3.2613 \(\int \frac{1}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=45 \[ \frac{20 \sqrt{5 x+3}}{363 \sqrt{1-2 x}}+\frac{2 \sqrt{5 x+3}}{33 (1-2 x)^{3/2}} \]

[Out]

(2*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) + (20*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x])

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Rubi [A]  time = 0.0049057, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {45, 37} \[ \frac{20 \sqrt{5 x+3}}{363 \sqrt{1-2 x}}+\frac{2 \sqrt{5 x+3}}{33 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) + (20*Sqrt[3 + 5*x])/(363*Sqrt[1 - 2*x])

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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}{(1-2 x)^{5/2} \sqrt{3+5 x}} \, dx &=\frac{2 \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}+\frac{10}{33} \int \frac{1}{(1-2 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{33 (1-2 x)^{3/2}}+\frac{20 \sqrt{3+5 x}}{363 \sqrt{1-2 x}}\\ \end{align*}

Mathematica [A]  time = 0.0061376, size = 27, normalized size = 0.6 \[ -\frac{2 \sqrt{5 x+3} (20 x-21)}{363 (1-2 x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[3 + 5*x]*(-21 + 20*x))/(363*(1 - 2*x)^(3/2))

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Maple [A]  time = 0.002, size = 22, normalized size = 0.5 \begin{align*} -{\frac{-42+40\,x}{363}\sqrt{3+5\,x} \left ( 1-2\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/363*(3+5*x)^(1/2)*(-21+20*x)/(1-2*x)^(3/2)

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Maxima [A]  time = 1.79904, size = 65, normalized size = 1.44 \begin{align*} \frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{33 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} - \frac{20 \, \sqrt{-10 \, x^{2} - x + 3}}{363 \,{\left (2 \, x - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/33*sqrt(-10*x^2 - x + 3)/(4*x^2 - 4*x + 1) - 20/363*sqrt(-10*x^2 - x + 3)/(2*x - 1)

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Fricas [A]  time = 1.47487, size = 90, normalized size = 2. \begin{align*} -\frac{2 \,{\left (20 \, x - 21\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{363 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/363*(20*x - 21)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [B]  time = 4.14442, size = 177, normalized size = 3.93 \begin{align*} \begin{cases} \frac{100 \sqrt{10} \left (x + \frac{3}{5}\right )}{3630 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right ) - 3993 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}} - \frac{165 \sqrt{10}}{3630 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right ) - 3993 \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{100 \sqrt{10} i \left (x + \frac{3}{5}\right )}{3630 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right ) - 3993 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}} + \frac{165 \sqrt{10} i}{3630 \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}} \left (x + \frac{3}{5}\right ) - 3993 \sqrt{1 - \frac{11}{10 \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/(1-2*x)**(5/2)/(3+5*x)**(1/2),x)

[Out]

Piecewise((100*sqrt(10)*(x + 3/5)/(3630*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5) - 3993*sqrt(-1 + 11/(10*(x + 3/
5)))) - 165*sqrt(10)/(3630*sqrt(-1 + 11/(10*(x + 3/5)))*(x + 3/5) - 3993*sqrt(-1 + 11/(10*(x + 3/5)))), 11/(10
*Abs(x + 3/5)) > 1), (-100*sqrt(10)*I*(x + 3/5)/(3630*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5) - 3993*sqrt(1 - 11
/(10*(x + 3/5)))) + 165*sqrt(10)*I/(3630*sqrt(1 - 11/(10*(x + 3/5)))*(x + 3/5) - 3993*sqrt(1 - 11/(10*(x + 3/5
)))), True))

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Giac [A]  time = 2.32866, size = 53, normalized size = 1.18 \begin{align*} -\frac{2 \,{\left (4 \, \sqrt{5}{\left (5 \, x + 3\right )} - 33 \, \sqrt{5}\right )} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1815 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1815*(4*sqrt(5)*(5*x + 3) - 33*sqrt(5))*sqrt(5*x + 3)*sqrt(-10*x + 5)/(2*x - 1)^2