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

Optimal. Leaf size=43 \[ \frac {1421}{27 (3 x+2)}+\frac {343}{54 (3 x+2)^2}-\frac {7189}{27} \log (3 x+2)+\frac {1331}{5} \log (5 x+3) \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 43, 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 {1421}{27 (3 x+2)}+\frac {343}{54 (3 x+2)^2}-\frac {7189}{27} \log (3 x+2)+\frac {1331}{5} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

343/(54*(2 + 3*x)^2) + 1421/(27*(2 + 3*x)) - (7189*Log[2 + 3*x])/27 + (1331*Log[3 + 5*x])/5

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)^3 (3+5 x)} \, dx &=\int \left (-\frac {343}{9 (2+3 x)^3}-\frac {1421}{9 (2+3 x)^2}-\frac {7189}{9 (2+3 x)}+\frac {1331}{3+5 x}\right ) \, dx\\ &=\frac {343}{54 (2+3 x)^2}+\frac {1421}{27 (2+3 x)}-\frac {7189}{27} \log (2+3 x)+\frac {1331}{5} \log (3+5 x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 39, normalized size = 0.91 \begin {gather*} \frac {49 (58 x+41)}{18 (3 x+2)^2}-\frac {7189}{27} \log (5 (3 x+2))+\frac {1331}{5} \log (5 x+3) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(49*(41 + 58*x))/(18*(2 + 3*x)^2) - (7189*Log[5*(2 + 3*x)])/27 + (1331*Log[3 + 5*x])/5

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(1-2 x)^3}{(2+3 x)^3 (3+5 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.35, size = 55, normalized size = 1.28 \begin {gather*} \frac {71874 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (5 \, x + 3\right ) - 71890 \, {\left (9 \, x^{2} + 12 \, x + 4\right )} \log \left (3 \, x + 2\right ) + 42630 \, x + 30135}{270 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/270*(71874*(9*x^2 + 12*x + 4)*log(5*x + 3) - 71890*(9*x^2 + 12*x + 4)*log(3*x + 2) + 42630*x + 30135)/(9*x^2
+ 12*x + 4)

________________________________________________________________________________________

giac [A]  time = 0.95, size = 33, normalized size = 0.77 \begin {gather*} \frac {49 \, {\left (58 \, x + 41\right )}}{18 \, {\left (3 \, x + 2\right )}^{2}} + \frac {1331}{5} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - \frac {7189}{27} \, \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)^3/(3+5*x),x, algorithm="giac")

[Out]

49/18*(58*x + 41)/(3*x + 2)^2 + 1331/5*log(abs(5*x + 3)) - 7189/27*log(abs(3*x + 2))

________________________________________________________________________________________

maple [A]  time = 0.01, size = 36, normalized size = 0.84 \begin {gather*} -\frac {7189 \ln \left (3 x +2\right )}{27}+\frac {1331 \ln \left (5 x +3\right )}{5}+\frac {343}{54 \left (3 x +2\right )^{2}}+\frac {1421}{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)^3/(5*x+3),x)

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.55, size = 36, normalized size = 0.84 \begin {gather*} \frac {49 \, {\left (58 \, x + 41\right )}}{18 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac {1331}{5} \, \log \left (5 \, x + 3\right ) - \frac {7189}{27} \, \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)^3/(3+5*x),x, algorithm="maxima")

[Out]

49/18*(58*x + 41)/(9*x^2 + 12*x + 4) + 1331/5*log(5*x + 3) - 7189/27*log(3*x + 2)

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 29, normalized size = 0.67 \begin {gather*} \frac {1331\,\ln \left (x+\frac {3}{5}\right )}{5}-\frac {7189\,\ln \left (x+\frac {2}{3}\right )}{27}+\frac {\frac {1421\,x}{81}+\frac {2009}{162}}{x^2+\frac {4\,x}{3}+\frac {4}{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1331*log(x + 3/5))/5 - (7189*log(x + 2/3))/27 + ((1421*x)/81 + 2009/162)/((4*x)/3 + x^2 + 4/9)

________________________________________________________________________________________

sympy [A]  time = 0.16, size = 36, normalized size = 0.84 \begin {gather*} - \frac {- 2842 x - 2009}{162 x^{2} + 216 x + 72} + \frac {1331 \log {\left (x + \frac {3}{5} \right )}}{5} - \frac {7189 \log {\left (x + \frac {2}{3} \right )}}{27} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**3/(2+3*x)**3/(3+5*x),x)

[Out]

-(-2842*x - 2009)/(162*x**2 + 216*x + 72) + 1331*log(x + 3/5)/5 - 7189*log(x + 2/3)/27

________________________________________________________________________________________