3.1314 \(\int \frac{(1-2 x)^2}{(2+3 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=39 \[ -\frac{49}{3 (3 x+2)}-\frac{121}{5 (5 x+3)}+154 \log (3 x+2)-154 \log (5 x+3) \]

[Out]

-49/(3*(2 + 3*x)) - 121/(5*(3 + 5*x)) + 154*Log[2 + 3*x] - 154*Log[3 + 5*x]

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Rubi [A]  time = 0.0177119, antiderivative size = 39, 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} \[ -\frac{49}{3 (3 x+2)}-\frac{121}{5 (5 x+3)}+154 \log (3 x+2)-154 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-49/(3*(2 + 3*x)) - 121/(5*(3 + 5*x)) + 154*Log[2 + 3*x] - 154*Log[3 + 5*x]

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)^2}{(2+3 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac{49}{(2+3 x)^2}+\frac{462}{2+3 x}+\frac{121}{(3+5 x)^2}-\frac{770}{3+5 x}\right ) \, dx\\ &=-\frac{49}{3 (2+3 x)}-\frac{121}{5 (3+5 x)}+154 \log (2+3 x)-154 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0303634, size = 61, normalized size = 1.56 \[ -\frac{-2310 \left (15 x^2+19 x+6\right ) \log (5 (3 x+2))+2310 \left (15 x^2+19 x+6\right ) \log (5 x+3)+2314 x+1461}{15 (3 x+2) (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(1461 + 2314*x - 2310*(6 + 19*x + 15*x^2)*Log[5*(2 + 3*x)] + 2310*(6 + 19*x + 15*x^2)*Log[3 + 5*x])/(15*(2 +
3*x)*(3 + 5*x))

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Maple [A]  time = 0.008, size = 36, normalized size = 0.9 \begin{align*} -{\frac{49}{6+9\,x}}-{\frac{121}{15+25\,x}}+154\,\ln \left ( 2+3\,x \right ) -154\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/3/(2+3*x)-121/5/(3+5*x)+154*ln(2+3*x)-154*ln(3+5*x)

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Maxima [A]  time = 1.09228, size = 49, normalized size = 1.26 \begin{align*} -\frac{2314 \, x + 1461}{15 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} - 154 \, \log \left (5 \, x + 3\right ) + 154 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(2314*x + 1461)/(15*x^2 + 19*x + 6) - 154*log(5*x + 3) + 154*log(3*x + 2)

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Fricas [A]  time = 1.512, size = 166, normalized size = 4.26 \begin{align*} -\frac{2310 \,{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (5 \, x + 3\right ) - 2310 \,{\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (3 \, x + 2\right ) + 2314 \, x + 1461}{15 \,{\left (15 \, x^{2} + 19 \, x + 6\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*(2310*(15*x^2 + 19*x + 6)*log(5*x + 3) - 2310*(15*x^2 + 19*x + 6)*log(3*x + 2) + 2314*x + 1461)/(15*x^2
+ 19*x + 6)

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Sympy [A]  time = 0.131609, size = 31, normalized size = 0.79 \begin{align*} - \frac{2314 x + 1461}{225 x^{2} + 285 x + 90} - 154 \log{\left (x + \frac{3}{5} \right )} + 154 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2314*x + 1461)/(225*x**2 + 285*x + 90) - 154*log(x + 3/5) + 154*log(x + 2/3)

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Giac [A]  time = 1.62428, size = 51, normalized size = 1.31 \begin{align*} -\frac{121}{5 \,{\left (5 \, x + 3\right )}} + \frac{245}{\frac{1}{5 \, x + 3} + 3} + 154 \, \log \left ({\left | -\frac{1}{5 \, x + 3} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-121/5/(5*x + 3) + 245/(1/(5*x + 3) + 3) + 154*log(abs(-1/(5*x + 3) - 3))