3.15.25 \(\int \frac {(1-2 x)^3}{(3+5 x)^3} \, dx\) [1425]

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

[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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Rubi [A]
time = 0.01, 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 = {45} \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 45

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 = 0.02, size = 37, normalized size = 0.97 \begin {gather*} \frac {\frac {5 \left (677+1548 x-280 x^2-400 x^3\right )}{(3+5 x)^2}+264 \log (6+10 x)}{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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Maple [A]
time = 0.09, size = 31, normalized size = 0.82

method	result	size
risch	\(-\frac {8 x}{125}+\frac {\frac {726 x}{125}+\frac {121}{50}}{\left (3+5 x \right )^{2}}+\frac {132 \ln \left (3+5 x \right )}{625}\)	\(27\)
default	\(-\frac {8 x}{125}-\frac {1331}{1250 \left (3+5 x \right )^{2}}+\frac {726}{625 \left (3+5 x \right )}+\frac {132 \ln \left (3+5 x \right )}{625}\)	\(31\)
norman	\(\frac {-\frac {1063}{375} x -\frac {3889}{450} x^{2}-\frac {8}{5} x^{3}}{\left (3+5 x \right )^{2}}+\frac {132 \ln \left (3+5 x \right )}{625}\)	\(32\)
meijerg	\(\frac {x \left (\frac {5 x}{3}+2\right )}{54 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {x^{2}}{9 \left (1+\frac {5 x}{3}\right )^{2}}-\frac {2 x \left (15 x +6\right )}{75 \left (1+\frac {5 x}{3}\right )^{2}}+\frac {132 \ln \left (1+\frac {5 x}{3}\right )}{625}-\frac {2 x \left (\frac {100}{9} x^{2}+30 x +12\right )}{125 \left (1+\frac {5 x}{3}\right )^{2}}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.28, 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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Fricas [A]
time = 0.41, 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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Sympy [A]
time = 0.04, 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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Giac [A]
time = 0.63, 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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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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