3.24.22 \(\int \frac {\sqrt {1-2 x}}{(3+5 x)^{5/2}} \, dx\) [2322]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {2 (1-2 x)^{3/2}}{33 (5 x+3)^{3/2}} \end {gather*}

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*}

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Mathematica [A]
time = 0.05, size = 22, normalized size = 1.00 \begin {gather*} -\frac {2 (1-2 x)^{3/2}}{33 (3+5 x)^{3/2}} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. \(33\) vs. \(2(16)=32\).
time = 0.15, size = 34, normalized size = 1.55

method	result	size
gosper	\(-\frac {2 \left (1-2 x \right )^{\frac {3}{2}}}{33 \left (3+5 x \right )^{\frac {3}{2}}}\)	\(17\)
default	\(-\frac {2 \sqrt {1-2 x}}{15 \left (3+5 x \right )^{\frac {3}{2}}}+\frac {4 \sqrt {1-2 x}}{165 \sqrt {3+5 x}}\)	\(34\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/15*(1-2*x)^(1/2)/(3+5*x)^(3/2)+4/165*(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 48 vs. \(2 (16) = 32\).
time = 0.54, size = 48, normalized size = 2.18 \begin {gather*} -\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 {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 33 vs. \(2 (16) = 32\).
time = 0.60, size = 33, normalized size = 1.50 \begin {gather*} \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 {gather*}

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 [C] Result contains complex when optimal does not.
time = 0.93, size = 99, normalized size = 4.50 \begin {gather*} \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 {1}{\left |{x + \frac {3}{5}}\right |} > \frac {10}{11} \\\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 {gather*}

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
)), 1/Abs(x + 3/5) > 10/11), (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] Leaf count of result is larger than twice the leaf count of optimal. 125 vs. \(2 (16) = 32\).
time = 2.07, size = 125, normalized size = 5.68 \begin {gather*} -\frac {1}{13200} \, \sqrt {5} {\left (\sqrt {2} {\left (\frac {{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} - \frac {12 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}}{\sqrt {5 \, x + 3}}\right )} + \frac {16 \, \sqrt {2} {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (\frac {3 \, {\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{{\left (\sqrt {2} \sqrt {-10 \, x + 5} - \sqrt {22}\right )}^{3}}\right )} \end {gather*}

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

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Mupad [B]
time = 2.06, size = 38, normalized size = 1.73 \begin {gather*} \frac {\sqrt {5\,x+3}\,\left (\frac {4\,x\,\sqrt {1-2\,x}}{825}-\frac {2\,\sqrt {1-2\,x}}{825}\right )}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*x + 3)^(1/2)*((4*x*(1 - 2*x)^(1/2))/825 - (2*(1 - 2*x)^(1/2))/825))/((6*x)/5 + x^2 + 9/25)

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