3.2427 \(\int \frac{(1-2 x)^{5/2} (2+3 x)}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=116 \[ -\frac{3}{40} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{49 \sqrt{5 x+3} (1-2 x)^{5/2}}{1200}+\frac{539 \sqrt{5 x+3} (1-2 x)^{3/2}}{4800}+\frac{5929 \sqrt{5 x+3} \sqrt{1-2 x}}{16000}+\frac{65219 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16000 \sqrt{10}} \]

[Out]

(5929*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/16000 + (539*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/4800 + (49*(1 - 2*x)^(5/2)*Sqrt
[3 + 5*x])/1200 - (3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/40 + (65219*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(16000*Sqrt[
10])

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Rubi [A]  time = 0.0293981, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {80, 50, 54, 216} \[ -\frac{3}{40} \sqrt{5 x+3} (1-2 x)^{7/2}+\frac{49 \sqrt{5 x+3} (1-2 x)^{5/2}}{1200}+\frac{539 \sqrt{5 x+3} (1-2 x)^{3/2}}{4800}+\frac{5929 \sqrt{5 x+3} \sqrt{1-2 x}}{16000}+\frac{65219 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{16000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5929*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/16000 + (539*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/4800 + (49*(1 - 2*x)^(5/2)*Sqrt
[3 + 5*x])/1200 - (3*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/40 + (65219*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(16000*Sqrt[
10])

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)^{5/2} (2+3 x)}{\sqrt{3+5 x}} \, dx &=-\frac{3}{40} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{49}{80} \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{49 (1-2 x)^{5/2} \sqrt{3+5 x}}{1200}-\frac{3}{40} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{539}{480} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{539 (1-2 x)^{3/2} \sqrt{3+5 x}}{4800}+\frac{49 (1-2 x)^{5/2} \sqrt{3+5 x}}{1200}-\frac{3}{40} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{5929 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{3200}\\ &=\frac{5929 \sqrt{1-2 x} \sqrt{3+5 x}}{16000}+\frac{539 (1-2 x)^{3/2} \sqrt{3+5 x}}{4800}+\frac{49 (1-2 x)^{5/2} \sqrt{3+5 x}}{1200}-\frac{3}{40} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{65219 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{32000}\\ &=\frac{5929 \sqrt{1-2 x} \sqrt{3+5 x}}{16000}+\frac{539 (1-2 x)^{3/2} \sqrt{3+5 x}}{4800}+\frac{49 (1-2 x)^{5/2} \sqrt{3+5 x}}{1200}-\frac{3}{40} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{65219 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{16000 \sqrt{5}}\\ &=\frac{5929 \sqrt{1-2 x} \sqrt{3+5 x}}{16000}+\frac{539 (1-2 x)^{3/2} \sqrt{3+5 x}}{4800}+\frac{49 (1-2 x)^{5/2} \sqrt{3+5 x}}{1200}-\frac{3}{40} (1-2 x)^{7/2} \sqrt{3+5 x}+\frac{65219 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{16000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0355378, size = 74, normalized size = 0.64 \[ \frac{-10 \sqrt{5 x+3} \left (57600 x^4-99520 x^3+41320 x^2+40094 x-21537\right )-195657 \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{480000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-10*Sqrt[3 + 5*x]*(-21537 + 40094*x + 41320*x^2 - 99520*x^3 + 57600*x^4) - 195657*Sqrt[10 - 20*x]*ArcSin[Sqrt
[5/11]*Sqrt[1 - 2*x]])/(480000*Sqrt[1 - 2*x])

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Maple [A]  time = 0.007, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{960000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 576000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-707200\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+195657\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +59600\,x\sqrt{-10\,{x}^{2}-x+3}+430740\,\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)^(5/2)*(2+3*x)/(3+5*x)^(1/2),x)

[Out]

1/960000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(576000*x^3*(-10*x^2-x+3)^(1/2)-707200*x^2*(-10*x^2-x+3)^(1/2)+195657*10^
(1/2)*arcsin(20/11*x+1/11)+59600*x*(-10*x^2-x+3)^(1/2)+430740*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 3.56195, size = 101, normalized size = 0.87 \begin{align*} \frac{3}{5} \, \sqrt{-10 \, x^{2} - x + 3} x^{3} - \frac{221}{300} \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + \frac{149}{2400} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{65219}{320000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{7179}{16000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/5*sqrt(-10*x^2 - x + 3)*x^3 - 221/300*sqrt(-10*x^2 - x + 3)*x^2 + 149/2400*sqrt(-10*x^2 - x + 3)*x - 65219/3
20000*sqrt(10)*arcsin(-20/11*x - 1/11) + 7179/16000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.73735, size = 246, normalized size = 2.12 \begin{align*} \frac{1}{48000} \,{\left (28800 \, x^{3} - 35360 \, x^{2} + 2980 \, x + 21537\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{65219}{320000} \, \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)^(5/2)*(2+3*x)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

1/48000*(28800*x^3 - 35360*x^2 + 2980*x + 21537)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 65219/320000*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 = 129.575, size = 296, normalized size = 2.55 \begin{align*} - \frac{7 \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{3 \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 )}{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-7*sqrt(2)*Piecewise((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 + 3*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 - 2000*(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)))/2

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Giac [B]  time = 1.91873, size = 274, normalized size = 2.36 \begin{align*} \frac{1}{800000} \, \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}{30000} \, \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}{400} \, \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{1}{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)^(5/2)*(2+3*x)/(3+5*x)^(1/2),x, algorithm="giac")

[Out]

1/800000*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))) - 1/30000*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))) - 1/400*sqrt(5)*(2*(20*x - 23)*sqr
t(5*x + 3)*sqrt(-10*x + 5) - 143*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/25*sqrt(5)*(11*sqrt(2)*arcsi
n(1/11*sqrt(22)*sqrt(5*x + 3)) + 2*sqrt(5*x + 3)*sqrt(-10*x + 5))