3.1462 \(\int \frac{(3+5 x)^2}{(1-2 x) (2+3 x)^3} \, dx\)

Optimal. Leaf size=43 \[ \frac{68}{441 (3 x+2)}-\frac{1}{126 (3 x+2)^2}-\frac{121}{343} \log (1-2 x)+\frac{121}{343} \log (3 x+2) \]

[Out]

-1/(126*(2 + 3*x)^2) + 68/(441*(2 + 3*x)) - (121*Log[1 - 2*x])/343 + (121*Log[2 + 3*x])/343

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Rubi [A]  time = 0.0191789, 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} \[ \frac{68}{441 (3 x+2)}-\frac{1}{126 (3 x+2)^2}-\frac{121}{343} \log (1-2 x)+\frac{121}{343} \log (3 x+2) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/(126*(2 + 3*x)^2) + 68/(441*(2 + 3*x)) - (121*Log[1 - 2*x])/343 + (121*Log[2 + 3*x])/343

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+3 x)^3} \, dx &=\int \left (-\frac{242}{343 (-1+2 x)}+\frac{1}{21 (2+3 x)^3}-\frac{68}{147 (2+3 x)^2}+\frac{363}{343 (2+3 x)}\right ) \, dx\\ &=-\frac{1}{126 (2+3 x)^2}+\frac{68}{441 (2+3 x)}-\frac{121}{343} \log (1-2 x)+\frac{121}{343} \log (2+3 x)\\ \end{align*}

Mathematica [A]  time = 0.0187307, size = 35, normalized size = 0.81 \[ \frac{\frac{7 (408 x+265)}{(3 x+2)^2}-2178 \log (1-2 x)+2178 \log (6 x+4)}{6174} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((7*(265 + 408*x))/(2 + 3*x)^2 - 2178*Log[1 - 2*x] + 2178*Log[4 + 6*x])/6174

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Maple [A]  time = 0.006, size = 36, normalized size = 0.8 \begin{align*} -{\frac{121\,\ln \left ( 2\,x-1 \right ) }{343}}-{\frac{1}{126\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{68}{882+1323\,x}}+{\frac{121\,\ln \left ( 2+3\,x \right ) }{343}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-121/343*ln(2*x-1)-1/126/(2+3*x)^2+68/441/(2+3*x)+121/343*ln(2+3*x)

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Maxima [A]  time = 1.10738, size = 49, normalized size = 1.14 \begin{align*} \frac{408 \, x + 265}{882 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{121}{343} \, \log \left (3 \, x + 2\right ) - \frac{121}{343} \, \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+3*x)^3,x, algorithm="maxima")

[Out]

1/882*(408*x + 265)/(9*x^2 + 12*x + 4) + 121/343*log(3*x + 2) - 121/343*log(2*x - 1)

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Fricas [A]  time = 1.25566, size = 163, normalized size = 3.79 \begin{align*} \frac{2178 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) - 2178 \,{\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (2 \, x - 1\right ) + 2856 \, x + 1855}{6174 \,{\left (9 \, x^{2} + 12 \, 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+3*x)^3,x, algorithm="fricas")

[Out]

1/6174*(2178*(9*x^2 + 12*x + 4)*log(3*x + 2) - 2178*(9*x^2 + 12*x + 4)*log(2*x - 1) + 2856*x + 1855)/(9*x^2 +
12*x + 4)

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Sympy [A]  time = 0.143161, size = 34, normalized size = 0.79 \begin{align*} \frac{408 x + 265}{7938 x^{2} + 10584 x + 3528} - \frac{121 \log{\left (x - \frac{1}{2} \right )}}{343} + \frac{121 \log{\left (x + \frac{2}{3} \right )}}{343} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(408*x + 265)/(7938*x**2 + 10584*x + 3528) - 121*log(x - 1/2)/343 + 121*log(x + 2/3)/343

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Giac [A]  time = 1.27075, size = 45, normalized size = 1.05 \begin{align*} \frac{408 \, x + 265}{882 \,{\left (3 \, x + 2\right )}^{2}} + \frac{121}{343} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) - \frac{121}{343} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/882*(408*x + 265)/(3*x + 2)^2 + 121/343*log(abs(3*x + 2)) - 121/343*log(abs(2*x - 1))