3.2322 \(\int \frac{\sqrt{1-2 x}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=22 \[ -\frac{2 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}} \]

[Out]

(-2*(1 - 2*x)^(3/2))/(33*(3 + 5*x)^(3/2))

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Rubi [A]  time = 0.0017629, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.053, Rules used = {37} \[ -\frac{2 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[1 - 2*x]/(3 + 5*x)^(5/2),x]

[Out]

(-2*(1 - 2*x)^(3/2))/(33*(3 + 5*x)^(3/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{1-2 x}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{3/2}}{33 (3+5 x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0047601, size = 22, normalized size = 1. \[ -\frac{2 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2))/(33*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.003, size = 17, normalized size = 0.8 \begin{align*} -{\frac{2}{33} \left ( 1-2\,x \right ) ^{{\frac{3}{2}}} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/33*(1-2*x)^(3/2)/(3+5*x)^(3/2)

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Maxima [B]  time = 1.61453, size = 65, normalized size = 2.95 \begin{align*} -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{15 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{4 \, \sqrt{-10 \, x^{2} - x + 3}}{165 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) + 4/165*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [B]  time = 1.73798, size = 88, normalized size = 4. \begin{align*} \frac{2 \, \sqrt{5 \, x + 3}{\left (2 \, x - 1\right )} \sqrt{-2 \, x + 1}}{33 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/33*sqrt(5*x + 3)*(2*x - 1)*sqrt(-2*x + 1)/(25*x^2 + 30*x + 9)

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Sympy [B]  time = 2.3908, size = 99, normalized size = 4.5 \begin{align*} \begin{cases} \frac{4 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{825} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{375 \left (x + \frac{3}{5}\right )} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\\frac{4 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{825} - \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{375 \left (x + \frac{3}{5}\right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((4*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/825 - 2*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/(375*(x + 3/5
)), 11/(10*Abs(x + 3/5)) > 1), (4*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/825 - 2*sqrt(10)*I*sqrt(1 - 11/(10*(x
+ 3/5)))/(375*(x + 3/5)), True))

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Giac [B]  time = 2.67971, size = 176, normalized size = 8. \begin{align*} -\frac{1}{13200} \, \sqrt{5}{\left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{12 \, \sqrt{2}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{16 \,{\left (\frac{3 \, \sqrt{2}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4 \, \sqrt{2}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/13200*sqrt(5)*(sqrt(2)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 12*sqrt(2)*(sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) + 16*(3*sqrt(2)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) - 4*sqrt(2
))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3)