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

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {(1-2 x)^2}{22 (5 x+3)^2} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 16, normalized size = 0.89 \begin {gather*} \frac {20 x+1}{50 (5 x+3)^2} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1-2 x}{(3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.64, size = 19, normalized size = 1.06 \begin {gather*} \frac {20 \, x + 1}{50 \, {\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)/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/50*(20*x + 1)/(25*x^2 + 30*x + 9)

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giac [A]  time = 1.18, size = 14, normalized size = 0.78 \begin {gather*} \frac {20 \, x + 1}{50 \, {\left (5 \, x + 3\right )}^{2}} \end {gather*}

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

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maple [A]  time = 0.01, size = 20, normalized size = 1.11 \begin {gather*} -\frac {11}{50 \left (5 x +3\right )^{2}}+\frac {2}{25 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.63, size = 19, normalized size = 1.06 \begin {gather*} \frac {20 \, x + 1}{50 \, {\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)/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/50*(20*x + 1)/(25*x^2 + 30*x + 9)

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mupad [B]  time = 1.10, size = 14, normalized size = 0.78 \begin {gather*} \frac {20\,x+1}{50\,{\left (5\,x+3\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)/(5*x + 3)^3,x)

[Out]

(20*x + 1)/(50*(5*x + 3)^2)

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sympy [A]  time = 0.10, size = 17, normalized size = 0.94 \begin {gather*} - \frac {- 20 x - 1}{1250 x^{2} + 1500 x + 450} \end {gather*}

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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