3.1505 \(\int \frac{(2+3 x)^4}{(1-2 x) (3+5 x)^2} \, dx\)

Optimal. Leaf size=44 \[ -\frac{81 x^2}{100}-\frac{1593 x}{500}-\frac{1}{6875 (5 x+3)}-\frac{2401}{968} \log (1-2 x)+\frac{134 \log (5 x+3)}{75625} \]

[Out]

(-1593*x)/500 - (81*x^2)/100 - 1/(6875*(3 + 5*x)) - (2401*Log[1 - 2*x])/968 + (134*Log[3 + 5*x])/75625

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Rubi [A]  time = 0.0195941, antiderivative size = 44, 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} \[ -\frac{81 x^2}{100}-\frac{1593 x}{500}-\frac{1}{6875 (5 x+3)}-\frac{2401}{968} \log (1-2 x)+\frac{134 \log (5 x+3)}{75625} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^4/((1 - 2*x)*(3 + 5*x)^2),x]

[Out]

(-1593*x)/500 - (81*x^2)/100 - 1/(6875*(3 + 5*x)) - (2401*Log[1 - 2*x])/968 + (134*Log[3 + 5*x])/75625

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)^4}{(1-2 x) (3+5 x)^2} \, dx &=\int \left (-\frac{1593}{500}-\frac{81 x}{50}-\frac{2401}{484 (-1+2 x)}+\frac{1}{1375 (3+5 x)^2}+\frac{134}{15125 (3+5 x)}\right ) \, dx\\ &=-\frac{1593 x}{500}-\frac{81 x^2}{100}-\frac{1}{6875 (3+5 x)}-\frac{2401}{968} \log (1-2 x)+\frac{134 \log (3+5 x)}{75625}\\ \end{align*}

Mathematica [A]  time = 0.0261691, size = 52, normalized size = 1.18 \[ -\frac{81}{400} (1-2 x)^2+\frac{999}{500} (1-2 x)-\frac{1}{6875 (5 x+3)}-\frac{2401}{968} \log (1-2 x)+\frac{134 \log (10 x+6)}{75625} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^4/((1 - 2*x)*(3 + 5*x)^2),x]

[Out]

(999*(1 - 2*x))/500 - (81*(1 - 2*x)^2)/400 - 1/(6875*(3 + 5*x)) - (2401*Log[1 - 2*x])/968 + (134*Log[6 + 10*x]
)/75625

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Maple [A]  time = 0.006, size = 35, normalized size = 0.8 \begin{align*} -{\frac{81\,{x}^{2}}{100}}-{\frac{1593\,x}{500}}-{\frac{2401\,\ln \left ( 2\,x-1 \right ) }{968}}-{\frac{1}{20625+34375\,x}}+{\frac{134\,\ln \left ( 3+5\,x \right ) }{75625}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^4/(1-2*x)/(3+5*x)^2,x)

[Out]

-81/100*x^2-1593/500*x-2401/968*ln(2*x-1)-1/6875/(3+5*x)+134/75625*ln(3+5*x)

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Maxima [A]  time = 1.1068, size = 46, normalized size = 1.05 \begin{align*} -\frac{81}{100} \, x^{2} - \frac{1593}{500} \, x - \frac{1}{6875 \,{\left (5 \, x + 3\right )}} + \frac{134}{75625} \, \log \left (5 \, x + 3\right ) - \frac{2401}{968} \, \log \left (2 \, x - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)/(3+5*x)^2,x, algorithm="maxima")

[Out]

-81/100*x^2 - 1593/500*x - 1/6875/(5*x + 3) + 134/75625*log(5*x + 3) - 2401/968*log(2*x - 1)

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Fricas [A]  time = 1.25154, size = 176, normalized size = 4. \begin{align*} -\frac{2450250 \, x^{3} + 11107800 \, x^{2} - 1072 \,{\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 1500625 \,{\left (5 \, x + 3\right )} \log \left (2 \, x - 1\right ) + 5782590 \, x + 88}{605000 \,{\left (5 \, x + 3\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/605000*(2450250*x^3 + 11107800*x^2 - 1072*(5*x + 3)*log(5*x + 3) + 1500625*(5*x + 3)*log(2*x - 1) + 5782590
*x + 88)/(5*x + 3)

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Sympy [A]  time = 0.144585, size = 37, normalized size = 0.84 \begin{align*} - \frac{81 x^{2}}{100} - \frac{1593 x}{500} - \frac{2401 \log{\left (x - \frac{1}{2} \right )}}{968} + \frac{134 \log{\left (x + \frac{3}{5} \right )}}{75625} - \frac{1}{34375 x + 20625} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**4/(1-2*x)/(3+5*x)**2,x)

[Out]

-81*x**2/100 - 1593*x/500 - 2401*log(x - 1/2)/968 + 134*log(x + 3/5)/75625 - 1/(34375*x + 20625)

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Giac [A]  time = 1.34675, size = 85, normalized size = 1.93 \begin{align*} -\frac{27}{2500} \,{\left (5 \, x + 3\right )}^{2}{\left (\frac{41}{5 \, x + 3} + 3\right )} - \frac{1}{6875 \,{\left (5 \, x + 3\right )}} + \frac{12393}{5000} \, \log \left (\frac{{\left | 5 \, x + 3 \right |}}{5 \,{\left (5 \, x + 3\right )}^{2}}\right ) - \frac{2401}{968} \, \log \left ({\left | -\frac{11}{5 \, x + 3} + 2 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^4/(1-2*x)/(3+5*x)^2,x, algorithm="giac")

[Out]

-27/2500*(5*x + 3)^2*(41/(5*x + 3) + 3) - 1/6875/(5*x + 3) + 12393/5000*log(1/5*abs(5*x + 3)/(5*x + 3)^2) - 24
01/968*log(abs(-11/(5*x + 3) + 2))