3.2362 \(\int \frac {(1-2 x)^{3/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=121 \[ -\frac {3}{40} (3 x+2) \sqrt {5 x+3} (1-2 x)^{5/2}-\frac {119}{800} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {301 \sqrt {5 x+3} (1-2 x)^{3/2}}{3200}+\frac {9933 \sqrt {5 x+3} \sqrt {1-2 x}}{32000}+\frac {109263 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{32000 \sqrt {10}} \]

[Out]

109263/320000*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)+301/3200*(1-2*x)^(3/2)*(3+5*x)^(1/2)-119/800*(1-2*x
)^(5/2)*(3+5*x)^(1/2)-3/40*(1-2*x)^(5/2)*(2+3*x)*(3+5*x)^(1/2)+9933/32000*(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} \[ -\frac {3}{40} (3 x+2) \sqrt {5 x+3} (1-2 x)^{5/2}-\frac {119}{800} \sqrt {5 x+3} (1-2 x)^{5/2}+\frac {301 \sqrt {5 x+3} (1-2 x)^{3/2}}{3200}+\frac {9933 \sqrt {5 x+3} \sqrt {1-2 x}}{32000}+\frac {109263 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{32000 \sqrt {10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9933*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/32000 + (301*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/3200 - (119*(1 - 2*x)^(5/2)*Sqr
t[3 + 5*x])/800 - (3*(1 - 2*x)^(5/2)*(2 + 3*x)*Sqrt[3 + 5*x])/40 + (109263*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(
32000*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 \frac {(1-2 x)^{3/2} (2+3 x)^2}{\sqrt {3+5 x}} \, dx &=-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}-\frac {1}{40} \int \frac {\left (-112-\frac {357 x}{2}\right ) (1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {301}{320} \int \frac {(1-2 x)^{3/2}}{\sqrt {3+5 x}} \, dx\\ &=\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {9933 \int \frac {\sqrt {1-2 x}}{\sqrt {3+5 x}} \, dx}{6400}\\ &=\frac {9933 \sqrt {1-2 x} \sqrt {3+5 x}}{32000}+\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {109263 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{64000}\\ &=\frac {9933 \sqrt {1-2 x} \sqrt {3+5 x}}{32000}+\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {109263 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{32000 \sqrt {5}}\\ &=\frac {9933 \sqrt {1-2 x} \sqrt {3+5 x}}{32000}+\frac {301 (1-2 x)^{3/2} \sqrt {3+5 x}}{3200}-\frac {119}{800} (1-2 x)^{5/2} \sqrt {3+5 x}-\frac {3}{40} (1-2 x)^{5/2} (2+3 x) \sqrt {3+5 x}+\frac {109263 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{32000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.06, size = 74, normalized size = 0.61 \[ \frac {10 \sqrt {5 x+3} \left (57600 x^4-9920 x^3-59480 x^2+18254 x+3383\right )+109263 \sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{320000 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[3 + 5*x]*(3383 + 18254*x - 59480*x^2 - 9920*x^3 + 57600*x^4) + 109263*Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5
/11]*Sqrt[-1 + 2*x]])/(320000*Sqrt[1 - 2*x])

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fricas [A]  time = 1.04, size = 72, normalized size = 0.60 \[ -\frac {1}{32000} \, {\left (28800 \, x^{3} + 9440 \, x^{2} - 25020 \, x - 3383\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {109263}{640000} \, \sqrt {10} \arctan \left (\frac {\sqrt {10} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32000*(28800*x^3 + 9440*x^2 - 25020*x - 3383)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 109263/640000*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 = 1.03, size = 203, normalized size = 1.68 \[ -\frac {3}{1600000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (8 \, {\left (60 \, x - 119\right )} {\left (5 \, x + 3\right )} + 6163\right )} {\left (5 \, x + 3\right )} - 66189\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 184305 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} - \frac {1}{8000} \, \sqrt {5} {\left (2 \, {\left (4 \, {\left (40 \, x - 59\right )} {\left (5 \, x + 3\right )} + 1293\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} + 4785 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {1}{500} \, \sqrt {5} {\left (2 \, {\left (20 \, x - 23\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 143 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} + \frac {2}{25} \, \sqrt {5} {\left (11 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) + 2 \, \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/1600000*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))) - 1/8000*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))) + 1/500*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))) + 2/25*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 \[ \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-576000 \sqrt {-10 x^{2}-x +3}\, x^{3}-188800 \sqrt {-10 x^{2}-x +3}\, x^{2}+500400 \sqrt {-10 x^{2}-x +3}\, x +109263 \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )+67660 \sqrt {-10 x^{2}-x +3}\right )}{640000 \sqrt {-10 x^{2}-x +3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/640000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-576000*(-10*x^2-x+3)^(1/2)*x^3-188800*(-10*x^2-x+3)^(1/2)*x^2+109263*1
0^(1/2)*arcsin(20/11*x+1/11)+500400*(-10*x^2-x+3)^(1/2)*x+67660*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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maxima [A]  time = 1.17, size = 75, normalized size = 0.62 \[ -\frac {9}{10} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {59}{200} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} + \frac {1251}{1600} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {109263}{640000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) + \frac {3383}{32000} \, \sqrt {-10 \, x^{2} - x + 3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-9/10*sqrt(-10*x^2 - x + 3)*x^3 - 59/200*sqrt(-10*x^2 - x + 3)*x^2 + 1251/1600*sqrt(-10*x^2 - x + 3)*x - 10926
3/640000*sqrt(10)*arcsin(-20/11*x - 1/11) + 3383/32000*sqrt(-10*x^2 - x + 3)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 176.47, size = 394, normalized size = 3.26 \[ - \frac {49 \sqrt {2} \left (\begin {cases} \frac {121 \sqrt {5} \left (\frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{968} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {3 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{8}\right )}{125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{4} + \frac {21 \sqrt {2} \left (\begin {cases} \frac {1331 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{7986} + \frac {3 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{1936} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {5 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{16}\right )}{625} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{2} - \frac {9 \sqrt {2} \left (\begin {cases} \frac {14641 \sqrt {5} \left (\frac {5 \sqrt {5} \left (1 - 2 x\right )^{\frac {3}{2}} \left (10 x + 6\right )^{\frac {3}{2}}}{3993} + \frac {7 \sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (20 x + 1\right )}{3872} + \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6} \left (12100 x - 2000 \left (1 - 2 x\right )^{3} + 6600 \left (1 - 2 x\right )^{2} - 4719\right )}{1874048} - \frac {\sqrt {5} \sqrt {1 - 2 x} \sqrt {10 x + 6}}{22} + \frac {35 \operatorname {asin}{\left (\frac {\sqrt {55} \sqrt {1 - 2 x}}{11} \right )}}{128}\right )}{3125} & \text {for}\: x \leq \frac {1}{2} \wedge x > - \frac {3}{5} \end {cases}\right )}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49*sqrt(2)*Piecewise((121*sqrt(5)*(sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/968 - sqrt(5)*sqrt(1 - 2*x
)*sqrt(10*x + 6)/22 + 3*asin(sqrt(55)*sqrt(1 - 2*x)/11)/8)/125, (x <= 1/2) & (x > -3/5)))/4 + 21*sqrt(2)*Piece
wise((1331*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/7986 + 3*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)
*(20*x + 1)/1936 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + 5*asin(sqrt(55)*sqrt(1 - 2*x)/11)/16)/625, (x <=
1/2) & (x > -3/5)))/2 - 9*sqrt(2)*Piecewise((14641*sqrt(5)*(5*sqrt(5)*(1 - 2*x)**(3/2)*(10*x + 6)**(3/2)/3993
+ 7*sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(20*x + 1)/3872 + sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)*(12100*x - 200
0*(1 - 2*x)**3 + 6600*(1 - 2*x)**2 - 4719)/1874048 - sqrt(5)*sqrt(1 - 2*x)*sqrt(10*x + 6)/22 + 35*asin(sqrt(55
)*sqrt(1 - 2*x)/11)/128)/3125, (x <= 1/2) & (x > -3/5)))/4

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