3.2275 \(\int \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx\)

Optimal. Leaf size=150 \[ -\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}-\frac {7 (1-2 x)^{3/2} (2256 x+3821) (5 x+3)^{5/2}}{32000}-\frac {953981 (1-2 x)^{3/2} (5 x+3)^{3/2}}{384000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {5 x+3}}{1024000}+\frac {115431701 \sqrt {1-2 x} \sqrt {5 x+3}}{10240000}+\frac {1269748711 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10240000 \sqrt {10}} \]

[Out]

-953981/384000*(1-2*x)^(3/2)*(3+5*x)^(3/2)-1/20*(1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(5/2)-7/32000*(1-2*x)^(3/2)*(3
+5*x)^(5/2)*(3821+2256*x)[phone]/102400000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-10493791/1024000*(
1-2*x)^(3/2)*(3+5*x)^(1/2)[phone]/10240000*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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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} \[ -\frac {1}{20} (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{5/2}-\frac {7 (1-2 x)^{3/2} (2256 x+3821) (5 x+3)^{5/2}}{32000}-\frac {953981 (1-2 x)^{3/2} (5 x+3)^{3/2}}{384000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {5 x+3}}{1024000}+\frac {115431701 \sqrt {1-2 x} \sqrt {5 x+3}}{10240000}+\frac {1269748711 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{10240000 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(115431701*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/10240000 - (10493791*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1024000 - (953981*
(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/384000 - ((1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(5/2))/20 - (7*(1 - 2*x)^(3/2
)*(3 + 5*x)^(5/2)*(3821 + 2256*x))/32000 + (1269748711*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(10240000*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 \sqrt {1-2 x} (2+3 x)^3 (3+5 x)^{3/2} \, dx &=-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {1}{60} \int \left (-315-\frac {987 x}{2}\right ) \sqrt {1-2 x} (2+3 x) (3+5 x)^{3/2} \, dx\\ &=-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {953981 \int \sqrt {1-2 x} (3+5 x)^{3/2} \, dx}{64000}\\ &=-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {10493791 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{256000}\\ &=-\frac {10493791 (1-2 x)^{3/2} \sqrt {3+5 x}}{1024000}-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {115431701 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{2048000}\\ &=\frac {115431701 \sqrt {1-2 x} \sqrt {3+5 x}}{10240000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {3+5 x}}{1024000}-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {1269748711 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{20480000}\\ &=\frac {115431701 \sqrt {1-2 x} \sqrt {3+5 x}}{10240000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {3+5 x}}{1024000}-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {1269748711 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{10240000 \sqrt {5}}\\ &=\frac {115431701 \sqrt {1-2 x} \sqrt {3+5 x}}{10240000}-\frac {10493791 (1-2 x)^{3/2} \sqrt {3+5 x}}{1024000}-\frac {953981 (1-2 x)^{3/2} (3+5 x)^{3/2}}{384000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{5/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{5/2} (3821+2256 x)}{32000}+\frac {1269748711 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{10240000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.19, size = 84, normalized size = 0.56 \[ \frac {3809246133 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (1382400000 x^6+3635712000 x^5+3038342400 x^4+97901120 x^3-1305876920 x^2-989489914 x+483864147\right )}{307200000 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(483864147 - 989489914*x - 1305876920*x^2 + 97901120*x^3 + 3038342400*x^4 + 3635712000*x^5
+ 1382400000*x^6) + 3809246133*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(307200000*Sqrt[1 - 2*x])

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fricas [A]  time = 1.03, size = 82, normalized size = 0.55 \[ \frac {1}{30720000} \, {\left (691200000 \, x^{5} + 2163456000 \, x^{4} + 2600899200 \, x^{3} + 1349400160 \, x^{2} + 21761620 \, x - 483864147\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {1269748711}{204800000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/30720000*(691200000*x^5 + 2163456000*x^4 + 2600899200*x^3 + 1349400160*x^2 + 21761620*x - 483864147)*sqrt(5*
x + 3)*sqrt(-2*x + 1) - 1269748711/204800000*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 = 1.40, size = 356, normalized size = 2.37 \[ \frac {9}{512000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (4 \, {\left (16 \, {\left (100 \, x - 311\right )} {\left (5 \, x + 3\right )} + 46071\right )} {\left (5 \, x + 3\right )} - 775911\right )} {\left (5 \, x + 3\right )} + 15385695\right )} {\left (5 \, x + 3\right )} - 99422145\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 220189365 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {9}{4000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {921}{3200000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {883}{60000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {141}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {36}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/512000000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 311)*(5*x + 3) + 46071)*(5*x + 3) - 775911)*(5*x + 3) + 15385695)
*(5*x + 3) - 99422145)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 220189365*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))
+ 9/4000000*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))) + 921/3200000*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(1/1
1*sqrt(22)*sqrt(5*x + 3))) + 883/60000*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))) + 141/500*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))) + 36/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sq
rt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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maple [A]  time = 0.01, size = 138, normalized size = 0.92 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (13824000000 \sqrt {-10 x^{2}-x +3}\, x^{5}[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-9677282940 \sqrt {-10 x^{2}-x +3}\right )}{614400000 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/614400000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(13824000000*(-10*x^2-x+3)^(1/2)*x^5+43269120000*(-10*x^2-x+3)^(1/2)*
x^4+52017984000*(-10*x^2-x+3)^(1/2)*x^3+26988003200*(-10*x^2-x+3)^(1/2)*x^2+3809246133*10^(1/2)*arcsin(20/11*x
+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x-9677282940*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.18, size = 104, normalized size = 0.69 \[ -\frac {9}{4} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {2727}{400} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {270711}{32000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {2147273}{384000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {10493791}{512000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {1269748711}{204800000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {10493791}{10240000} \, \sqrt {-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/4*(-10*x^2 - x + 3)^(3/2)*x^3 - 2727/400*(-10*x^2 - x + 3)^(3/2)*x^2 - 270711/32000*(-10*x^2 - x + 3)^(3/2)
*x - 2147273/384000*(-10*x^2 - x + 3)^(3/2) + 10493791/512000*sqrt(-10*x^2 - x + 3)*x - 1269748711/204800000*s
qrt(10)*arcsin(-20/11*x - 1/11) + 10493791/10240000*sqrt(-10*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 119.74, size = 694, normalized size = 4.63 \[ - \frac {3773 \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 )}{32} + \frac {3283 \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 {1071 \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 )}{8} + \frac {621 \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} - \frac {135 \sqrt {2} \left (\begin {cases} \frac {1771561 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {5}{2}} \left (10 x + 6\right )^{\frac {5}{2}}}{161051} + \frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}} \left (20 x + 1\right )^{3}}{170069856} - \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 )}{15488} - \frac {13 \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 )}{14992384} + \frac {21 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{1024}\right )}{15625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{32} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3773*sqrt(2)*Piecewise((121*sqrt(5)*(-sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/121 + asin(sqrt(55)*sqr
t(1 - 2*x)/11))/200, (x <= 1/2) & (x > -3/5)))/32 + 3283*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 - 1071*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)*s
qrt(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)))/8 + 621*sqrt(2)*Piecewise((161051*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(5/2)
*(10*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 - 135*sqrt(2)*P
iecewise((1771561*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(5/2)*(10*x + 6)**(5/2)/161051 + 5*sqrt(5)*(1 - 2*x)**(3/2)*(1
0*x + 6)**(3/2)*(20*x + 1)**3/170069856 - 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)/15488 - 13*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 2000*(1 - 2*x)**3 +
6600*(1 - 2*x)**2 - 4719)/14992384 + 21*asin(sqrt(55)*sqrt(1 - 2*x)/11)/1024)/15625, (x <= 1/2) & (x > -3/5))
)/32

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