3.1565 \(\int \frac{(3+5 x)^2}{(1-2 x)^2 (2+3 x)^3} \, dx\)

Optimal. Leaf size=54 \[ \frac{121}{343 (1-2 x)}+\frac{22}{343 (3 x+2)}-\frac{1}{294 (3 x+2)^2}-\frac{319 \log (1-2 x)}{2401}+\frac{319 \log (3 x+2)}{2401} \]

[Out]

121/(343*(1 - 2*x)) - 1/(294*(2 + 3*x)^2) + 22/(343*(2 + 3*x)) - (319*Log[1 - 2*x])/2401 + (319*Log[2 + 3*x])/
2401

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Rubi [A]  time = 0.0248051, 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} \[ \frac{121}{343 (1-2 x)}+\frac{22}{343 (3 x+2)}-\frac{1}{294 (3 x+2)^2}-\frac{319 \log (1-2 x)}{2401}+\frac{319 \log (3 x+2)}{2401} \]

Antiderivative was successfully verified.

[In]

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

[Out]

121/(343*(1 - 2*x)) - 1/(294*(2 + 3*x)^2) + 22/(343*(2 + 3*x)) - (319*Log[1 - 2*x])/2401 + (319*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)^2}{(1-2 x)^2 (2+3 x)^3} \, dx &=\int \left (\frac{242}{343 (-1+2 x)^2}-\frac{638}{2401 (-1+2 x)}+\frac{1}{49 (2+3 x)^3}-\frac{66}{343 (2+3 x)^2}+\frac{957}{2401 (2+3 x)}\right ) \, dx\\ &=\frac{121}{343 (1-2 x)}-\frac{1}{294 (2+3 x)^2}+\frac{22}{343 (2+3 x)}-\frac{319 \log (1-2 x)}{2401}+\frac{319 \log (2+3 x)}{2401}\\ \end{align*}

Mathematica [A]  time = 0.0260642, size = 47, normalized size = 0.87 \[ \frac{-\frac{7 \left (5742 x^2+8594 x+3161\right )}{(2 x-1) (3 x+2)^2}-1914 \log (1-2 x)+1914 \log (6 x+4)}{14406} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-7*(3161 + 8594*x + 5742*x^2))/((-1 + 2*x)*(2 + 3*x)^2) - 1914*Log[1 - 2*x] + 1914*Log[4 + 6*x])/14406

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Maple [A]  time = 0.01, size = 45, normalized size = 0.8 \begin{align*} -{\frac{121}{686\,x-343}}-{\frac{319\,\ln \left ( 2\,x-1 \right ) }{2401}}-{\frac{1}{294\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{22}{686+1029\,x}}+{\frac{319\,\ln \left ( 2+3\,x \right ) }{2401}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-121/343/(2*x-1)-319/2401*ln(2*x-1)-1/294/(2+3*x)^2+22/343/(2+3*x)+319/2401*ln(2+3*x)

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Maxima [A]  time = 1.44934, size = 62, normalized size = 1.15 \begin{align*} -\frac{5742 \, x^{2} + 8594 \, x + 3161}{2058 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )}} + \frac{319}{2401} \, \log \left (3 \, x + 2\right ) - \frac{319}{2401} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2058*(5742*x^2 + 8594*x + 3161)/(18*x^3 + 15*x^2 - 4*x - 4) + 319/2401*log(3*x + 2) - 319/2401*log(2*x - 1)

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Fricas [A]  time = 1.19211, size = 221, normalized size = 4.09 \begin{align*} -\frac{40194 \, x^{2} - 1914 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (3 \, x + 2\right ) + 1914 \,{\left (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\right )} \log \left (2 \, x - 1\right ) + 60158 \, x + 22127}{14406 \,{\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)^2/(1-2*x)^2/(2+3*x)^3,x, algorithm="fricas")

[Out]

-1/14406*(40194*x^2 - 1914*(18*x^3 + 15*x^2 - 4*x - 4)*log(3*x + 2) + 1914*(18*x^3 + 15*x^2 - 4*x - 4)*log(2*x
- 1) + 60158*x + 22127)/(18*x^3 + 15*x^2 - 4*x - 4)

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Sympy [A]  time = 0.149052, size = 44, normalized size = 0.81 \begin{align*} - \frac{5742 x^{2} + 8594 x + 3161}{37044 x^{3} + 30870 x^{2} - 8232 x - 8232} - \frac{319 \log{\left (x - \frac{1}{2} \right )}}{2401} + \frac{319 \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)**2/(1-2*x)**2/(2+3*x)**3,x)

[Out]

-(5742*x**2 + 8594*x + 3161)/(37044*x**3 + 30870*x**2 - 8232*x - 8232) - 319*log(x - 1/2)/2401 + 319*log(x + 2
/3)/2401

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Giac [A]  time = 2.55792, size = 69, normalized size = 1.28 \begin{align*} -\frac{121}{343 \,{\left (2 \, x - 1\right )}} - \frac{2 \,{\left (\frac{448}{2 \, x - 1} + 195\right )}}{2401 \,{\left (\frac{7}{2 \, x - 1} + 3\right )}^{2}} + \frac{319}{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)^2/(1-2*x)^2/(2+3*x)^3,x, algorithm="giac")

[Out]

-121/343/(2*x - 1) - 2/2401*(448/(2*x - 1) + 195)/(7/(2*x - 1) + 3)^2 + 319/2401*log(abs(-7/(2*x - 1) - 3))