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

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

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Rubi [A]  time = 0.02, antiderivative size = 38, 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 {8 x}{125}+\frac {726}{625 (5 x+3)}-\frac {1331}{1250 (5 x+3)^2}+\frac {132}{625} \log (5 x+3) \end {gather*}

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*}

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Mathematica [A]  time = 0.03, size = 37, normalized size = 0.97 \begin {gather*} \frac {\frac {5 \left (-400 x^3-280 x^2+1548 x+677\right )}{(5 x+3)^2}+264 \log (10 x+6)}{1250} \end {gather*}

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.94, size = 47, normalized size = 1.24 \begin {gather*} -\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 {gather*}

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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giac [A]  time = 0.93, size = 27, normalized size = 0.71 \begin {gather*} -\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 {gather*}

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))

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maple [A]  time = 0.01, size = 31, normalized size = 0.82 \begin {gather*} -\frac {8 x}{125}+\frac {132 \ln \left (5 x +3\right )}{625}-\frac {1331}{1250 \left (5 x +3\right )^{2}}+\frac {726}{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)^3,x)

[Out]

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

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maxima [A]  time = 0.42, size = 31, normalized size = 0.82 \begin {gather*} -\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 {gather*}

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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mupad [B]  time = 1.12, size = 26, normalized size = 0.68 \begin {gather*} \frac {132\,\ln \left (x+\frac {3}{5}\right )}{625}-\frac {8\,x}{125}+\frac {\frac {726\,x}{3125}+\frac {121}{1250}}{x^2+\frac {6\,x}{5}+\frac {9}{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)^3/(5*x + 3)^3,x)

[Out]

(132*log(x + 3/5))/625 - (8*x)/125 + ((726*x)/3125 + 121/1250)/((6*x)/5 + x^2 + 9/25)

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sympy [A]  time = 0.12, size = 31, normalized size = 0.82 \begin {gather*} - \frac {8 x}{125} - \frac {- 1452 x - 605}{6250 x^{2} + 7500 x + 2250} + \frac {132 \log {\left (5 x + 3 \right )}}{625} \end {gather*}

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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