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*ln(2+3*x)-154*ln(3+5*x)

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Rubi [A]  time = 0.02, antiderivative size = 39, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.05, 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]

-1/15*(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])/((2
+ 3*x)*(3 + 5*x))

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fricas [A]  time = 0.66, size = 55, normalized size = 1.41 \[ -\frac {2310 \, {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (5 \, x + 3\right ) - 2310 \, {\left (15 \, x^{2} + 19 \, x + 6\right )} \log \left (3 \, x + 2\right ) + 2314 \, x + 1461}{15 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} \]

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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giac [A]  time = 0.86, size = 38, normalized size = 0.97 \[ -\frac {121}{5 \, {\left (5 \, x + 3\right )}} + \frac {245}{\frac {1}{5 \, x + 3} + 3} + 154 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 36, normalized size = 0.92 \[ 154 \ln \left (3 x +2\right )-154 \ln \left (5 x +3\right )-\frac {49}{3 \left (3 x +2\right )}-\frac {121}{5 \left (5 x +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.56, size = 36, normalized size = 0.92 \[ -\frac {2314 \, x + 1461}{15 \, {\left (15 \, x^{2} + 19 \, x + 6\right )}} - 154 \, \log \left (5 \, x + 3\right ) + 154 \, \log \left (3 \, x + 2\right ) \]

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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mupad [B]  time = 0.04, size = 26, normalized size = 0.67 \[ 308\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {2314\,x}{225}+\frac {487}{75}}{x^2+\frac {19\,x}{15}+\frac {2}{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

308*atanh(30*x + 19) - ((2314*x)/225 + 487/75)/((19*x)/15 + x^2 + 2/5)

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sympy [A]  time = 0.14, size = 32, normalized size = 0.82 \[ \frac {- 2314 x - 1461}{225 x^{2} + 285 x + 90} - 154 \log {\left (x + \frac {3}{5} \right )} + 154 \log {\left (x + \frac {2}{3} \right )} \]

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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