3.1425 \(\int \frac{(1-2 x)^3}{(3+5 x)^3} \, dx\)

Optimal. Leaf size=38 \[ -\frac{8 x}{125}+\frac{726}{625 (5 x+3)}-\frac{1331}{1250 (5 x+3)^2}+\frac{132}{625} \log (5 x+3) \]

[Out]

(-8*x)/125 - 1331/(1250*(3 + 5*x)^2) + 726/(625*(3 + 5*x)) + (132*Log[3 + 5*x])/625

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Rubi [A]  time = 0.0161619, antiderivative size = 38, 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} \[ -\frac{8 x}{125}+\frac{726}{625 (5 x+3)}-\frac{1331}{1250 (5 x+3)^2}+\frac{132}{625} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^3/(3 + 5*x)^3,x]

[Out]

(-8*x)/125 - 1331/(1250*(3 + 5*x)^2) + 726/(625*(3 + 5*x)) + (132*Log[3 + 5*x])/625

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)^3}{(3+5 x)^3} \, dx &=\int \left (-\frac{8}{125}+\frac{1331}{125 (3+5 x)^3}-\frac{726}{125 (3+5 x)^2}+\frac{132}{125 (3+5 x)}\right ) \, dx\\ &=-\frac{8 x}{125}-\frac{1331}{1250 (3+5 x)^2}+\frac{726}{625 (3+5 x)}+\frac{132}{625} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0167205, size = 37, normalized size = 0.97 \[ \frac{\frac{5 \left (-400 x^3-280 x^2+1548 x+677\right )}{(5 x+3)^2}+264 \log (10 x+6)}{1250} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^3/(3 + 5*x)^3,x]

[Out]

((5*(677 + 1548*x - 280*x^2 - 400*x^3))/(3 + 5*x)^2 + 264*Log[6 + 10*x])/1250

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Maple [A]  time = 0.005, size = 31, normalized size = 0.8 \begin{align*} -{\frac{8\,x}{125}}-{\frac{1331}{1250\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{726}{1875+3125\,x}}+{\frac{132\,\ln \left ( 3+5\,x \right ) }{625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/125*x-1331/1250/(3+5*x)^2+726/625/(3+5*x)+132/625*ln(3+5*x)

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Maxima [A]  time = 1.03506, size = 42, normalized size = 1.11 \begin{align*} -\frac{8}{125} \, x + \frac{121 \,{\left (12 \, x + 5\right )}}{250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{132}{625} \, \log \left (5 \, x + 3\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/125*x + 121/250*(12*x + 5)/(25*x^2 + 30*x + 9) + 132/625*log(5*x + 3)

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Fricas [A]  time = 1.38171, size = 143, normalized size = 3.76 \begin{align*} -\frac{2000 \, x^{3} + 2400 \, x^{2} - 264 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 6540 \, x - 3025}{1250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1250*(2000*x^3 + 2400*x^2 - 264*(25*x^2 + 30*x + 9)*log(5*x + 3) - 6540*x - 3025)/(25*x^2 + 30*x + 9)

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Sympy [A]  time = 0.110429, size = 29, normalized size = 0.76 \begin{align*} - \frac{8 x}{125} + \frac{1452 x + 605}{6250 x^{2} + 7500 x + 2250} + \frac{132 \log{\left (5 x + 3 \right )}}{625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8*x/125 + (1452*x + 605)/(6250*x**2 + 7500*x + 2250) + 132*log(5*x + 3)/625

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Giac [A]  time = 2.6055, size = 36, normalized size = 0.95 \begin{align*} -\frac{8}{125} \, x + \frac{121 \,{\left (12 \, x + 5\right )}}{250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{132}{625} \, \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)^3/(3+5*x)^3,x, algorithm="giac")

[Out]

-8/125*x + 121/250*(12*x + 5)/(5*x + 3)^2 + 132/625*log(abs(5*x + 3))