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

Optimal. Leaf size=46 \[ \frac {49}{3 x+2}+\frac {154}{5 x+3}-\frac {121}{10 (5 x+3)^2}-707 \log (3 x+2)+707 \log (5 x+3) \]

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Rubi [A]  time = 0.02, antiderivative size = 46, 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 {49}{3 x+2}+\frac {154}{5 x+3}-\frac {121}{10 (5 x+3)^2}-707 \log (3 x+2)+707 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(2 + 3*x) - 121/(10*(3 + 5*x)^2) + 154/(3 + 5*x) - 707*Log[2 + 3*x] + 707*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)^3} \, dx &=\int \left (-\frac {147}{(2+3 x)^2}-\frac {2121}{2+3 x}+\frac {121}{(3+5 x)^3}-\frac {770}{(3+5 x)^2}+\frac {3535}{3+5 x}\right ) \, dx\\ &=\frac {49}{2+3 x}-\frac {121}{10 (3+5 x)^2}+\frac {154}{3+5 x}-707 \log (2+3 x)+707 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 1.04 \begin {gather*} \frac {49}{3 x+2}+\frac {154}{5 x+3}-\frac {121}{10 (5 x+3)^2}-707 \log (5 (3 x+2))+707 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

49/(2 + 3*x) - 121/(10*(3 + 5*x)^2) + 154/(3 + 5*x) - 707*Log[5*(2 + 3*x)] + 707*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)^2 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.13, size = 75, normalized size = 1.63 \begin {gather*} \frac {35350 \, x^{2} + 7070 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (5 \, x + 3\right ) - 7070 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (3 \, x + 2\right ) + 43597 \, x + 13408}{10 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*(35350*x^2 + 7070*(75*x^3 + 140*x^2 + 87*x + 18)*log(5*x + 3) - 7070*(75*x^3 + 140*x^2 + 87*x + 18)*log(3
*x + 2) + 43597*x + 13408)/(75*x^3 + 140*x^2 + 87*x + 18)

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giac [A]  time = 0.96, size = 49, normalized size = 1.07 \begin {gather*} \frac {49}{3 \, x + 2} - \frac {33 \, {\left (\frac {206}{3 \, x + 2} - 865\right )}}{2 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + 707 \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/(3*x + 2) - 33/2*(206/(3*x + 2) - 865)/(1/(3*x + 2) - 5)^2 + 707*log(abs(-1/(3*x + 2) + 5))

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maple [A]  time = 0.01, size = 45, normalized size = 0.98 \begin {gather*} -707 \ln \left (3 x +2\right )+707 \ln \left (5 x +3\right )+\frac {49}{3 x +2}-\frac {121}{10 \left (5 x +3\right )^{2}}+\frac {154}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/(3*x+2)-121/10/(5*x+3)^2+154/(5*x+3)-707*ln(3*x+2)+707*ln(5*x+3)

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maxima [A]  time = 0.49, size = 46, normalized size = 1.00 \begin {gather*} \frac {35350 \, x^{2} + 43597 \, x + 13408}{10 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} + 707 \, \log \left (5 \, x + 3\right ) - 707 \, \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)^2/(3+5*x)^3,x, algorithm="maxima")

[Out]

1/10*(35350*x^2 + 43597*x + 13408)/(75*x^3 + 140*x^2 + 87*x + 18) + 707*log(5*x + 3) - 707*log(3*x + 2)

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mupad [B]  time = 1.09, size = 35, normalized size = 0.76 \begin {gather*} \frac {\frac {707\,x^2}{15}+\frac {43597\,x}{750}+\frac {6704}{375}}{x^3+\frac {28\,x^2}{15}+\frac {29\,x}{25}+\frac {6}{25}}-1414\,\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)^2*(5*x + 3)^3),x)

[Out]

((43597*x)/750 + (707*x^2)/15 + 6704/375)/((29*x)/25 + (28*x^2)/15 + x^3 + 6/25) - 1414*atanh(30*x + 19)

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sympy [A]  time = 0.16, size = 41, normalized size = 0.89 \begin {gather*} \frac {35350 x^{2} + 43597 x + 13408}{750 x^{3} + 1400 x^{2} + 870 x + 180} + 707 \log {\left (x + \frac {3}{5} \right )} - 707 \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)**2/(3+5*x)**3,x)

[Out]

(35350*x**2 + 43597*x + 13408)/(750*x**3 + 1400*x**2 + 870*x + 180) + 707*log(x + 3/5) - 707*log(x + 2/3)

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