3.1504 \(\int \frac {(2+3 x)^5}{(1-2 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=51 \[ -\frac {81 x^3}{50}-\frac {6399 x^2}{1000}-\frac {69039 x}{5000}-\frac {1}{34375 (5 x+3)}-\frac {16807 \log (1-2 x)}{1936}+\frac {167 \log (5 x+3)}{378125} \]

[Out]

-69039/5000*x-6399/1000*x^2-81/50*x^3-1/34375/(3+5*x)-16807/1936*ln(1-2*x)+167/378125*ln(3+5*x)

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Rubi [A]  time = 0.02, antiderivative size = 51, 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} \[ -\frac {81 x^3}{50}-\frac {6399 x^2}{1000}-\frac {69039 x}{5000}-\frac {1}{34375 (5 x+3)}-\frac {16807 \log (1-2 x)}{1936}+\frac {167 \log (5 x+3)}{378125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-69039*x)/5000 - (6399*x^2)/1000 - (81*x^3)/50 - 1/(34375*(3 + 5*x)) - (16807*Log[1 - 2*x])/1936 + (167*Log[3
+ 5*x])/378125

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)^5}{(1-2 x) (3+5 x)^2} \, dx &=\int \left (-\frac {69039}{5000}-\frac {6399 x}{500}-\frac {243 x^2}{50}-\frac {16807}{968 (-1+2 x)}+\frac {1}{6875 (3+5 x)^2}+\frac {167}{75625 (3+5 x)}\right ) \, dx\\ &=-\frac {69039 x}{5000}-\frac {6399 x^2}{1000}-\frac {81 x^3}{50}-\frac {1}{34375 (3+5 x)}-\frac {16807 \log (1-2 x)}{1936}+\frac {167 \log (3+5 x)}{378125}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 50, normalized size = 0.98 \[ \frac {-\frac {11 \left (8910000 x^4+40540500 x^3+97059600 x^2-2318085 x-28730263\right )}{5 x+3}-105043750 \log (1-2 x)+5344 \log (10 x+6)}{12100000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-11*(-28730263 - 2318085*x + 97059600*x^2 + 40540500*x^3 + 8910000*x^4))/(3 + 5*x) - 105043750*Log[1 - 2*x]
+ 5344*Log[6 + 10*x])/12100000

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fricas [A]  time = 0.78, size = 55, normalized size = 1.08 \[ -\frac {49005000 \, x^{4} + 222972750 \, x^{3} + 533827800 \, x^{2} - 2672 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 52521875 \, {\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 250611570 \, x + 176}{6050000 \, {\left (5 \, x + 3\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6050000*(49005000*x^4 + 222972750*x^3 + 533827800*x^2 - 2672*(5*x + 3)*log(5*x + 3) + 52521875*(5*x + 3)*lo
g(2*x - 1) + 250611570*x + 176)/(5*x + 3)

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giac [A]  time = 0.95, size = 72, normalized size = 1.41 \[ -\frac {27}{25000} \, {\left (5 \, x + 3\right )}^{3} {\left (\frac {129}{5 \, x + 3} + \frac {1459}{{\left (5 \, x + 3\right )}^{2}} + 12\right )} - \frac {1}{34375 \, {\left (5 \, x + 3\right )}} + \frac {434043}{50000} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) - \frac {16807}{1936} \, \log \left ({\left | -\frac {11}{5 \, x + 3} + 2 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-27/25000*(5*x + 3)^3*(129/(5*x + 3) + 1459/(5*x + 3)^2 + 12) - 1/34375/(5*x + 3) + 434043/50000*log(1/5*abs(5
*x + 3)/(5*x + 3)^2) - 16807/1936*log(abs(-11/(5*x + 3) + 2))

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maple [A]  time = 0.01, size = 40, normalized size = 0.78 \[ -\frac {81 x^{3}}{50}-\frac {6399 x^{2}}{1000}-\frac {69039 x}{5000}-\frac {16807 \ln \left (2 x -1\right )}{1936}+\frac {167 \ln \left (5 x +3\right )}{378125}-\frac {1}{34375 \left (5 x +3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/50*x^3-6399/1000*x^2-69039/5000*x-1/34375/(5*x+3)+167/378125*ln(5*x+3)-16807/1936*ln(2*x-1)

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maxima [A]  time = 0.55, size = 39, normalized size = 0.76 \[ -\frac {81}{50} \, x^{3} - \frac {6399}{1000} \, x^{2} - \frac {69039}{5000} \, x - \frac {1}{34375 \, {\left (5 \, x + 3\right )}} + \frac {167}{378125} \, \log \left (5 \, x + 3\right ) - \frac {16807}{1936} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/50*x^3 - 6399/1000*x^2 - 69039/5000*x - 1/34375/(5*x + 3) + 167/378125*log(5*x + 3) - 16807/1936*log(2*x -
1)

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mupad [B]  time = 0.04, size = 35, normalized size = 0.69 \[ \frac {167\,\ln \left (x+\frac {3}{5}\right )}{378125}-\frac {16807\,\ln \left (x-\frac {1}{2}\right )}{1936}-\frac {69039\,x}{5000}-\frac {1}{171875\,\left (x+\frac {3}{5}\right )}-\frac {6399\,x^2}{1000}-\frac {81\,x^3}{50} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(167*log(x + 3/5))/378125 - (16807*log(x - 1/2))/1936 - (69039*x)/5000 - 1/(171875*(x + 3/5)) - (6399*x^2)/100
0 - (81*x^3)/50

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sympy [A]  time = 0.16, size = 44, normalized size = 0.86 \[ - \frac {81 x^{3}}{50} - \frac {6399 x^{2}}{1000} - \frac {69039 x}{5000} - \frac {16807 \log {\left (x - \frac {1}{2} \right )}}{1936} + \frac {167 \log {\left (x + \frac {3}{5} \right )}}{378125} - \frac {1}{171875 x + 103125} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81*x**3/50 - 6399*x**2/1000 - 69039*x/5000 - 16807*log(x - 1/2)/1936 + 167*log(x + 3/5)/378125 - 1/(171875*x
+ 103125)

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