∫ (1−2x)3 ________________________

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

Optimal. Leaf size=97 \[ \frac {6618975}{3 x+2}+\frac {3584625}{5 x+3}+\frac {953535}{2 (3 x+2)^2}-\frac {166375}{2 (5 x+3)^2}+\frac {42878}{(3 x+2)^3}+\frac {3927}{(3 x+2)^4}+\frac {1617}{5 (3 x+2)^5}+\frac {343}{18 (3 x+2)^6}-43848750 \log (3 x+2)+43848750 \log (5 x+3) \]

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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 {6618975}{3 x+2}+\frac {3584625}{5 x+3}+\frac {953535}{2 (3 x+2)^2}-\frac {166375}{2 (5 x+3)^2}+\frac {42878}{(3 x+2)^3}+\frac {3927}{(3 x+2)^4}+\frac {1617}{5 (3 x+2)^5}+\frac {343}{18 (3 x+2)^6}-43848750 \log (3 x+2)+43848750 \log (5 x+3) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

343/(18*(2 + 3*x)^6) + 1617/(5*(2 + 3*x)^5) + 3927/(2 + 3*x)^4 + 42878/(2 + 3*x)^3 + 953535/(2*(2 + 3*x)^2) +

6618975/(2 + 3*x) - 166375/(2*(3 + 5*x)^2) + 3584625/(3 + 5*x) - 43848750*Log[2 + 3*x] + 43848750*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)^3} \, dx &=\int \left (-\frac {343}{(2+3 x)^7}-\frac {4851}{(2+3 x)^6}-\frac {47124}{(2+3 x)^5}-\frac {385902}{(2+3 x)^4}-\frac {2860605}{(2+3 x)^3}-\frac {19856925}{(2+3 x)^2}-\frac {131546250}{2+3 x}+\frac {831875}{(3+5 x)^3}-\frac {17923125}{(3+5 x)^2}+\frac {219243750}{3+5 x}\right ) \, dx\\ &=\frac {343}{18 (2+3 x)^6}+\frac {1617}{5 (2+3 x)^5}+\frac {3927}{(2+3 x)^4}+\frac {42878}{(2+3 x)^3}+\frac {953535}{2 (2+3 x)^2}+\frac {6618975}{2+3 x}-\frac {166375}{2 (3+5 x)^2}+\frac {3584625}{3+5 x}-43848750 \log (2+3 x)+43848750 \log (3+5 x)\\ \end {align*}

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Mathematica [A] time = 0.16, size = 99, normalized size = 1.02 \begin {gather*} \frac {6618975}{3 x+2}+\frac {3584625}{5 x+3}+\frac {953535}{2 (3 x+2)^2}-\frac {166375}{2 (5 x+3)^2}+\frac {42878}{(3 x+2)^3}+\frac {3927}{(3 x+2)^4}+\frac {1617}{5 (3 x+2)^5}+\frac {343}{18 (3 x+2)^6}-43848750 \log (5 (3 x+2))+43848750 \log (5 x+3) \end {gather*}

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

343/(18*(2 + 3*x)^6) + 1617/(5*(2 + 3*x)^5) + 3927/(2 + 3*x)^4 + 42878/(2 + 3*x)^3 + 953535/(2*(2 + 3*x)^2) +

6618975/(2 + 3*x) - 166375/(2*(3 + 5*x)^2) + 3584625/(3 + 5*x) - 43848750*Log[5*(2 + 3*x)] + 43848750*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)^3} \, dx \end {gather*}

Veriﬁcation is not applicable to the result.

[In]

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

[Out]

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

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fricas [A] time = 1.03, size = 175, normalized size = 1.80 \begin {gather*} \frac {4794860812500 \, x^{7} + 21896531043750 \, x^{6} + 42841193422500 \, x^{5} + 46551705341625 \, x^{4} + 30340145968110 \, x^{3} + 11860532030465 \, x^{2} + 3946387500 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )} \log \left (5 \, x + 3\right ) - 3946387500 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )} \log \left (3 \, x + 2\right ) + 2574943269792 \, x + 239497011063}{90 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\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)^3,x, algorithm="fricas")

[Out]

1/90*(4794860812500*x^7 + 21896531043750*x^6 + 42841193422500*x^5 + 46551705341625*x^4 + 30340145968110*x^3 +

11860532030465*x^2 + 3946387500*(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 3

8320*x^2 + 7104*x + 576)*log(5*x + 3) - 3946387500*(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340*x

^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576)*log(3*x + 2) + 2574943269792*x + 239497011063)/(18225*x^8 + 94770*

x^7 + 215541*x^6 + 280044*x^5 + 227340*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576)

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giac [A] time = 0.98, size = 70, normalized size = 0.72 \begin {gather*} \frac {4794860812500 \, x^{7} + 21896531043750 \, x^{6} + 42841193422500 \, x^{5} + 46551705341625 \, x^{4} + 30340145968110 \, x^{3} + 11860532030465 \, x^{2} + 2574943269792 \, x + 239497011063}{90 \, {\left (5 \, x + 3\right )}^{2} {\left (3 \, x + 2\right )}^{6}} + 43848750 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 43848750 \, \log \left ({\left | 3 \, x + 2 \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)^3,x, algorithm="giac")

[Out]

1/90*(4794860812500*x^7 + 21896531043750*x^6 + 42841193422500*x^5 + 46551705341625*x^4 + 30340145968110*x^3 +

11860532030465*x^2 + 2574943269792*x + 239497011063)/((5*x + 3)^2*(3*x + 2)^6) + 43848750*log(abs(5*x + 3)) -

43848750*log(abs(3*x + 2))

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maple [A] time = 0.01, size = 90, normalized size = 0.93 \begin {gather*} -43848750 \ln \left (3 x +2\right )+43848750 \ln \left (5 x +3\right )+\frac {343}{18 \left (3 x +2\right )^{6}}+\frac {1617}{5 \left (3 x +2\right )^{5}}+\frac {3927}{\left (3 x +2\right )^{4}}+\frac {42878}{\left (3 x +2\right )^{3}}+\frac {953535}{2 \left (3 x +2\right )^{2}}+\frac {6618975}{3 x +2}-\frac {166375}{2 \left (5 x +3\right )^{2}}+\frac {3584625}{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)^3,x)

[Out]

343/18/(3*x+2)^6+1617/5/(3*x+2)^5+3927/(3*x+2)^4+42878/(3*x+2)^3+953535/2/(3*x+2)^2+6618975/(3*x+2)-166375/2/(

5*x+3)^2+3584625/(5*x+3)-43848750*ln(3*x+2)+43848750*ln(5*x+3)

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maxima [A] time = 0.48, size = 96, normalized size = 0.99 \begin {gather*} \frac {4794860812500 \, x^{7} + 21896531043750 \, x^{6} + 42841193422500 \, x^{5} + 46551705341625 \, x^{4} + 30340145968110 \, x^{3} + 11860532030465 \, x^{2} + 2574943269792 \, x + 239497011063}{90 \, {\left (18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576\right )}} + 43848750 \, \log \left (5 \, x + 3\right ) - 43848750 \, \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)^3,x, algorithm="maxima")

[Out]

1/90*(4794860812500*x^7 + 21896531043750*x^6 + 42841193422500*x^5 + 46551705341625*x^4 + 30340145968110*x^3 +

11860532030465*x^2 + 2574943269792*x + 239497011063)/(18225*x^8 + 94770*x^7 + 215541*x^6 + 280044*x^5 + 227340

*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576) + 43848750*log(5*x + 3) - 43848750*log(3*x + 2)

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mupad [B] time = 1.13, size = 85, normalized size = 0.88 \begin {gather*} \frac {2923250\,x^7+\frac {40048525\,x^6}{3}+\frac {705204830\,x^5}{27}+\frac {170285159\,x^4}{6}+\frac {12485656777\,x^3}{675}+\frac {2372106406093\,x^2}{328050}+\frac {429157211632\,x}{273375}+\frac {2956753223}{20250}}{x^8+\frac {26\,x^7}{5}+\frac {887\,x^6}{75}+\frac {10372\,x^5}{675}+\frac {1684\,x^4}{135}+\frac {2624\,x^3}{405}+\frac {7664\,x^2}{3645}+\frac {2368\,x}{6075}+\frac {64}{2025}}-87697500\,\mathrm {atanh}\left (30\,x+19\right ) \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)^3),x)

[Out]

((429157211632*x)/273375 + (2372106406093*x^2)/328050 + (12485656777*x^3)/675 + (170285159*x^4)/6 + (705204830

*x^5)/27 + (40048525*x^6)/3 + 2923250*x^7 + 2956753223/20250)/((2368*x)/6075 + (7664*x^2)/3645 + (2624*x^3)/40

5 + (1684*x^4)/135 + (10372*x^5)/675 + (887*x^6)/75 + (26*x^7)/5 + x^8 + 64/2025) - 87697500*atanh(30*x + 19)

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sympy [A] time = 0.25, size = 94, normalized size = 0.97 \begin {gather*} - \frac {- 4794860812500 x^{7} - 21896531043750 x^{6} - 42841193422500 x^{5} - 46551705341625 x^{4} - 30340145968110 x^{3} - 11860532030465 x^{2} - 2574943269792 x - 239497011063}{1640250 x^{8} + 8529300 x^{7} + 19398690 x^{6} + 25203960 x^{5} + 20460600 x^{4} + 10627200 x^{3} + 3448800 x^{2} + 639360 x + 51840} + 43848750 \log {\left (x + \frac {3}{5} \right )} - 43848750 \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)**3,x)

[Out]

-(-4794860812500*x**7 - 21896531043750*x**6 - 42841193422500*x**5 - 46551705341625*x**4 - 30340145968110*x**3

- 11860532030465*x**2 - 2574943269792*x - 239497011063)/(1640250*x**8 + 8529300*x**7 + 19398690*x**6 + 2520396

0*x**5 + 20460600*x**4 + 10627200*x**3 + 3448800*x**2 + 639360*x + 51840) + 43848750*log(x + 3/5) - 43848750*l

og(x + 2/3)

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