3.1312 \(\int \frac{(1-2 x)^2}{(3+5 x)^2} \, dx\)

Optimal. Leaf size=27 \[ \frac{4 x}{25}-\frac{121}{125 (5 x+3)}-\frac{44}{125} \log (5 x+3) \]

[Out]

(4*x)/25 - 121/(125*(3 + 5*x)) - (44*Log[3 + 5*x])/125

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Rubi [A]  time = 0.0096519, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {43} \[ \frac{4 x}{25}-\frac{121}{125 (5 x+3)}-\frac{44}{125} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2/(3 + 5*x)^2,x]

[Out]

(4*x)/25 - 121/(125*(3 + 5*x)) - (44*Log[3 + 5*x])/125

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{(1-2 x)^2}{(3+5 x)^2} \, dx &=\int \left (\frac{4}{25}+\frac{121}{25 (3+5 x)^2}-\frac{44}{25 (3+5 x)}\right ) \, dx\\ &=\frac{4 x}{25}-\frac{121}{125 (3+5 x)}-\frac{44}{125} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0103306, size = 34, normalized size = 1.26 \[ \frac{100 x^2+10 x-44 (5 x+3) \log (10 x+6)-151}{125 (5 x+3)} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2/(3 + 5*x)^2,x]

[Out]

(-151 + 10*x + 100*x^2 - 44*(3 + 5*x)*Log[6 + 10*x])/(125*(3 + 5*x))

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Maple [A]  time = 0.004, size = 22, normalized size = 0.8 \begin{align*}{\frac{4\,x}{25}}-{\frac{121}{375+625\,x}}-{\frac{44\,\ln \left ( 3+5\,x \right ) }{125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/25*x-121/125/(3+5*x)-44/125*ln(3+5*x)

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Maxima [A]  time = 1.11889, size = 28, normalized size = 1.04 \begin{align*} \frac{4}{25} \, x - \frac{121}{125 \,{\left (5 \, x + 3\right )}} - \frac{44}{125} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/25*x - 121/125/(5*x + 3) - 44/125*log(5*x + 3)

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Fricas [A]  time = 1.55409, size = 92, normalized size = 3.41 \begin{align*} \frac{100 \, x^{2} - 44 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 60 \, x - 121}{125 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/125*(100*x^2 - 44*(5*x + 3)*log(5*x + 3) + 60*x - 121)/(5*x + 3)

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Sympy [A]  time = 0.091352, size = 20, normalized size = 0.74 \begin{align*} \frac{4 x}{25} - \frac{44 \log{\left (5 x + 3 \right )}}{125} - \frac{121}{625 x + 375} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*x/25 - 44*log(5*x + 3)/125 - 121/(625*x + 375)

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Giac [A]  time = 2.12781, size = 43, normalized size = 1.59 \begin{align*} \frac{4}{25} \, x - \frac{121}{125 \,{\left (5 \, x + 3\right )}} + \frac{44}{125} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) + \frac{12}{125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/25*x - 121/125/(5*x + 3) + 44/125*log(1/5*abs(5*x + 3)/(5*x + 3)^2) + 12/125