3.1397 \(\int \frac{(1-2 x)^3}{(2+3 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=43 \[ \frac{1421}{27 (3 x+2)}+\frac{343}{54 (3 x+2)^2}-\frac{7189}{27} \log (3 x+2)+\frac{1331}{5} \log (5 x+3) \]

[Out]

343/(54*(2 + 3*x)^2) + 1421/(27*(2 + 3*x)) - (7189*Log[2 + 3*x])/27 + (1331*Log[3 + 5*x])/5

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Rubi [A]  time = 0.0170406, 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} \[ \frac{1421}{27 (3 x+2)}+\frac{343}{54 (3 x+2)^2}-\frac{7189}{27} \log (3 x+2)+\frac{1331}{5} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

343/(54*(2 + 3*x)^2) + 1421/(27*(2 + 3*x)) - (7189*Log[2 + 3*x])/27 + (1331*Log[3 + 5*x])/5

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 (3+5 x)} \, dx &=\int \left (-\frac{343}{9 (2+3 x)^3}-\frac{1421}{9 (2+3 x)^2}-\frac{7189}{9 (2+3 x)}+\frac{1331}{3+5 x}\right ) \, dx\\ &=\frac{343}{54 (2+3 x)^2}+\frac{1421}{27 (2+3 x)}-\frac{7189}{27} \log (2+3 x)+\frac{1331}{5} \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0233453, size = 39, normalized size = 0.91 \[ \frac{49 (58 x+41)}{18 (3 x+2)^2}-\frac{7189}{27} \log (5 (3 x+2))+\frac{1331}{5} \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(49*(41 + 58*x))/(18*(2 + 3*x)^2) - (7189*Log[5*(2 + 3*x)])/27 + (1331*Log[3 + 5*x])/5

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Maple [A]  time = 0.007, size = 36, normalized size = 0.8 \begin{align*}{\frac{343}{54\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{1421}{54+81\,x}}-{\frac{7189\,\ln \left ( 2+3\,x \right ) }{27}}+{\frac{1331\,\ln \left ( 3+5\,x \right ) }{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

343/54/(2+3*x)^2+1421/27/(2+3*x)-7189/27*ln(2+3*x)+1331/5*ln(3+5*x)

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Maxima [A]  time = 1.00372, size = 49, normalized size = 1.14 \begin{align*} \frac{49 \,{\left (58 \, x + 41\right )}}{18 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{1331}{5} \, \log \left (5 \, x + 3\right ) - \frac{7189}{27} \, \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/(3+5*x),x, algorithm="maxima")

[Out]

49/18*(58*x + 41)/(9*x^2 + 12*x + 4) + 1331/5*log(5*x + 3) - 7189/27*log(3*x + 2)

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Fricas [A]  time = 1.35897, size = 167, normalized size = 3.88 \begin{align*} \frac{71874 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (5 \, x + 3\right ) - 71890 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 42630 \, x + 30135}{270 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/270*(71874*(9*x^2 + 12*x + 4)*log(5*x + 3) - 71890*(9*x^2 + 12*x + 4)*log(3*x + 2) + 42630*x + 30135)/(9*x^2
+ 12*x + 4)

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Sympy [A]  time = 0.147467, size = 34, normalized size = 0.79 \begin{align*} \frac{2842 x + 2009}{162 x^{2} + 216 x + 72} + \frac{1331 \log{\left (x + \frac{3}{5} \right )}}{5} - \frac{7189 \log{\left (x + \frac{2}{3} \right )}}{27} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2842*x + 2009)/(162*x**2 + 216*x + 72) + 1331*log(x + 3/5)/5 - 7189*log(x + 2/3)/27

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Giac [A]  time = 2.62177, size = 45, normalized size = 1.05 \begin{align*} \frac{49 \,{\left (58 \, x + 41\right )}}{18 \,{\left (3 \, x + 2\right )}^{2}} + \frac{1331}{5} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{7189}{27} \, \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/(3+5*x),x, algorithm="giac")

[Out]

49/18*(58*x + 41)/(3*x + 2)^2 + 1331/5*log(abs(5*x + 3)) - 7189/27*log(abs(3*x + 2))