3.13.77 \(\int \frac {(1-2 x)^3}{(2+3 x)^7 (3+5 x)^2} \, dx\)

Optimal. Leaf size=88 \[ -\frac {617100}{3 x+2}-\frac {166375}{5 x+3}-\frac {103455}{2 (3 x+2)^2}-\frac {5566}{(3 x+2)^3}-\frac {2541}{4 (3 x+2)^4}-\frac {3136}{45 (3 x+2)^5}-\frac {343}{54 (3 x+2)^6}+3584625 \log (3 x+2)-3584625 \log (5 x+3) \]

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Rubi [A]  time = 0.05, antiderivative size = 88, 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 {617100}{3 x+2}-\frac {166375}{5 x+3}-\frac {103455}{2 (3 x+2)^2}-\frac {5566}{(3 x+2)^3}-\frac {2541}{4 (3 x+2)^4}-\frac {3136}{45 (3 x+2)^5}-\frac {343}{54 (3 x+2)^6}+3584625 \log (3 x+2)-3584625 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^3/((2 + 3*x)^7*(3 + 5*x)^2),x]

[Out]

-343/(54*(2 + 3*x)^6) - 3136/(45*(2 + 3*x)^5) - 2541/(4*(2 + 3*x)^4) - 5566/(2 + 3*x)^3 - 103455/(2*(2 + 3*x)^
2) - 617100/(2 + 3*x) - 166375/(3 + 5*x) + 3584625*Log[2 + 3*x] - 3584625*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)^7 (3+5 x)^2} \, dx &=\int \left (\frac {343}{3 (2+3 x)^7}+\frac {3136}{3 (2+3 x)^6}+\frac {7623}{(2+3 x)^5}+\frac {50094}{(2+3 x)^4}+\frac {310365}{(2+3 x)^3}+\frac {1851300}{(2+3 x)^2}+\frac {10753875}{2+3 x}+\frac {831875}{(3+5 x)^2}-\frac {17923125}{3+5 x}\right ) \, dx\\ &=-\frac {343}{54 (2+3 x)^6}-\frac {3136}{45 (2+3 x)^5}-\frac {2541}{4 (2+3 x)^4}-\frac {5566}{(2+3 x)^3}-\frac {103455}{2 (2+3 x)^2}-\frac {617100}{2+3 x}-\frac {166375}{3+5 x}+3584625 \log (2+3 x)-3584625 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.12, size = 90, normalized size = 1.02 \begin {gather*} -\frac {617100}{3 x+2}-\frac {166375}{5 x+3}-\frac {103455}{2 (3 x+2)^2}-\frac {5566}{(3 x+2)^3}-\frac {2541}{4 (3 x+2)^4}-\frac {3136}{45 (3 x+2)^5}-\frac {343}{54 (3 x+2)^6}+3584625 \log (5 (3 x+2))-3584625 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^3/((2 + 3*x)^7*(3 + 5*x)^2),x]

[Out]

-343/(54*(2 + 3*x)^6) - 3136/(45*(2 + 3*x)^5) - 2541/(4*(2 + 3*x)^4) - 5566/(2 + 3*x)^3 - 103455/(2*(2 + 3*x)^
2) - 617100/(2 + 3*x) - 166375/(3 + 5*x) + 3584625*Log[5*(2 + 3*x)] - 3584625*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)^7 (3+5 x)^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^3/((2 + 3*x)^7*(3 + 5*x)^2),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^3/((2 + 3*x)^7*(3 + 5*x)^2), x]

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fricas [A]  time = 1.42, size = 155, normalized size = 1.76 \begin {gather*} -\frac {470374492500 \, x^{6} + 1865818820250 \, x^{5} + 3083217691950 \, x^{4} + 2716778541015 \, x^{3} + 1346292632205 \, x^{2} + 1935697500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (5 \, x + 3\right ) - 1935697500 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )} \log \left (3 \, x + 2\right ) + 355739265638 \, x + 39157648662}{540 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3/(2+3*x)^7/(3+5*x)^2,x, algorithm="fricas")

[Out]

-1/540*(470374492500*x^6 + 1865818820250*x^5 + 3083217691950*x^4 + 2716778541015*x^3 + 1346292632205*x^2 + 193
5697500*(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(5*x + 3) - 19
35697500*(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192)*log(3*x + 2) + 3
55739265638*x + 39157648662)/(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 1
92)

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giac [A]  time = 1.12, size = 85, normalized size = 0.97 \begin {gather*} -\frac {166375}{5 \, x + 3} + \frac {125 \, {\left (\frac {246075138}{5 \, x + 3} + \frac {181716633}{{\left (5 \, x + 3\right )}^{2}} + \frac {68296076}{{\left (5 \, x + 3\right )}^{3}} + \frac {13169954}{{\left (5 \, x + 3\right )}^{4}} + \frac {1059036}{{\left (5 \, x + 3\right )}^{5}} + 135033993\right )}}{4 \, {\left (\frac {1}{5 \, x + 3} + 3\right )}^{6}} + 3584625 \, \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)^7/(3+5*x)^2,x, algorithm="giac")

[Out]

-166375/(5*x + 3) + 125/4*(246075138/(5*x + 3) + 181716633/(5*x + 3)^2 + 68296076/(5*x + 3)^3 + 13169954/(5*x
+ 3)^4 + 1059036/(5*x + 3)^5 + 135033993)/(1/(5*x + 3) + 3)^6 + 3584625*log(abs(-1/(5*x + 3) - 3))

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maple [A]  time = 0.01, size = 81, normalized size = 0.92 \begin {gather*} 3584625 \ln \left (3 x +2\right )-3584625 \ln \left (5 x +3\right )-\frac {343}{54 \left (3 x +2\right )^{6}}-\frac {3136}{45 \left (3 x +2\right )^{5}}-\frac {2541}{4 \left (3 x +2\right )^{4}}-\frac {5566}{\left (3 x +2\right )^{3}}-\frac {103455}{2 \left (3 x +2\right )^{2}}-\frac {617100}{3 x +2}-\frac {166375}{5 x +3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3/(3*x+2)^7/(5*x+3)^2,x)

[Out]

-343/54/(3*x+2)^6-3136/45/(3*x+2)^5-2541/4/(3*x+2)^4-5566/(3*x+2)^3-103455/2/(3*x+2)^2-617100/(3*x+2)-166375/(
5*x+3)+3584625*ln(3*x+2)-3584625*ln(5*x+3)

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maxima [A]  time = 0.48, size = 86, normalized size = 0.98 \begin {gather*} -\frac {470374492500 \, x^{6} + 1865818820250 \, x^{5} + 3083217691950 \, x^{4} + 2716778541015 \, x^{3} + 1346292632205 \, x^{2} + 355739265638 \, x + 39157648662}{540 \, {\left (3645 \, x^{7} + 16767 \, x^{6} + 33048 \, x^{5} + 36180 \, x^{4} + 23760 \, x^{3} + 9360 \, x^{2} + 2048 \, x + 192\right )}} - 3584625 \, \log \left (5 \, x + 3\right ) + 3584625 \, \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)^7/(3+5*x)^2,x, algorithm="maxima")

[Out]

-1/540*(470374492500*x^6 + 1865818820250*x^5 + 3083217691950*x^4 + 2716778541015*x^3 + 1346292632205*x^2 + 355
739265638*x + 39157648662)/(3645*x^7 + 16767*x^6 + 33048*x^5 + 36180*x^4 + 23760*x^3 + 9360*x^2 + 2048*x + 192
) - 3584625*log(5*x + 3) + 3584625*log(3*x + 2)

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mupad [B]  time = 1.15, size = 76, normalized size = 0.86 \begin {gather*} 7169250\,\mathrm {atanh}\left (30\,x+19\right )-\frac {238975\,x^6+\frac {5687605\,x^5}{6}+\frac {84587591\,x^4}{54}+\frac {248447969\,x^3}{180}+\frac {29917614049\,x^2}{43740}+\frac {177869632819\,x}{984150}+\frac {6526274777}{328050}}{x^7+\frac {23\,x^6}{5}+\frac {136\,x^5}{15}+\frac {268\,x^4}{27}+\frac {176\,x^3}{27}+\frac {208\,x^2}{81}+\frac {2048\,x}{3645}+\frac {64}{1215}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*x - 1)^3/((3*x + 2)^7*(5*x + 3)^2),x)

[Out]

7169250*atanh(30*x + 19) - ((177869632819*x)/984150 + (29917614049*x^2)/43740 + (248447969*x^3)/180 + (8458759
1*x^4)/54 + (5687605*x^5)/6 + 238975*x^6 + 6526274777/328050)/((2048*x)/3645 + (208*x^2)/81 + (176*x^3)/27 + (
268*x^4)/27 + (136*x^5)/15 + (23*x^6)/5 + x^7 + 64/1215)

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sympy [A]  time = 0.22, size = 82, normalized size = 0.93 \begin {gather*} - \frac {470374492500 x^{6} + 1865818820250 x^{5} + 3083217691950 x^{4} + 2716778541015 x^{3} + 1346292632205 x^{2} + 355739265638 x + 39157648662}{1968300 x^{7} + 9054180 x^{6} + 17845920 x^{5} + 19537200 x^{4} + 12830400 x^{3} + 5054400 x^{2} + 1105920 x + 103680} - 3584625 \log {\left (x + \frac {3}{5} \right )} + 3584625 \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)**7/(3+5*x)**2,x)

[Out]

-(470374492500*x**6 + 1865818820250*x**5 + 3083217691950*x**4 + 2716778541015*x**3 + 1346292632205*x**2 + 3557
39265638*x + 39157648662)/(1968300*x**7 + 9054180*x**6 + 17845920*x**5 + 19537200*x**4 + 12830400*x**3 + 50544
00*x**2 + 1105920*x + 103680) - 3584625*log(x + 3/5) + 3584625*log(x + 2/3)

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