3.22.37 \(\int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=143 \[ -\frac {153}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}-\frac {9007 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9600}-\frac {99077 (1-2 x)^{3/2} \sqrt {5 x+3}}{25600}+\frac {1089847 \sqrt {1-2 x} \sqrt {5 x+3}}{256000}+\frac {11988317 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{256000 \sqrt {10}} \]

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Rubi [A]  time = 0.04, antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {90, 80, 50, 54, 216} \begin {gather*} -\frac {153}{800} (1-2 x)^{3/2} (5 x+3)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (3 x+2) (5 x+3)^{5/2}-\frac {9007 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9600}-\frac {99077 (1-2 x)^{3/2} \sqrt {5 x+3}}{25600}+\frac {1089847 \sqrt {1-2 x} \sqrt {5 x+3}}{256000}+\frac {11988317 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{256000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1089847*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/256000 - (99077*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/25600 - (9007*(1 - 2*x)^(
3/2)*(3 + 5*x)^(3/2))/9600 - (153*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/800 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x
)^(5/2))/50 + (11988317*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(256000*Sqrt[10])

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

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

Rule 80

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

Rule 90

Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a + b*
x)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 3)), x] + Dist[1/(d*f*(n + p + 3)), Int[(c + d*x)^n*(e +
f*x)^p*Simp[a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(n + p + 4) - b*(d*e*(
n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]

Rule 216

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

Rubi steps

\begin {align*} \int \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2} \, dx &=-\frac {3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}-\frac {1}{50} \int \left (-248-\frac {765 x}{2}\right ) \sqrt {1-2 x} (3+5 x)^{3/2} \, dx\\ &=-\frac {153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac {9007 \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx}{1600}\\ &=-\frac {9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac {153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac {99077 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{6400}\\ &=-\frac {99077 (1-2 x)^{3/2} \sqrt {3+5 x}}{25600}-\frac {9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac {153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac {1089847 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{51200}\\ &=\frac {1089847 \sqrt {1-2 x} \sqrt {3+5 x}}{256000}-\frac {99077 (1-2 x)^{3/2} \sqrt {3+5 x}}{25600}-\frac {9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac {153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac {11988317 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{512000}\\ &=\frac {1089847 \sqrt {1-2 x} \sqrt {3+5 x}}{256000}-\frac {99077 (1-2 x)^{3/2} \sqrt {3+5 x}}{25600}-\frac {9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac {153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac {11988317 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{256000 \sqrt {5}}\\ &=\frac {1089847 \sqrt {1-2 x} \sqrt {3+5 x}}{256000}-\frac {99077 (1-2 x)^{3/2} \sqrt {3+5 x}}{25600}-\frac {9007 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9600}-\frac {153}{800} (1-2 x)^{3/2} (3+5 x)^{5/2}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{5/2}+\frac {11988317 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{256000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 79, normalized size = 0.55 \begin {gather*} \frac {35964951 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (13824000 x^5+26668800 x^4+11035840 x^3-8808040 x^2-10584158 x+4015809\right )}{7680000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(4015809 - 10584158*x - 8808040*x^2 + 11035840*x^3 + 26668800*x^4 + 13824000*x^5) + 3596495
1*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(7680000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.22, size = 141, normalized size = 0.99 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {16888125 (1-2 x)^4}{(5 x+3)^4}+\frac {31524500 (1-2 x)^3}{(5 x+3)^3}+\frac {22996480 (1-2 x)^2}{(5 x+3)^2}+\frac {7500080 (1-2 x)}{5 x+3}-432336\right )}{768000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^5}-\frac {11988317 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{256000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*Sqrt[1 - 2*x]*(-432336 + (16888125*(1 - 2*x)^4)/(3 + 5*x)^4 + (31524500*(1 - 2*x)^3)/(3 + 5*x)^3 + (229
96480*(1 - 2*x)^2)/(3 + 5*x)^2 + (7500080*(1 - 2*x))/(3 + 5*x)))/(768000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 +
5*x))^5) - (11988317*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(256000*Sqrt[10])

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fricas [A]  time = 1.55, size = 77, normalized size = 0.54 \begin {gather*} \frac {1}{768000} \, {\left (6912000 \, x^{4} + 16790400 \, x^{3} + 13913120 \, x^{2} + 2552540 \, x - 4015809\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {11988317}{5120000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/768000*(6912000*x^4 + 16790400*x^3 + 13913120*x^2 + 2552540*x - 4015809)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1198
8317/5120000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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giac [B]  time = 1.26, size = 275, normalized size = 1.92 \begin {gather*} \frac {3}{12800000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {19}{320000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {541}{120000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {57}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {18}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/12800000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqrt(
5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 19/320000*sqrt(5)*(2*(4*(8*
(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 184305*sqrt(2)*arcsin(1/11*s
qrt(22)*sqrt(5*x + 3))) + 541/120000*sqrt(5)*(2*(4*(40*x - 59)*(5*x + 3) + 1293)*sqrt(5*x + 3)*sqrt(-10*x + 5)
+ 4785*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 57/500*sqrt(5)*(2*(20*x - 23)*sqrt(5*x + 3)*sqrt(-10*x
+ 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 18/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(
5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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maple [A]  time = 0.01, size = 121, normalized size = 0.85 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (138240000 \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-80316180 \sqrt {-10 x^{2}-x +3}\right )}{15360000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15360000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(138240000*(-10*x^2-x+3)^(1/2)*x^4+335808000*(-10*x^2-x+3)^(1/2)*x^3+2
78262400*(-10*x^2-x+3)^(1/2)*x^2+35964951*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x-8031618
0*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.25, size = 87, normalized size = 0.61 \begin {gather*} -\frac {9}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {1677}{800} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {17971}{9600} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {99077}{12800} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {11988317}{5120000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {99077}{256000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/10*(-10*x^2 - x + 3)^(3/2)*x^2 - 1677/800*(-10*x^2 - x + 3)^(3/2)*x - 17971/9600*(-10*x^2 - x + 3)^(3/2) +
99077/12800*sqrt(-10*x^2 - x + 3)*x - 11988317/5120000*sqrt(10)*arcsin(-20/11*x - 1/11) + 99077/256000*sqrt(-1
0*x^2 - x + 3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \sqrt {1-2\,x}\,{\left (3\,x+2\right )}^2\,{\left (5\,x+3\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 68.65, size = 488, normalized size = 3.41 \begin {gather*} - \frac {539 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{121} + \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}\right )}{200} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{16} + \frac {707 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{16} - \frac {309 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (- \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{3872} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{16} + \frac {45 \sqrt {2} \left (\begin {cases} \frac {161051 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{322102} - \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{7744} - \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{3748096} + \frac {7 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{256}\right )}{3125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{16} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-539*sqrt(2)*Piecewise((121*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/121 + asin(sqrt(55)*sqrt
(1 - 2*x)/11))/200, (x <= 1/2) & (x > -3/5)))/16 + 707*sqrt(2)*Piecewise((1331*sqrt(5)*(-5*sqrt(5)*(1 - 2*x)**
(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/1936 + asin(sqrt(55)*sqrt(1 - 2
*x)/11)/16)/125, (x <= 1/2) & (x > -3/5)))/16 - 309*sqrt(2)*Piecewise((14641*sqrt(5)*(-5*sqrt(5)*(1 - 2*x)**(3
/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/3872 - sqrt(5)*sqrt(1 - 2*x)*sqrt
(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/1874048 + 5*asin(sqrt(55)*sqrt(1 - 2*x)/11
)/128)/625, (x <= 1/2) & (x > -3/5)))/16 + 45*sqrt(2)*Piecewise((161051*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(5/2)*(1
0*x + 6)**(5/2)/322102 - 5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x +
6)*(20*x + 1)/7744 - 3*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2
- 4719)/3748096 + 7*asin(sqrt(55)*sqrt(1 - 2*x)/11)/256)/3125, (x <= 1/2) & (x > -3/5)))/16

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