3.1638 \(\int \frac{3+5 x}{(1-2 x)^3} \, dx\)

Optimal. Leaf size=18 \[ \frac{(5 x+3)^2}{22 (1-2 x)^2} \]

[Out]

(3 + 5*x)^2/(22*(1 - 2*x)^2)

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Rubi [A]  time = 0.0022027, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {37} \[ \frac{(5 x+3)^2}{22 (1-2 x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)/(1 - 2*x)^3,x]

[Out]

(3 + 5*x)^2/(22*(1 - 2*x)^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{3+5 x}{(1-2 x)^3} \, dx &=\frac{(3+5 x)^2}{22 (1-2 x)^2}\\ \end{align*}

Mathematica [A]  time = 0.0035941, size = 16, normalized size = 0.89 \[ \frac{20 x+1}{8 (1-2 x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)/(1 - 2*x)^3,x]

[Out]

(1 + 20*x)/(8*(1 - 2*x)^2)

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Maple [A]  time = 0.003, size = 20, normalized size = 1.1 \begin{align*}{\frac{11}{8\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{5}{8\,x-4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

11/8/(2*x-1)^2+5/4/(2*x-1)

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Maxima [A]  time = 1.16529, size = 26, normalized size = 1.44 \begin{align*} \frac{20 \, x + 1}{8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(20*x + 1)/(4*x^2 - 4*x + 1)

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Fricas [A]  time = 1.22962, size = 46, normalized size = 2.56 \begin{align*} \frac{20 \, x + 1}{8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(20*x + 1)/(4*x^2 - 4*x + 1)

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Sympy [A]  time = 0.098572, size = 14, normalized size = 0.78 \begin{align*} \frac{20 x + 1}{32 x^{2} - 32 x + 8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(20*x + 1)/(32*x**2 - 32*x + 8)

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Giac [A]  time = 7.60148, size = 19, normalized size = 1.06 \begin{align*} \frac{20 \, x + 1}{8 \,{\left (2 \, x - 1\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(20*x + 1)/(2*x - 1)^2