3.1623 \(\int \frac{(2+3 x)^2}{(1-2 x)^2 (3+5 x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{49}{1331 (1-2 x)}-\frac{14}{1331 (5 x+3)}-\frac{1}{1210 (5 x+3)^2}-\frac{273 \log (1-2 x)}{14641}+\frac{273 \log (5 x+3)}{14641} \]

[Out]

49/(1331*(1 - 2*x)) - 1/(1210*(3 + 5*x)^2) - 14/(1331*(3 + 5*x)) - (273*Log[1 - 2*x])/14641 + (273*Log[3 + 5*x
])/14641

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Rubi [A]  time = 0.0232598, antiderivative size = 54, 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} \[ \frac{49}{1331 (1-2 x)}-\frac{14}{1331 (5 x+3)}-\frac{1}{1210 (5 x+3)^2}-\frac{273 \log (1-2 x)}{14641}+\frac{273 \log (5 x+3)}{14641} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(1331*(1 - 2*x)) - 1/(1210*(3 + 5*x)^2) - 14/(1331*(3 + 5*x)) - (273*Log[1 - 2*x])/14641 + (273*Log[3 + 5*x
])/14641

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)^3} \, dx &=\int \left (\frac{98}{1331 (-1+2 x)^2}-\frac{546}{14641 (-1+2 x)}+\frac{1}{121 (3+5 x)^3}+\frac{70}{1331 (3+5 x)^2}+\frac{1365}{14641 (3+5 x)}\right ) \, dx\\ &=\frac{49}{1331 (1-2 x)}-\frac{1}{1210 (3+5 x)^2}-\frac{14}{1331 (3+5 x)}-\frac{273 \log (1-2 x)}{14641}+\frac{273 \log (3+5 x)}{14641}\\ \end{align*}

Mathematica [A]  time = 0.0258601, size = 47, normalized size = 0.87 \[ \frac{-\frac{11 \left (13650 x^2+14862 x+3979\right )}{(2 x-1) (5 x+3)^2}-2730 \log (1-2 x)+2730 \log (10 x+6)}{146410} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(3979 + 14862*x + 13650*x^2))/((-1 + 2*x)*(3 + 5*x)^2) - 2730*Log[1 - 2*x] + 2730*Log[6 + 10*x])/146410

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Maple [A]  time = 0.007, size = 45, normalized size = 0.8 \begin{align*} -{\frac{49}{2662\,x-1331}}-{\frac{273\,\ln \left ( 2\,x-1 \right ) }{14641}}-{\frac{1}{1210\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{14}{3993+6655\,x}}+{\frac{273\,\ln \left ( 3+5\,x \right ) }{14641}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/1331/(2*x-1)-273/14641*ln(2*x-1)-1/1210/(3+5*x)^2-14/1331/(3+5*x)+273/14641*ln(3+5*x)

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Maxima [A]  time = 1.02256, size = 62, normalized size = 1.15 \begin{align*} -\frac{13650 \, x^{2} + 14862 \, x + 3979}{13310 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} + \frac{273}{14641} \, \log \left (5 \, x + 3\right ) - \frac{273}{14641} \, \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)^2/(3+5*x)^3,x, algorithm="maxima")

[Out]

-1/13310*(13650*x^2 + 14862*x + 3979)/(50*x^3 + 35*x^2 - 12*x - 9) + 273/14641*log(5*x + 3) - 273/14641*log(2*
x - 1)

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Fricas [A]  time = 1.29238, size = 230, normalized size = 4.26 \begin{align*} -\frac{150150 \, x^{2} - 2730 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (5 \, x + 3\right ) + 2730 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )} \log \left (2 \, x - 1\right ) + 163482 \, x + 43769}{146410 \,{\left (50 \, x^{3} + 35 \, x^{2} - 12 \, x - 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/146410*(150150*x^2 - 2730*(50*x^3 + 35*x^2 - 12*x - 9)*log(5*x + 3) + 2730*(50*x^3 + 35*x^2 - 12*x - 9)*log
(2*x - 1) + 163482*x + 43769)/(50*x^3 + 35*x^2 - 12*x - 9)

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Sympy [A]  time = 0.150312, size = 44, normalized size = 0.81 \begin{align*} - \frac{13650 x^{2} + 14862 x + 3979}{665500 x^{3} + 465850 x^{2} - 159720 x - 119790} - \frac{273 \log{\left (x - \frac{1}{2} \right )}}{14641} + \frac{273 \log{\left (x + \frac{3}{5} \right )}}{14641} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(13650*x**2 + 14862*x + 3979)/(665500*x**3 + 465850*x**2 - 159720*x - 119790) - 273*log(x - 1/2)/14641 + 273*
log(x + 3/5)/14641

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Giac [A]  time = 2.49705, size = 69, normalized size = 1.28 \begin{align*} -\frac{49}{1331 \,{\left (2 \, x - 1\right )}} + \frac{2 \,{\left (\frac{792}{2 \, x - 1} + 355\right )}}{14641 \,{\left (\frac{11}{2 \, x - 1} + 5\right )}^{2}} + \frac{273}{14641} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-49/1331/(2*x - 1) + 2/14641*(792/(2*x - 1) + 355)/(11/(2*x - 1) + 5)^2 + 273/14641*log(abs(-11/(2*x - 1) - 5)
)