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

Optimal. Leaf size=150 \[ -\frac {1}{20} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {(5 x+3)^{3/2} (63120 x+88987) (1-2 x)^{5/2}}{160000}-\frac {339983 \sqrt {5 x+3} (1-2 x)^{5/2}}{384000}+\frac {3739813 \sqrt {5 x+3} (1-2 x)^{3/2}}{7680000}+\frac {41137943 \sqrt {5 x+3} \sqrt {1-2 x}}{25600000}+\frac {452517373 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25600000 \sqrt {10}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 150, 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 = {100, 147, 50, 54, 216} \begin {gather*} -\frac {1}{20} (3 x+2)^2 (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac {(5 x+3)^{3/2} (63120 x+88987) (1-2 x)^{5/2}}{160000}-\frac {339983 \sqrt {5 x+3} (1-2 x)^{5/2}}{384000}+\frac {3739813 \sqrt {5 x+3} (1-2 x)^{3/2}}{7680000}+\frac {41137943 \sqrt {5 x+3} \sqrt {1-2 x}}{25600000}+\frac {452517373 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{25600000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(41137943*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/25600000 + (3739813*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/7680000 - (339983*(1
- 2*x)^(5/2)*Sqrt[3 + 5*x])/384000 - ((1 - 2*x)^(5/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/20 - ((1 - 2*x)^(5/2)*(3 +
5*x)^(3/2)*(88987 + 63120*x))/160000 + (452517373*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(25600000*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 (1-2 x)^{3/2} (2+3 x)^3 \sqrt {3+5 x} \, dx &=-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {1}{60} \int \left (-249-\frac {789 x}{2}\right ) (1-2 x)^{3/2} (2+3 x) \sqrt {3+5 x} \, dx\\ &=-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {339983 \int (1-2 x)^{3/2} \sqrt {3+5 x} \, dx}{64000}\\ &=-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {3739813 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{768000}\\ &=\frac {3739813 (1-2 x)^{3/2} \sqrt {3+5 x}}{7680000}-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {41137943 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{5120000}\\ &=\frac {41137943 \sqrt {1-2 x} \sqrt {3+5 x}}{25600000}+\frac {3739813 (1-2 x)^{3/2} \sqrt {3+5 x}}{7680000}-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {452517373 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{51200000}\\ &=\frac {41137943 \sqrt {1-2 x} \sqrt {3+5 x}}{25600000}+\frac {3739813 (1-2 x)^{3/2} \sqrt {3+5 x}}{7680000}-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {452517373 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{25600000 \sqrt {5}}\\ &=\frac {41137943 \sqrt {1-2 x} \sqrt {3+5 x}}{25600000}+\frac {3739813 (1-2 x)^{3/2} \sqrt {3+5 x}}{7680000}-\frac {339983 (1-2 x)^{5/2} \sqrt {3+5 x}}{384000}-\frac {1}{20} (1-2 x)^{5/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {(1-2 x)^{5/2} (3+5 x)^{3/2} (88987+63120 x)}{160000}+\frac {452517373 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{25600000 \sqrt {10}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.17, size = 84, normalized size = 0.56 \begin {gather*} \frac {10 \sqrt {5 x+3} \left (1382400000 x^6+1810944000 x^5-634003200 x^4-1555668160 x^3-125580440 x^2+537385502 x-81405921\right )[phone] \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{768000000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(-81405921 + 537385502*x - 125580440*x^2 - 1555668160*x^3 - 634003200*x^4 + 1810944000*x^5 +
1382400000*x^6) + 1357552119*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(768000000*Sqrt[1 - 2*x])

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.28, size = 157, normalized size = 1.05 \begin {gather*} -\frac {1331 \sqrt {1-2 x} \left (\frac {3187340625 (1-2 x)^5}{(5 x+3)^5}+\frac {7214398750 (1-2 x)^4}{(5 x+3)^4}+\frac {6595266600 (1-2 x)^3}{(5 x+3)^3}+\frac {2575814640 (1-2 x)^2}{(5 x+3)^2}-\frac {462376880 (1-2 x)}{5 x+3}-32638368\right )}{76800000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^6}-\frac {452517373 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{25600000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-1331*Sqrt[1 - 2*x]*(-32638368 + (3187340625*(1 - 2*x)^5)/(3 + 5*x)^5 + (7214398750*(1 - 2*x)^4)/(3 + 5*x)^4
+ (6595266600*(1 - 2*x)^3)/(3 + 5*x)^3 + (2575814640*(1 - 2*x)^2)/(3 + 5*x)^2 - (462376880*(1 - 2*x))/(3 + 5*x
)))/(76800000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^6) - (452517373*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqr
t[3 + 5*x]])/(25600000*Sqrt[10])

________________________________________________________________________________________

fricas [A]  time = 1.50, size = 82, normalized size = 0.55 \begin {gather*} -\frac {1}{76800000} \, {\left (691200000 \, x^{5} + 1251072000 \, x^{4} + 308534400 \, x^{3} - 623566880 \, x^{2} - 374573660 \, x + 81405921\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {452517373}{512000000} \, \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)^3*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/76800000*(691200000*x^5 + 1251072000*x^4 + 308534400*x^3 - 623566880*x^2 - 374573660*x + 81405921)*sqrt(5*x
+ 3)*sqrt(-2*x + 1) - 452517373/512000000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x +
1)/(10*x^2 + x - 3))

________________________________________________________________________________________

giac [B]  time = 1.16, size = 356, normalized size = 2.37 \begin {gather*} -\frac {9}{1280000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {189}{320000000} \, \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 {111}{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 {23}{60000} \, \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 {1}{20} \, \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 {12}{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)^3*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

-9/1280000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 1538569
5)*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))
) - 189/320000000*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))) - 111/3200000*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)*arcs
in(1/11*sqrt(22)*sqrt(5*x + 3))) + 23/60000*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))) + 1/20*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))) + 12/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*
sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 138, normalized size = 0.92 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-13824000000 \sqrt {-10 x^{2}-x +3}\, x^{5}-25021440000 \sqrt {-10 x^{2}-x +3}\, x^{4}-6170688000 \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 )-1628118420 \sqrt {-10 x^{2}-x +3}\right )}{1536000000 \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)^3*(5*x+3)^(1/2),x)

[Out]

1/1536000000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-13824000000*(-10*x^2-x+3)^(1/2)*x^5-25021440000*(-10*x^2-x+3)^(1/2
)*x^4-6170688000*(-10*x^2-x+3)^(1/2)*x^3+12471337600*(-10*x^2-x+3)^(1/2)*x^2+1357552119*10^(1/2)*arcsin(20/11*
x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x-1628118420*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

maxima [A]  time = 1.27, size = 104, normalized size = 0.69 \begin {gather*} \frac {9}{10} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} + \frac {1539}{1000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {41427}{80000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {385939}{960000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3739813}{1280000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {452517373}{512000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3739813}{25600000} \, \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)^3*(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

9/10*(-10*x^2 - x + 3)^(3/2)*x^3 + 1539/1000*(-10*x^2 - x + 3)^(3/2)*x^2 + 41427/80000*(-10*x^2 - x + 3)^(3/2)
*x - 385939/960000*(-10*x^2 - x + 3)^(3/2) + 3739813/1280000*sqrt(-10*x^2 - x + 3)*x - 452517373/512000000*sqr
t(10)*arcsin(-20/11*x - 1/11) + 3739813/25600000*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

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 )}^3\,\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)^3*(5*x + 3)^(1/2),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 124.09, size = 695, normalized size = 4.63 \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 )}{15625} + \frac {194 \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 )}{15625} + \frac {558 \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 )}{15625} + \frac {486 \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 )}{15625} - \frac {108 \sqrt {5} \left (\begin {cases} \frac {1771561 \sqrt {2} \left (\frac {4 \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 (- 20 x - 1\right )^{3} \left (5 x + 3\right )^{\frac {3}{2}}}{85034928} - \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}}{15488} - \frac {13 \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 )}{14992384} + \frac {21 \operatorname {asin}{\left (\frac {\sqrt {22} \sqrt {5 x + 3}}{11} \right )}}{1024}\right )}{64} & \text {for}\: x \geq - \frac {3}{5} \wedge x < \frac {1}{2} \end {cases}\right )}{15625} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(3/2)*(2+3*x)**3*(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)))/15625 + 194*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)))/15625 + 558*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)*sqr
t(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)))/15625 + 486*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)*sqrt(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)))/15625 - 108*sqrt(5)*P
iecewise((1771561*sqrt(2)*(4*sqrt(2)*(5 - 10*x)**(5/2)*(5*x + 3)**(5/2)/805255 + sqrt(2)*(5 - 10*x)**(3/2)*(-2
0*x - 1)**3*(5*x + 3)**(3/2)/85034928 - 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)/15488 - 13*sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x + 3)*(-12100*x - 128*(5*x + 3)**3 + 10
56*(5*x + 3)**2 - 5929)/14992384 + 21*asin(sqrt(22)*sqrt(5*x + 3)/11)/1024)/64, (x >= -3/5) & (x < 1/2)))/1562
5

________________________________________________________________________________________