3.1415 \(\int \frac{(1-2 x)^3}{(2+3 x)^6 (3+5 x)^2} \, dx\)

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

[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]

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Rubi [A]  time = 0.0404303, antiderivative size = 75, 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} \[ -\frac{103455}{3 x+2}-\frac{33275}{5 x+3}-\frac{8349}{(3 x+2)^2}-\frac{847}{(3 x+2)^3}-\frac{784}{9 (3 x+2)^4}-\frac{343}{45 (3 x+2)^5}+617100 \log (3 x+2)-617100 \log (5 x+3) \]

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*}

Mathematica [A]  time = 0.0563579, size = 62, normalized size = 0.83 \[ -\frac{2249329500 x^5+7422787350 x^4+9795413430 x^3+6461351715 x^2+2130399775 x+280877649}{45 (3 x+2)^5 (5 x+3)}+617100 \log (5 (3 x+2))-617100 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.009, size = 72, normalized size = 1. \begin{align*} -{\frac{343}{45\, \left ( 2+3\,x \right ) ^{5}}}-{\frac{784}{9\, \left ( 2+3\,x \right ) ^{4}}}-847\, \left ( 2+3\,x \right ) ^{-3}-8349\, \left ( 2+3\,x \right ) ^{-2}-103455\, \left ( 2+3\,x \right ) ^{-1}-33275\, \left ( 3+5\,x \right ) ^{-1}+617100\,\ln \left ( 2+3\,x \right ) -617100\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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*ln(2+3*x)-6
17100*ln(3+5*x)

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Maxima [A]  time = 1.07706, size = 103, normalized size = 1.37 \begin{align*} -\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{align*}

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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Fricas [A]  time = 1.29769, size = 478, normalized size = 6.37 \begin{align*} -\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{align*}

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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Sympy [A]  time = 0.201401, size = 71, normalized size = 0.95 \begin{align*} - \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{align*}

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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Giac [A]  time = 2.77874, size = 103, normalized size = 1.37 \begin{align*} -\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{align*}

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))