3.2329 \(\int (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=116 \[ -\frac{3}{40} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{23}{96} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{253 \sqrt{5 x+3} (1-2 x)^{3/2}}{1920}+\frac{2783 \sqrt{5 x+3} \sqrt{1-2 x}}{6400}+\frac{30613 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

[Out]

(2783*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6400 + (253*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1920 - (23*(1 - 2*x)^(5/2)*Sqrt[
3 + 5*x])/96 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/40 + (30613*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*Sqrt[10
])

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Rubi [A]  time = 0.0291132, 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} \[ -\frac{3}{40} (5 x+3)^{3/2} (1-2 x)^{5/2}-\frac{23}{96} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{253 \sqrt{5 x+3} (1-2 x)^{3/2}}{1920}+\frac{2783 \sqrt{5 x+3} \sqrt{1-2 x}}{6400}+\frac{30613 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6400 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2783*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6400 + (253*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1920 - (23*(1 - 2*x)^(5/2)*Sqrt[
3 + 5*x])/96 - (3*(1 - 2*x)^(5/2)*(3 + 5*x)^(3/2))/40 + (30613*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6400*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 (1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x} \, dx &=-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{23}{16} \int (1-2 x)^{3/2} \sqrt{3+5 x} \, dx\\ &=-\frac{23}{96} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{253}{192} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=\frac{253 (1-2 x)^{3/2} \sqrt{3+5 x}}{1920}-\frac{23}{96} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{2783 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{1280}\\ &=\frac{2783 \sqrt{1-2 x} \sqrt{3+5 x}}{6400}+\frac{253 (1-2 x)^{3/2} \sqrt{3+5 x}}{1920}-\frac{23}{96} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{30613 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{12800}\\ &=\frac{2783 \sqrt{1-2 x} \sqrt{3+5 x}}{6400}+\frac{253 (1-2 x)^{3/2} \sqrt{3+5 x}}{1920}-\frac{23}{96} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{30613 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{6400 \sqrt{5}}\\ &=\frac{2783 \sqrt{1-2 x} \sqrt{3+5 x}}{6400}+\frac{253 (1-2 x)^{3/2} \sqrt{3+5 x}}{1920}-\frac{23}{96} (1-2 x)^{5/2} \sqrt{3+5 x}-\frac{3}{40} (1-2 x)^{5/2} (3+5 x)^{3/2}+\frac{30613 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{6400 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0336978, size = 65, normalized size = 0.56 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (-28800 x^3-6880 x^2+23420 x+1959\right )-91839 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{192000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(1959 + 23420*x - 6880*x^2 - 28800*x^3) - 91839*Sqrt[10]*ArcSin[Sqrt[5/11]*Sqr
t[1 - 2*x]])/192000

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Maple [A]  time = 0.008, size = 104, normalized size = 0.9 \begin{align*}{\frac{1}{384000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( -576000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-137600\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+91839\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +468400\,x\sqrt{-10\,{x}^{2}-x+3}+39180\,\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)*(3+5*x)^(1/2),x)

[Out]

1/384000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-576000*x^3*(-10*x^2-x+3)^(1/2)-137600*x^2*(-10*x^2-x+3)^(1/2)+91839*10^
(1/2)*arcsin(20/11*x+1/11)+468400*x*(-10*x^2-x+3)^(1/2)+39180*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 4.18823, size = 95, normalized size = 0.82 \begin{align*} \frac{3}{20} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{1}{48} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{253}{320} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{30613}{128000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{253}{6400} \, \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)*(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

3/20*(-10*x^2 - x + 3)^(3/2)*x + 1/48*(-10*x^2 - x + 3)^(3/2) + 253/320*sqrt(-10*x^2 - x + 3)*x - 30613/128000
*sqrt(10)*arcsin(-20/11*x - 1/11) + 253/6400*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.55923, size = 246, normalized size = 2.12 \begin{align*} -\frac{1}{19200} \,{\left (28800 \, x^{3} + 6880 \, x^{2} - 23420 \, x - 1959\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{30613}{128000} \, \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)*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

-1/19200*(28800*x^3 + 6880*x^2 - 23420*x - 1959)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 30613/128000*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 = 28.6448, size = 316, normalized size = 2.72 \begin{align*} \frac{22 \sqrt{5} \left (\begin{cases} \frac{121 \sqrt{2} \left (- \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{121} + \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}\right )}{32} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} + \frac{62 \sqrt{5} \left (\begin{cases} \frac{1331 \sqrt{2} \left (- \frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{1936} + \frac{\operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{16}\right )}{8} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} - \frac{12 \sqrt{5} \left (\begin{cases} \frac{14641 \sqrt{2} \left (- \frac{\sqrt{2} \left (5 - 10 x\right )^{\frac{3}{2}} \left (5 x + 3\right )^{\frac{3}{2}}}{3993} - \frac{\sqrt{2} \sqrt{5 - 10 x} \left (- 20 x - 1\right ) \sqrt{5 x + 3}}{3872} - \frac{\sqrt{2} \sqrt{5 - 10 x} \sqrt{5 x + 3} \left (- 12100 x - 128 \left (5 x + 3\right )^{3} + 1056 \left (5 x + 3\right )^{2} - 5929\right )}{1874048} + \frac{5 \operatorname{asin}{\left (\frac{\sqrt{22} \sqrt{5 x + 3}}{11} \right )}}{128}\right )}{16} & \text{for}\: x \geq - \frac{3}{5} \wedge x < \frac{1}{2} \end{cases}\right )}{625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

22*sqrt(5)*Piecewise((121*sqrt(2)*(-sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/121 + asin(sqrt(22)*sqrt(
5*x + 3)/11))/32, (x >= -3/5) & (x < 1/2)))/625 + 62*sqrt(5)*Piecewise((1331*sqrt(2)*(-sqrt(2)*(5 - 10*x)**(3/
2)*(5*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/1936 + asin(sqrt(22)*sqrt(5*x + 3)
/11)/16)/8, (x >= -3/5) & (x < 1/2)))/625 - 12*sqrt(5)*Piecewise((14641*sqrt(2)*(-sqrt(2)*(5 - 10*x)**(3/2)*(5
*x + 3)**(3/2)/3993 - sqrt(2)*sqrt(5 - 10*x)*(-20*x - 1)*sqrt(5*x + 3)/3872 - sqrt(2)*sqrt(5 - 10*x)*sqrt(5*x
+ 3)*(-12100*x - 128*(5*x + 3)**3 + 1056*(5*x + 3)**2 - 5929)/1874048 + 5*asin(sqrt(22)*sqrt(5*x + 3)/11)/128)
/16, (x >= -3/5) & (x < 1/2)))/625

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Giac [A]  time = 2.45594, size = 220, normalized size = 1.9 \begin{align*} -\frac{1}{320000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (8 \,{\left (60 \, x - 71\right )}{\left (5 \, x + 3\right )} + 2179\right )}{\left (5 \, x + 3\right )} - 4125\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 45375 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} - \frac{1}{24000} \, \sqrt{5}{\left (2 \,{\left (4 \,{\left (40 \, x - 23\right )}{\left (5 \, x + 3\right )} + 33\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} - 363 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right )\right )} + \frac{1}{200} \, \sqrt{5}{\left (2 \,{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + 121 \, \sqrt{2} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, 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)*(3+5*x)^(1/2),x, algorithm="giac")

[Out]

-1/320000*sqrt(5)*(2*(4*(8*(60*x - 71)*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 453
75*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 1/24000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x +
3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 1/200*sqrt(5)*(2*(20*x + 1)*sqrt(5*x
+ 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))