3.1296 \(\int \frac{(1-2 x)^2}{3+5 x} \, dx\)

Optimal. Leaf size=23 \[ \frac{2 x^2}{5}-\frac{32 x}{25}+\frac{121}{125} \log (5 x+3) \]

[Out]

(-32*x)/25 + (2*x^2)/5 + (121*Log[3 + 5*x])/125

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Rubi [A]  time = 0.0078704, antiderivative size = 23, 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} \[ \frac{2 x^2}{5}-\frac{32 x}{25}+\frac{121}{125} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2/(3 + 5*x),x]

[Out]

(-32*x)/25 + (2*x^2)/5 + (121*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} \, dx &=\int \left (-\frac{32}{25}+\frac{4 x}{5}+\frac{121}{25 (3+5 x)}\right ) \, dx\\ &=-\frac{32 x}{25}+\frac{2 x^2}{5}+\frac{121}{125} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0055031, size = 22, normalized size = 0.96 \[ \frac{1}{125} \left (50 x^2-160 x+121 \log (5 x+3)-114\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2/(3 + 5*x),x]

[Out]

(-114 - 160*x + 50*x^2 + 121*Log[3 + 5*x])/125

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Maple [A]  time = 0.003, size = 18, normalized size = 0.8 \begin{align*} -{\frac{32\,x}{25}}+{\frac{2\,{x}^{2}}{5}}+{\frac{121\,\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),x)

[Out]

-32/25*x+2/5*x^2+121/125*ln(3+5*x)

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Maxima [A]  time = 1.02713, size = 23, normalized size = 1. \begin{align*} \frac{2}{5} \, x^{2} - \frac{32}{25} \, x + \frac{121}{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),x, algorithm="maxima")

[Out]

2/5*x^2 - 32/25*x + 121/125*log(5*x + 3)

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Fricas [A]  time = 1.56635, size = 57, normalized size = 2.48 \begin{align*} \frac{2}{5} \, x^{2} - \frac{32}{25} \, x + \frac{121}{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),x, algorithm="fricas")

[Out]

2/5*x^2 - 32/25*x + 121/125*log(5*x + 3)

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Sympy [A]  time = 0.076658, size = 20, normalized size = 0.87 \begin{align*} \frac{2 x^{2}}{5} - \frac{32 x}{25} + \frac{121 \log{\left (5 x + 3 \right )}}{125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*x**2/5 - 32*x/25 + 121*log(5*x + 3)/125

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Giac [A]  time = 2.24137, size = 24, normalized size = 1.04 \begin{align*} \frac{2}{5} \, x^{2} - \frac{32}{25} \, x + \frac{121}{125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*x^2 - 32/25*x + 121/125*log(abs(5*x + 3))