(d^
17*e^8) + (2*d^(1/2)*((2*d^(1/2)*((65536*(44*a^(5/2)*b^7*c^(11/2)*d^2*f^6 - 4*a^(1/2)*b^9*c^(15/2)*f^6 - 4*a^(
7/2)*b^6*c^(9/2)*d^3*f^6 - 24*a^(9/2)*b^5*c^(7/2)*d^4*f^6 + 28*a^(11/2)*b^4*c^(5/2)*d^5*f^6 - 16*a^(13/2)*b^3*
c^(3/2)*d^6*f^6 - 24*a^(3/2)*b^8*c^(13/2)*d*f^6 + 51*a^(1/2)*b^9*c^(13/2)*d*e*f^5 + 16*a^(3/2)*b^8*c^(11/2)*d^
2*e*f^5 - 150*a^(5/2)*b^7*c^(9/2)*d^3*e*f^5 + 116*a^(7/2)*b^6*c^(7/2)*d^4*e*f^5 - 85*a^(9/2)*b^5*c^(5/2)*d^5*e
*f^5 + 36*a^(11/2)*b^4*c^(3/2)*d^6*e*f^5 + 16*a^(13/2)*b^3*c^(1/2)*d^7*e*f^5 - 36*a^(1/2)*b^9*c^(7/2)*d^4*e^4*
f^2 + 36*a^(3/2)*b^8*c^(5/2)*d^5*e^4*f^2 + 36*a^(5/2)*b^7*c^(3/2)*d^6*e^4*f^2 - 36*a^(7/2)*b^6*c^(1/2)*d^7*e^4
*f^2 + 127*a^(1/2)*b^9*c^(9/2)*d^3*e^3*f^3 - 180*a^(3/2)*b^8*c^(7/2)*d^4*e^3*f^3 + 66*a^(5/2)*b^7*c^(5/2)*d^5*
e^3*f^3 - 100*a^(7/2)*b^6*c^(3/2)*d^6*e^3*f^3 + 87*a^(9/2)*b^5*c^(1/2)*d^7*e^3*f^3 - 138*a^(1/2)*b^9*c^(11/2)*
d^2*e^2*f^4 + 152*a^(3/2)*b^8*c^(9/2)*d^3*e^2*f^4 + 4*a^(5/2)*b^7*c^(7/2)*d^4*e^2*f^4 + 24*a^(7/2)*b^6*c^(5/2)
*d^5*e^2*f^4 + 22*a^(9/2)*b^5*c^(3/2)*d^6*e^2*f^4 - 64*a^(11/2)*b^4*c^(1/2)*d^7*e^2*f^4))/(d^17*e^8) + (2*d^(1
/2)*((65536*(4*b^10*c^7*e*f^6 - 4*a*b^9*c^7*f^7 + 12*a^3*b^7*c^5*d^2*f^7 - 44*a^4*b^6*c^4*d^3*f^7 + 24*a^5*b^5
*c^3*d^4*f^7 + 3*a^4*b^6*d^7*e^4*f^3 - 3*a^5*b^5*d^7*e^3*f^4 + 3*b^10*c^4*d^3*e^4*f^3 - 2*b^10*c^5*d^2*e^3*f^4
+ 12*a^2*b^8*c^6*d*f^7 - 5*b^10*c^6*d*e^2*f^5 + 12*a*b^9*c^3*d^4*e^4*f^3 - 39*a*b^9*c^4*d^3*e^3*f^4 + 38*a*b^
9*c^5*d^2*e^2*f^5 - 48*a^2*b^8*c^5*d^2*e*f^6 + 12*a^3*b^7*c*d^6*e^4*f^3 + 50*a^3*b^7*c^4*d^3*e*f^6 - 42*a^4*b^
6*c*d^6*e^3*f^4 + 52*a^4*b^6*c^3*d^4*e*f^6 + 30*a^5*b^5*c*d^6*e^2*f^5 - 51*a^5*b^5*c^2*d^5*e*f^6 - 7*a*b^9*c^6
*d*e*f^6 - 30*a^2*b^8*c^2*d^5*e^4*f^3 + 36*a^2*b^8*c^3*d^4*e^3*f^4 + 30*a^2*b^8*c^4*d^3*e^2*f^5 + 50*a^3*b^7*c
^2*d^5*e^3*f^4 - 124*a^3*b^7*c^3*d^4*e^2*f^5 + 31*a^4*b^6*c^2*d^5*e^2*f^5))/(d^17*e^8) + (2*d^(1/2)*((65536*(4
*a^(3/2)*b^9*c^(13/2)*f^8 - 4*a^(7/2)*b^7*c^(9/2)*d^2*f^8 - 4*a^(9/2)*b^6*c^(7/2)*d^3*f^8 + 4*a^(13/2)*b^4*c^(
3/2)*d^5*f^8 - 4*a^(1/2)*b^10*c^(13/2)*e*f^7 - 23*a^(3/2)*b^9*c^(11/2)*d*e*f^7 + 23*a^(1/2)*b^10*c^(11/2)*d*e^
2*f^6 + 32*a^(5/2)*b^8*c^(9/2)*d^2*e*f^7 - 10*a^(7/2)*b^7*c^(7/2)*d^3*e*f^7 + 32*a^(9/2)*b^6*c^(5/2)*d^4*e*f^7
- 23*a^(11/2)*b^5*c^(3/2)*d^5*e*f^7 - 4*a^(13/2)*b^4*c^(1/2)*d^6*e*f^7 + 12*a^(1/2)*b^10*c^(7/2)*d^3*e^4*f^4
- 12*a^(3/2)*b^9*c^(5/2)*d^4*e^4*f^4 - 12*a^(5/2)*b^8*c^(3/2)*d^5*e^4*f^4 + 12*a^(7/2)*b^7*c^(1/2)*d^6*e^4*f^4
- 31*a^(1/2)*b^10*c^(9/2)*d^2*e^3*f^5 + 28*a^(3/2)*b^9*c^(7/2)*d^3*e^3*f^5 + 6*a^(5/2)*b^8*c^(5/2)*d^4*e^3*f^
5 + 28*a^(7/2)*b^7*c^(3/2)*d^5*e^3*f^5 - 31*a^(9/2)*b^6*c^(1/2)*d^6*e^3*f^5 + 3*a^(3/2)*b^9*c^(9/2)*d^2*e^2*f^
6 - 26*a^(5/2)*b^8*c^(7/2)*d^3*e^2*f^6 - 26*a^(7/2)*b^7*c^(5/2)*d^4*e^2*f^6 + 3*a^(9/2)*b^6*c^(3/2)*d^5*e^2*f^
6 + 23*a^(11/2)*b^5*c^(1/2)*d^6*e^2*f^6))/(d^17*e^8) - (65536*((a + b*x)^(1/2) - a^(1/2))*(8*a*b^9*c^7*f^8 - 8
*b^10*c^7*e*f^7 + 72*a^3*b^7*c^5*d^2*f^8 - 100*a^4*b^6*c^4*d^3*f^8 + 72*a^5*b^5*c^3*d^4*f^8 - 30*a^6*b^4*c^2*d
^5*f^8 + 3*a^4*b^6*d^7*e^4*f^4 - 13*a^5*b^5*d^7*e^3*f^5 + 18*a^6*b^4*d^7*e^2*f^6 + 3*b^10*c^4*d^3*e^4*f^4 - 13
*b^10*c^5*d^2*e^3*f^5 - 30*a^2*b^8*c^6*d*f^8 + 8*a^7*b^3*c*d^6*f^8 - 8*a^7*b^3*d^7*e*f^7 + 18*b^10*c^6*d*e^2*f
^6 + 12*a^6*b^4*c*d^6*e*f^7 + 12*a*b^9*c^3*d^4*e^4*f^4 - 9*a*b^9*c^4*d^3*e^3*f^5 - 23*a*b^9*c^5*d^2*e^2*f^6 -
36*a^2*b^8*c^5*d^2*e*f^7 + 12*a^3*b^7*c*d^6*e^4*f^4 + 32*a^3*b^7*c^4*d^3*e*f^7 - 9*a^4*b^6*c*d^6*e^3*f^5 + 32*
a^4*b^6*c^3*d^4*e*f^7 - 23*a^5*b^5*c*d^6*e^2*f^6 - 36*a^5*b^5*c^2*d^5*e*f^7 + 12*a*b^9*c^6*d*e*f^7 - 30*a^2*b^
8*c^2*d^5*e^4*f^4 + 22*a^2*b^8*c^3*d^4*e^3*f^5 + 74*a^2*b^8*c^4*d^3*e^2*f^6 + 22*a^3*b^7*c^2*d^5*e^3*f^5 - 138
*a^3*b^7*c^3*d^4*e^2*f^6 + 74*a^4*b^6*c^2*d^5*e^2*f^6))/(d^17*e^8*((c + d*x)^(1/2) - c^(1/2)))))/(b^(1/2)*f) +
(65536*((a + b*x)^(1/2) - a^(1/2))*(96*a^(7/2)*b^6*c^(9/2)*d^3*f^7 - 72*a^(5/2)*b^7*c^(11/2)*d^2*f^7 - 40*a^(
9/2)*b^5*c^(7/2)*d^4*f^7 + 16*a^(3/2)*b^8*c^(13/2)*d*f^7 - 16*a^(1/2)*b^9*c^(13/2)*d*e*f^6 + 52*a^(3/2)*b^8*c^
(11/2)*d^2*e*f^6 + 36*a^(5/2)*b^7*c^(9/2)*d^3*e...

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