3.1582 \(\int \frac{(3+5 x)^3}{(1-2 x)^2 (2+3 x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{1331}{686 (1-2 x)}-\frac{101}{3087 (3 x+2)}+\frac{1}{882 (3 x+2)^2}+\frac{363 \log (1-2 x)}{2401}-\frac{363 \log (3 x+2)}{2401} \]

[Out]

1331/(686*(1 - 2*x)) + 1/(882*(2 + 3*x)^2) - 101/(3087*(2 + 3*x)) + (363*Log[1 - 2*x])/2401 - (363*Log[2 + 3*x
])/2401

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Rubi [A]  time = 0.0233992, 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} \[ \frac{1331}{686 (1-2 x)}-\frac{101}{3087 (3 x+2)}+\frac{1}{882 (3 x+2)^2}+\frac{363 \log (1-2 x)}{2401}-\frac{363 \log (3 x+2)}{2401} \]

Antiderivative was successfully verified.

[In]

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

[Out]

1331/(686*(1 - 2*x)) + 1/(882*(2 + 3*x)^2) - 101/(3087*(2 + 3*x)) + (363*Log[1 - 2*x])/2401 - (363*Log[2 + 3*x
])/2401

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{(3+5 x)^3}{(1-2 x)^2 (2+3 x)^3} \, dx &=\int \left (\frac{1331}{343 (-1+2 x)^2}+\frac{726}{2401 (-1+2 x)}-\frac{1}{147 (2+3 x)^3}+\frac{101}{1029 (2+3 x)^2}-\frac{1089}{2401 (2+3 x)}\right ) \, dx\\ &=\frac{1331}{686 (1-2 x)}+\frac{1}{882 (2+3 x)^2}-\frac{101}{3087 (2+3 x)}+\frac{363 \log (1-2 x)}{2401}-\frac{363 \log (2+3 x)}{2401}\\ \end{align*}

Mathematica [A]  time = 0.0294321, size = 48, normalized size = 0.89 \[ \frac{\frac{83853}{1-2 x}-\frac{1414}{3 x+2}+\frac{49}{(3 x+2)^2}+6534 \log (1-2 x)-6534 \log (6 x+4)}{43218} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(83853/(1 - 2*x) + 49/(2 + 3*x)^2 - 1414/(2 + 3*x) + 6534*Log[1 - 2*x] - 6534*Log[4 + 6*x])/43218

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Maple [A]  time = 0.007, size = 45, normalized size = 0.8 \begin{align*} -{\frac{1331}{1372\,x-686}}+{\frac{363\,\ln \left ( 2\,x-1 \right ) }{2401}}+{\frac{1}{882\, \left ( 2+3\,x \right ) ^{2}}}-{\frac{101}{6174+9261\,x}}-{\frac{363\,\ln \left ( 2+3\,x \right ) }{2401}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1331/686/(2*x-1)+363/2401*ln(2*x-1)+1/882/(2+3*x)^2-101/3087/(2+3*x)-363/2401*ln(2+3*x)

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Maxima [A]  time = 1.10034, size = 62, normalized size = 1.15 \begin{align*} -\frac{109023 \, x^{2} + 143936 \, x + 47519}{6174 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} - \frac{363}{2401} \, \log \left (3 \, x + 2\right ) + \frac{363}{2401} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6174*(109023*x^2 + 143936*x + 47519)/(18*x^3 + 15*x^2 - 4*x - 4) - 363/2401*log(3*x + 2) + 363/2401*log(2*x
- 1)

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Fricas [A]  time = 1.46515, size = 227, normalized size = 4.2 \begin{align*} -\frac{763161 \, x^{2} + 6534 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (3 \, x + 2\right ) - 6534 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (2 \, x - 1\right ) + 1007552 \, x + 332633}{43218 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/43218*(763161*x^2 + 6534*(18*x^3 + 15*x^2 - 4*x - 4)*log(3*x + 2) - 6534*(18*x^3 + 15*x^2 - 4*x - 4)*log(2*
x - 1) + 1007552*x + 332633)/(18*x^3 + 15*x^2 - 4*x - 4)

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Sympy [A]  time = 0.153694, size = 44, normalized size = 0.81 \begin{align*} - \frac{109023 x^{2} + 143936 x + 47519}{111132 x^{3} + 92610 x^{2} - 24696 x - 24696} + \frac{363 \log{\left (x - \frac{1}{2} \right )}}{2401} - \frac{363 \log{\left (x + \frac{2}{3} \right )}}{2401} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(109023*x**2 + 143936*x + 47519)/(111132*x**3 + 92610*x**2 - 24696*x - 24696) + 363*log(x - 1/2)/2401 - 363*l
og(x + 2/3)/2401

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Giac [A]  time = 3.12242, size = 69, normalized size = 1.28 \begin{align*} -\frac{1331}{686 \,{\left (2 \, x - 1\right )}} + \frac{2 \,{\left (\frac{231}{2 \, x - 1} + 100\right )}}{2401 \,{\left (\frac{7}{2 \, x - 1} + 3\right )}^{2}} - \frac{363}{2401} \, \log \left ({\left | -\frac{7}{2 \, x - 1} - 3 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1331/686/(2*x - 1) + 2/2401*(231/(2*x - 1) + 100)/(7/(2*x - 1) + 3)^2 - 363/2401*log(abs(-7/(2*x - 1) - 3))