3.1300 \(\int \frac{(1-2 x)^2}{(2+3 x)^4 (3+5 x)} \, dx\)

Optimal. Leaf size=48 \[ \frac{121}{3 x+2}+\frac{217}{18 (3 x+2)^2}+\frac{49}{27 (3 x+2)^3}-605 \log (3 x+2)+605 \log (5 x+3) \]

[Out]

49/(27*(2 + 3*x)^3) + 217/(18*(2 + 3*x)^2) + 121/(2 + 3*x) - 605*Log[2 + 3*x] + 605*Log[3 + 5*x]

________________________________________________________________________________________

Rubi [A]  time = 0.0195223, antiderivative size = 48, 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} \[ \frac{121}{3 x+2}+\frac{217}{18 (3 x+2)^2}+\frac{49}{27 (3 x+2)^3}-605 \log (3 x+2)+605 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

49/(27*(2 + 3*x)^3) + 217/(18*(2 + 3*x)^2) + 121/(2 + 3*x) - 605*Log[2 + 3*x] + 605*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)^2}{(2+3 x)^4 (3+5 x)} \, dx &=\int \left (-\frac{49}{3 (2+3 x)^4}-\frac{217}{3 (2+3 x)^3}-\frac{363}{(2+3 x)^2}-\frac{1815}{2+3 x}+\frac{3025}{3+5 x}\right ) \, dx\\ &=\frac{49}{27 (2+3 x)^3}+\frac{217}{18 (2+3 x)^2}+\frac{121}{2+3 x}-605 \log (2+3 x)+605 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0248894, size = 40, normalized size = 0.83 \[ \frac{58806 x^2+80361 x+27536}{54 (3 x+2)^3}-605 \log (5 (3 x+2))+605 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(27536 + 80361*x + 58806*x^2)/(54*(2 + 3*x)^3) - 605*Log[5*(2 + 3*x)] + 605*Log[3 + 5*x]

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 45, normalized size = 0.9 \begin{align*}{\frac{49}{27\, \left ( 2+3\,x \right ) ^{3}}}+{\frac{217}{18\, \left ( 2+3\,x \right ) ^{2}}}+121\, \left ( 2+3\,x \right ) ^{-1}-605\,\ln \left ( 2+3\,x \right ) +605\,\ln \left ( 3+5\,x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

49/27/(2+3*x)^3+217/18/(2+3*x)^2+121/(2+3*x)-605*ln(2+3*x)+605*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 1.01985, size = 62, normalized size = 1.29 \begin{align*} \frac{58806 \, x^{2} + 80361 \, x + 27536}{54 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + 605 \, \log \left (5 \, x + 3\right ) - 605 \, \log \left (3 \, x + 2\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/54*(58806*x^2 + 80361*x + 27536)/(27*x^3 + 54*x^2 + 36*x + 8) + 605*log(5*x + 3) - 605*log(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.5478, size = 223, normalized size = 4.65 \begin{align*} \frac{58806 \, x^{2} + 32670 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (5 \, x + 3\right ) - 32670 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} \log \left (3 \, x + 2\right ) + 80361 \, x + 27536}{54 \,{\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)^2/(2+3*x)^4/(3+5*x),x, algorithm="fricas")

[Out]

1/54*(58806*x^2 + 32670*(27*x^3 + 54*x^2 + 36*x + 8)*log(5*x + 3) - 32670*(27*x^3 + 54*x^2 + 36*x + 8)*log(3*x
+ 2) + 80361*x + 27536)/(27*x^3 + 54*x^2 + 36*x + 8)

________________________________________________________________________________________

Sympy [A]  time = 0.146873, size = 41, normalized size = 0.85 \begin{align*} \frac{58806 x^{2} + 80361 x + 27536}{1458 x^{3} + 2916 x^{2} + 1944 x + 432} + 605 \log{\left (x + \frac{3}{5} \right )} - 605 \log{\left (x + \frac{2}{3} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(58806*x**2 + 80361*x + 27536)/(1458*x**3 + 2916*x**2 + 1944*x + 432) + 605*log(x + 3/5) - 605*log(x + 2/3)

________________________________________________________________________________________

Giac [A]  time = 1.69258, size = 51, normalized size = 1.06 \begin{align*} \frac{58806 \, x^{2} + 80361 \, x + 27536}{54 \,{\left (3 \, x + 2\right )}^{3}} + 605 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 605 \, \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)^2/(2+3*x)^4/(3+5*x),x, algorithm="giac")

[Out]

1/54*(58806*x^2 + 80361*x + 27536)/(3*x + 2)^3 + 605*log(abs(5*x + 3)) - 605*log(abs(3*x + 2))