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

Optimal. Leaf size=65 \[ \frac {144}{14641 (1-2 x)}-\frac {195}{14641 (5 x+3)}+\frac {7}{1331 (1-2 x)^2}-\frac {5}{2662 (5 x+3)^2}-\frac {1110 \log (1-2 x)}{161051}+\frac {1110 \log (5 x+3)}{161051} \]

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Rubi [A]  time = 0.03, antiderivative size = 65, 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} \begin {gather*} \frac {144}{14641 (1-2 x)}-\frac {195}{14641 (5 x+3)}+\frac {7}{1331 (1-2 x)^2}-\frac {5}{2662 (5 x+3)^2}-\frac {1110 \log (1-2 x)}{161051}+\frac {1110 \log (5 x+3)}{161051} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

7/(1331*(1 - 2*x)^2) + 144/(14641*(1 - 2*x)) - 5/(2662*(3 + 5*x)^2) - 195/(14641*(3 + 5*x)) - (1110*Log[1 - 2*
x])/161051 + (1110*Log[3 + 5*x])/161051

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)^3} \, dx &=\int \left (-\frac {28}{1331 (-1+2 x)^3}+\frac {288}{14641 (-1+2 x)^2}-\frac {2220}{161051 (-1+2 x)}+\frac {25}{1331 (3+5 x)^3}+\frac {975}{14641 (3+5 x)^2}+\frac {5550}{161051 (3+5 x)}\right ) \, dx\\ &=\frac {7}{1331 (1-2 x)^2}+\frac {144}{14641 (1-2 x)}-\frac {5}{2662 (3+5 x)^2}-\frac {195}{14641 (3+5 x)}-\frac {1110 \log (1-2 x)}{161051}+\frac {1110 \log (3+5 x)}{161051}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 48, normalized size = 0.74 \begin {gather*} \frac {\frac {11 \left (-22200 x^3-3330 x^2+11026 x+2753\right )}{\left (10 x^2+x-3\right )^2}-2220 \log (1-2 x)+2220 \log (5 x+3)}{322102} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((11*(2753 + 11026*x - 3330*x^2 - 22200*x^3))/(-3 + x + 10*x^2)^2 - 2220*Log[1 - 2*x] + 2220*Log[3 + 5*x])/322
102

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2+3 x}{(1-2 x)^3 (3+5 x)^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.36, size = 95, normalized size = 1.46 \begin {gather*} -\frac {244200 \, x^{3} + 36630 \, x^{2} - 2220 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (5 \, x + 3\right ) + 2220 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \log \left (2 \, x - 1\right ) - 121286 \, x - 30283}{322102 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/322102*(244200*x^3 + 36630*x^2 - 2220*(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*log(5*x + 3) + 2220*(100*x^4 +
20*x^3 - 59*x^2 - 6*x + 9)*log(2*x - 1) - 121286*x - 30283)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)

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giac [A]  time = 1.22, size = 46, normalized size = 0.71 \begin {gather*} -\frac {22200 \, x^{3} + 3330 \, x^{2} - 11026 \, x - 2753}{29282 \, {\left (10 \, x^{2} + x - 3\right )}^{2}} + \frac {1110}{161051} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {1110}{161051} \, \log \left ({\left | 2 \, x - 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29282*(22200*x^3 + 3330*x^2 - 11026*x - 2753)/(10*x^2 + x - 3)^2 + 1110/161051*log(abs(5*x + 3)) - 1110/161
051*log(abs(2*x - 1))

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maple [A]  time = 0.01, size = 54, normalized size = 0.83 \begin {gather*} -\frac {1110 \ln \left (2 x -1\right )}{161051}+\frac {1110 \ln \left (5 x +3\right )}{161051}-\frac {5}{2662 \left (5 x +3\right )^{2}}-\frac {195}{14641 \left (5 x +3\right )}+\frac {7}{1331 \left (2 x -1\right )^{2}}-\frac {144}{14641 \left (2 x -1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/2662/(5*x+3)^2-195/14641/(5*x+3)+1110/161051*ln(5*x+3)+7/1331/(2*x-1)^2-144/14641/(2*x-1)-1110/161051*ln(2*
x-1)

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maxima [A]  time = 0.54, size = 56, normalized size = 0.86 \begin {gather*} -\frac {22200 \, x^{3} + 3330 \, x^{2} - 11026 \, x - 2753}{29282 \, {\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )}} + \frac {1110}{161051} \, \log \left (5 \, x + 3\right ) - \frac {1110}{161051} \, \log \left (2 \, x - 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/29282*(22200*x^3 + 3330*x^2 - 11026*x - 2753)/(100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9) + 1110/161051*log(5*x +
3) - 1110/161051*log(2*x - 1)

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mupad [B]  time = 0.04, size = 45, normalized size = 0.69 \begin {gather*} \frac {2220\,\mathrm {atanh}\left (\frac {20\,x}{11}+\frac {1}{11}\right )}{161051}+\frac {-\frac {111\,x^3}{14641}-\frac {333\,x^2}{292820}+\frac {5513\,x}{1464100}+\frac {2753}{2928200}}{x^4+\frac {x^3}{5}-\frac {59\,x^2}{100}-\frac {3\,x}{50}+\frac {9}{100}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2220*atanh((20*x)/11 + 1/11))/161051 + ((5513*x)/1464100 - (333*x^2)/292820 - (111*x^3)/14641 + 2753/2928200)
/(x^3/5 - (59*x^2)/100 - (3*x)/50 + x^4 + 9/100)

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sympy [A]  time = 0.18, size = 54, normalized size = 0.83 \begin {gather*} - \frac {22200 x^{3} + 3330 x^{2} - 11026 x - 2753}{2928200 x^{4} + 585640 x^{3} - 1727638 x^{2} - 175692 x + 263538} - \frac {1110 \log {\left (x - \frac {1}{2} \right )}}{161051} + \frac {1110 \log {\left (x + \frac {3}{5} \right )}}{161051} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(22200*x**3 + 3330*x**2 - 11026*x - 2753)/(2928200*x**4 + 585640*x**3 - 1727638*x**2 - 175692*x + 263538) - 1
110*log(x - 1/2)/161051 + 1110*log(x + 3/5)/161051

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