3.26.84 \(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{5/2}} \, dx\) [2584]

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

[Out]

2/33*(3+5*x)^(3/2)/(1-2*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 (5 x+3)^{3/2}}{33 (1-2 x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + 5*x]/(1 - 2*x)^(5/2),x]

[Out]

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

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

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(3 + 5*x)^(3/2))/(33*(1 - 2*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 = 0.08, size = 34, normalized size = 1.55

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3+5*x)^(1/2)/(1-2*x)^(3/2)-5/33*(3+5*x)^(1/2)/(1-2*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 = 0.53, size = 48, normalized size = 2.18 \begin {gather*} \frac {\sqrt {-10 \, x^{2} - x + 3}}{3 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {5 \, \sqrt {-10 \, x^{2} - x + 3}}{33 \, {\left (2 \, x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.46, size = 28, normalized size = 1.27 \begin {gather*} \frac {2 \, {\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}{33 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C] Result contains complex when optimal does not.
time = 0.93, size = 80, normalized size = 3.64 \begin {gather*} \begin {cases} \frac {250 i \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{330 \left (x + \frac {3}{5}\right ) \sqrt {10 x - 5} - 363 \sqrt {10 x - 5}} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\- \frac {250 \left (x + \frac {3}{5}\right )^{\frac {3}{2}}}{330 \sqrt {5 - 10 x} \left (x + \frac {3}{5}\right ) - 363 \sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((250*I*(x + 3/5)**(3/2)/(330*(x + 3/5)*sqrt(10*x - 5) - 363*sqrt(10*x - 5)), Abs(x + 3/5) > 11/10),
(-250*(x + 3/5)**(3/2)/(330*sqrt(5 - 10*x)*(x + 3/5) - 363*sqrt(5 - 10*x)), True))

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Giac [A]
time = 0.69, size = 26, normalized size = 1.18 \begin {gather*} \frac {2 \, \sqrt {5} {\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {-10 \, x + 5}}{165 \, {\left (2 \, x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/165*sqrt(5)*(5*x + 3)^(3/2)*sqrt(-10*x + 5)/(2*x - 1)^2

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Mupad [B]
time = 2.36, size = 35, normalized size = 1.59 \begin {gather*} -\frac {\sqrt {5\,x+3}\,\left (\frac {5\,x}{33}+\frac {1}{11}\right )}{x\,\sqrt {1-2\,x}-\frac {\sqrt {1-2\,x}}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((5*x + 3)^(1/2)*((5*x)/33 + 1/11))/(x*(1 - 2*x)^(1/2) - (1 - 2*x)^(1/2)/2)

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