3.2076 \(\int \frac{3+5 x}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=27 \[ \frac{5}{2} \sqrt{1-2 x}+\frac{11}{2 \sqrt{1-2 x}} \]

[Out]

11/(2*Sqrt[1 - 2*x]) + (5*Sqrt[1 - 2*x])/2

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Rubi [A]  time = 0.0059016, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{5}{2} \sqrt{1-2 x}+\frac{11}{2 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/(1 - 2*x)^(3/2),x]

[Out]

11/(2*Sqrt[1 - 2*x]) + (5*Sqrt[1 - 2*x])/2

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.0059007, size = 15, normalized size = 0.56 \[ \frac{8-5 x}{\sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(8 - 5*x)/Sqrt[1 - 2*x]

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Maple [A]  time = 0.002, size = 15, normalized size = 0.6 \begin{align*} -{(-8+5\,x){\frac{1}{\sqrt{1-2\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(-8+5*x)/(1-2*x)^(1/2)

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Maxima [A]  time = 1.07534, size = 26, normalized size = 0.96 \begin{align*} \frac{5}{2} \, \sqrt{-2 \, x + 1} + \frac{11}{2 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*sqrt(-2*x + 1) + 11/2/sqrt(-2*x + 1)

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Fricas [A]  time = 1.6033, size = 49, normalized size = 1.81 \begin{align*} \frac{{\left (5 \, x - 8\right )} \sqrt{-2 \, x + 1}}{2 \, x - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*x - 8)*sqrt(-2*x + 1)/(2*x - 1)

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Sympy [A]  time = 0.342798, size = 20, normalized size = 0.74 \begin{align*} - \frac{5 x}{\sqrt{1 - 2 x}} + \frac{8}{\sqrt{1 - 2 x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5*x/sqrt(1 - 2*x) + 8/sqrt(1 - 2*x)

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Giac [A]  time = 1.32777, size = 26, normalized size = 0.96 \begin{align*} \frac{5}{2} \, \sqrt{-2 \, x + 1} + \frac{11}{2 \, \sqrt{-2 \, x + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/2*sqrt(-2*x + 1) + 11/2/sqrt(-2*x + 1)