3.2263 \(\int \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x} \, dx\)

Optimal. Leaf size=128 \[ -\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2-\frac {21 (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)}{16000}-\frac {323491 (1-2 x)^{3/2} \sqrt {5 x+3}}{128000}+\frac {3558401 \sqrt {1-2 x} \sqrt {5 x+3}}{1280000}+\frac {39142411 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1280000 \sqrt {10}} \]

[Out]

-3/50*(1-2*x)^(3/2)*(2+3*x)^2*(3+5*x)^(3/2)-21/16000*(1-2*x)^(3/2)*(3+5*x)^(3/2)*(731+444*x)[phone]/12800000
*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)-323491/128000*(1-2*x)^(3/2)*(3+5*x)^(1/2)+3558401/1280000*(1-2*x
)^(1/2)*(3+5*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 128, 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 = {100, 147, 50, 54, 216} \[ -\frac {3}{50} (1-2 x)^{3/2} (5 x+3)^{3/2} (3 x+2)^2-\frac {21 (1-2 x)^{3/2} (5 x+3)^{3/2} (444 x+731)}{16000}-\frac {323491 (1-2 x)^{3/2} \sqrt {5 x+3}}{128000}+\frac {3558401 \sqrt {1-2 x} \sqrt {5 x+3}}{1280000}+\frac {39142411 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{1280000 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3558401*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1280000 - (323491*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/128000 - (3*(1 - 2*x)^(
3/2)*(2 + 3*x)^2*(3 + 5*x)^(3/2))/50 - (21*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)*(731 + 444*x))/16000 + (39142411*Ar
cSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(1280000*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 \sqrt {3+5 x} \, dx &=-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {1}{50} \int \left (-245-\frac {777 x}{2}\right ) \sqrt {1-2 x} (2+3 x) \sqrt {3+5 x} \, dx\\ &=-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {323491 \int \sqrt {1-2 x} \sqrt {3+5 x} \, dx}{32000}\\ &=-\frac {323491 (1-2 x)^{3/2} \sqrt {3+5 x}}{128000}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {3558401 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{256000}\\ &=\frac {3558401 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000}-\frac {323491 (1-2 x)^{3/2} \sqrt {3+5 x}}{128000}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {39142411 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{2560000}\\ &=\frac {3558401 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000}-\frac {323491 (1-2 x)^{3/2} \sqrt {3+5 x}}{128000}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {39142411 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{1280000 \sqrt {5}}\\ &=\frac {3558401 \sqrt {1-2 x} \sqrt {3+5 x}}{1280000}-\frac {323491 (1-2 x)^{3/2} \sqrt {3+5 x}}{128000}-\frac {3}{50} (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}-\frac {21 (1-2 x)^{3/2} (3+5 x)^{3/2} (731+444 x)}{16000}+\frac {39142411 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{1280000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 79, normalized size = 0.62 \[ \frac {39142411 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )-10 \sqrt {5 x+3} \left (13824000 x^5+27820800 x^4+12527040 x^3-8941640 x^2-11567238 x+4282349\right )}{12800000 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(4282349 - 11567238*x - 8941640*x^2 + 12527040*x^3 + 27820800*x^4 + 13824000*x^5) + 3914241
1*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(12800000*Sqrt[1 - 2*x])

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fricas [A]  time = 1.08, size = 77, normalized size = 0.60 \[ \frac {1}{1280000} \, {\left (6912000 \, x^{4} + 17366400 \, x^{3} + 14946720 \, x^{2} + 3002540 \, x - 4282349\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {39142411}{25600000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1280000*(6912000*x^4 + 17366400*x^3 + 14946720*x^2 + 3002540*x - 4282349)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 391
42411/25600000*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 = 1.10, size = 275, normalized size = 2.15 \[ \frac {9}{64000000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (12 \, {\left (80 \, x - 203\right )} {\left (5 \, x + 3\right )} + 19073\right )} {\left (5 \, x + 3\right )} - 506185\right )} {\left (5 \, x + 3\right )} + 4031895\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 10392195 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {117}{3200000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {57}{20000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {37}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {12}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/64000000*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))) + 117/3200000*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(1/11
*sqrt(22)*sqrt(5*x + 3))) + 57/20000*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))) + 37/500*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))) + 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))

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maple [A]  time = 0.01, size = 121, normalized size = 0.95 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (138240000 \sqrt {-10 x^{2}-x +3}\, x^{4}[phone] \sqrt {-10 x^{2}-x +3}\, x^{3}[phone] \sqrt {-10 x^{2}-x +3}\, x^{2}[phone] \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-85646980 \sqrt {-10 x^{2}-x +3}\right )}{25600000 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/25600000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(138240000*(-10*x^2-x+3)^(1/2)*x^4+347328000*(-10*x^2-x+3)^(1/2)*x^3+2
98934400*(-10*x^2-x+3)^(1/2)*x^2+39142411*10^(1/2)*arcsin(20/11*x+1/11)[phone]*(-10*x^2-x+3)^(1/2)*x-8564698
0*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.35, size = 87, normalized size = 0.68 \[ -\frac {27}{50} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x^{2} - \frac {5211}{4000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {19191}{16000} \, {\left (-10 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {323491}{64000} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {39142411}{25600000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {323491}{1280000} \, \sqrt {-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/50*(-10*x^2 - x + 3)^(3/2)*x^2 - 5211/4000*(-10*x^2 - x + 3)^(3/2)*x - 19191/16000*(-10*x^2 - x + 3)^(3/2)
+ 323491/64000*sqrt(-10*x^2 - x + 3)*x - 39142411/25600000*sqrt(10)*arcsin(-20/11*x - 1/11) + 323491/1280000*
sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 13.16, size = 881, normalized size = 6.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((22297589*((1 - 2*x)^(1/2) - 1))/(12207031250*(3^(1/2) - (5*x + 3)^(1/2))) + (369826027*((1 - 2*x)^(1/2) - 1)
^3)/(4882812500*(3^(1/2) - (5*x + 3)^(1/2))^3) - (4945417109*((1 - 2*x)^(1/2) - 1)^5)/(2441406250*(3^(1/2) - (
5*x + 3)^(1/2))^5) + (1598593169*((1 - 2*x)^(1/2) - 1)^7)/(195312500*(3^(1/2) - (5*x + 3)^(1/2))^7) - (9149019
53*((1 - 2*x)^(1/2) - 1)^9)/(156250000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (914901953*((1 - 2*x)^(1/2) - 1)^11)/(
62500000*(3^(1/2) - (5*x + 3)^(1/2))^11) - (1598593169*((1 - 2*x)^(1/2) - 1)^13)/(12500000*(3^(1/2) - (5*x + 3
)^(1/2))^13) + (4945417109*((1 - 2*x)^(1/2) - 1)^15)/(25000000*(3^(1/2) - (5*x + 3)^(1/2))^15) - (369826027*((
1 - 2*x)^(1/2) - 1)^17)/(8000000*(3^(1/2) - (5*x + 3)^(1/2))^17) - (22297589*((1 - 2*x)^(1/2) - 1)^19)/(320000
0*(3^(1/2) - (5*x + 3)^(1/2))^19) - (8192*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)^(1/2))
^2) + (90112*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(9765625*(3^(1/2) - (5*x + 3)^(1/2))^4) + (8316928*3^(1/2)*((1 -
2*x)^(1/2) - 1)^6)/(9765625*(3^(1/2) - (5*x + 3)^(1/2))^6) - (216457216*3^(1/2)*((1 - 2*x)^(1/2) - 1)^8)/(976
5625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (58587136*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(1953125*(3^(1/2) - (5*x + 3
)^(1/2))^10) - (54114304*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^12) + (519808*3
^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^14) + (1408*3^(1/2)*((1 - 2*x)^(1/2) - 1)^
16)/(625*(3^(1/2) - (5*x + 3)^(1/2))^16) - (32*3^(1/2)*((1 - 2*x)^(1/2) - 1)^18)/(3^(1/2) - (5*x + 3)^(1/2))^1
8)/((1024*((1 - 2*x)^(1/2) - 1)^2)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (2304*((1 - 2*x)^(1/2) - 1)^4)/(78
125*(3^(1/2) - (5*x + 3)^(1/2))^4) + (3072*((1 - 2*x)^(1/2) - 1)^6)/(15625*(3^(1/2) - (5*x + 3)^(1/2))^6) + (2
688*((1 - 2*x)^(1/2) - 1)^8)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^8) + (8064*((1 - 2*x)^(1/2) - 1)^10)/(3125*(3^(
1/2) - (5*x + 3)^(1/2))^10) + (672*((1 - 2*x)^(1/2) - 1)^12)/(125*(3^(1/2) - (5*x + 3)^(1/2))^12) + (192*((1 -
2*x)^(1/2) - 1)^14)/(25*(3^(1/2) - (5*x + 3)^(1/2))^14) + (36*((1 - 2*x)^(1/2) - 1)^16)/(5*(3^(1/2) - (5*x +
3)^(1/2))^16) + (4*((1 - 2*x)^(1/2) - 1)^18)/(3^(1/2) - (5*x + 3)^(1/2))^18 + ((1 - 2*x)^(1/2) - 1)^20/(3^(1/2
) - (5*x + 3)^(1/2))^20 + 1024/9765625) + (39142411*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2)
- (5*x + 3)^(1/2)))))/6400000

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

[Out]

Timed out

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