3.1592 \(\int \frac{(2+3 x)^4}{(1-2 x)^2 (3+5 x)} \, dx\)

Optimal. Leaf size=44 \[ \frac{81 x^2}{40}+\frac{621 x}{50}+\frac{2401}{176 (1-2 x)}+\frac{33271 \log (1-2 x)}{1936}+\frac{\log (5 x+3)}{15125} \]

[Out]

2401/(176*(1 - 2*x)) + (621*x)/50 + (81*x^2)/40 + (33271*Log[1 - 2*x])/1936 + Log[3 + 5*x]/15125

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Rubi [A]  time = 0.020464, 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} \[ \frac{81 x^2}{40}+\frac{621 x}{50}+\frac{2401}{176 (1-2 x)}+\frac{33271 \log (1-2 x)}{1936}+\frac{\log (5 x+3)}{15125} \]

Antiderivative was successfully verified.

[In]

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

[Out]

2401/(176*(1 - 2*x)) + (621*x)/50 + (81*x^2)/40 + (33271*Log[1 - 2*x])/1936 + 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)^4}{(1-2 x)^2 (3+5 x)} \, dx &=\int \left (\frac{621}{50}+\frac{81 x}{20}+\frac{2401}{88 (-1+2 x)^2}+\frac{33271}{968 (-1+2 x)}+\frac{1}{3025 (3+5 x)}\right ) \, dx\\ &=\frac{2401}{176 (1-2 x)}+\frac{621 x}{50}+\frac{81 x^2}{40}+\frac{33271 \log (1-2 x)}{1936}+\frac{\log (3+5 x)}{15125}\\ \end{align*}

Mathematica [A]  time = 0.0217436, size = 45, normalized size = 1.02 \[ \frac{81 x^2}{40}+\frac{621 x}{50}+\frac{2401}{176-352 x}+\frac{33271 \log (5-10 x)}{1936}+\frac{\log (5 x+3)}{15125}+\frac{6723}{1000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

6723/1000 + 2401/(176 - 352*x) + (621*x)/50 + (81*x^2)/40 + (33271*Log[5 - 10*x])/1936 + Log[3 + 5*x]/15125

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Maple [A]  time = 0.006, size = 35, normalized size = 0.8 \begin{align*}{\frac{81\,{x}^{2}}{40}}+{\frac{621\,x}{50}}-{\frac{2401}{352\,x-176}}+{\frac{33271\,\ln \left ( 2\,x-1 \right ) }{1936}}+{\frac{\ln \left ( 3+5\,x \right ) }{15125}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81/40*x^2+621/50*x-2401/176/(2*x-1)+33271/1936*ln(2*x-1)+1/15125*ln(3+5*x)

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Maxima [A]  time = 1.10237, size = 46, normalized size = 1.05 \begin{align*} \frac{81}{40} \, x^{2} + \frac{621}{50} \, x - \frac{2401}{176 \,{\left (2 \, x - 1\right )}} + \frac{1}{15125} \, \log \left (5 \, x + 3\right ) + \frac{33271}{1936} \, \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)^2/(3+5*x),x, algorithm="maxima")

[Out]

81/40*x^2 + 621/50*x - 2401/176/(2*x - 1) + 1/15125*log(5*x + 3) + 33271/1936*log(2*x - 1)

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Fricas [A]  time = 1.31864, size = 176, normalized size = 4. \begin{align*} \frac{980100 \, x^{3} + 5521230 \, x^{2} + 16 \,{\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) + 4158875 \,{\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 3005640 \, x - 3301375}{242000 \,{\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)^2/(3+5*x),x, algorithm="fricas")

[Out]

1/242000*(980100*x^3 + 5521230*x^2 + 16*(2*x - 1)*log(5*x + 3) + 4158875*(2*x - 1)*log(2*x - 1) - 3005640*x -
3301375)/(2*x - 1)

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Sympy [A]  time = 0.133265, size = 36, normalized size = 0.82 \begin{align*} \frac{81 x^{2}}{40} + \frac{621 x}{50} + \frac{33271 \log{\left (x - \frac{1}{2} \right )}}{1936} + \frac{\log{\left (x + \frac{3}{5} \right )}}{15125} - \frac{2401}{352 x - 176} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

81*x**2/40 + 621*x/50 + 33271*log(x - 1/2)/1936 + log(x + 3/5)/15125 - 2401/(352*x - 176)

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Giac [A]  time = 2.16522, size = 85, normalized size = 1.93 \begin{align*} \frac{27}{800} \,{\left (2 \, x - 1\right )}^{2}{\left (\frac{214}{2 \, x - 1} + 15\right )} - \frac{2401}{176 \,{\left (2 \, x - 1\right )}} - \frac{34371}{2000} \, \log \left (\frac{{\left | 2 \, x - 1 \right |}}{2 \,{\left (2 \, x - 1\right )}^{2}}\right ) + \frac{1}{15125} \, \log \left ({\left | -\frac{11}{2 \, x - 1} - 5 \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/800*(2*x - 1)^2*(214/(2*x - 1) + 15) - 2401/176/(2*x - 1) - 34371/2000*log(1/2*abs(2*x - 1)/(2*x - 1)^2) +
1/15125*log(abs(-11/(2*x - 1) - 5))