3.2502 \(\int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=22 \[ -\frac{2 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]

[Out]

(-2*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.0018495, 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} \[ -\frac{2 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x])

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{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx &=-\frac{2 \sqrt{1-2 x}}{11 \sqrt{3+5 x}}\\ \end{align*}

Mathematica [A]  time = 0.0034236, size = 22, normalized size = 1. \[ -\frac{2 \sqrt{1-2 x}}{11 \sqrt{5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/(11*Sqrt[3 + 5*x])

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Maple [A]  time = 0.001, size = 17, normalized size = 0.8 \begin{align*} -{\frac{2}{11}\sqrt{1-2\,x}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]  time = 3.45186, size = 28, normalized size = 1.27 \begin{align*} -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{11 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 1.66171, size = 49, normalized size = 2.23 \begin{align*} -\frac{2 \, \sqrt{-2 \, x + 1}}{11 \, \sqrt{5 \, x + 3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/11*sqrt(-2*x + 1)/sqrt(5*x + 3)

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Sympy [A]  time = 1.06604, size = 53, normalized size = 2.41 \begin{align*} \begin{cases} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{55} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{55} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/55, 11/(10*Abs(x + 3/5)) > 1), (-2*sqrt(10)*I*sqrt(1 - 11/
(10*(x + 3/5)))/55, True))

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Giac [B]  time = 1.84967, size = 82, normalized size = 3.73 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{110 \, \sqrt{5 \, x + 3}} + \frac{2 \, \sqrt{10} \sqrt{5 \, x + 3}}{55 \,{\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/(3+5*x)^(3/2)/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

-1/110*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 2/55*sqrt(10)*sqrt(5*x + 3)/(sqrt(2)*sqrt
(-10*x + 5) - sqrt(22))