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

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Rubi [A]  time = 0.02, antiderivative size = 44, 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} \[ \frac {81 x^2}{40}+\frac {621 x}{50}+\frac {2401}{176 (1-2 x)}+\frac {33271 \log (1-2 x)}{1936}+\frac {\log (5 x+3)}{15125} \]

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*}

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Mathematica [A]  time = 0.03, 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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fricas [A]  time = 0.58, size = 50, normalized size = 1.14 \[ \frac {980100 \, x^{3} + 5521230 \, x^{2} + 16 \, {\left (2 \, x - 1\right )} \log \left (5 \, x + 3\right ) + 4158875 \, {\left (2 \, x - 1\right )} \log \left (2 \, x - 1\right ) - 3005640 \, x - 3301375}{242000 \, {\left (2 \, x - 1\right )}} \]

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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giac [A]  time = 1.21, size = 63, normalized size = 1.43 \[ \frac {27}{800} \, {\left (2 \, x - 1\right )}^{2} {\left (\frac {214}{2 \, x - 1} + 15\right )} - \frac {2401}{176 \, {\left (2 \, x - 1\right )}} - \frac {34371}{2000} \, \log \left (\frac {{\left | 2 \, x - 1 \right |}}{2 \, {\left (2 \, x - 1\right )}^{2}}\right ) + \frac {1}{15125} \, \log \left ({\left | -\frac {11}{2 \, x - 1} - 5 \right |}\right ) \]

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))

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maple [A]  time = 0.01, size = 35, normalized size = 0.80 \[ \frac {81 x^{2}}{40}+\frac {621 x}{50}+\frac {33271 \ln \left (2 x -1\right )}{1936}+\frac {\ln \left (5 x +3\right )}{15125}-\frac {2401}{176 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.66, size = 34, normalized size = 0.77 \[ \frac {81}{40} \, x^{2} + \frac {621}{50} \, x - \frac {2401}{176 \, {\left (2 \, x - 1\right )}} + \frac {1}{15125} \, \log \left (5 \, x + 3\right ) + \frac {33271}{1936} \, \log \left (2 \, x - 1\right ) \]

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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mupad [B]  time = 1.09, size = 30, normalized size = 0.68 \[ \frac {621\,x}{50}+\frac {33271\,\ln \left (x-\frac {1}{2}\right )}{1936}+\frac {\ln \left (x+\frac {3}{5}\right )}{15125}-\frac {2401}{352\,\left (x-\frac {1}{2}\right )}+\frac {81\,x^2}{40} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 0.15, size = 36, normalized size = 0.82 \[ \frac {81 x^{2}}{40} + \frac {621 x}{50} + \frac {33271 \log {\left (x - \frac {1}{2} \right )}}{1936} + \frac {\log {\left (x + \frac {3}{5} \right )}}{15125} - \frac {2401}{352 x - 176} \]

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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