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

Optimal. Leaf size=34 \[ -\frac {4 x^2}{25}+\frac {108 x}{125}-\frac {1331}{625 (5 x+3)}-\frac {726}{625} \log (5 x+3) \]

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.00, 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} \begin {gather*} -\frac {4 x^2}{25}+\frac {108 x}{125}-\frac {1331}{625 (5 x+3)}-\frac {726}{625} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^3/(3 + 5*x)^2,x]

[Out]

(108*x)/125 - (4*x^2)/25 - 1331/(625*(3 + 5*x)) - (726*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)^2} \, dx &=\int \left (\frac {108}{125}-\frac {8 x}{25}+\frac {1331}{125 (3+5 x)^2}-\frac {726}{125 (3+5 x)}\right ) \, dx\\ &=\frac {108 x}{125}-\frac {4 x^2}{25}-\frac {1331}{625 (3+5 x)}-\frac {726}{625} \log (3+5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 39, normalized size = 1.15 \begin {gather*} \frac {-500 x^3+2400 x^2+395 x-726 (5 x+3) \log (10 x+6)-2066}{625 (5 x+3)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^3/(3 + 5*x)^2,x]

[Out]

(-2066 + 395*x + 2400*x^2 - 500*x^3 - 726*(3 + 5*x)*Log[6 + 10*x])/(625*(3 + 5*x))

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^3}{(3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^3/(3 + 5*x)^2,x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^3/(3 + 5*x)^2, x]

________________________________________________________________________________________

fricas [A]  time = 1.25, size = 37, normalized size = 1.09 \begin {gather*} -\frac {500 \, x^{3} - 2400 \, x^{2} + 726 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) - 1620 \, x + 1331}{625 \, {\left (5 \, x + 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/625*(500*x^3 - 2400*x^2 + 726*(5*x + 3)*log(5*x + 3) - 1620*x + 1331)/(5*x + 3)

________________________________________________________________________________________

giac [A]  time = 0.98, size = 48, normalized size = 1.41 \begin {gather*} \frac {4}{625} \, {\left (5 \, x + 3\right )}^{2} {\left (\frac {33}{5 \, x + 3} - 1\right )} - \frac {1331}{625 \, {\left (5 \, x + 3\right )}} + \frac {726}{625} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/625*(5*x + 3)^2*(33/(5*x + 3) - 1) - 1331/625/(5*x + 3) + 726/625*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 27, normalized size = 0.79 \begin {gather*} -\frac {4 x^{2}}{25}+\frac {108 x}{125}-\frac {726 \ln \left (5 x +3\right )}{625}-\frac {1331}{625 \left (5 x +3\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

108/125*x-4/25*x^2-1331/625/(5*x+3)-726/625*ln(5*x+3)

________________________________________________________________________________________

maxima [A]  time = 0.46, size = 26, normalized size = 0.76 \begin {gather*} -\frac {4}{25} \, x^{2} + \frac {108}{125} \, x - \frac {1331}{625 \, {\left (5 \, x + 3\right )}} - \frac {726}{625} \, \log \left (5 \, x + 3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/25*x^2 + 108/125*x - 1331/625/(5*x + 3) - 726/625*log(5*x + 3)

________________________________________________________________________________________

mupad [B]  time = 0.03, size = 24, normalized size = 0.71 \begin {gather*} \frac {108\,x}{125}-\frac {726\,\ln \left (x+\frac {3}{5}\right )}{625}-\frac {1331}{3125\,\left (x+\frac {3}{5}\right )}-\frac {4\,x^2}{25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)^3/(5*x + 3)^2,x)

[Out]

(108*x)/125 - (726*log(x + 3/5))/625 - 1331/(3125*(x + 3/5)) - (4*x^2)/25

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 27, normalized size = 0.79 \begin {gather*} - \frac {4 x^{2}}{25} + \frac {108 x}{125} - \frac {726 \log {\left (5 x + 3 \right )}}{625} - \frac {1331}{3125 x + 1875} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*x**2/25 + 108*x/125 - 726*log(5*x + 3)/625 - 1331/(3125*x + 1875)

________________________________________________________________________________________