3.49 \(\int \frac{3+2 x}{(7+6 x)^3} \, dx\)

Optimal. Leaf size=18 \[ -\frac{(2 x+3)^2}{8 (6 x+7)^2} \]

[Out]

-(3 + 2*x)^2/(8*(7 + 6*x)^2)

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Rubi [A]  time = 0.0015898, 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} \[ -\frac{(2 x+3)^2}{8 (6 x+7)^2} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x)/(7 + 6*x)^3,x]

[Out]

-(3 + 2*x)^2/(8*(7 + 6*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+2 x}{(7+6 x)^3} \, dx &=-\frac{(3+2 x)^2}{8 (7+6 x)^2}\\ \end{align*}

Mathematica [A]  time = 0.003006, size = 16, normalized size = 0.89 \[ -\frac{3 x+4}{9 (6 x+7)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x)/(7 + 6*x)^3,x]

[Out]

-(4 + 3*x)/(9*(7 + 6*x)^2)

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Maple [A]  time = 0.004, size = 20, normalized size = 1.1 \begin{align*} -{\frac{1}{126+108\,x}}-{\frac{1}{18\, \left ( 7+6\,x \right ) ^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+2*x)/(7+6*x)^3,x)

[Out]

-1/18/(7+6*x)-1/18/(7+6*x)^2

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Maxima [A]  time = 0.958868, size = 26, normalized size = 1.44 \begin{align*} -\frac{3 \, x + 4}{9 \,{\left (36 \, x^{2} + 84 \, x + 49\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(7+6*x)^3,x, algorithm="maxima")

[Out]

-1/9*(3*x + 4)/(36*x^2 + 84*x + 49)

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Fricas [A]  time = 0.407813, size = 50, normalized size = 2.78 \begin{align*} -\frac{3 \, x + 4}{9 \,{\left (36 \, x^{2} + 84 \, x + 49\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(7+6*x)^3,x, algorithm="fricas")

[Out]

-1/9*(3*x + 4)/(36*x^2 + 84*x + 49)

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Sympy [A]  time = 0.095918, size = 15, normalized size = 0.83 \begin{align*} - \frac{3 x + 4}{324 x^{2} + 756 x + 441} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(7+6*x)**3,x)

[Out]

-(3*x + 4)/(324*x**2 + 756*x + 441)

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Giac [A]  time = 1.07244, size = 19, normalized size = 1.06 \begin{align*} -\frac{3 \, x + 4}{9 \,{\left (6 \, x + 7\right )}^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+2*x)/(7+6*x)^3,x, algorithm="giac")

[Out]

-1/9*(3*x + 4)/(6*x + 7)^2