3.2301 \(\int \frac {\sqrt {1-2 x} (2+3 x)^3}{\sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=106 \[ -\frac {3}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2-\frac {21 (1-2 x)^{3/2} \sqrt {5 x+3} (216 x+335)}{6400}+\frac {47761 \sqrt {1-2 x} \sqrt {5 x+3}}{64000}+\frac {525371 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64000 \sqrt {10}} \]

[Out]

525371/640000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-3/40*(1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(1/2)-21/6400*
(1-2*x)^(3/2)*(335+216*x)*(3+5*x)^(1/2)+47761/64000*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac {3}{40} (1-2 x)^{3/2} \sqrt {5 x+3} (3 x+2)^2-\frac {21 (1-2 x)^{3/2} \sqrt {5 x+3} (216 x+335)}{6400}+\frac {47761 \sqrt {1-2 x} \sqrt {5 x+3}}{64000}+\frac {525371 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{64000 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(47761*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/64000 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^2*Sqrt[3 + 5*x])/40 - (21*(1 - 2*x)^(
3/2)*Sqrt[3 + 5*x]*(335 + 216*x))/6400 + (525371*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(64000*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 100

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))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 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 \frac {\sqrt {1-2 x} (2+3 x)^3}{\sqrt {3+5 x}} \, dx &=-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {1}{40} \int \frac {\left (-175-\frac {567 x}{2}\right ) \sqrt {1-2 x} (2+3 x)}{\sqrt {3+5 x}} \, dx\\ &=-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (335+216 x)}{6400}+\frac {47761 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{12800}\\ &=\frac {47761 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (335+216 x)}{6400}+\frac {525371 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{128000}\\ &=\frac {47761 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (335+216 x)}{6400}+\frac {525371 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{64000 \sqrt {5}}\\ &=\frac {47761 \sqrt {1-2 x} \sqrt {3+5 x}}{64000}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x)^2 \sqrt {3+5 x}-\frac {21 (1-2 x)^{3/2} \sqrt {3+5 x} (335+216 x)}{6400}+\frac {525371 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{64000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 74, normalized size = 0.70 \[ \frac {525371 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (172800 x^4+239040 x^3-10440 x^2-159718 x+41789\right )}{640000 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(41789 - 159718*x - 10440*x^2 + 239040*x^3 + 172800*x^4) + 525371*Sqrt[-10 + 20*x]*ArcSinh[
Sqrt[5/11]*Sqrt[-1 + 2*x]])/(640000*Sqrt[1 - 2*x])

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fricas [A]  time = 0.75, size = 72, normalized size = 0.68 \[ \frac {1}{64000} \, {\left (86400 \, x^{3} + 162720 \, x^{2} + 76140 \, x - 41789\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {525371}{1280000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/64000*(86400*x^3 + 162720*x^2 + 76140*x - 41789)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 525371/1280000*sqrt(10)*arct
an(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 = 1.18, size = 203, normalized size = 1.92 \[ \frac {9}{3200000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{20000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {4}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/3200000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 1
84305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/20000*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))) + 9/500*sqrt(5)*(2*(20*x - 23)*sq
rt(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 4/25*sqrt(5)*(11*sqrt(2)*arcs
in(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 = 104, normalized size = 0.98 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (1728000 \sqrt {-10 x^{2}-x +3}\, x^{3}+3254400 \sqrt {-10 x^{2}-x +3}\, x^{2}+1522800 \sqrt {-10 x^{2}-x +3}\, x +525371 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-835780 \sqrt {-10 x^{2}-x +3}\right )}{1280000 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1280000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(1728000*(-10*x^2-x+3)^(1/2)*x^3+3254400*(-10*x^2-x+3)^(1/2)*x^2+525371
*10^(1/2)*arcsin(20/11*x+1/11)+1522800*(-10*x^2-x+3)^(1/2)*x-835780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.13, size = 73, normalized size = 0.69 \[ -\frac {27}{200} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {525371}{1280000} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {963}{4000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {21663}{16000} \, \sqrt {-10 \, x^{2} - x + 3} x + \frac {887}{12800} \, \sqrt {-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/200*(-10*x^2 - x + 3)^(3/2)*x + 525371/1280000*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - 963/4000*(-10*x^2
- x + 3)^(3/2) + 21663/16000*sqrt(-10*x^2 - x + 3)*x + 887/12800*sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 11.18, size = 708, normalized size = 6.68 \[ \frac {525371\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{320000}-\frac {\frac {204911\,{\left (\sqrt {1-2\,x}-1\right )}^3}{39062500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {498629\,\left (\sqrt {1-2\,x}-1\right )}{97656250\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}+\frac {116866071\,{\left (\sqrt {1-2\,x}-1\right )}^5}{78125000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {264903917\,{\left (\sqrt {1-2\,x}-1\right )}^7}{31250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {264903917\,{\left (\sqrt {1-2\,x}-1\right )}^9}{12500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}-\frac {116866071\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{5000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}-\frac {204911\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{400000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}+\frac {498629\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{160000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {2048\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {86016\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {623616\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {1223168\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {155904\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}-\frac {5376\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {32\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}}{\frac {1024\,{\left (\sqrt {1-2\,x}-1\right )}^2}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^6}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {224\,{\left (\sqrt {1-2\,x}-1\right )}^8}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {448\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {112\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {16\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{16}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}+\frac {256}{390625}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(525371*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/320000 - ((204911*((1
- 2*x)^(1/2) - 1)^3)/(39062500*(3^(1/2) - (5*x + 3)^(1/2))^3) - (498629*((1 - 2*x)^(1/2) - 1))/(97656250*(3^(
1/2) - (5*x + 3)^(1/2))) + (116866071*((1 - 2*x)^(1/2) - 1)^5)/(78125000*(3^(1/2) - (5*x + 3)^(1/2))^5) - (264
903917*((1 - 2*x)^(1/2) - 1)^7)/(31250000*(3^(1/2) - (5*x + 3)^(1/2))^7) + (264903917*((1 - 2*x)^(1/2) - 1)^9)
/(12500000*(3^(1/2) - (5*x + 3)^(1/2))^9) - (116866071*((1 - 2*x)^(1/2) - 1)^11)/(5000000*(3^(1/2) - (5*x + 3)
^(1/2))^11) - (204911*((1 - 2*x)^(1/2) - 1)^13)/(400000*(3^(1/2) - (5*x + 3)^(1/2))^13) + (498629*((1 - 2*x)^(
1/2) - 1)^15)/(160000*(3^(1/2) - (5*x + 3)^(1/2))^15) + (2048*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(78125*(3^(1/2)
- (5*x + 3)^(1/2))^2) - (86016*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (623
616*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (1223168*3^(1/2)*((1 - 2*x)^(1/2
) - 1)^8)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (155904*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(15625*(3^(1/2) -
(5*x + 3)^(1/2))^10) - (5376*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(625*(3^(1/2) - (5*x + 3)^(1/2))^12) + (32*3^(
1/2)*((1 - 2*x)^(1/2) - 1)^14)/(5*(3^(1/2) - (5*x + 3)^(1/2))^14))/((1024*((1 - 2*x)^(1/2) - 1)^2)/(78125*(3^(
1/2) - (5*x + 3)^(1/2))^2) + (1792*((1 - 2*x)^(1/2) - 1)^4)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (1792*((1
- 2*x)^(1/2) - 1)^6)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^6) + (224*((1 - 2*x)^(1/2) - 1)^8)/(125*(3^(1/2) - (5*x
+ 3)^(1/2))^8) + (448*((1 - 2*x)^(1/2) - 1)^10)/(125*(3^(1/2) - (5*x + 3)^(1/2))^10) + (112*((1 - 2*x)^(1/2)
- 1)^12)/(25*(3^(1/2) - (5*x + 3)^(1/2))^12) + (16*((1 - 2*x)^(1/2) - 1)^14)/(5*(3^(1/2) - (5*x + 3)^(1/2))^14
) + ((1 - 2*x)^(1/2) - 1)^16/(3^(1/2) - (5*x + 3)^(1/2))^16 + 256/390625)

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sympy [A]  time = 36.20, size = 462, normalized size = 4.36 \[ - \frac {343 \sqrt {2} \left (\begin {cases} \frac {11 \sqrt {5} \left (- \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {\operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{2}\right )}{25} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{8} + \frac {441 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (\frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{968} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{8}\right )}{125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{8} - \frac {189 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} + \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{1936} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{8} + \frac {27 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{3872} + \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{3125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-343*sqrt(2)*Piecewise((11*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + asin(sqrt(55)*sqrt(1 - 2*x)/11)
/2)/25, (x <= 1/2) & (x > -3/5)))/8 + 441*sqrt(2)*Piecewise((121*sqrt(5)*(sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)
*(20*x + 1)/968 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + 3*asin(sqrt(55)*sqrt(1 - 2*x)/11)/8)/125, (x <= 1/
2) & (x > -3/5)))/8 - 189*sqrt(2)*Piecewise((1331*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 +
3*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/1936 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + 5*asin(sqr
t(55)*sqrt(1 - 2*x)/11)/16)/625, (x <= 1/2) & (x > -3/5)))/8 + 27*sqrt(2)*Piecewise((14641*sqrt(5)*(5*sqrt(5)*
(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/3993 + 7*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/3872 + sqrt(5)*sqr
t(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/1874048 - sqrt(5)*sqrt(1 -
2*x)*sqrt(10*x + 6)/22 + 35*asin(sqrt(55)*sqrt(1 - 2*x)/11)/128)/3125, (x <= 1/2) & (x > -3/5)))/8

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