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]

(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])

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Rubi [A]  time = 0.0310465, antiderivative size = 121, normalized size of antiderivative = 1., 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 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 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 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 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*}

Mathematica [A]  time = 0.0380532, size = 74, normalized size = 0.61 \[ \frac{10 \sqrt{5 x+3} \left (57600 x^4-9920 x^3-59480 x^2+18254 x+3383\right )-109263 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{320000 \sqrt{1-2 x}} \]

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]*ArcSin[Sqrt[5/1
1]*Sqrt[1 - 2*x]])/(320000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.009, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{640000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -576000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-188800\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+109263\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +500400\,x\sqrt{-10\,{x}^{2}-x+3}+67660\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.55174, size = 101, normalized size = 0.83 \begin{align*} -\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} \end{align*}

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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Fricas [A]  time = 1.50327, size = 247, normalized size = 2.04 \begin{align*} -\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 ) \end{align*}

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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Sympy [A]  time = 110.684, size = 394, normalized size = 3.26 \begin{align*} - \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} \end{align*}

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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Giac [B]  time = 2.53876, size = 274, normalized size = 2.26 \begin{align*} -\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 )} \end{align*}

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))