3.1398 \(\int \frac{(1-2 x)^3}{(2+3 x)^4 (3+5 x)} \, dx\)

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

[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]

________________________________________________________________________________________

Rubi [A]  time = 0.0217267, antiderivative size = 50, normalized size of antiderivative = 1., 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} \[ \frac{7189}{27 (3 x+2)}+\frac{1421}{54 (3 x+2)^2}+\frac{343}{81 (3 x+2)^3}-1331 \log (3 x+2)+1331 \log (5 x+3) \]

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*}

Mathematica [A]  time = 0.0251482, size = 40, normalized size = 0.8 \[ \frac{7 \left (55458 x^2+75771 x+25964\right )}{162 (3 x+2)^3}-1331 \log (5 (3 x+2))+1331 \log (5 x+3) \]

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]

________________________________________________________________________________________

Maple [A]  time = 0.007, size = 45, normalized size = 0.9 \begin{align*}{\frac{343}{81\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{1421}{54\, \left ( 2+3\,x \right ) ^{2}}}+{\frac{7189}{54+81\,x}}-1331\,\ln \left ( 2+3\,x \right ) +1331\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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*ln(2+3*x)+1331*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 2.38946, size = 62, normalized size = 1.24 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Fricas [A]  time = 1.29852, size = 231, normalized size = 4.62 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Sympy [A]  time = 0.150603, size = 41, normalized size = 0.82 \begin{align*} \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{align*}

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)

________________________________________________________________________________________

Giac [A]  time = 2.37139, size = 51, normalized size = 1.02 \begin{align*} \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{align*}

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))