3.13.59 \(\int \frac {(1-2 x)^3}{(2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=50 \[ \frac {7189}{27 (3 x+2)}+\frac {1421}{54 (3 x+2)^2}+\frac {343}{81 (3 x+2)^3}-1331 \log (3 x+2)+1331 \log (5 x+3) \]

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Rubi [A]  time = 0.02, antiderivative size = 50, 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 {7189}{27 (3 x+2)}+\frac {1421}{54 (3 x+2)^2}+\frac {343}{81 (3 x+2)^3}-1331 \log (3 x+2)+1331 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^3/((2 + 3*x)^4*(3 + 5*x)),x]

[Out]

343/(81*(2 + 3*x)^3) + 1421/(54*(2 + 3*x)^2) + 7189/(27*(2 + 3*x)) - 1331*Log[2 + 3*x] + 1331*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)^4 (3+5 x)} \, dx &=\int \left (-\frac {343}{9 (2+3 x)^4}-\frac {1421}{9 (2+3 x)^3}-\frac {7189}{9 (2+3 x)^2}-\frac {3993}{2+3 x}+\frac {6655}{3+5 x}\right ) \, dx\\ &=\frac {343}{81 (2+3 x)^3}+\frac {1421}{54 (2+3 x)^2}+\frac {7189}{27 (2+3 x)}-1331 \log (2+3 x)+1331 \log (3+5 x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 40, normalized size = 0.80 \begin {gather*} \frac {7 \left (55458 x^2+75771 x+25964\right )}{162 (3 x+2)^3}-1331 \log (5 (3 x+2))+1331 \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^3/((2 + 3*x)^4*(3 + 5*x)),x]

[Out]

(7*(25964 + 75771*x + 55458*x^2))/(162*(2 + 3*x)^3) - 1331*Log[5*(2 + 3*x)] + 1331*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)^4 (3+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[(1 - 2*x)^3/((2 + 3*x)^4*(3 + 5*x)),x]

[Out]

IntegrateAlgebraic[(1 - 2*x)^3/((2 + 3*x)^4*(3 + 5*x)), x]

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fricas [A]  time = 1.26, size = 75, normalized size = 1.50 \begin {gather*} \frac {388206 \, x^{2} + 215622 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (5 \, x + 3\right ) - 215622 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 530397 \, x + 181748}{162 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^3/(2+3*x)^4/(3+5*x),x, algorithm="fricas")

[Out]

1/162*(388206*x^2 + 215622*(27*x^3 + 54*x^2 + 36*x + 8)*log(5*x + 3) - 215622*(27*x^3 + 54*x^2 + 36*x + 8)*log
(3*x + 2) + 530397*x + 181748)/(27*x^3 + 54*x^2 + 36*x + 8)

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giac [A]  time = 0.81, size = 38, normalized size = 0.76 \begin {gather*} \frac {7 \, {\left (55458 \, x^{2} + 75771 \, x + 25964\right )}}{162 \, {\left (3 \, x + 2\right )}^{3}} + 1331 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 1331 \, \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)^4/(3+5*x),x, algorithm="giac")

[Out]

7/162*(55458*x^2 + 75771*x + 25964)/(3*x + 2)^3 + 1331*log(abs(5*x + 3)) - 1331*log(abs(3*x + 2))

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maple [A]  time = 0.01, size = 45, normalized size = 0.90 \begin {gather*} -1331 \ln \left (3 x +2\right )+1331 \ln \left (5 x +3\right )+\frac {343}{81 \left (3 x +2\right )^{3}}+\frac {1421}{54 \left (3 x +2\right )^{2}}+\frac {7189}{27 \left (3 x +2\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^3/(3*x+2)^4/(5*x+3),x)

[Out]

343/81/(3*x+2)^3+1421/54/(3*x+2)^2+7189/27/(3*x+2)-1331*ln(3*x+2)+1331*ln(5*x+3)

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maxima [A]  time = 0.48, size = 46, normalized size = 0.92 \begin {gather*} \frac {7 \, {\left (55458 \, x^{2} + 75771 \, x + 25964\right )}}{162 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + 1331 \, \log \left (5 \, x + 3\right ) - 1331 \, \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)^4/(3+5*x),x, algorithm="maxima")

[Out]

7/162*(55458*x^2 + 75771*x + 25964)/(27*x^3 + 54*x^2 + 36*x + 8) + 1331*log(5*x + 3) - 1331*log(3*x + 2)

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mupad [B]  time = 1.12, size = 35, normalized size = 0.70 \begin {gather*} \frac {\frac {7189\,x^2}{81}+\frac {58933\,x}{486}+\frac {90874}{2187}}{x^3+2\,x^2+\frac {4\,x}{3}+\frac {8}{27}}-2662\,\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)^4*(5*x + 3)),x)

[Out]

((58933*x)/486 + (7189*x^2)/81 + 90874/2187)/((4*x)/3 + 2*x^2 + x^3 + 8/27) - 2662*atanh(30*x + 19)

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sympy [A]  time = 0.16, size = 42, normalized size = 0.84 \begin {gather*} - \frac {- 388206 x^{2} - 530397 x - 181748}{4374 x^{3} + 8748 x^{2} + 5832 x + 1296} + 1331 \log {\left (x + \frac {3}{5} \right )} - 1331 \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)**4/(3+5*x),x)

[Out]

-(-388206*x**2 - 530397*x - 181748)/(4374*x**3 + 8748*x**2 + 5832*x + 1296) + 1331*log(x + 3/5) - 1331*log(x +
2/3)

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