3.12.94 \(\int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^3} \, dx\)

Optimal. Leaf size=68 \[ \frac {6934}{3 x+2}+\frac {7480}{5 x+3}+\frac {707}{2 (3 x+2)^2}-\frac {605}{2 (5 x+3)^2}+\frac {49}{3 (3 x+2)^3}-57110 \log (3 x+2)+57110 \log (5 x+3) \]

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Rubi [A]  time = 0.03, antiderivative size = 68, 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} \begin {gather*} \frac {6934}{3 x+2}+\frac {7480}{5 x+3}+\frac {707}{2 (3 x+2)^2}-\frac {605}{2 (5 x+3)^2}+\frac {49}{3 (3 x+2)^3}-57110 \log (3 x+2)+57110 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^2/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

49/(3*(2 + 3*x)^3) + 707/(2*(2 + 3*x)^2) + 6934/(2 + 3*x) - 605/(2*(3 + 5*x)^2) + 7480/(3 + 5*x) - 57110*Log[2
+ 3*x] + 57110*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)^4 (3+5 x)^3} \, dx &=\int \left (-\frac {147}{(2+3 x)^4}-\frac {2121}{(2+3 x)^3}-\frac {20802}{(2+3 x)^2}-\frac {171330}{2+3 x}+\frac {3025}{(3+5 x)^3}-\frac {37400}{(3+5 x)^2}+\frac {285550}{3+5 x}\right ) \, dx\\ &=\frac {49}{3 (2+3 x)^3}+\frac {707}{2 (2+3 x)^2}+\frac {6934}{2+3 x}-\frac {605}{2 (3+5 x)^2}+\frac {7480}{3+5 x}-57110 \log (2+3 x)+57110 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 70, normalized size = 1.03 \begin {gather*} \frac {6934}{3 x+2}+\frac {7480}{5 x+3}+\frac {707}{2 (3 x+2)^2}-\frac {605}{2 (5 x+3)^2}+\frac {49}{3 (3 x+2)^3}-57110 \log (5 (3 x+2))+57110 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^2/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

49/(3*(2 + 3*x)^3) + 707/(2*(2 + 3*x)^2) + 6934/(2 + 3*x) - 605/(2*(3 + 5*x)^2) + 7480/(3 + 5*x) - 57110*Log[5
*(2 + 3*x)] + 57110*Log[3 + 5*x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^2/((2 + 3*x)^4*(3 + 5*x)^3),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^2/((2 + 3*x)^4*(3 + 5*x)^3), x]

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fricas [A]  time = 1.19, size = 115, normalized size = 1.69 \begin {gather*} \frac {15419700 \, x^{4} + 39577230 \, x^{3} + 38058104 \, x^{2} + 342660 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (5 \, x + 3\right ) - 342660 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )} \log \left (3 \, x + 2\right ) + 16250079 \, x + 2599404}{6 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^4/(3+5*x)^3,x, algorithm="fricas")

[Out]

1/6*(15419700*x^4 + 39577230*x^3 + 38058104*x^2 + 342660*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 7
2)*log(5*x + 3) - 342660*(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)*log(3*x + 2) + 16250079*x + 2
599404)/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*x^2 + 564*x + 72)

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giac [A]  time = 0.88, size = 55, normalized size = 0.81 \begin {gather*} \frac {15419700 \, x^{4} + 39577230 \, x^{3} + 38058104 \, x^{2} + 16250079 \, x + 2599404}{6 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{3}} + 57110 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 57110 \, \log \left ({\left | 3 \, x + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^4/(3+5*x)^3,x, algorithm="giac")

[Out]

1/6*(15419700*x^4 + 39577230*x^3 + 38058104*x^2 + 16250079*x + 2599404)/((5*x + 3)^2*(3*x + 2)^3) + 57110*log(
abs(5*x + 3)) - 57110*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 63, normalized size = 0.93 \begin {gather*} -57110 \ln \left (3 x +2\right )+57110 \ln \left (5 x +3\right )+\frac {49}{3 \left (3 x +2\right )^{3}}+\frac {707}{2 \left (3 x +2\right )^{2}}+\frac {6934}{3 x +2}-\frac {605}{2 \left (5 x +3\right )^{2}}+\frac {7480}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^2/(3*x+2)^4/(5*x+3)^3,x)

[Out]

49/3/(3*x+2)^3+707/2/(3*x+2)^2+6934/(3*x+2)-605/2/(5*x+3)^2+7480/(5*x+3)-57110*ln(3*x+2)+57110*ln(5*x+3)

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maxima [A]  time = 0.54, size = 66, normalized size = 0.97 \begin {gather*} \frac {15419700 \, x^{4} + 39577230 \, x^{3} + 38058104 \, x^{2} + 16250079 \, x + 2599404}{6 \, {\left (675 \, x^{5} + 2160 \, x^{4} + 2763 \, x^{3} + 1766 \, x^{2} + 564 \, x + 72\right )}} + 57110 \, \log \left (5 \, x + 3\right ) - 57110 \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^2/(2+3*x)^4/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/6*(15419700*x^4 + 39577230*x^3 + 38058104*x^2 + 16250079*x + 2599404)/(675*x^5 + 2160*x^4 + 2763*x^3 + 1766*
x^2 + 564*x + 72) + 57110*log(5*x + 3) - 57110*log(3*x + 2)

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mupad [B]  time = 0.04, size = 55, normalized size = 0.81 \begin {gather*} \frac {\frac {11422\,x^4}{3}+\frac {439747\,x^3}{45}+\frac {19029052\,x^2}{2025}+\frac {5416693\,x}{1350}+\frac {433234}{675}}{x^5+\frac {16\,x^4}{5}+\frac {307\,x^3}{75}+\frac {1766\,x^2}{675}+\frac {188\,x}{225}+\frac {8}{75}}-114220\,\mathrm {atanh}\left (30\,x+19\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*x - 1)^2/((3*x + 2)^4*(5*x + 3)^3),x)

[Out]

((5416693*x)/1350 + (19029052*x^2)/2025 + (439747*x^3)/45 + (11422*x^4)/3 + 433234/675)/((188*x)/225 + (1766*x
^2)/675 + (307*x^3)/75 + (16*x^4)/5 + x^5 + 8/75) - 114220*atanh(30*x + 19)

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sympy [A]  time = 0.19, size = 61, normalized size = 0.90 \begin {gather*} \frac {15419700 x^{4} + 39577230 x^{3} + 38058104 x^{2} + 16250079 x + 2599404}{4050 x^{5} + 12960 x^{4} + 16578 x^{3} + 10596 x^{2} + 3384 x + 432} + 57110 \log {\left (x + \frac {3}{5} \right )} - 57110 \log {\left (x + \frac {2}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**2/(2+3*x)**4/(3+5*x)**3,x)

[Out]

(15419700*x**4 + 39577230*x**3 + 38058104*x**2 + 16250079*x + 2599404)/(4050*x**5 + 12960*x**4 + 16578*x**3 +
10596*x**2 + 3384*x + 432) + 57110*log(x + 3/5) - 57110*log(x + 2/3)

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