3.13.76 \(\int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^2} \, dx\)

Optimal. Leaf size=75 \[ -\frac {103455}{3 x+2}-\frac {33275}{5 x+3}-\frac {8349}{(3 x+2)^2}-\frac {847}{(3 x+2)^3}-\frac {784}{9 (3 x+2)^4}-\frac {343}{45 (3 x+2)^5}+617100 \log (3 x+2)-617100 \log (5 x+3) \]

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Rubi [A]  time = 0.04, antiderivative size = 75, 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} \begin {gather*} -\frac {103455}{3 x+2}-\frac {33275}{5 x+3}-\frac {8349}{(3 x+2)^2}-\frac {847}{(3 x+2)^3}-\frac {784}{9 (3 x+2)^4}-\frac {343}{45 (3 x+2)^5}+617100 \log (3 x+2)-617100 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^3/((2 + 3*x)^6*(3 + 5*x)^2),x]

[Out]

-343/(45*(2 + 3*x)^5) - 784/(9*(2 + 3*x)^4) - 847/(2 + 3*x)^3 - 8349/(2 + 3*x)^2 - 103455/(2 + 3*x) - 33275/(3
+ 5*x) + 617100*Log[2 + 3*x] - 617100*Log[3 + 5*x]

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 {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^2} \, dx &=\int \left (\frac {343}{3 (2+3 x)^6}+\frac {3136}{3 (2+3 x)^5}+\frac {7623}{(2+3 x)^4}+\frac {50094}{(2+3 x)^3}+\frac {310365}{(2+3 x)^2}+\frac {1851300}{2+3 x}+\frac {166375}{(3+5 x)^2}-\frac {3085500}{3+5 x}\right ) \, dx\\ &=-\frac {343}{45 (2+3 x)^5}-\frac {784}{9 (2+3 x)^4}-\frac {847}{(2+3 x)^3}-\frac {8349}{(2+3 x)^2}-\frac {103455}{2+3 x}-\frac {33275}{3+5 x}+617100 \log (2+3 x)-617100 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.10, size = 62, normalized size = 0.83 \begin {gather*} -\frac {2249329500 x^5+7422787350 x^4+9795413430 x^3+6461351715 x^2+2130399775 x+280877649}{45 (3 x+2)^5 (5 x+3)}+617100 \log (5 (3 x+2))-617100 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^3/((2 + 3*x)^6*(3 + 5*x)^2),x]

[Out]

-1/45*(280877649 + 2130399775*x + 6461351715*x^2 + 9795413430*x^3 + 7422787350*x^4 + 2249329500*x^5)/((2 + 3*x
)^5*(3 + 5*x)) + 617100*Log[5*(2 + 3*x)] - 617100*Log[3 + 5*x]

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^3}{(2+3 x)^6 (3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^3/((2 + 3*x)^6*(3 + 5*x)^2),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^3/((2 + 3*x)^6*(3 + 5*x)^2), x]

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fricas [A]  time = 1.35, size = 135, normalized size = 1.80 \begin {gather*} -\frac {2249329500 \, x^{5} + 7422787350 \, x^{4} + 9795413430 \, x^{3} + 6461351715 \, x^{2} + 27769500 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (5 \, x + 3\right ) - 27769500 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )} \log \left (3 \, x + 2\right ) + 2130399775 \, x + 280877649}{45 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3/(2+3*x)^6/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/45*(2249329500*x^5 + 7422787350*x^4 + 9795413430*x^3 + 6461351715*x^2 + 27769500*(1215*x^6 + 4779*x^5 + 783
0*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96)*log(5*x + 3) - 27769500*(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 +
3360*x^2 + 880*x + 96)*log(3*x + 2) + 2130399775*x + 280877649)/(1215*x^6 + 4779*x^5 + 7830*x^4 + 6840*x^3 +
3360*x^2 + 880*x + 96)

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giac [A]  time = 0.91, size = 76, normalized size = 1.01 \begin {gather*} -\frac {33275}{5 \, x + 3} + \frac {25 \, {\left (\frac {13068279}{5 \, x + 3} + \frac {7369449}{{\left (5 \, x + 3\right )}^{2}} + \frac {1895648}{{\left (5 \, x + 3\right )}^{3}} + \frac {190707}{{\left (5 \, x + 3\right )}^{4}} + 8846550\right )}}{{\left (\frac {1}{5 \, x + 3} + 3\right )}^{5}} + 617100 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3/(2+3*x)^6/(3+5*x)^2,x, algorithm="giac")

[Out]

-33275/(5*x + 3) + 25*(13068279/(5*x + 3) + 7369449/(5*x + 3)^2 + 1895648/(5*x + 3)^3 + 190707/(5*x + 3)^4 + 8
846550)/(1/(5*x + 3) + 3)^5 + 617100*log(abs(-1/(5*x + 3) - 3))

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maple [A]  time = 0.01, size = 72, normalized size = 0.96 \begin {gather*} 617100 \ln \left (3 x +2\right )-617100 \ln \left (5 x +3\right )-\frac {343}{45 \left (3 x +2\right )^{5}}-\frac {784}{9 \left (3 x +2\right )^{4}}-\frac {847}{\left (3 x +2\right )^{3}}-\frac {8349}{\left (3 x +2\right )^{2}}-\frac {103455}{3 x +2}-\frac {33275}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3/(3*x+2)^6/(5*x+3)^2,x)

[Out]

-343/45/(3*x+2)^5-784/9/(3*x+2)^4-847/(3*x+2)^3-8349/(3*x+2)^2-103455/(3*x+2)-33275/(5*x+3)+617100*ln(3*x+2)-6
17100*ln(5*x+3)

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maxima [A]  time = 0.56, size = 76, normalized size = 1.01 \begin {gather*} -\frac {2249329500 \, x^{5} + 7422787350 \, x^{4} + 9795413430 \, x^{3} + 6461351715 \, x^{2} + 2130399775 \, x + 280877649}{45 \, {\left (1215 \, x^{6} + 4779 \, x^{5} + 7830 \, x^{4} + 6840 \, x^{3} + 3360 \, x^{2} + 880 \, x + 96\right )}} - 617100 \, \log \left (5 \, x + 3\right ) + 617100 \, \log \left (3 \, x + 2\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3/(2+3*x)^6/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/45*(2249329500*x^5 + 7422787350*x^4 + 9795413430*x^3 + 6461351715*x^2 + 2130399775*x + 280877649)/(1215*x^6
+ 4779*x^5 + 7830*x^4 + 6840*x^3 + 3360*x^2 + 880*x + 96) - 617100*log(5*x + 3) + 617100*log(3*x + 2)

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mupad [B]  time = 0.05, size = 66, normalized size = 0.88 \begin {gather*} 1234200\,\mathrm {atanh}\left (30\,x+19\right )-\frac {41140\,x^5+135762\,x^4+\frac {24186206\,x^3}{135}+\frac {430756781\,x^2}{3645}+\frac {85215991\,x}{2187}+\frac {93625883}{18225}}{x^6+\frac {59\,x^5}{15}+\frac {58\,x^4}{9}+\frac {152\,x^3}{27}+\frac {224\,x^2}{81}+\frac {176\,x}{243}+\frac {32}{405}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)^3/((3*x + 2)^6*(5*x + 3)^2),x)

[Out]

1234200*atanh(30*x + 19) - ((85215991*x)/2187 + (430756781*x^2)/3645 + (24186206*x^3)/135 + 135762*x^4 + 41140
*x^5 + 93625883/18225)/((176*x)/243 + (224*x^2)/81 + (152*x^3)/27 + (58*x^4)/9 + (59*x^5)/15 + x^6 + 32/405)

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sympy [A]  time = 0.21, size = 71, normalized size = 0.95 \begin {gather*} - \frac {2249329500 x^{5} + 7422787350 x^{4} + 9795413430 x^{3} + 6461351715 x^{2} + 2130399775 x + 280877649}{54675 x^{6} + 215055 x^{5} + 352350 x^{4} + 307800 x^{3} + 151200 x^{2} + 39600 x + 4320} - 617100 \log {\left (x + \frac {3}{5} \right )} + 617100 \log {\left (x + \frac {2}{3} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3/(2+3*x)**6/(3+5*x)**2,x)

[Out]

-(2249329500*x**5 + 7422787350*x**4 + 9795413430*x**3 + 6461351715*x**2 + 2130399775*x + 280877649)/(54675*x**
6 + 215055*x**5 + 352350*x**4 + 307800*x**3 + 151200*x**2 + 39600*x + 4320) - 617100*log(x + 3/5) + 617100*log
(x + 2/3)

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