3.1678 \(\int \frac {2+3 x}{(1-2 x)^3 (3+5 x)} \, dx\)

Optimal. Leaf size=43 \[ \frac {1}{121 (1-2 x)}+\frac {7}{44 (1-2 x)^2}-\frac {5 \log (1-2 x)}{1331}+\frac {5 \log (5 x+3)}{1331} \]

[Out]

7/44/(1-2*x)^2+1/121/(1-2*x)-5/1331*ln(1-2*x)+5/1331*ln(3+5*x)

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Rubi [A]  time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {77} \[ \frac {1}{121 (1-2 x)}+\frac {7}{44 (1-2 x)^2}-\frac {5 \log (1-2 x)}{1331}+\frac {5 \log (5 x+3)}{1331} \]

Antiderivative was successfully verified.

[In]

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

[Out]

7/(44*(1 - 2*x)^2) + 1/(121*(1 - 2*x)) - (5*Log[1 - 2*x])/1331 + (5*Log[3 + 5*x])/1331

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rubi steps

\begin {align*} \int \frac {2+3 x}{(1-2 x)^3 (3+5 x)} \, dx &=\int \left (-\frac {7}{11 (-1+2 x)^3}+\frac {2}{121 (-1+2 x)^2}-\frac {10}{1331 (-1+2 x)}+\frac {25}{1331 (3+5 x)}\right ) \, dx\\ &=\frac {7}{44 (1-2 x)^2}+\frac {1}{121 (1-2 x)}-\frac {5 \log (1-2 x)}{1331}+\frac {5 \log (3+5 x)}{1331}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 46, normalized size = 1.07 \[ \frac {-88 x-20 (1-2 x)^2 \log (1-2 x)+20 (1-2 x)^2 \log (10 x+6)+891}{5324 (1-2 x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(891 - 88*x - 20*(1 - 2*x)^2*Log[1 - 2*x] + 20*(1 - 2*x)^2*Log[6 + 10*x])/(5324*(1 - 2*x)^2)

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fricas [A]  time = 0.52, size = 55, normalized size = 1.28 \[ \frac {20 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (5 \, x + 3\right ) - 20 \, {\left (4 \, x^{2} - 4 \, x + 1\right )} \log \left (2 \, x - 1\right ) - 88 \, x + 891}{5324 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5324*(20*(4*x^2 - 4*x + 1)*log(5*x + 3) - 20*(4*x^2 - 4*x + 1)*log(2*x - 1) - 88*x + 891)/(4*x^2 - 4*x + 1)

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giac [A]  time = 1.17, size = 33, normalized size = 0.77 \[ -\frac {8 \, x - 81}{484 \, {\left (2 \, x - 1\right )}^{2}} + \frac {5}{1331} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {5}{1331} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/484*(8*x - 81)/(2*x - 1)^2 + 5/1331*log(abs(5*x + 3)) - 5/1331*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 36, normalized size = 0.84 \[ -\frac {5 \ln \left (2 x -1\right )}{1331}+\frac {5 \ln \left (5 x +3\right )}{1331}+\frac {7}{44 \left (2 x -1\right )^{2}}-\frac {1}{121 \left (2 x -1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/1331*ln(5*x+3)+7/44/(2*x-1)^2-1/121/(2*x-1)-5/1331*ln(2*x-1)

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maxima [A]  time = 0.50, size = 36, normalized size = 0.84 \[ -\frac {8 \, x - 81}{484 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} + \frac {5}{1331} \, \log \left (5 \, x + 3\right ) - \frac {5}{1331} \, \log \left (2 \, x - 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/484*(8*x - 81)/(4*x^2 - 4*x + 1) + 5/1331*log(5*x + 3) - 5/1331*log(2*x - 1)

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mupad [B]  time = 0.04, size = 26, normalized size = 0.60 \[ \frac {10\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{1331}-\frac {\frac {x}{242}-\frac {81}{1936}}{x^2-x+\frac {1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(10*atanh((20*x)/11 + 1/11))/1331 - (x/242 - 81/1936)/(x^2 - x + 1/4)

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sympy [A]  time = 0.14, size = 34, normalized size = 0.79 \[ - \frac {8 x - 81}{1936 x^{2} - 1936 x + 484} - \frac {5 \log {\left (x - \frac {1}{2} \right )}}{1331} + \frac {5 \log {\left (x + \frac {3}{5} \right )}}{1331} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(8*x - 81)/(1936*x**2 - 1936*x + 484) - 5*log(x - 1/2)/1331 + 5*log(x + 3/5)/1331

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