3.1609 \(\int \frac {(2+3 x)^2}{(1-2 x)^2 (3+5 x)^2} \, dx\)

Optimal. Leaf size=43 \[ \frac {49}{242 (1-2 x)}-\frac {1}{605 (5 x+3)}-\frac {14 \log (1-2 x)}{1331}+\frac {14 \log (5 x+3)}{1331} \]

[Out]

49/242/(1-2*x)-1/605/(3+5*x)-14/1331*ln(1-2*x)+14/1331*ln(3+5*x)

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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 = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {88} \[ \frac {49}{242 (1-2 x)}-\frac {1}{605 (5 x+3)}-\frac {14 \log (1-2 x)}{1331}+\frac {14 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(242*(1 - 2*x)) - 1/(605*(3 + 5*x)) - (14*Log[1 - 2*x])/1331 + (14*Log[3 + 5*x])/1331

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 {(2+3 x)^2}{(1-2 x)^2 (3+5 x)^2} \, dx &=\int \left (\frac {49}{121 (-1+2 x)^2}-\frac {28}{1331 (-1+2 x)}+\frac {1}{121 (3+5 x)^2}+\frac {70}{1331 (3+5 x)}\right ) \, dx\\ &=\frac {49}{242 (1-2 x)}-\frac {1}{605 (3+5 x)}-\frac {14 \log (1-2 x)}{1331}+\frac {14 \log (3+5 x)}{1331}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 38, normalized size = 0.88 \[ \frac {-\frac {11 (1229 x+733)}{10 x^2+x-3}+140 \log (-5 x-3)-140 \log (1-2 x)}{13310} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(733 + 1229*x))/(-3 + x + 10*x^2) + 140*Log[-3 - 5*x] - 140*Log[1 - 2*x])/13310

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fricas [A]  time = 0.70, size = 49, normalized size = 1.14 \[ \frac {140 \, {\left (10 \, x^{2} + x - 3\right )} \log \left (5 \, x + 3\right ) - 140 \, {\left (10 \, x^{2} + x - 3\right )} \log \left (2 \, x - 1\right ) - 13519 \, x - 8063}{13310 \, {\left (10 \, x^{2} + x - 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/13310*(140*(10*x^2 + x - 3)*log(5*x + 3) - 140*(10*x^2 + x - 3)*log(2*x - 1) - 13519*x - 8063)/(10*x^2 + x -
3)

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giac [A]  time = 1.20, size = 40, normalized size = 0.93 \[ -\frac {1}{605 \, {\left (5 \, x + 3\right )}} + \frac {245}{1331 \, {\left (\frac {11}{5 \, x + 3} - 2\right )}} - \frac {14}{1331} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/605/(5*x + 3) + 245/1331/(11/(5*x + 3) - 2) - 14/1331*log(abs(-11/(5*x + 3) + 2))

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maple [A]  time = 0.01, size = 36, normalized size = 0.84 \[ -\frac {14 \ln \left (2 x -1\right )}{1331}+\frac {14 \ln \left (5 x +3\right )}{1331}-\frac {1}{605 \left (5 x +3\right )}-\frac {49}{242 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/605/(5*x+3)+14/1331*ln(5*x+3)-49/242/(2*x-1)-14/1331*ln(2*x-1)

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maxima [A]  time = 0.64, size = 34, normalized size = 0.79 \[ -\frac {1229 \, x + 733}{1210 \, {\left (10 \, x^{2} + x - 3\right )}} + \frac {14}{1331} \, \log \left (5 \, x + 3\right ) - \frac {14}{1331} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1210*(1229*x + 733)/(10*x^2 + x - 3) + 14/1331*log(5*x + 3) - 14/1331*log(2*x - 1)

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mupad [B]  time = 0.04, size = 26, normalized size = 0.60 \[ \frac {28\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{1331}-\frac {\frac {1229\,x}{12100}+\frac {733}{12100}}{x^2+\frac {x}{10}-\frac {3}{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(28*atanh((20*x)/11 + 1/11))/1331 - ((1229*x)/12100 + 733/12100)/(x/10 + x^2 - 3/10)

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sympy [A]  time = 0.14, size = 36, normalized size = 0.84 \[ \frac {- 1229 x - 733}{12100 x^{2} + 1210 x - 3630} - \frac {14 \log {\left (x - \frac {1}{2} \right )}}{1331} + \frac {14 \log {\left (x + \frac {3}{5} \right )}}{1331} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1229*x - 733)/(12100*x**2 + 1210*x - 3630) - 14*log(x - 1/2)/1331 + 14*log(x + 3/5)/1331

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