3.2512 \(\int \frac{1}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=45 \[ -\frac{8 \sqrt{1-2 x}}{363 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{33 (5 x+3)^{3/2}} \]

[Out]

(-2*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) - (8*Sqrt[1 - 2*x])/(363*Sqrt[3 + 5*x])

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Rubi [A]  time = 0.0050273, 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} \[ -\frac{8 \sqrt{1-2 x}}{363 \sqrt{5 x+3}}-\frac{2 \sqrt{1-2 x}}{33 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x])/(33*(3 + 5*x)^(3/2)) - (8*Sqrt[1 - 2*x])/(363*Sqrt[3 + 5*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}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{1-2 x}}{33 (3+5 x)^{3/2}}+\frac{4}{33} \int \frac{1}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{1-2 x}}{33 (3+5 x)^{3/2}}-\frac{8 \sqrt{1-2 x}}{363 \sqrt{3+5 x}}\\ \end{align*}

Mathematica [A]  time = 0.0059106, size = 27, normalized size = 0.6 \[ -\frac{2 \sqrt{1-2 x} (20 x+23)}{363 (5 x+3)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*(23 + 20*x))/(363*(3 + 5*x)^(3/2))

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Maple [A]  time = 0.003, size = 22, normalized size = 0.5 \begin{align*} -{\frac{46+40\,x}{363}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/363*(23+20*x)/(3+5*x)^(3/2)*(1-2*x)^(1/2)

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Maxima [A]  time = 1.8694, size = 65, normalized size = 1.44 \begin{align*} -\frac{2 \, \sqrt{-10 \, x^{2} - x + 3}}{33 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} - \frac{8 \, \sqrt{-10 \, x^{2} - x + 3}}{363 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/33*sqrt(-10*x^2 - x + 3)/(25*x^2 + 30*x + 9) - 8/363*sqrt(-10*x^2 - x + 3)/(5*x + 3)

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Fricas [A]  time = 1.71167, size = 93, normalized size = 2.07 \begin{align*} -\frac{2 \,{\left (20 \, x + 23\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{363 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/363*(20*x + 23)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 4.09515, size = 102, normalized size = 2.27 \begin{align*} \begin{cases} - \frac{8 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1815} - \frac{2 \sqrt{10} \sqrt{-1 + \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{825 \left (x + \frac{3}{5}\right )} & \text{for}\: \frac{11}{10 \left |{x + \frac{3}{5}}\right |} > 1 \\- \frac{8 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{1815} - \frac{2 \sqrt{10} i \sqrt{1 - \frac{11}{10 \left (x + \frac{3}{5}\right )}}}{825 \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/(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Piecewise((-8*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/1815 - 2*sqrt(10)*sqrt(-1 + 11/(10*(x + 3/5)))/(825*(x + 3
/5)), 11/(10*Abs(x + 3/5)) > 1), (-8*sqrt(10)*I*sqrt(1 - 11/(10*(x + 3/5)))/1815 - 2*sqrt(10)*I*sqrt(1 - 11/(1
0*(x + 3/5)))/(825*(x + 3/5)), True))

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Giac [B]  time = 2.17013, size = 170, normalized size = 3.78 \begin{align*} -\frac{\sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}}{29040 \,{\left (5 \, x + 3\right )}^{\frac{3}{2}}} - \frac{3 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{2420 \, \sqrt{5 \, x + 3}} + \frac{{\left (\frac{9 \, \sqrt{10}{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} + 4 \, \sqrt{10}\right )}{\left (5 \, x + 3\right )}^{\frac{3}{2}}}{1815 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29040*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3/(5*x + 3)^(3/2) - 3/2420*sqrt(10)*(sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) + 1/1815*(9*sqrt(10)*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^2/(5*x + 3) + 4*sqrt
(10))*(5*x + 3)^(3/2)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))^3