3.2373 \(\int \frac{(1-2 x)^{3/2} (2+3 x)}{(3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=94 \[ -\frac{2 (1-2 x)^{5/2}}{55 \sqrt{5 x+3}}+\frac{1}{22} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{3}{20} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{33 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20 \sqrt{10}} \]

[Out]

(-2*(1 - 2*x)^(5/2))/(55*Sqrt[3 + 5*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20 + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])
/22 + (33*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20*Sqrt[10])

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Rubi [A]  time = 0.022091, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {78, 50, 54, 216} \[ -\frac{2 (1-2 x)^{5/2}}{55 \sqrt{5 x+3}}+\frac{1}{22} \sqrt{5 x+3} (1-2 x)^{3/2}+\frac{3}{20} \sqrt{5 x+3} \sqrt{1-2 x}+\frac{33 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{20 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2))/(55*Sqrt[3 + 5*x]) + (3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/20 + ((1 - 2*x)^(3/2)*Sqrt[3 + 5*x])
/22 + (33*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(20*Sqrt[10])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

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)}{(3+5 x)^{3/2}} \, dx &=-\frac{2 (1-2 x)^{5/2}}{55 \sqrt{3+5 x}}+\frac{5}{11} \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{55 \sqrt{3+5 x}}+\frac{1}{22} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{3}{4} \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{55 \sqrt{3+5 x}}+\frac{3}{20} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{1}{22} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{33}{40} \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2}}{55 \sqrt{3+5 x}}+\frac{3}{20} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{1}{22} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{33 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{20 \sqrt{5}}\\ &=-\frac{2 (1-2 x)^{5/2}}{55 \sqrt{3+5 x}}+\frac{3}{20} \sqrt{1-2 x} \sqrt{3+5 x}+\frac{1}{22} (1-2 x)^{3/2} \sqrt{3+5 x}+\frac{33 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{20 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0277082, size = 78, normalized size = 0.83 \[ \frac{10 \left (24 x^3-46 x^2-5 x+11\right )-33 \sqrt{10-20 x} \sqrt{5 x+3} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{200 \sqrt{1-2 x} \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*(11 - 5*x - 46*x^2 + 24*x^3) - 33*Sqrt[10 - 20*x]*Sqrt[3 + 5*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(200*Sqr
t[1 - 2*x]*Sqrt[3 + 5*x])

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Maple [A]  time = 0.01, size = 99, normalized size = 1.1 \begin{align*}{\frac{1}{400} \left ( 165\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) x-240\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}+99\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) +340\,x\sqrt{-10\,{x}^{2}-x+3}+220\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \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)^(3/2),x)

[Out]

1/400*(165*10^(1/2)*arcsin(20/11*x+1/11)*x-240*x^2*(-10*x^2-x+3)^(1/2)+99*10^(1/2)*arcsin(20/11*x+1/11)+340*x*
(-10*x^2-x+3)^(1/2)+220*(-10*x^2-x+3)^(1/2))*(1-2*x)^(1/2)/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 3.43588, size = 131, normalized size = 1.39 \begin{align*} \frac{33}{400} \, \sqrt{5} \sqrt{2} \arcsin \left (\frac{20}{11} \, x + \frac{1}{11}\right ) + \frac{99}{500} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{25 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{50 \,{\left (5 \, x + 3\right )}} - \frac{33 \, \sqrt{-10 \, x^{2} - x + 3}}{125 \,{\left (5 \, x + 3\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)^(3/2),x, algorithm="maxima")

[Out]

33/400*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) + 99/500*sqrt(-10*x^2 - x + 3) + 1/25*(-10*x^2 - x + 3)^(3/2)/(2
5*x^2 + 30*x + 9) + 3/50*(-10*x^2 - x + 3)^(3/2)/(5*x + 3) - 33/125*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 1.5199, size = 238, normalized size = 2.53 \begin{align*} -\frac{33 \, \sqrt{10}{\left (5 \, x + 3\right )} \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 ) + 20 \,{\left (12 \, x^{2} - 17 \, x - 11\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{400 \,{\left (5 \, x + 3\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)^(3/2),x, algorithm="fricas")

[Out]

-1/400*(33*sqrt(10)*(5*x + 3)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) +
20*(12*x^2 - 17*x - 11)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(5*x + 3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (1 - 2 x\right )^{\frac{3}{2}} \left (3 x + 2\right )}{\left (5 x + 3\right )^{\frac{3}{2}}}\, dx \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)**(3/2),x)

[Out]

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

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Giac [A]  time = 2.12591, size = 150, normalized size = 1.6 \begin{align*} -\frac{1}{2500} \,{\left (12 \, \sqrt{5}{\left (5 \, x + 3\right )} - 157 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5} + \frac{33}{200} \, \sqrt{10} \arcsin \left (\frac{1}{11} \, \sqrt{22} \sqrt{5 \, x + 3}\right ) - \frac{11 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{1250 \, \sqrt{5 \, x + 3}} + \frac{22 \, \sqrt{10} \sqrt{5 \, x + 3}}{625 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\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)^(3/2),x, algorithm="giac")

[Out]

-1/2500*(12*sqrt(5)*(5*x + 3) - 157*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5) + 33/200*sqrt(10)*arcsin(1/11*sqrt(
22)*sqrt(5*x + 3)) - 11/1250*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 22/625*sqrt(10)*sqr
t(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))