3.1229 \(\int \frac{1-2 x}{(3+5 x)^3} \, dx\)

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

[Out]

-(1 - 2*x)^2/(22*(3 + 5*x)^2)

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Rubi [A]  time = 0.0020572, 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} \[ -\frac{(1-2 x)^2}{22 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)/(3 + 5*x)^3,x]

[Out]

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

Mathematica [A]  time = 0.0028611, size = 16, normalized size = 0.89 \[ \frac{20 x+1}{50 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)/(3 + 5*x)^3,x]

[Out]

(1 + 20*x)/(50*(3 + 5*x)^2)

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Maple [A]  time = 0.004, size = 20, normalized size = 1.1 \begin{align*} -{\frac{11}{50\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{2}{75+125\,x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-11/50/(3+5*x)^2+2/25/(3+5*x)

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Maxima [A]  time = 1.05051, size = 26, normalized size = 1.44 \begin{align*} \frac{20 \, x + 1}{50 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50*(20*x + 1)/(25*x^2 + 30*x + 9)

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Fricas [A]  time = 1.40836, size = 50, normalized size = 2.78 \begin{align*} \frac{20 \, x + 1}{50 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50*(20*x + 1)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.096869, size = 14, normalized size = 0.78 \begin{align*} \frac{20 x + 1}{1250 x^{2} + 1500 x + 450} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(20*x + 1)/(1250*x**2 + 1500*x + 450)

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Giac [A]  time = 2.26591, size = 19, normalized size = 1.06 \begin{align*} \frac{20 \, x + 1}{50 \,{\left (5 \, x + 3\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50*(20*x + 1)/(5*x + 3)^2