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

Optimal. Leaf size=165 \[ -\frac {1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac {7189 \sqrt {5 x+3} (1-2 x)^{7/2}}{32000}+\frac {79079 \sqrt {5 x+3} (1-2 x)^{5/2}}{960000}+\frac {869869 \sqrt {5 x+3} (1-2 x)^{3/2}}{3840000}+\frac {9568559 \sqrt {5 x+3} \sqrt {1-2 x}}{12800000}+\frac {105254149 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800000 \sqrt {10}} \]

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Rubi [A]  time = 0.05, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 8, 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 {1}{20} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{7/2}-\frac {193 (5 x+3)^{3/2} (1-2 x)^{7/2}}{2000}-\frac {7189 \sqrt {5 x+3} (1-2 x)^{7/2}}{32000}+\frac {79079 \sqrt {5 x+3} (1-2 x)^{5/2}}{960000}+\frac {869869 \sqrt {5 x+3} (1-2 x)^{3/2}}{3840000}+\frac {9568559 \sqrt {5 x+3} \sqrt {1-2 x}}{12800000}+\frac {105254149 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(9568559*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800000 + (869869*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/3840000 + (79079*(1 -
2*x)^(5/2)*Sqrt[3 + 5*x])/960000 - (7189*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/32000 - (193*(1 - 2*x)^(7/2)*(3 + 5*x)
^(3/2))/2000 - ((1 - 2*x)^(7/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/20 + (105254149*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(
12800000*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)^{5/2} (2+3 x)^2 \sqrt {3+5 x} \, dx &=-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}-\frac {1}{60} \int \left (-186-\frac {579 x}{2}\right ) (1-2 x)^{5/2} \sqrt {3+5 x} \, dx\\ &=-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {7189 \int (1-2 x)^{5/2} \sqrt {3+5 x} \, dx}{4000}\\ &=-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {79079 \int \frac {(1-2 x)^{5/2}}{\sqrt {3+5 x}} \, dx}{64000}\\ &=\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {869869 \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx}{384000}\\ &=\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {9568559 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{2560000}\\ &=\frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{25600000}\\ &=\frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{12800000 \sqrt {5}}\\ &=\frac {9568559 \sqrt {1-2 x} \sqrt {3+5 x}}{12800000}+\frac {869869 (1-2 x)^{3/2} \sqrt {3+5 x}}{3840000}+\frac {79079 (1-2 x)^{5/2} \sqrt {3+5 x}}{960000}-\frac {7189 (1-2 x)^{7/2} \sqrt {3+5 x}}{32000}-\frac {193 (1-2 x)^{7/2} (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{7/2} (2+3 x) (3+5 x)^{3/2}+\frac {105254149 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.09, size = 84, normalized size = 0.51 \begin {gather*} \frac {315762447 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (460800000 x^6-41472000 x^5-568838400 x^4+114662080 x^3+266309720 x^2-83915726 x-9303927\right )}{384000000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-9303927 - 83915726*x + 266309720*x^2 + 114662080*x^3 - 568838400*x^4 - 41472000*x^5 + 460
800000*x^6) + 315762447*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(384000000*Sqrt[1 - 2*x])

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IntegrateAlgebraic [A]  time = 0.28, size = 157, normalized size = 0.95 \begin {gather*} -\frac {14641 \sqrt {1-2 x} \left (\frac {67396875 (1-2 x)^5}{(5 x+3)^5}+\frac {147646250 (1-2 x)^4}{(5 x+3)^4}+\frac {95647800 (1-2 x)^3}{(5 x+3)^3}-\frac {56936880 (1-2 x)^2}{(5 x+3)^2}-\frac {9777040 (1-2 x)}{5 x+3}-690144\right )}{38400000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^6}-\frac {105254149 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{12800000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-14641*Sqrt[1 - 2*x]*(-690144 + (67396875*(1 - 2*x)^5)/(3 + 5*x)^5 + (147646250*(1 - 2*x)^4)/(3 + 5*x)^4 + (9
5647800*(1 - 2*x)^3)/(3 + 5*x)^3 - (56936880*(1 - 2*x)^2)/(3 + 5*x)^2 - (9777040*(1 - 2*x))/(3 + 5*x)))/(38400
000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^6) - (105254149*ArcTan[(Sqrt[5/2]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]
])/(12800000*Sqrt[10])

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fricas [A]  time = 1.32, size = 82, normalized size = 0.50 \begin {gather*} \frac {1}{38400000} \, {\left (230400000 \, x^{5} + 94464000 \, x^{4} - 237187200 \, x^{3} - 61262560 \, x^{2} + 102523580 \, x + 9303927\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {105254149}{256000000} \, \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)^(5/2)*(2+3*x)^2*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/38400000*(230400000*x^5 + 94464000*x^4 - 237187200*x^3 - 61262560*x^2 + 102523580*x + 9303927)*sqrt(5*x + 3)
*sqrt(-2*x + 1) - 105254149/256000000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(1
0*x^2 + x - 3))

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giac [B]  time = 1.32, size = 356, normalized size = 2.16 \begin {gather*} \frac {3}{640000000} \, \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 {7}{40000000} \, \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 {79}{9600000} \, \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 {89}{120000} \, \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}{250} \, \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)^(5/2)*(2+3*x)^2*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

3/640000000*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)))
+ 7/40000000*sqrt(5)*(2*(4*(8*(12*(80*x - 203)*(5*x + 3) + 19073)*(5*x + 3) - 506185)*(5*x + 3) + 4031895)*sqr
t(5*x + 3)*sqrt(-10*x + 5) + 10392195*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 79/9600000*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))) - 89/120000*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))) + 1/250*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 = 138, normalized size = 0.84 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (4608000000 \sqrt {-10 x^{2}-x +3}\, x^{5}[phone] \sqrt {-10 x^{2}-x +3}\, x^{4}-4743744000 \sqrt {-10 x^{2}-x +3}\, x^{3}-1225251200 \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 )[phone] \sqrt {-10 x^{2}-x +3}\right )}{768000000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/768000000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(4608000000*(-10*x^2-x+3)^(1/2)*x^5+1889280000*(-10*x^2-x+3)^(1/2)*x^
4-4743744000*(-10*x^2-x+3)^(1/2)*x^3-1225251200*(-10*x^2-x+3)^(1/2)*x^2+315762447*10^(1/2)*arcsin(20/11*x+1/11
)[phone]*(-10*x^2-x+3)^(1/2)*x+186078540*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.28, size = 104, normalized size = 0.63 \begin {gather*} -\frac {3}{5} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {93}{500} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} + \frac {18251}{40000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x + \frac {27893}{480000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {869869}{640000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {105254149}{256000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {869869}{12800000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/5*(-10*x^2 - x + 3)^(3/2)*x^3 - 93/500*(-10*x^2 - x + 3)^(3/2)*x^2 + 18251/40000*(-10*x^2 - x + 3)^(3/2)*x
+ 27893/480000*(-10*x^2 - x + 3)^(3/2) + 869869/640000*sqrt(-10*x^2 - x + 3)*x - 105254149/256000000*sqrt(10)*
arcsin(-20/11*x - 1/11) + 869869/12800000*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 )}^{5/2}\,{\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)^(5/2)*(3*x + 2)^2*(5*x + 3)^(1/2),x)

[Out]

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

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sympy [A]  time = 129.59, size = 695, normalized size = 4.21 \begin {gather*} \frac {242 \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 {1364 \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 {1658 \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 {744 \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 {72 \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)**(5/2)*(2+3*x)**2*(3+5*x)**(1/2),x)

[Out]

242*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 + 1364*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 + 1658*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)))/15625 - 744*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 + 72*sqrt(5)
*Piecewise((1771561*sqrt(2)*(4*sqrt(2)*(5 - 10*x)**(5/2)*(5*x + 3)**(5/2)/805255 + sqrt(2)*(5 - 10*x)**(3/2)*(
-20*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 - 1
0*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 +
1056*(5*x + 3)**2 - 5929)/14992384 + 21*asin(sqrt(22)*sqrt(5*x + 3)/11)/1024)/64, (x >= -3/5) & (x < 1/2)))/15
625

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