3.1.49 \(\int \frac {3+2 x}{(7+6 x)^3} \, dx\) [49]

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

[Out]

-1/8*(3+2*x)^2/(7+6*x)^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 {(2 x+3)^2}{8 (6 x+7)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(3 + 2*x)/(7 + 6*x)^3,x]

[Out]

-1/8*(3 + 2*x)^2/(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*}

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Mathematica [A]
time = 0.00, size = 16, normalized size = 0.89 \begin {gather*} -\frac {4+3 x}{9 (7+6 x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(3 + 2*x)/(7 + 6*x)^3,x]

[Out]

-1/9*(4 + 3*x)/(7 + 6*x)^2

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Maple [A]
time = 0.06, size = 20, normalized size = 1.11

method	result	size
norman	\(\frac {-\frac {x}{3}-\frac {4}{9}}{\left (7+6 x \right )^{2}}\)	\(14\)
gosper	\(-\frac {3 x +4}{9 \left (7+6 x \right )^{2}}\)	\(15\)
risch	\(\frac {-\frac {x}{3}-\frac {4}{9}}{\left (7+6 x \right )^{2}}\)	\(15\)
default	\(-\frac {1}{18 \left (7+6 x \right )}-\frac {1}{18 \left (7+6 x \right )^{2}}\)	\(20\)
meijerg	\(\frac {3 x \left (\frac {6 x}{7}+2\right )}{686 \left (1+\frac {6 x}{7}\right )^{2}}+\frac {x^{2}}{343 \left (1+\frac {6 x}{7}\right )^{2}}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 1.34, size = 19, normalized size = 1.06 \begin {gather*} -\frac {3 \, x + 4}{9 \, {\left (36 \, x^{2} + 84 \, x + 49\right )}} \end {gather*}

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 = 2.03, size = 19, normalized size = 1.06 \begin {gather*} -\frac {3 \, x + 4}{9 \, {\left (36 \, x^{2} + 84 \, x + 49\right )}} \end {gather*}

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 = 0.03, size = 15, normalized size = 0.83 \begin {gather*} \frac {- 3 x - 4}{324 x^{2} + 756 x + 441} \end {gather*}

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 = 0.48, size = 14, normalized size = 0.78 \begin {gather*} -\frac {3 \, x + 4}{9 \, {\left (6 \, x + 7\right )}^{2}} \end {gather*}

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

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Mupad [B]
time = 0.04, size = 14, normalized size = 0.78 \begin {gather*} -\frac {3\,x+4}{9\,{\left (6\,x+7\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x + 3)/(6*x + 7)^3,x)

[Out]

-(3*x + 4)/(9*(6*x + 7)^2)

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