3.13.78 \(\int \frac {(1-2 x)^3}{(2+3 x)^8 (3+5 x)^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac {3584625}{3 x+2}-\frac {831875}{5 x+3}-\frac {308550}{(3 x+2)^2}-\frac {34485}{(3 x+2)^3}-\frac {8349}{2 (3 x+2)^4}-\frac {2541}{5 (3 x+2)^5}-\frac {1568}{27 (3 x+2)^6}-\frac {49}{9 (3 x+2)^7}[phone] \log (3 x+2)-20418750 \log (5 x+3) \]

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Rubi [A]  time = 0.05, antiderivative size = 97, 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 {3584625}{3 x+2}-\frac {831875}{5 x+3}-\frac {308550}{(3 x+2)^2}-\frac {34485}{(3 x+2)^3}-\frac {8349}{2 (3 x+2)^4}-\frac {2541}{5 (3 x+2)^5}-\frac {1568}{27 (3 x+2)^6}-\frac {49}{9 (3 x+2)^7}[phone] \log (3 x+2)-20418750 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^3/((2 + 3*x)^8*(3 + 5*x)^2),x]

[Out]

-49/(9*(2 + 3*x)^7) - 1568/(27*(2 + 3*x)^6) - 2541/(5*(2 + 3*x)^5) - 8349/(2*(2 + 3*x)^4) - 34485/(2 + 3*x)^3
- 308550/(2 + 3*x)^2 - 3584625/(2 + 3*x) - 831875/(3 + 5*x) + 20418750*Log[2 + 3*x] - 20418750*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)^8 (3+5 x)^2} \, dx &=\int \left (\frac {343}{3 (2+3 x)^8}+\frac {3136}{3 (2+3 x)^7}+\frac {7623}{(2+3 x)^6}+\frac {50094}{(2+3 x)^5}+\frac {310365}{(2+3 x)^4}+\frac {1851300}{(2+3 x)^3}+\frac {10753875}{(2+3 x)^2}+\frac {61256250}{2+3 x}+\frac {4159375}{(3+5 x)^2}-\frac {102093750}{3+5 x}\right ) \, dx\\ &=-\frac {49}{9 (2+3 x)^7}-\frac {1568}{27 (2+3 x)^6}-\frac {2541}{5 (2+3 x)^5}-\frac {8349}{2 (2+3 x)^4}-\frac {34485}{(2+3 x)^3}-\frac {308550}{(2+3 x)^2}-\frac {3584625}{2+3 x}-\frac {831875}{3+5 x}[phone] \log (2+3 x)-20418750 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 99, normalized size = 1.02 \begin {gather*} -\frac {3584625}{3 x+2}-\frac {831875}{5 x+3}-\frac {308550}{(3 x+2)^2}-\frac {34485}{(3 x+2)^3}-\frac {8349}{2 (3 x+2)^4}-\frac {2541}{5 (3 x+2)^5}-\frac {1568}{27 (3 x+2)^6}-\frac {49}{9 (3 x+2)^7}[phone] \log (5 (3 x+2))-20418750 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^3/((2 + 3*x)^8*(3 + 5*x)^2),x]

[Out]

-49/(9*(2 + 3*x)^7) - 1568/(27*(2 + 3*x)^6) - 2541/(5*(2 + 3*x)^5) - 8349/(2*(2 + 3*x)^4) - 34485/(2 + 3*x)^3
- 308550/(2 + 3*x)^2 - 3584625/(2 + 3*x) - 831875/(3 + 5*x) + 20418750*Log[5*(2 + 3*x)] - 20418750*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)^8 (3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^3/((2 + 3*x)^8*(3 + 5*x)^2),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^3/((2 + 3*x)^8*(3 + 5*x)^2), x]

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fricas [A]  time = 1.45, size = 175, normalized size = 1.80 \begin {gather*} -\frac {4019022562500 \, x^{7} + 18621471206250 \, x^{6} + 36972030521250 \, x^{5} + 40775613627375 \, x^{4} + 26978454053595 \, x^{3} + 10708299857748 \, x^{2} + 5513062500 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )} \log \left (5 \, x + 3\right ) - 5513062500 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )} \log \left (3 \, x + 2\right ) + 2360937751874 \, x + 223049897418}{270 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3/(2+3*x)^8/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/270*(4019022562500*x^7 + 18621471206250*x^6 + 36972030521250*x^5 + 40775613627375*x^4 + 26978454053595*x^3
+ 10708299857748*x^2 + 5513062500*(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x^3 +
24864*x^2 + 4672*x + 384)*log(5*x + 3) - 5513062500*(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 143640*
x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384)*log(3*x + 2) + 2360937751874*x + 223049897418)/(10935*x^8 + 57591*
x^7 + 132678*x^6 + 174636*x^5 + 143640*x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384)

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giac [A]  time = 0.86, size = 94, normalized size = 0.97 \begin {gather*} -\frac {831875}{5 \, x + 3} + \frac {625 \, {\left (\frac {537521373}{5 \, x + 3} + \frac {489712095}{{\left (5 \, x + 3\right )}^{2}} + \frac {241051911}{{\left (5 \, x + 3\right )}^{3}} + \frac {67932770}{{\left (5 \, x + 3\right )}^{4}} + \frac {10476370}{{\left (5 \, x + 3\right )}^{5}} + \frac {701580}{{\left (5 \, x + 3\right )}^{6}} + 248285331\right )}}{2 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{7}} + 20418750 \, \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)^8/(3+5*x)^2,x, algorithm="giac")

[Out]

-831875/(5*x + 3) + 625/2*(537521373/(5*x + 3) + 489712095/(5*x + 3)^2 + 241051911/(5*x + 3)^3 + 67932770/(5*x
+ 3)^4 + 10476370/(5*x + 3)^5 + 701580/(5*x + 3)^6 + 248285331)/(1/(5*x + 3) + 3)^7 + 20418750*log(abs(-1/(5*
x + 3) - 3))

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maple [A]  time = 0.01, size = 90, normalized size = 0.93 \begin {gather*} 20418750 \ln \left (3 x +2\right )-20418750 \ln \left (5 x +3\right )-\frac {49}{9 \left (3 x +2\right )^{7}}-\frac {1568}{27 \left (3 x +2\right )^{6}}-\frac {2541}{5 \left (3 x +2\right )^{5}}-\frac {8349}{2 \left (3 x +2\right )^{4}}-\frac {34485}{\left (3 x +2\right )^{3}}-\frac {308550}{\left (3 x +2\right )^{2}}-\frac {3584625}{3 x +2}-\frac {831875}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3/(3*x+2)^8/(5*x+3)^2,x)

[Out]

-49/9/(3*x+2)^7-1568/27/(3*x+2)^6-2541/5/(3*x+2)^5-8349/2/(3*x+2)^4-34485/(3*x+2)^3-308550/(3*x+2)^2-3584625/(
3*x+2)-831875/(5*x+3)[phone]*ln(3*x+2)-20418750*ln(5*x+3)

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maxima [A]  time = 0.52, size = 96, normalized size = 0.99 \begin {gather*} -\frac {4019022562500 \, x^{7} + 18621471206250 \, x^{6} + 36972030521250 \, x^{5} + 40775613627375 \, x^{4} + 26978454053595 \, x^{3} + 10708299857748 \, x^{2} + 2360937751874 \, x + 223049897418}{270 \, {\left (10935 \, x^{8} + 57591 \, x^{7} + 132678 \, x^{6} + 174636 \, x^{5} + 143640 \, x^{4} + 75600 \, x^{3} + 24864 \, x^{2} + 4672 \, x + 384\right )}} - 20418750 \, \log \left (5 \, x + 3\right ) + 20418750 \, \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)^8/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/270*(4019022562500*x^7 + 18621471206250*x^6 + 36972030521250*x^5 + 40775613627375*x^4 + 26978454053595*x^3
+ 10708299857748*x^2 + 2360937751874*x + 223049897418)/(10935*x^8 + 57591*x^7 + 132678*x^6 + 174636*x^5 + 1436
40*x^4 + 75600*x^3 + 24864*x^2 + 4672*x + 384) - 20418750*log(5*x + 3) + 20418750*log(3*x + 2)

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mupad [B]  time = 1.16, size = 86, normalized size = 0.89 \begin {gather*} 40837500\,\mathrm {atanh}\left (30\,x+19\right )-\frac {1361250\,x^7+6307125\,x^6+\frac {37567475\,x^5}{3}+\frac {248593895\,x^4}{18}+\frac {274129493\,x^3}{30}+\frac {1784716642958\,x^2}{492075}+\frac {1180468875937\,x}{1476225}+\frac {37174982903}{492075}}{x^8+\frac {79\,x^7}{15}+\frac {182\,x^6}{15}+\frac {2156\,x^5}{135}+\frac {1064\,x^4}{81}+\frac {560\,x^3}{81}+\frac {8288\,x^2}{3645}+\frac {4672\,x}{10935}+\frac {128}{3645}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)^3/((3*x + 2)^8*(5*x + 3)^2),x)

[Out]

40837500*atanh(30*x + 19) - ((1180468875937*x)/1476225 + (1784716642958*x^2)/492075 + (274129493*x^3)/30 + (24
8593895*x^4)/18 + (37567475*x^5)/3 + 6307125*x^6 + 1361250*x^7 + 37174982903/492075)/((4672*x)/10935 + (8288*x
^2)/3645 + (560*x^3)/81 + (1064*x^4)/81 + (2156*x^5)/135 + (182*x^6)/15 + (79*x^7)/15 + x^8 + 128/3645)

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sympy [A]  time = 0.24, size = 92, normalized size = 0.95 \begin {gather*} - \frac {4019022562500 x^{7} + 18621471206250 x^{6} + 36972030521250 x^{5} + 40775613627375 x^{4} + 26978454053595 x^{3} + 10708299857748 x^{2} + 2360937751874 x + 223049897418}{2952450 x^{8} + 15549570 x^{7} + 35823060 x^{6} + 47151720 x^{5} + 38782800 x^{4} + 20412000 x^{3} + 6713280 x^{2} + 1261440 x + 103680} - 20418750 \log {\left (x + \frac {3}{5} \right )} + 20418750 \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)**8/(3+5*x)**2,x)

[Out]

-(4019022562500*x**7 + 18621471206250*x**6 + 36972030521250*x**5 + 40775613627375*x**4 + 26978454053595*x**3 +
10708299857748*x**2 + 2360937751874*x + 223049897418)/(2952450*x**8 + 15549570*x**7 + 35823060*x**6 + 4715172
0*x**5 + 38782800*x**4 + 20412000*x**3 + 6713280*x**2 + 1261440*x + 103680) - 20418750*log(x + 3/5) + 20418750
*log(x + 2/3)

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