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]

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

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Rubi [A]  time = 0.02, antiderivative size = 37, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.03, size = 37, normalized size = 1.00 \[ \frac {16335 (1-2 x)-\frac {44}{5 x+3}-42875 \log (1-2 x)+404 \log (10 x+6)}{60500} \]

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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fricas [A]  time = 1.02, size = 45, normalized size = 1.22 \[ -\frac {163350 \, x^{2} - 404 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 42875 \, {\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 98010 \, x + 44}{60500 \, {\left (5 \, x + 3\right )}} \]

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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giac [A]  time = 0.96, size = 47, normalized size = 1.27 \[ -\frac {27}{50} \, x - \frac {1}{1375 \, {\left (5 \, x + 3\right )}} + \frac {351}{500} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {343}{484} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) - \frac {81}{250} \]

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

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maple [A]  time = 0.01, size = 30, normalized size = 0.81 \[ -\frac {27 x}{50}-\frac {343 \ln \left (2 x -1\right )}{484}+\frac {101 \ln \left (5 x +3\right )}{15125}-\frac {1}{1375 \left (5 x +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.64, size = 29, normalized size = 0.78 \[ -\frac {27}{50} \, x - \frac {1}{1375 \, {\left (5 \, x + 3\right )}} + \frac {101}{15125} \, \log \left (5 \, x + 3\right ) - \frac {343}{484} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 1.12, size = 25, normalized size = 0.68 \[ \frac {101\,\ln \left (x+\frac {3}{5}\right )}{15125}-\frac {343\,\ln \left (x-\frac {1}{2}\right )}{484}-\frac {27\,x}{50}-\frac {1}{6875\,\left (x+\frac {3}{5}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.15, size = 31, normalized size = 0.84 \[ - \frac {27 x}{50} - \frac {343 \log {\left (x - \frac {1}{2} \right )}}{484} + \frac {101 \log {\left (x + \frac {3}{5} \right )}}{15125} - \frac {1}{6875 x + 4125} \]

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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