3.1704 \(\int \frac{(2+3 x)^2}{(1-2 x)^3 (3+5 x)^3} \, dx\)

Optimal. Leaf size=65 \[ \frac{273}{14641 (1-2 x)}-\frac{72}{14641 (5 x+3)}+\frac{49}{2662 (1-2 x)^2}-\frac{1}{2662 (5 x+3)^2}-\frac{1509 \log (1-2 x)}{161051}+\frac{1509 \log (5 x+3)}{161051} \]

[Out]

49/(2662*(1 - 2*x)^2) + 273/(14641*(1 - 2*x)) - 1/(2662*(3 + 5*x)^2) - 72/(14641*(3 + 5*x)) - (1509*Log[1 - 2*
x])/161051 + (1509*Log[3 + 5*x])/161051

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Rubi [A]  time = 0.0307276, antiderivative size = 65, 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} \[ \frac{273}{14641 (1-2 x)}-\frac{72}{14641 (5 x+3)}+\frac{49}{2662 (1-2 x)^2}-\frac{1}{2662 (5 x+3)^2}-\frac{1509 \log (1-2 x)}{161051}+\frac{1509 \log (5 x+3)}{161051} \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(2662*(1 - 2*x)^2) + 273/(14641*(1 - 2*x)) - 1/(2662*(3 + 5*x)^2) - 72/(14641*(3 + 5*x)) - (1509*Log[1 - 2*
x])/161051 + (1509*Log[3 + 5*x])/161051

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)^3} \, dx &=\int \left (-\frac{98}{1331 (-1+2 x)^3}+\frac{546}{14641 (-1+2 x)^2}-\frac{3018}{161051 (-1+2 x)}+\frac{5}{1331 (3+5 x)^3}+\frac{360}{14641 (3+5 x)^2}+\frac{7545}{161051 (3+5 x)}\right ) \, dx\\ &=\frac{49}{2662 (1-2 x)^2}+\frac{273}{14641 (1-2 x)}-\frac{1}{2662 (3+5 x)^2}-\frac{72}{14641 (3+5 x)}-\frac{1509 \log (1-2 x)}{161051}+\frac{1509 \log (3+5 x)}{161051}\\ \end{align*}

Mathematica [A]  time = 0.0280077, size = 48, normalized size = 0.74 \[ \frac{-\frac{11 \left (30180 x^3+4527 x^2-23774 x-9322\right )}{\left (10 x^2+x-3\right )^2}+3018 \log (-5 x-3)-3018 \log (1-2 x)}{322102} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(-9322 - 23774*x + 4527*x^2 + 30180*x^3))/(-3 + x + 10*x^2)^2 + 3018*Log[-3 - 5*x] - 3018*Log[1 - 2*x])/
322102

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Maple [A]  time = 0.008, size = 54, normalized size = 0.8 \begin{align*}{\frac{49}{2662\, \left ( 2\,x-1 \right ) ^{2}}}-{\frac{273}{29282\,x-14641}}-{\frac{1509\,\ln \left ( 2\,x-1 \right ) }{161051}}-{\frac{1}{2662\, \left ( 3+5\,x \right ) ^{2}}}-{\frac{72}{43923+73205\,x}}+{\frac{1509\,\ln \left ( 3+5\,x \right ) }{161051}} \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)^3,x)

[Out]

49/2662/(2*x-1)^2-273/14641/(2*x-1)-1509/161051*ln(2*x-1)-1/2662/(3+5*x)^2-72/14641/(3+5*x)+1509/161051*ln(3+5
*x)

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Maxima [A]  time = 2.36457, size = 76, normalized size = 1.17 \begin{align*} -\frac{30180 \, x^{3} + 4527 \, x^{2} - 23774 \, x - 9322}{29282 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac{1509}{161051} \, \log \left (5 \, x + 3\right ) - \frac{1509}{161051} \, \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)^3,x, algorithm="maxima")

[Out]

-1/29282*(30180*x^3 + 4527*x^2 - 23774*x - 9322)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 1509/161051*log(5*x +
3) - 1509/161051*log(2*x - 1)

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Fricas [A]  time = 1.54265, size = 284, normalized size = 4.37 \begin{align*} -\frac{331980 \, x^{3} + 49797 \, x^{2} - 3018 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 3018 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 261514 \, x - 102542}{322102 \,{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\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)^3,x, algorithm="fricas")

[Out]

-1/322102*(331980*x^3 + 49797*x^2 - 3018*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) + 3018*(100*x^4 +
20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) - 261514*x - 102542)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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Sympy [A]  time = 0.168781, size = 54, normalized size = 0.83 \begin{align*} - \frac{30180 x^{3} + 4527 x^{2} - 23774 x - 9322}{2928200 x^{4} + 585640 x^{3} - 1727638 x^{2} - 175692 x + 263538} - \frac{1509 \log{\left (x - \frac{1}{2} \right )}}{161051} + \frac{1509 \log{\left (x + \frac{3}{5} \right )}}{161051} \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)**3,x)

[Out]

-(30180*x**3 + 4527*x**2 - 23774*x - 9322)/(2928200*x**4 + 585640*x**3 - 1727638*x**2 - 175692*x + 263538) - 1
509*log(x - 1/2)/161051 + 1509*log(x + 3/5)/161051

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Giac [A]  time = 2.89683, size = 62, normalized size = 0.95 \begin{align*} -\frac{30180 \, x^{3} + 4527 \, x^{2} - 23774 \, x - 9322}{29282 \,{\left (10 \, x^{2} + x - 3\right )}^{2}} + \frac{1509}{161051} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac{1509}{161051} \, \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)^3,x, algorithm="giac")

[Out]

-1/29282*(30180*x^3 + 4527*x^2 - 23774*x - 9322)/(10*x^2 + x - 3)^2 + 1509/161051*log(abs(5*x + 3)) - 1509/161
051*log(abs(2*x - 1))