3.2459 \(\int \frac {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\)

Optimal. Leaf size=106 \[ -\frac {3}{40} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2} (408 x+865)}{1280}-\frac {61547 \sqrt {1-2 x} \sqrt {5 x+3}}{5120}+\frac {677017 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5120 \sqrt {10}} \]

[Out]

677017/51200*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-3/40*(2+3*x)^2*(3+5*x)^(3/2)*(1-2*x)^(1/2)-3/1280*(3
+5*x)^(3/2)*(865+408*x)*(1-2*x)^(1/2)-61547/5120*(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} \[ -\frac {3}{40} \sqrt {1-2 x} (5 x+3)^{3/2} (3 x+2)^2-\frac {3 \sqrt {1-2 x} (5 x+3)^{3/2} (408 x+865)}{1280}-\frac {61547 \sqrt {1-2 x} \sqrt {5 x+3}}{5120}+\frac {677017 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{5120 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-61547*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/5120 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^2*(3 + 5*x)^(3/2))/40 - (3*Sqrt[1 - 2*x
]*(3 + 5*x)^(3/2)*(865 + 408*x))/1280 + (677017*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(5120*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 {(2+3 x)^3 \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx &=-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {1}{40} \int \frac {\left (-241-\frac {765 x}{2}\right ) (2+3 x) \sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx\\ &=-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (865+408 x)}{1280}+\frac {61547 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}} \, dx}{2560}\\ &=-\frac {61547 \sqrt {1-2 x} \sqrt {3+5 x}}{5120}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (865+408 x)}{1280}+\frac {677017 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{10240}\\ &=-\frac {61547 \sqrt {1-2 x} \sqrt {3+5 x}}{5120}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (865+408 x)}{1280}+\frac {677017 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{5120 \sqrt {5}}\\ &=-\frac {61547 \sqrt {1-2 x} \sqrt {3+5 x}}{5120}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^2 (3+5 x)^{3/2}-\frac {3 \sqrt {1-2 x} (3+5 x)^{3/2} (865+408 x)}{1280}+\frac {677017 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{5120 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 83, normalized size = 0.78 \[ -\frac {\sqrt {1-2 x} \left (10 \sqrt {2 x-1} \sqrt {5 x+3} \left (17280 x^3+57888 x^2+88092 x+97295\right )+677017 \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{51200 \sqrt {2 x-1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/51200*(Sqrt[1 - 2*x]*(10*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(97295 + 88092*x + 57888*x^2 + 17280*x^3) + 677017*Sq
rt[10]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]]))/Sqrt[-1 + 2*x]

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fricas [A]  time = 0.97, size = 72, normalized size = 0.68 \[ -\frac {1}{5120} \, {\left (17280 \, x^{3} + 57888 \, x^{2} + 88092 \, x + 97295\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {677017}{102400} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/5120*(17280*x^3 + 57888*x^2 + 88092*x + 97295)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 677017/102400*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 [A]  time = 1.34, size = 63, normalized size = 0.59 \[ -\frac {1}{1280000} \, \sqrt {5} {\left (2 \, {\left (36 \, {\left (24 \, {\left (20 \, x + 43\right )} {\left (5 \, x + 3\right )} + 5179\right )} {\left (5 \, x + 3\right )} + 1538675\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 16925425 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1280000*sqrt(5)*(2*(36*(24*(20*x + 43)*(5*x + 3) + 5179)*(5*x + 3) + 1538675)*sqrt(5*x + 3)*sqrt(-10*x + 5)
- 16925425*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))

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maple [A]  time = 0.01, size = 104, normalized size = 0.98 \[ \frac {\sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (-345600 \sqrt {-10 x^{2}-x +3}\, x^{3}-1157760 \sqrt {-10 x^{2}-x +3}\, x^{2}-1761840 \sqrt {-10 x^{2}-x +3}\, x +677017 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-1945900 \sqrt {-10 x^{2}-x +3}\right )}{102400 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/102400*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(-345600*(-10*x^2-x+3)^(1/2)*x^3-1157760*(-10*x^2-x+3)^(1/2)*x^2+677017*
10^(1/2)*arcsin(20/11*x+1/11)-1761840*(-10*x^2-x+3)^(1/2)*x-1945900*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.29, size = 73, normalized size = 0.69 \[ \frac {27}{80} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {677017}{102400} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) + \frac {351}{320} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} - \frac {4383}{256} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {114143}{5120} \, \sqrt {-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/80*(-10*x^2 - x + 3)^(3/2)*x + 677017/102400*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 351/320*(-10*x^2 - x
+ 3)^(3/2) - 4383/256*sqrt(-10*x^2 - x + 3)*x - 114143/5120*sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 11.58, size = 708, normalized size = 6.68 \[ \frac {677017\,\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 )}{25600}-\frac {\frac {431257\,\left (\sqrt {1-2\,x}-1\right )}{7812500\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {418991\,{\left (\sqrt {1-2\,x}-1\right )}^3}{625000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {284249727\,{\left (\sqrt {1-2\,x}-1\right )}^5}{31250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {157157861\,{\left (\sqrt {1-2\,x}-1\right )}^7}{12500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {157157861\,{\left (\sqrt {1-2\,x}-1\right )}^9}{5000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {284249727\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{2000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {418991\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{6400\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}-\frac {431257\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{12800\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {75776\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {2039808\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {8020992\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {5040128\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {2005248\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {127488\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {1184\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{25\,{\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(((3*x + 2)^3*(5*x + 3)^(1/2))/(1 - 2*x)^(1/2),x)

[Out]

(677017*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/25600 - ((431257*((1
- 2*x)^(1/2) - 1))/(7812500*(3^(1/2) - (5*x + 3)^(1/2))) - (418991*((1 - 2*x)^(1/2) - 1)^3)/(625000*(3^(1/2) -
(5*x + 3)^(1/2))^3) - (284249727*((1 - 2*x)^(1/2) - 1)^5)/(31250000*(3^(1/2) - (5*x + 3)^(1/2))^5) + (1571578
61*((1 - 2*x)^(1/2) - 1)^7)/(12500000*(3^(1/2) - (5*x + 3)^(1/2))^7) - (157157861*((1 - 2*x)^(1/2) - 1)^9)/(50
00000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (284249727*((1 - 2*x)^(1/2) - 1)^11)/(2000000*(3^(1/2) - (5*x + 3)^(1/2
))^11) + (418991*((1 - 2*x)^(1/2) - 1)^13)/(6400*(3^(1/2) - (5*x + 3)^(1/2))^13) - (431257*((1 - 2*x)^(1/2) -
1)^15)/(12800*(3^(1/2) - (5*x + 3)^(1/2))^15) + (75776*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*
x + 3)^(1/2))^2) + (2039808*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (8020992
*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (5040128*3^(1/2)*((1 - 2*x)^(1/2) -
1)^8)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^8) + (2005248*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(15625*(3^(1/2) - (5
*x + 3)^(1/2))^10) + (127488*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(625*(3^(1/2) - (5*x + 3)^(1/2))^12) + (1184*3^
(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(25*(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 = 35.26, size = 466, normalized size = 4.40 \[ \frac {2 \sqrt {5} \left (\begin {cases} \frac {11 \sqrt {2} \left (- \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {\operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{2}\right )}{4} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{625} + \frac {18 \sqrt {5} \left (\begin {cases} \frac {121 \sqrt {2} \left (\frac {\sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{968} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{8}\right )}{8} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{625} + \frac {54 \sqrt {5} \left (\begin {cases} \frac {1331 \sqrt {2} \left (\frac {\sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {3 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{1936} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{16}\right )}{16} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{625} + \frac {54 \sqrt {5} \left (\begin {cases} \frac {14641 \sqrt {2} \left (\frac {2 \sqrt {2} \left (5 - 10 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {2} \sqrt {5 - 10 x} \left (- 20 x - 1\right ) \sqrt {5 x + 3}}{3872} + \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} - \frac {\sqrt {2} \sqrt {5 - 10 x} \sqrt {5 x + 3}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{128}\right )}{32} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{625} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*sqrt(5)*Piecewise((11*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + asin(sqrt(22)*sqrt(5*x + 3)/11)/2)
/4, (x >= -3/5) & (x < 1/2)))/625 + 18*sqrt(5)*Piecewise((121*sqrt(2)*(sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt
(5*x + 3)/968 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 3*asin(sqrt(22)*sqrt(5*x + 3)/11)/8)/8, (x >= -3/5)
& (x < 1/2)))/625 + 54*sqrt(5)*Piecewise((1331*sqrt(2)*(sqrt(2)*(5 - 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 3*sq
rt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)/22 + 5*asin(sqrt(22
)*sqrt(5*x + 3)/11)/16)/16, (x >= -3/5) & (x < 1/2)))/625 + 54*sqrt(5)*Piecewise((14641*sqrt(2)*(2*sqrt(2)*(5
- 10*x)**(3/2)*(5*x + 3)**(3/2)/3993 + 7*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 - sqrt(2)*sqrt(5 - 10
*x)*sqrt(5*x + 3)/22 + 35*asin(sqrt(22)*sqrt(5*x + 3)/11)/128)/32, (x >= -3/5) & (x < 1/2)))/625

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