3.1506 \(\int \frac{(2+3 x)^3}{(1-2 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=37 \[ -\frac{27 x}{50}-\frac{1}{1375 (5 x+3)}-\frac{343}{484} \log (1-2 x)+\frac{101 \log (5 x+3)}{15125} \]

[Out]

(-27*x)/50 - 1/(1375*(3 + 5*x)) - (343*Log[1 - 2*x])/484 + (101*Log[3 + 5*x])/15125

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Rubi [A]  time = 0.0153583, antiderivative size = 37, 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} \[ -\frac{27 x}{50}-\frac{1}{1375 (5 x+3)}-\frac{343}{484} \log (1-2 x)+\frac{101 \log (5 x+3)}{15125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-27*x)/50 - 1/(1375*(3 + 5*x)) - (343*Log[1 - 2*x])/484 + (101*Log[3 + 5*x])/15125

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)^3}{(1-2 x) (3+5 x)^2} \, dx &=\int \left (-\frac{27}{50}-\frac{343}{242 (-1+2 x)}+\frac{1}{275 (3+5 x)^2}+\frac{101}{3025 (3+5 x)}\right ) \, dx\\ &=-\frac{27 x}{50}-\frac{1}{1375 (3+5 x)}-\frac{343}{484} \log (1-2 x)+\frac{101 \log (3+5 x)}{15125}\\ \end{align*}

Mathematica [A]  time = 0.0210582, size = 37, normalized size = 1. \[ \frac{16335 (1-2 x)-\frac{44}{5 x+3}-42875 \log (1-2 x)+404 \log (10 x+6)}{60500} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(16335*(1 - 2*x) - 44/(3 + 5*x) - 42875*Log[1 - 2*x] + 404*Log[6 + 10*x])/60500

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Maple [A]  time = 0.006, size = 30, normalized size = 0.8 \begin{align*} -{\frac{27\,x}{50}}-{\frac{343\,\ln \left ( 2\,x-1 \right ) }{484}}-{\frac{1}{4125+6875\,x}}+{\frac{101\,\ln \left ( 3+5\,x \right ) }{15125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/50*x-343/484*ln(2*x-1)-1/1375/(3+5*x)+101/15125*ln(3+5*x)

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Maxima [A]  time = 2.45823, size = 39, normalized size = 1.05 \begin{align*} -\frac{27}{50} \, x - \frac{1}{1375 \,{\left (5 \, x + 3\right )}} + \frac{101}{15125} \, \log \left (5 \, x + 3\right ) - \frac{343}{484} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/50*x - 1/1375/(5*x + 3) + 101/15125*log(5*x + 3) - 343/484*log(2*x - 1)

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Fricas [A]  time = 1.56202, size = 146, normalized size = 3.95 \begin{align*} -\frac{163350 \, x^{2} - 404 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 42875 \,{\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 98010 \, x + 44}{60500 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/60500*(163350*x^2 - 404*(5*x + 3)*log(5*x + 3) + 42875*(5*x + 3)*log(2*x - 1) + 98010*x + 44)/(5*x + 3)

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Sympy [A]  time = 0.139723, size = 31, normalized size = 0.84 \begin{align*} - \frac{27 x}{50} - \frac{343 \log{\left (x - \frac{1}{2} \right )}}{484} + \frac{101 \log{\left (x + \frac{3}{5} \right )}}{15125} - \frac{1}{6875 x + 4125} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27*x/50 - 343*log(x - 1/2)/484 + 101*log(x + 3/5)/15125 - 1/(6875*x + 4125)

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Giac [A]  time = 1.38479, size = 63, normalized size = 1.7 \begin{align*} -\frac{27}{50} \, x - \frac{1}{1375 \,{\left (5 \, x + 3\right )}} + \frac{351}{500} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) - \frac{343}{484} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) - \frac{81}{250} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/50*x - 1/1375/(5*x + 3) + 351/500*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 343/484*log(abs(-11/(5*x + 3) + 2))
- 81/250