3.25.65 \(\int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx\) [2465]

Optimal. Leaf size=93 \[ -\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}+\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}-\frac {451 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{196 \sqrt {7}} \]

[Out]

-451/1372*arctan(1/7*(1-2*x)^(1/2)*7^(1/2)/(3+5*x)^(1/2))*7^(1/2)+3/14*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^2-4
1/196*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)

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Rubi [A]
time = 0.01, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {98, 96, 95, 210} \begin {gather*} -\frac {451 \text {ArcTan}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {5 x+3}}\right )}{196 \sqrt {7}}+\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2}}{14 (3 x+2)^2}-\frac {41 \sqrt {1-2 x} \sqrt {5 x+3}}{196 (3 x+2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + 5*x]/(Sqrt[1 - 2*x]*(2 + 3*x)^3),x]

[Out]

(-41*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(196*(2 + 3*x)) + (3*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(14*(2 + 3*x)^2) - (451*
ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(196*Sqrt[7])

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 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 98

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 210

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

Rubi steps

\begin {align*} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^3} \, dx &=\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}+\frac {41}{28} \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^2} \, dx\\ &=-\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}+\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}+\frac {451}{392} \int \frac {1}{\sqrt {1-2 x} (2+3 x) \sqrt {3+5 x}} \, dx\\ &=-\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}+\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}+\frac {451}{196} \text {Subst}\left (\int \frac {1}{-7-x^2} \, dx,x,\frac {\sqrt {1-2 x}}{\sqrt {3+5 x}}\right )\\ &=-\frac {41 \sqrt {1-2 x} \sqrt {3+5 x}}{196 (2+3 x)}+\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2}}{14 (2+3 x)^2}-\frac {451 \tan ^{-1}\left (\frac {\sqrt {1-2 x}}{\sqrt {7} \sqrt {3+5 x}}\right )}{196 \sqrt {7}}\\ \end {align*}

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Mathematica [A]
time = 1.47, size = 135, normalized size = 1.45 \begin {gather*} \frac {\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (44+87 x)}{(2+3 x)^2}+451 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {2 \left (34+\sqrt {1155}\right )} \sqrt {3+5 x}}{-\sqrt {11}+\sqrt {5-10 x}}\right )+451 \sqrt {7} \tan ^{-1}\left (\frac {\sqrt {6+10 x}}{\sqrt {34+\sqrt {1155}} \left (-\sqrt {11}+\sqrt {5-10 x}\right )}\right )}{1372} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[3 + 5*x]/(Sqrt[1 - 2*x]*(2 + 3*x)^3),x]

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(44 + 87*x))/(2 + 3*x)^2 + 451*Sqrt[7]*ArcTan[(Sqrt[2*(34 + Sqrt[1155])]*Sqrt[
3 + 5*x])/(-Sqrt[11] + Sqrt[5 - 10*x])] + 451*Sqrt[7]*ArcTan[Sqrt[6 + 10*x]/(Sqrt[34 + Sqrt[1155]]*(-Sqrt[11]
+ Sqrt[5 - 10*x]))])/1372

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(153\) vs. \(2(72)=144\).
time = 0.08, size = 154, normalized size = 1.66

method	result	size
risch	\(-\frac {\sqrt {3+5 x}\, \left (-1+2 x \right ) \left (44+87 x \right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{196 \left (2+3 x \right )^{2} \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right )}\, \sqrt {1-2 x}}+\frac {451 \sqrt {7}\, \arctan \left (\frac {9 \left (\frac {20}{3}+\frac {37 x}{3}\right ) \sqrt {7}}{14 \sqrt {-90 \left (\frac {2}{3}+x \right )^{2}+67+111 x}}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{2744 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(119\)
default	\(\frac {\sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (4059 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x^{2}+5412 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right ) x +1804 \sqrt {7}\, \arctan \left (\frac {\left (37 x +20\right ) \sqrt {7}}{14 \sqrt {-10 x^{2}-x +3}}\right )+1218 x \sqrt {-10 x^{2}-x +3}+616 \sqrt {-10 x^{2}-x +3}\right )}{2744 \sqrt {-10 x^{2}-x +3}\, \left (2+3 x \right )^{2}}\)	\(154\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2744*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(4059*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+5412*7
^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+1804*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^
2-x+3)^(1/2))+1218*x*(-10*x^2-x+3)^(1/2)+616*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)/(2+3*x)^2

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Maxima [A]
time = 0.66, size = 76, normalized size = 0.82 \begin {gather*} \frac {451}{2744} \, \sqrt {7} \arcsin \left (\frac {37 \, x}{11 \, {\left | 3 \, x + 2 \right |}} + \frac {20}{11 \, {\left | 3 \, x + 2 \right |}}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{14 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {29 \, \sqrt {-10 \, x^{2} - x + 3}}{196 \, {\left (3 \, x + 2\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

451/2744*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) - 1/14*sqrt(-10*x^2 - x + 3)/(9*x^2 + 12*x
+ 4) + 29/196*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]
time = 0.40, size = 86, normalized size = 0.92 \begin {gather*} -\frac {451 \, \sqrt {7} {\left (9 \, x^{2} + 12 \, x + 4\right )} \arctan \left (\frac {\sqrt {7} {\left (37 \, x + 20\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{14 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \, {\left (87 \, x + 44\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{2744 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/2744*(451*sqrt(7)*(9*x^2 + 12*x + 4)*arctan(1/14*sqrt(7)*(37*x + 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 +
x - 3)) - 14*(87*x + 44)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(9*x^2 + 12*x + 4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {5 x + 3}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)**(1/2)/(2+3*x)**3/(1-2*x)**(1/2),x)

[Out]

Integral(sqrt(5*x + 3)/(sqrt(1 - 2*x)*(3*x + 2)**3), x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (72) = 144\).
time = 1.59, size = 252, normalized size = 2.71 \begin {gather*} \frac {451}{27440} \, \sqrt {70} \sqrt {10} {\left (\pi + 2 \, \arctan \left (-\frac {\sqrt {70} \sqrt {5 \, x + 3} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}\right )\right )} - \frac {11 \, \sqrt {10} {\left (41 \, {\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{3} - \frac {7000 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}} + \frac {28000 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}}{98 \, {\left ({\left (\frac {\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}{\sqrt {5 \, x + 3}} - \frac {4 \, \sqrt {5 \, x + 3}}{\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}}\right )}^{2} + 280\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(1/2)/(2+3*x)^3/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

451/27440*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))
^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 11/98*sqrt(10)*(41*((sqrt(2)*sqrt(-10*x + 5) - sqrt
(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 - 7000*(sqrt(2)*sqrt(-10*x + 5)
- sqrt(22))/sqrt(5*x + 3) + 28000*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))/(((sqrt(2)*sqrt(-10*x +
5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^2

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Mupad [B]
time = 13.13, size = 1037, normalized size = 11.15 \begin {gather*} \frac {\frac {199\,{\left (\sqrt {1-2\,x}-1\right )}^5}{245\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {398\,{\left (\sqrt {1-2\,x}-1\right )}^3}{1225\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {314\,\left (\sqrt {1-2\,x}-1\right )}{30625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {157\,{\left (\sqrt {1-2\,x}-1\right )}^7}{980\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {2197\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{30625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {4276\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{30625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {2197\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{4900\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}}{\frac {544\,{\left (\sqrt {1-2\,x}-1\right )}^2}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {1764\,{\left (\sqrt {1-2\,x}-1\right )}^4}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {136\,{\left (\sqrt {1-2\,x}-1\right )}^6}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^8}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}-\frac {96\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^3}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}+\frac {48\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^5}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {12\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^7}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {96\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{625\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {16}{625}}-\frac {451\,\sqrt {7}\,\mathrm {atan}\left (\frac {\frac {451\,\sqrt {7}\,\left (\frac {2706\,\sqrt {3}}{6125}+\frac {1353\,\left (\sqrt {1-2\,x}-1\right )}{6125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {1353\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1225\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,451{}\mathrm {i}}{2744}\right )}{2744}+\frac {451\,\sqrt {7}\,\left (\frac {2706\,\sqrt {3}}{6125}+\frac {1353\,\left (\sqrt {1-2\,x}-1\right )}{6125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {1353\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1225\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,451{}\mathrm {i}}{2744}\right )}{2744}}{\frac {203401\,{\left (\sqrt {1-2\,x}-1\right )}^2}{480200\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {203401}{1200500}+\frac {\sqrt {7}\,\left (\frac {2706\,\sqrt {3}}{6125}+\frac {1353\,\left (\sqrt {1-2\,x}-1\right )}{6125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {1353\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1225\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,451{}\mathrm {i}}{2744}\right )\,451{}\mathrm {i}}{2744}-\frac {\sqrt {7}\,\left (\frac {2706\,\sqrt {3}}{6125}+\frac {1353\,\left (\sqrt {1-2\,x}-1\right )}{6125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {1353\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{1225\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {\sqrt {7}\,\left (\frac {212\,{\left (\sqrt {1-2\,x}-1\right )}^2}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {888\,\sqrt {3}\,\left (\sqrt {1-2\,x}-1\right )}{125\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {536}{125}\right )\,451{}\mathrm {i}}{2744}\right )\,451{}\mathrm {i}}{2744}}\right )}{1372} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x + 3)^(1/2)/((1 - 2*x)^(1/2)*(3*x + 2)^3),x)

[Out]

((199*((1 - 2*x)^(1/2) - 1)^5)/(245*(3^(1/2) - (5*x + 3)^(1/2))^5) - (398*((1 - 2*x)^(1/2) - 1)^3)/(1225*(3^(1
/2) - (5*x + 3)^(1/2))^3) - (314*((1 - 2*x)^(1/2) - 1))/(30625*(3^(1/2) - (5*x + 3)^(1/2))) + (157*((1 - 2*x)^
(1/2) - 1)^7)/(980*(3^(1/2) - (5*x + 3)^(1/2))^7) + (2197*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(30625*(3^(1/2) - (
5*x + 3)^(1/2))^2) - (4276*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(30625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (2197*3^(1
/2)*((1 - 2*x)^(1/2) - 1)^6)/(4900*(3^(1/2) - (5*x + 3)^(1/2))^6))/((544*((1 - 2*x)^(1/2) - 1)^2)/(625*(3^(1/2
) - (5*x + 3)^(1/2))^2) - (1764*((1 - 2*x)^(1/2) - 1)^4)/(625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (136*((1 - 2*x)
^(1/2) - 1)^6)/(25*(3^(1/2) - (5*x + 3)^(1/2))^6) + ((1 - 2*x)^(1/2) - 1)^8/(3^(1/2) - (5*x + 3)^(1/2))^8 - (9
6*3^(1/2)*((1 - 2*x)^(1/2) - 1)^3)/(625*(3^(1/2) - (5*x + 3)^(1/2))^3) + (48*3^(1/2)*((1 - 2*x)^(1/2) - 1)^5)/
(125*(3^(1/2) - (5*x + 3)^(1/2))^5) + (12*3^(1/2)*((1 - 2*x)^(1/2) - 1)^7)/(5*(3^(1/2) - (5*x + 3)^(1/2))^7) -
(96*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(625*(3^(1/2) - (5*x + 3)^(1/2))) + 16/625) - (451*7^(1/2)*atan(((451*7^(1
/2)*((2706*3^(1/2))/6125 + (1353*((1 - 2*x)^(1/2) - 1))/(6125*(3^(1/2) - (5*x + 3)^(1/2))) - (7^(1/2)*((212*((
1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2)
- (5*x + 3)^(1/2))) - 536/125)*451i)/2744 - (1353*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(1225*(3^(1/2) - (5*x + 3)^
(1/2))^2)))/2744 + (451*7^(1/2)*((2706*3^(1/2))/6125 + (1353*((1 - 2*x)^(1/2) - 1))/(6125*(3^(1/2) - (5*x + 3)
^(1/2))) + (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x
)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*451i)/2744 - (1353*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)
/(1225*(3^(1/2) - (5*x + 3)^(1/2))^2)))/2744)/((7^(1/2)*((2706*3^(1/2))/6125 + (1353*((1 - 2*x)^(1/2) - 1))/(6
125*(3^(1/2) - (5*x + 3)^(1/2))) - (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3^(1/2) - (5*x + 3)^(1/2))^2)
+ (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125)*451i)/2744 - (1353*3^(1/2)*
((1 - 2*x)^(1/2) - 1)^2)/(1225*(3^(1/2) - (5*x + 3)^(1/2))^2))*451i)/2744 - (7^(1/2)*((2706*3^(1/2))/6125 + (1
353*((1 - 2*x)^(1/2) - 1))/(6125*(3^(1/2) - (5*x + 3)^(1/2))) + (7^(1/2)*((212*((1 - 2*x)^(1/2) - 1)^2)/(25*(3
^(1/2) - (5*x + 3)^(1/2))^2) + (888*3^(1/2)*((1 - 2*x)^(1/2) - 1))/(125*(3^(1/2) - (5*x + 3)^(1/2))) - 536/125
)*451i)/2744 - (1353*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(1225*(3^(1/2) - (5*x + 3)^(1/2))^2))*451i)/2744 + (2034
01*((1 - 2*x)^(1/2) - 1)^2)/(480200*(3^(1/2) - (5*x + 3)^(1/2))^2) + 203401/1200500)))/1372

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