3.22.23 \(\int \sqrt {1-2 x} (2+3 x)^4 \sqrt {3+5 x} \, dx\)

Optimal. Leaf size=157 \[ -\frac {1}{20} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^3-\frac {333 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2}{2000}-\frac {7 (1-2 x)^{3/2} (5 x+3)^{3/2} (140652 x+231223)}{640000}-\frac {34069301 (1-2 x)^{3/2} \sqrt {5 x+3}}{5120000}+\frac {374762311 \sqrt {1-2 x} \sqrt {5 x+3}}{51200000}+\frac {4122385421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200000 \sqrt {10}} \]

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Rubi [A]  time = 0.06, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {100, 153, 147, 50, 54, 216} \begin {gather*} -\frac {1}{20} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^3-\frac {333 (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2}{2000}-\frac {7 (1-2 x)^{3/2} (5 x+3)^{3/2} (140652 x+231223)}{640000}-\frac {34069301 (1-2 x)^{3/2} \sqrt {5 x+3}}{5120000}+\frac {374762311 \sqrt {1-2 x} \sqrt {5 x+3}}{51200000}+\frac {4122385421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{51200000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x],x]

[Out]

(374762311*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/51200000 - (34069301*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/5120000 - (333*(1
- 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/2000 - ((1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(3/2))/20 - (7*(1 - 2*
x)^(3/2)*(3 + 5*x)^(3/2)*(231223 + 140652*x))/640000 + (4122385421*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(51200000
*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 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegerQ[m]

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)^4 \sqrt {3+5 x} \, dx &=-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {1}{60} \int \left (-312-\frac {999 x}{2}\right ) \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx\\ &=-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}+\frac {\int \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \left (\frac {77385}{2}+\frac {246141 x}{4}\right ) \, dx}{3000}\\ &=-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {34069301 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{1280000}\\ &=-\frac {34069301 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {374762311 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{10240000}\\ &=\frac {374762311 \sqrt {1-2 x} \sqrt {3+5 x}}{51200000}-\frac {34069301 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {4122385421 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{102400000}\\ &=\frac {374762311 \sqrt {1-2 x} \sqrt {3+5 x}}{51200000}-\frac {34069301 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {4122385421 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{51200000 \sqrt {5}}\\ &=\frac {374762311 \sqrt {1-2 x} \sqrt {3+5 x}}{51200000}-\frac {34069301 (1-2 x)^{3/2} \sqrt {3+5 x}}{5120000}-\frac {333 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}}{2000}-\frac {1}{20} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}-\frac {7 (1-2 x)^{3/2} (3+5 x)^{3/2} (231223+140652 x)}{640000}+\frac {4122385421 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{51200000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.11, size = 84, normalized size = 0.54 \begin {gather*} \frac {4122385421 \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+3746304000 x^5+3260908800 x^4+198117440 x^3-1377410040 x^2-1082027818 x+518122939\right )}{512000000 \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x],x]

[Out]

(-10*Sqrt[3 + 5*x]*(518122939 - 1082027818*x - 1377410040*x^2 + 198117440*x^3 + 3260908800*x^4 + 3746304000*x^
5 + 1382400000*x^6) + 4122385421*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(512000000*Sqrt[1 - 2*x]
)

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IntegrateAlgebraic [A]  time = 0.29, size = 157, normalized size = 1.00 \begin {gather*} -\frac {121 \sqrt {1-2 x} \left (\frac {106466565625 (1-2 x)^5}{(5 x+3)^5}+\frac {241317388750 (1-2 x)^4}{(5 x+3)^4}+\frac {224737783400 (1-2 x)^3}{(5 x+3)^3}+\frac {108503125360 (1-2 x)^2}{(5 x+3)^2}+\frac {25532316880 (1-2 x)}{5 x+3}-1090217632\right )}{51200000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^6}-\frac {4122385421 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{51200000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[1 - 2*x]*(2 + 3*x)^4*Sqrt[3 + 5*x],x]

[Out]

(-121*Sqrt[1 - 2*x]*(-1090217632 + (106466565625*(1 - 2*x)^5)/(3 + 5*x)^5 + (241317388750*(1 - 2*x)^4)/(3 + 5*
x)^4 + (224737783400*(1 - 2*x)^3)/(3 + 5*x)^3 + (108503125360*(1 - 2*x)^2)/(3 + 5*x)^2 + (25532316880*(1 - 2*x
))/(3 + 5*x)))/(51200000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^6) - (4122385421*ArcTan[(Sqrt[5/2]*Sqrt[1
- 2*x])/Sqrt[3 + 5*x]])/(51200000*Sqrt[10])

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fricas [A]  time = 1.50, size = 82, normalized size = 0.52 \begin {gather*} \frac {1}{51200000} \, {\left (691200000 \, x^{5} + 2218752000 \, x^{4} + 2739830400 \, x^{3} + 1468973920 \, x^{2} + 45781940 \, x - 518122939\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {4122385421}{1024000000} \, \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((2+3*x)^4*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/51200000*(691200000*x^5 + 2218752000*x^4 + 2739830400*x^3 + 1468973920*x^2 + 45781940*x - 518122939)*sqrt(5*
x + 3)*sqrt(-2*x + 1) - 4122385421/1024000000*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 = 1.32, size = 356, normalized size = 2.27 \begin {gather*} \frac {27}{2560000000} \, \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 {441}{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 {9}{50000} \, \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 {47}{5000} \, \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 {23}{125} \, \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 {24}{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((2+3*x)^4*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

27/2560000000*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))
) + 441/320000000*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))) + 9/50000*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
/11*sqrt(22)*sqrt(5*x + 3))) + 47/5000*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))) + 23/125*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))) + 24/25*sqrt(5)*(11*sqrt(2)*arcsin(1/11*sqrt(22)*sqr
t(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))

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maple [A]  time = 0.02, size = 138, normalized size = 0.88 \begin {gather*} \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 )-10362458780 \sqrt {-10 x^{2}-x +3}\right )}{1024000000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3*x+2)^4*(-2*x+1)^(1/2)*(5*x+3)^(1/2),x)

[Out]

1/1024000000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(13824000000*x^5*(-10*x^2-x+3)^(1/2)[phone]*x^4*(-10*x^2-x+3)^(
1/2)[phone]*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*
x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-10362458780*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.14, size = 104, normalized size = 0.66 \begin {gather*} -\frac {27}{20} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{3} - \frac {8397}{2000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {853821}{160000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {2300801}{640000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {34069301}{2560000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {4122385421}{1024000000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {34069301}{51200000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4*(1-2*x)^(1/2)*(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

-27/20*(-10*x^2 - x + 3)^(3/2)*x^3 - 8397/2000*(-10*x^2 - x + 3)^(3/2)*x^2 - 853821/160000*(-10*x^2 - x + 3)^(
3/2)*x - 2300801/640000*(-10*x^2 - x + 3)^(3/2) + 34069301/2560000*sqrt(-10*x^2 - x + 3)*x - 4122385421/102400
0000*sqrt(10)*arcsin(-20/11*x - 1/11) + 34069301/51200000*sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 15.54, size = 1056, normalized size = 6.73

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - 2*x)^(1/2)*(3*x + 2)^4*(5*x + 3)^(1/2),x)

[Out]

(4122385421*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/256000000 - ((215
2576553931*((1 - 2*x)^(1/2) - 1)^5)/(2441406250000*(3^(1/2) - (5*x + 3)^(1/2))^5) - (13952215351*((1 - 2*x)^(1
/2) - 1)^3)/(244140625000*(3^(1/2) - (5*x + 3)^(1/2))^3) - (792814579*((1 - 2*x)^(1/2) - 1))/(3051757812500*(3
^(1/2) - (5*x + 3)^(1/2))) + (3076029438827*((1 - 2*x)^(1/2) - 1)^7)/(976562500000*(3^(1/2) - (5*x + 3)^(1/2))
^7) - (11028639133187*((1 - 2*x)^(1/2) - 1)^9)/(195312500000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (3093660904217*(
(1 - 2*x)^(1/2) - 1)^11)/(15625000000*(3^(1/2) - (5*x + 3)^(1/2))^11) - (3093660904217*((1 - 2*x)^(1/2) - 1)^1
3)/(6250000000*(3^(1/2) - (5*x + 3)^(1/2))^13) + (11028639133187*((1 - 2*x)^(1/2) - 1)^15)/(12500000000*(3^(1/
2) - (5*x + 3)^(1/2))^15) - (3076029438827*((1 - 2*x)^(1/2) - 1)^17)/(10000000000*(3^(1/2) - (5*x + 3)^(1/2))^
17) - (2152576553931*((1 - 2*x)^(1/2) - 1)^19)/(4000000000*(3^(1/2) - (5*x + 3)^(1/2))^19) + (13952215351*((1
- 2*x)^(1/2) - 1)^21)/(64000000*(3^(1/2) - (5*x + 3)^(1/2))^21) + (792814579*((1 - 2*x)^(1/2) - 1)^23)/(128000
000*(3^(1/2) - (5*x + 3)^(1/2))^23) + (2228224*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(244140625*(3^(1/2) - (5*x + 3
)^(1/2))^2) + (5308416*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(48828125*(3^(1/2) - (5*x + 3)^(1/2))^4) - (389513216*
3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(244140625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (4601470976*3^(1/2)*((1 - 2*x)^(1
/2) - 1)^8)/(244140625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (1286299648*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(2441406
25*(3^(1/2) - (5*x + 3)^(1/2))^10) + (5601267712*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(48828125*(3^(1/2) - (5*x +
3)^(1/2))^12) + (321574912*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(9765625*(3^(1/2) - (5*x + 3)^(1/2))^14) + (2875
91936*3^(1/2)*((1 - 2*x)^(1/2) - 1)^16)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^16) - (6086144*3^(1/2)*((1 - 2*x)^
(1/2) - 1)^18)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^18) + (20736*3^(1/2)*((1 - 2*x)^(1/2) - 1)^20)/(125*(3^(1/2)
- (5*x + 3)^(1/2))^20) + (2176*3^(1/2)*((1 - 2*x)^(1/2) - 1)^22)/(25*(3^(1/2) - (5*x + 3)^(1/2))^22))/((24576
*((1 - 2*x)^(1/2) - 1)^2)/(48828125*(3^(1/2) - (5*x + 3)^(1/2))^2) + (67584*((1 - 2*x)^(1/2) - 1)^4)/(9765625*
(3^(1/2) - (5*x + 3)^(1/2))^4) + (22528*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (253
44*((1 - 2*x)^(1/2) - 1)^8)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^8) + (101376*((1 - 2*x)^(1/2) - 1)^10)/(78125*(
3^(1/2) - (5*x + 3)^(1/2))^10) + (59136*((1 - 2*x)^(1/2) - 1)^12)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^12) + (25
344*((1 - 2*x)^(1/2) - 1)^14)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^14) + (1584*((1 - 2*x)^(1/2) - 1)^16)/(125*(3^
(1/2) - (5*x + 3)^(1/2))^16) + (352*((1 - 2*x)^(1/2) - 1)^18)/(25*(3^(1/2) - (5*x + 3)^(1/2))^18) + (264*((1 -
2*x)^(1/2) - 1)^20)/(25*(3^(1/2) - (5*x + 3)^(1/2))^20) + (24*((1 - 2*x)^(1/2) - 1)^22)/(5*(3^(1/2) - (5*x +
3)^(1/2))^22) + ((1 - 2*x)^(1/2) - 1)^24/(3^(1/2) - (5*x + 3)^(1/2))^24 + 4096/244140625)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4*(1-2*x)**(1/2)*(3+5*x)**(1/2),x)

[Out]

Timed out

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