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

Optimal. Leaf size=48 \[ \frac {121}{3 x+2}+\frac {217}{18 (3 x+2)^2}+\frac {49}{27 (3 x+2)^3}-605 \log (3 x+2)+605 \log (5 x+3) \]

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Rubi [A]  time = 0.02, antiderivative size = 48, 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 {121}{3 x+2}+\frac {217}{18 (3 x+2)^2}+\frac {49}{27 (3 x+2)^3}-605 \log (3 x+2)+605 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(27*(2 + 3*x)^3) + 217/(18*(2 + 3*x)^2) + 121/(2 + 3*x) - 605*Log[2 + 3*x] + 605*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)} \, dx &=\int \left (-\frac {49}{3 (2+3 x)^4}-\frac {217}{3 (2+3 x)^3}-\frac {363}{(2+3 x)^2}-\frac {1815}{2+3 x}+\frac {3025}{3+5 x}\right ) \, dx\\ &=\frac {49}{27 (2+3 x)^3}+\frac {217}{18 (2+3 x)^2}+\frac {121}{2+3 x}-605 \log (2+3 x)+605 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 40, normalized size = 0.83 \begin {gather*} \frac {58806 x^2+80361 x+27536}{54 (3 x+2)^3}-605 \log (5 (3 x+2))+605 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(27536 + 80361*x + 58806*x^2)/(54*(2 + 3*x)^3) - 605*Log[5*(2 + 3*x)] + 605*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)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.81, size = 75, normalized size = 1.56 \begin {gather*} \frac {58806 \, x^{2} + 32670 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (5 \, x + 3\right ) - 32670 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 80361 \, x + 27536}{54 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\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),x, algorithm="fricas")

[Out]

1/54*(58806*x^2 + 32670*(27*x^3 + 54*x^2 + 36*x + 8)*log(5*x + 3) - 32670*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x
+ 2) + 80361*x + 27536)/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 0.98, size = 38, normalized size = 0.79 \begin {gather*} \frac {58806 \, x^{2} + 80361 \, x + 27536}{54 \, {\left (3 \, x + 2\right )}^{3}} + 605 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 605 \, \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),x, algorithm="giac")

[Out]

1/54*(58806*x^2 + 80361*x + 27536)/(3*x + 2)^3 + 605*log(abs(5*x + 3)) - 605*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 45, normalized size = 0.94 \begin {gather*} -605 \ln \left (3 x +2\right )+605 \ln \left (5 x +3\right )+\frac {49}{27 \left (3 x +2\right )^{3}}+\frac {217}{18 \left (3 x +2\right )^{2}}+\frac {121}{3 x +2} \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),x)

[Out]

49/27/(3*x+2)^3+217/18/(3*x+2)^2+121/(3*x+2)-605*ln(3*x+2)+605*ln(5*x+3)

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maxima [A]  time = 0.60, size = 46, normalized size = 0.96 \begin {gather*} \frac {58806 \, x^{2} + 80361 \, x + 27536}{54 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + 605 \, \log \left (5 \, x + 3\right ) - 605 \, \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),x, algorithm="maxima")

[Out]

1/54*(58806*x^2 + 80361*x + 27536)/(27*x^3 + 54*x^2 + 36*x + 8) + 605*log(5*x + 3) - 605*log(3*x + 2)

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mupad [B]  time = 0.04, size = 35, normalized size = 0.73 \begin {gather*} \frac {\frac {121\,x^2}{3}+\frac {8929\,x}{162}+\frac {13768}{729}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}}-1210\,\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)),x)

[Out]

((8929*x)/162 + (121*x^2)/3 + 13768/729)/((4*x)/3 + 2*x^2 + x^3 + 8/27) - 1210*atanh(30*x + 19)

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sympy [A]  time = 0.16, size = 41, normalized size = 0.85 \begin {gather*} \frac {58806 x^{2} + 80361 x + 27536}{1458 x^{3} + 2916 x^{2} + 1944 x + 432} + 605 \log {\left (x + \frac {3}{5} \right )} - 605 \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),x)

[Out]

(58806*x**2 + 80361*x + 27536)/(1458*x**3 + 2916*x**2 + 1944*x + 432) + 605*log(x + 3/5) - 605*log(x + 2/3)

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