3.2264 \(\int \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx\)

Optimal. Leaf size=121 \[ -\frac {3}{40} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {37}{160} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {1313 \sqrt {5 x+3} (1-2 x)^{3/2}}{1280}+\frac {14443 \sqrt {5 x+3} \sqrt {1-2 x}}{12800}+\frac {158873 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800 \sqrt {10}} \]

[Out]

-37/160*(1-2*x)^(3/2)*(3+5*x)^(3/2)-3/40*(1-2*x)^(3/2)*(2+3*x)*(3+5*x)^(3/2)+158873/128000*arcsin(1/11*22^(1/2
)*(3+5*x)^(1/2))*10^(1/2)-1313/1280*(1-2*x)^(3/2)*(3+5*x)^(1/2)+14443/12800*(1-2*x)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.03, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {3}{40} (3 x+2) (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {37}{160} (5 x+3)^{3/2} (1-2 x)^{3/2}-\frac {1313 \sqrt {5 x+3} (1-2 x)^{3/2}}{1280}+\frac {14443 \sqrt {5 x+3} \sqrt {1-2 x}}{12800}+\frac {158873 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{12800 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14443*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/12800 - (1313*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1280 - (37*(1 - 2*x)^(3/2)*(3
+ 5*x)^(3/2))/160 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)*(3 + 5*x)^(3/2))/40 + (158873*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]])/(12800*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 \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x} \, dx &=-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}-\frac {1}{40} \int \left (-178-\frac {555 x}{2}\right ) \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {1313}{320} \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx\\ &=-\frac {1313 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {14443 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{2560}\\ &=\frac {14443 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1313 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {158873 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{25600}\\ &=\frac {14443 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1313 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {158873 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{12800 \sqrt {5}}\\ &=\frac {14443 \sqrt {1-2 x} \sqrt {3+5 x}}{12800}-\frac {1313 (1-2 x)^{3/2} \sqrt {3+5 x}}{1280}-\frac {37}{160} (1-2 x)^{3/2} (3+5 x)^{3/2}-\frac {3}{40} (1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}+\frac {158873 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{12800 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 74, normalized size = 0.61 \[ \frac {158873 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (57600 x^4+74560 x^3-6680 x^2-49154 x+13327\right )}{128000 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(13327 - 49154*x - 6680*x^2 + 74560*x^3 + 57600*x^4) + 158873*Sqrt[-10 + 20*x]*ArcSinh[Sqrt
[5/11]*Sqrt[-1 + 2*x]])/(128000*Sqrt[1 - 2*x])

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fricas [A]  time = 1.13, size = 72, normalized size = 0.60 \[ \frac {1}{12800} \, {\left (28800 \, x^{3} + 51680 \, x^{2} + 22500 \, x - 13327\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {158873}{256000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12800*(28800*x^3 + 51680*x^2 + 22500*x - 13327)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 158873/256000*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 = 1.00, size = 203, normalized size = 1.68 \[ \frac {3}{640000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {29}{40000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {7}{250} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {6}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/640000*sqrt(5)*(2*(4*(8*(60*x - 119)*(5*x + 3) + 6163)*(5*x + 3) - 66189)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 18
4305*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 29/40000*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))) + 7/250*sqrt(5)*(2*(20*x - 23)*sq
rt(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)*arcs
in(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 = 104, normalized size = 0.86 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (576000 \sqrt {-10 x^{2}-x +3}\, x^{3}+1033600 \sqrt {-10 x^{2}-x +3}\, x^{2}+450000 \sqrt {-10 x^{2}-x +3}\, x +158873 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-266540 \sqrt {-10 x^{2}-x +3}\right )}{256000 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/256000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(576000*(-10*x^2-x+3)^(1/2)*x^3+1033600*(-10*x^2-x+3)^(1/2)*x^2+158873*1
0^(1/2)*arcsin(20/11*x+1/11)+450000*(-10*x^2-x+3)^(1/2)*x-266540*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.28, size = 70, normalized size = 0.58 \[ -\frac {9}{40} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {61}{160} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {1313}{640} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {158873}{256000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {1313}{12800} \, \sqrt {-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/40*(-10*x^2 - x + 3)^(3/2)*x - 61/160*(-10*x^2 - x + 3)^(3/2) + 1313/640*sqrt(-10*x^2 - x + 3)*x - 158873/2
56000*sqrt(10)*arcsin(-20/11*x - 1/11) + 1313/12800*sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 10.53, size = 708, normalized size = 5.85 \[ \frac {158873\,\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 )}{64000}-\frac {\frac {38070349\,{\left (\sqrt {1-2\,x}-1\right )}^5}{15625000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}-\frac {3947\,{\left (\sqrt {1-2\,x}-1\right )}^3}{7812500\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {148327\,\left (\sqrt {1-2\,x}-1\right )}{19531250\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {88735647\,{\left (\sqrt {1-2\,x}-1\right )}^7}{6250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}+\frac {88735647\,{\left (\sqrt {1-2\,x}-1\right )}^9}{2500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}-\frac {38070349\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{1000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {3947\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{80000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}+\frac {148327\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{32000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {16384\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}-\frac {137728\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1014272\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {364288\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {253568\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}-\frac {8608\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {256\,\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((1 - 2*x)^(1/2)*(3*x + 2)^2*(5*x + 3)^(1/2),x)

[Out]

(158873*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/64000 - ((38070349*((
1 - 2*x)^(1/2) - 1)^5)/(15625000*(3^(1/2) - (5*x + 3)^(1/2))^5) - (3947*((1 - 2*x)^(1/2) - 1)^3)/(7812500*(3^(
1/2) - (5*x + 3)^(1/2))^3) - (148327*((1 - 2*x)^(1/2) - 1))/(19531250*(3^(1/2) - (5*x + 3)^(1/2))) - (88735647
*((1 - 2*x)^(1/2) - 1)^7)/(6250000*(3^(1/2) - (5*x + 3)^(1/2))^7) + (88735647*((1 - 2*x)^(1/2) - 1)^9)/(250000
0*(3^(1/2) - (5*x + 3)^(1/2))^9) - (38070349*((1 - 2*x)^(1/2) - 1)^11)/(1000000*(3^(1/2) - (5*x + 3)^(1/2))^11
) + (3947*((1 - 2*x)^(1/2) - 1)^13)/(80000*(3^(1/2) - (5*x + 3)^(1/2))^13) + (148327*((1 - 2*x)^(1/2) - 1)^15)
/(32000*(3^(1/2) - (5*x + 3)^(1/2))^15) + (16384*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)
^(1/2))^2) - (137728*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (1014272*3^(1/2
)*((1 - 2*x)^(1/2) - 1)^6)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (364288*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(
78125*(3^(1/2) - (5*x + 3)^(1/2))^8) + (253568*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(15625*(3^(1/2) - (5*x + 3)^(
1/2))^10) - (8608*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(625*(3^(1/2) - (5*x + 3)^(1/2))^12) + (256*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 [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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