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

Optimal. Leaf size=143 \[ -\frac {3}{50} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {567 (5 x+3)^{3/2} (1-2 x)^{5/2}}{4000}-\frac {4123 \sqrt {5 x+3} (1-2 x)^{5/2}}{9600}+\frac {45353 \sqrt {5 x+3} (1-2 x)^{3/2}}{192000}+\frac {498883 \sqrt {5 x+3} \sqrt {1-2 x}}{640000}+\frac {5487713 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{640000 \sqrt {10}} \]

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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 {3}{50} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {567 (5 x+3)^{3/2} (1-2 x)^{5/2}}{4000}-\frac {4123 \sqrt {5 x+3} (1-2 x)^{5/2}}{9600}+\frac {45353 \sqrt {5 x+3} (1-2 x)^{3/2}}{192000}+\frac {498883 \sqrt {5 x+3} \sqrt {1-2 x}}{640000}+\frac {5487713 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{640000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(498883*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/640000 + (45353*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/192000 - (4123*(1 - 2*x)^(
5/2)*Sqrt[3 + 5*x])/9600 - (567*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/4000 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*(3 + 5*x)
^(3/2))/50 + (5487713*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(640000*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 (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x} \, dx &=-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}-\frac {1}{50} \int \left (-182-\frac {567 x}{2}\right ) (1-2 x)^{3/2} \sqrt {3+5 x} \, dx\\ &=-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {4123 \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx}{1600}\\ &=-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {45353 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{19200}\\ &=\frac {45353 (1-2 x)^{3/2} \sqrt {3+5 x}}{192000}-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {498883 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{128000}\\ &=\frac {498883 \sqrt {1-2 x} \sqrt {3+5 x}}{640000}+\frac {45353 (1-2 x)^{3/2} \sqrt {3+5 x}}{192000}-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {5487713 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1280000}\\ &=\frac {498883 \sqrt {1-2 x} \sqrt {3+5 x}}{640000}+\frac {45353 (1-2 x)^{3/2} \sqrt {3+5 x}}{192000}-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {5487713 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{640000 \sqrt {5}}\\ &=\frac {498883 \sqrt {1-2 x} \sqrt {3+5 x}}{640000}+\frac {45353 (1-2 x)^{3/2} \sqrt {3+5 x}}{192000}-\frac {4123 (1-2 x)^{5/2} \sqrt {3+5 x}}{9600}-\frac {567 (1-2 x)^{5/2} (3+5 x)^{3/2}}{4000}-\frac {3}{50} (1-2 x)^{5/2} (2+3 x) (3+5 x)^{3/2}+\frac {5487713 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{640000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 79, normalized size = 0.55 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (13824000 x^5+7660800 x^4-13568960 x^3-6603640 x^2+5636662 x-382101\right )[phone] \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{19200000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-382101 + 5636662*x - 6603640*x^2 - 13568960*x^3 + 7660800*x^4 + 13824000*x^5) + 16463139*S
qrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(19200000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.24, size = 141, normalized size = 0.99 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {7730625 (1-2 x)^4}{(5 x+3)^4}+\frac {14174500 (1-2 x)^3}{(5 x+3)^3}+\frac {8261120 (1-2 x)^2}{(5 x+3)^2}-\frac {2308880 (1-2 x)}{5 x+3}-197904\right )}{1920000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^5}-\frac {5487713 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{640000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*Sqrt[1 - 2*x]*(-197904 + (7730625*(1 - 2*x)^4)/(3 + 5*x)^4 + (14174500*(1 - 2*x)^3)/(3 + 5*x)^3 + (8261
120*(1 - 2*x)^2)/(3 + 5*x)^2 - (2308880*(1 - 2*x))/(3 + 5*x)))/(1920000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 +
5*x))^5) - (5487713*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(640000*Sqrt[10])

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fricas [A]  time = 1.46, size = 77, normalized size = 0.54 \begin {gather*} -\frac {1}{1920000} \, {\left (6912000 \, x^{4} + 7286400 \, x^{3} - 3141280 \, x^{2} - 4872460 \, x + 382101\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {5487713}{12800000} \, \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((1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/1920000*(6912000*x^4 + 7286400*x^3 - 3141280*x^2 - 4872460*x + 382101)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 54877
13/12800000*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 = 1.36, size = 275, normalized size = 1.92 \begin {gather*} -\frac {3}{32000000} \, \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 {43}{3200000} \, \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 {1}{4800} \, \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 {2}{125} \, \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 {6}{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((1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

-3/32000000*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))) - 43/3200000*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
*sqrt(22)*sqrt(5*x + 3))) - 1/4800*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))) + 2/125*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))) + 6/25*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}-145728000 \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 )-7642020 \sqrt {-10 x^{2}-x +3}\right )}{38400000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/38400000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-138240000*(-10*x^2-x+3)^(1/2)*x^4-145728000*(-10*x^2-x+3)^(1/2)*x^3+
62825600*(-10*x^2-x+3)^(1/2)*x^2+16463139*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x-7642020
*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.06, size = 87, normalized size = 0.61 \begin {gather*} \frac {9}{25} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {687}{2000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {2159}{24000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {45353}{32000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {5487713}{12800000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {45353}{640000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/25*(-10*x^2 - x + 3)^(3/2)*x^2 + 687/2000*(-10*x^2 - x + 3)^(3/2)*x - 2159/24000*(-10*x^2 - x + 3)^(3/2) + 4
5353/32000*sqrt(-10*x^2 - x + 3)*x - 5487713/12800000*sqrt(10)*arcsin(-20/11*x - 1/11) + 45353/640000*sqrt(-10
*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 70.68, size = 490, normalized size = 3.43 \begin {gather*} \frac {22 \sqrt {5} \left (\begin {cases} \frac {121 \sqrt {2} \left (- \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{121} + \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}\right )}{32} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} + \frac {128 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (- \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} + \frac {\operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{8} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} + \frac {174 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \left (- \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{16} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} - \frac {36 \sqrt {5} \left (\begin {cases} \frac {161051 \sqrt {2} \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {5}{2}} \left (5 x + 3\right )^{\frac {5}{2}}}{805255} - \frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} - \frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{7744} - \frac {3 \sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{3748096} + \frac {7 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{256}\right )}{32} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{3125} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

22*sqrt(5)*Piecewise((121*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/121 + asin(sqrt(22)*sqrt(
5*x + 3)/11))/32, (x >= -3/5) & (x < 1/2)))/3125 + 128*sqrt(5)*Piecewise((1331*sqrt(2)*(-sqrt(2)*(5 - 10*x)**(
3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 + asin(sqrt(22)*sqrt(5*x +
3)/11)/16)/8, (x >= -3/5) & (x < 1/2)))/3125 + 174*sqrt(5)*Piecewise((14641*sqrt(2)*(-sqrt(2)*(5 - 10*x)**(3/2
)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/3872 - sqrt(2)*sqrt(5 - 10*x)*sqrt(
5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 + 5*asin(sqrt(22)*sqrt(5*x + 3)/11)/
128)/16, (x >= -3/5) & (x < 1/2)))/3125 - 36*sqrt(5)*Piecewise((161051*sqrt(2)*(2*sqrt(2)*(5 - 10*x)**(5/2)*(5
*x + 3)**(5/2)/805255 - sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*s
qrt(5*x + 3)/7744 - 3*sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 -
5929)/3748096 + 7*asin(sqrt(22)*sqrt(5*x + 3)/11)/256)/32, (x >= -3/5) & (x < 1/2)))/3125

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