3.1426 \(\int \frac{(1-2 x)^3}{(2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=43 \[ \frac{4719}{125 (5 x+3)}-\frac{1331}{250 (5 x+3)^2}-\frac{343}{3} \log (3 x+2)+\frac{14289}{125} \log (5 x+3) \]

[Out]

-1331/(250*(3 + 5*x)^2) + 4719/(125*(3 + 5*x)) - (343*Log[2 + 3*x])/3 + (14289*Log[3 + 5*x])/125

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Rubi [A]  time = 0.0195802, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {88} \[ \frac{4719}{125 (5 x+3)}-\frac{1331}{250 (5 x+3)^2}-\frac{343}{3} \log (3 x+2)+\frac{14289}{125} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1331/(250*(3 + 5*x)^2) + 4719/(125*(3 + 5*x)) - (343*Log[2 + 3*x])/3 + (14289*Log[3 + 5*x])/125

Rule 88

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandI
ntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] &&
(IntegerQ[p] || (GtQ[m, 0] && GeQ[n, -1]))

Rubi steps

\begin{align*} \int \frac{(1-2 x)^3}{(2+3 x) (3+5 x)^3} \, dx &=\int \left (-\frac{343}{2+3 x}+\frac{1331}{25 (3+5 x)^3}-\frac{4719}{25 (3+5 x)^2}+\frac{14289}{25 (3+5 x)}\right ) \, dx\\ &=-\frac{1331}{250 (3+5 x)^2}+\frac{4719}{125 (3+5 x)}-\frac{343}{3} \log (2+3 x)+\frac{14289}{125} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0229934, size = 44, normalized size = 1.02 \[ \frac{11 \left (4290 x+2598 (5 x+3)^2 \log (-3 (5 x+3))+2453\right )}{250 (5 x+3)^2}-\frac{343}{3} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-343*Log[2 + 3*x])/3 + (11*(2453 + 4290*x + 2598*(3 + 5*x)^2*Log[-3*(3 + 5*x)]))/(250*(3 + 5*x)^2)

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \begin{align*} -{\frac{1331}{250\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{4719}{375+625\,x}}-{\frac{343\,\ln \left ( 2+3\,x \right ) }{3}}+{\frac{14289\,\ln \left ( 3+5\,x \right ) }{125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1331/250/(3+5*x)^2+4719/125/(3+5*x)-343/3*ln(2+3*x)+14289/125*ln(3+5*x)

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Maxima [A]  time = 1.02351, size = 49, normalized size = 1.14 \begin{align*} \frac{121 \,{\left (390 \, x + 223\right )}}{250 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + \frac{14289}{125} \, \log \left (5 \, x + 3\right ) - \frac{343}{3} \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/250*(390*x + 223)/(25*x^2 + 30*x + 9) + 14289/125*log(5*x + 3) - 343/3*log(3*x + 2)

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Fricas [A]  time = 1.25902, size = 173, normalized size = 4.02 \begin{align*} \frac{85734 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 85750 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (3 \, x + 2\right ) + 141570 \, x + 80949}{750 \,{\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/(2+3*x)/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/750*(85734*(25*x^2 + 30*x + 9)*log(5*x + 3) - 85750*(25*x^2 + 30*x + 9)*log(3*x + 2) + 141570*x + 80949)/(25
*x^2 + 30*x + 9)

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Sympy [A]  time = 0.149424, size = 34, normalized size = 0.79 \begin{align*} \frac{47190 x + 26983}{6250 x^{2} + 7500 x + 2250} + \frac{14289 \log{\left (x + \frac{3}{5} \right )}}{125} - \frac{343 \log{\left (x + \frac{2}{3} \right )}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(47190*x + 26983)/(6250*x**2 + 7500*x + 2250) + 14289*log(x + 3/5)/125 - 343*log(x + 2/3)/3

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Giac [A]  time = 2.09283, size = 45, normalized size = 1.05 \begin{align*} \frac{121 \,{\left (390 \, x + 223\right )}}{250 \,{\left (5 \, x + 3\right )}^{2}} + \frac{14289}{125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{343}{3} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/250*(390*x + 223)/(5*x + 3)^2 + 14289/125*log(abs(5*x + 3)) - 343/3*log(abs(3*x + 2))