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

Optimal. Leaf size=43 \[ \frac {7}{121 (1-2 x)}-\frac {1}{121 (5 x+3)}-\frac {37 \log (1-2 x)}{1331}+\frac {37 \log (5 x+3)}{1331} \]

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \begin {gather*} \frac {7}{121 (1-2 x)}-\frac {1}{121 (5 x+3)}-\frac {37 \log (1-2 x)}{1331}+\frac {37 \log (5 x+3)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

7/(121*(1 - 2*x)) - 1/(121*(3 + 5*x)) - (37*Log[1 - 2*x])/1331 + (37*Log[3 + 5*x])/1331

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {2+3 x}{(1-2 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac {14}{121 (-1+2 x)^2}-\frac {74}{1331 (-1+2 x)}+\frac {5}{121 (3+5 x)^2}+\frac {185}{1331 (3+5 x)}\right ) \, dx\\ &=\frac {7}{121 (1-2 x)}-\frac {1}{121 (3+5 x)}-\frac {37 \log (1-2 x)}{1331}+\frac {37 \log (3+5 x)}{1331}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 40, normalized size = 0.93 \begin {gather*} \frac {-37 x-20}{121 \left (10 x^2+x-3\right )}-\frac {37 \log (1-2 x)}{1331}+\frac {37 \log (5 x+3)}{1331} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-20 - 37*x)/(121*(-3 + x + 10*x^2)) - (37*Log[1 - 2*x])/1331 + (37*Log[3 + 5*x])/1331

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.50, size = 49, normalized size = 1.14 \begin {gather*} \frac {37 \, {\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) - 37 \, {\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 407 \, x - 220}{1331 \, {\left (10 \, x^{2} + x - 3\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1331*(37*(10*x^2 + x - 3)*log(5*x + 3) - 37*(10*x^2 + x - 3)*log(2*x - 1) - 407*x - 220)/(10*x^2 + x - 3)

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giac [A]  time = 1.19, size = 40, normalized size = 0.93 \begin {gather*} -\frac {1}{121 \, {\left (5 \, x + 3\right )}} + \frac {70}{1331 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}} - \frac {37}{1331} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/121/(5*x + 3) + 70/1331/(11/(5*x + 3) - 2) - 37/1331*log(abs(-11/(5*x + 3) + 2))

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maple [A]  time = 0.01, size = 36, normalized size = 0.84 \begin {gather*} -\frac {37 \ln \left (2 x -1\right )}{1331}+\frac {37 \ln \left (5 x +3\right )}{1331}-\frac {1}{121 \left (5 x +3\right )}-\frac {7}{121 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/121/(5*x+3)+37/1331*ln(5*x+3)-7/121/(2*x-1)-37/1331*ln(2*x-1)

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maxima [A]  time = 0.64, size = 34, normalized size = 0.79 \begin {gather*} -\frac {37 \, x + 20}{121 \, {\left (10 \, x^{2} + x - 3\right )}} + \frac {37}{1331} \, \log \left (5 \, x + 3\right ) - \frac {37}{1331} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/121*(37*x + 20)/(10*x^2 + x - 3) + 37/1331*log(5*x + 3) - 37/1331*log(2*x - 1)

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mupad [B]  time = 0.04, size = 26, normalized size = 0.60 \begin {gather*} \frac {74\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{1331}-\frac {\frac {37\,x}{1210}+\frac {2}{121}}{x^2+\frac {x}{10}-\frac {3}{10}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(74*atanh((20*x)/11 + 1/11))/1331 - ((37*x)/1210 + 2/121)/(x/10 + x^2 - 3/10)

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sympy [A]  time = 0.14, size = 36, normalized size = 0.84 \begin {gather*} \frac {- 37 x - 20}{1210 x^{2} + 121 x - 363} - \frac {37 \log {\left (x - \frac {1}{2} \right )}}{1331} + \frac {37 \log {\left (x + \frac {3}{5} \right )}}{1331} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-37*x - 20)/(1210*x**2 + 121*x - 363) - 37*log(x - 1/2)/1331 + 37*log(x + 3/5)/1331

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