3.1677 \(\int \frac{(2+3 x)^2}{(1-2 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=43 \[ -\frac{217}{484 (1-2 x)}+\frac{49}{88 (1-2 x)^2}-\frac{\log (1-2 x)}{1331}+\frac{\log (5 x+3)}{1331} \]

[Out]

49/(88*(1 - 2*x)^2) - 217/(484*(1 - 2*x)) - Log[1 - 2*x]/1331 + Log[3 + 5*x]/1331

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Rubi [A]  time = 0.0178942, antiderivative size = 43, normalized size of antiderivative = 1., 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} \[ -\frac{217}{484 (1-2 x)}+\frac{49}{88 (1-2 x)^2}-\frac{\log (1-2 x)}{1331}+\frac{\log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(88*(1 - 2*x)^2) - 217/(484*(1 - 2*x)) - Log[1 - 2*x]/1331 + 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)^3 (3+5 x)} \, dx &=\int \left (-\frac{49}{22 (-1+2 x)^3}-\frac{217}{242 (-1+2 x)^2}-\frac{2}{1331 (-1+2 x)}+\frac{5}{1331 (3+5 x)}\right ) \, dx\\ &=\frac{49}{88 (1-2 x)^2}-\frac{217}{484 (1-2 x)}-\frac{\log (1-2 x)}{1331}+\frac{\log (3+5 x)}{1331}\\ \end{align*}

Mathematica [A]  time = 0.0206069, size = 35, normalized size = 0.81 \[ \frac{\frac{77 (124 x+15)}{(1-2 x)^2}-8 \log (5-10 x)+8 \log (5 x+3)}{10648} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((77*(15 + 124*x))/(1 - 2*x)^2 - 8*Log[5 - 10*x] + 8*Log[3 + 5*x])/10648

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*}{\frac{49}{88\, \left ( 2\,x-1 \right ) ^{2}}}+{\frac{217}{968\,x-484}}-{\frac{\ln \left ( 2\,x-1 \right ) }{1331}}+{\frac{\ln \left ( 3+5\,x \right ) }{1331}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/88/(2*x-1)^2+217/484/(2*x-1)-1/1331*ln(2*x-1)+1/1331*ln(3+5*x)

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Maxima [A]  time = 1.01137, size = 49, normalized size = 1.14 \begin{align*} \frac{7 \,{\left (124 \, x + 15\right )}}{968 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac{1}{1331} \, \log \left (5 \, x + 3\right ) - \frac{1}{1331} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/968*(124*x + 15)/(4*x^2 - 4*x + 1) + 1/1331*log(5*x + 3) - 1/1331*log(2*x - 1)

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Fricas [A]  time = 1.57349, size = 153, normalized size = 3.56 \begin{align*} \frac{8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) - 8 \,{\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) + 9548 \, x + 1155}{10648 \,{\left (4 \, x^{2} - 4 \, x + 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10648*(8*(4*x^2 - 4*x + 1)*log(5*x + 3) - 8*(4*x^2 - 4*x + 1)*log(2*x - 1) + 9548*x + 1155)/(4*x^2 - 4*x + 1
)

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Sympy [A]  time = 0.137814, size = 31, normalized size = 0.72 \begin{align*} \frac{868 x + 105}{3872 x^{2} - 3872 x + 968} - \frac{\log{\left (x - \frac{1}{2} \right )}}{1331} + \frac{\log{\left (x + \frac{3}{5} \right )}}{1331} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(868*x + 105)/(3872*x**2 - 3872*x + 968) - log(x - 1/2)/1331 + log(x + 3/5)/1331

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Giac [A]  time = 1.84607, size = 45, normalized size = 1.05 \begin{align*} \frac{7 \,{\left (124 \, x + 15\right )}}{968 \,{\left (2 \, x - 1\right )}^{2}} + \frac{1}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{1}{1331} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

7/968*(124*x + 15)/(2*x - 1)^2 + 1/1331*log(abs(5*x + 3)) - 1/1331*log(abs(2*x - 1))