3.1330 \(\int \frac{(1-2 x)^2}{(2+3 x) (3+5 x)^3} \, dx\)

Optimal. Leaf size=39 \[ \frac{407}{25 (5 x+3)}-\frac{121}{50 (5 x+3)^2}-49 \log (3 x+2)+49 \log (5 x+3) \]

[Out]

-121/(50*(3 + 5*x)^2) + 407/(25*(3 + 5*x)) - 49*Log[2 + 3*x] + 49*Log[3 + 5*x]

________________________________________________________________________________________

Rubi [A]  time = 0.0181024, antiderivative size = 39, 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} \[ \frac{407}{25 (5 x+3)}-\frac{121}{50 (5 x+3)^2}-49 \log (3 x+2)+49 \log (5 x+3) \]

Antiderivative was successfully verified.

[In]

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

[Out]

-121/(50*(3 + 5*x)^2) + 407/(25*(3 + 5*x)) - 49*Log[2 + 3*x] + 49*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) (3+5 x)^3} \, dx &=\int \left (-\frac{147}{2+3 x}+\frac{121}{5 (3+5 x)^3}-\frac{407}{5 (3+5 x)^2}+\frac{245}{3+5 x}\right ) \, dx\\ &=-\frac{121}{50 (3+5 x)^2}+\frac{407}{25 (3+5 x)}-49 \log (2+3 x)+49 \log (3+5 x)\\ \end{align*}

Mathematica [A]  time = 0.0178352, size = 48, normalized size = 1.23 \[ \frac{4070 x-2450 (5 x+3)^2 \log (3 x+2)+2450 (5 x+3)^2 \log (-3 (5 x+3))+2321}{50 (5 x+3)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2321 + 4070*x - 2450*(3 + 5*x)^2*Log[2 + 3*x] + 2450*(3 + 5*x)^2*Log[-3*(3 + 5*x)])/(50*(3 + 5*x)^2)

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 36, normalized size = 0.9 \begin{align*} -{\frac{121}{50\, \left ( 3+5\,x \right ) ^{2}}}+{\frac{407}{75+125\,x}}-49\,\ln \left ( 2+3\,x \right ) +49\,\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)/(3+5*x)^3,x)

[Out]

-121/50/(3+5*x)^2+407/25/(3+5*x)-49*ln(2+3*x)+49*ln(3+5*x)

________________________________________________________________________________________

Maxima [A]  time = 1.11537, size = 49, normalized size = 1.26 \begin{align*} \frac{11 \,{\left (370 \, x + 211\right )}}{50 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} + 49 \, \log \left (5 \, x + 3\right ) - 49 \, \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)/(3+5*x)^3,x, algorithm="maxima")

[Out]

11/50*(370*x + 211)/(25*x^2 + 30*x + 9) + 49*log(5*x + 3) - 49*log(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.34134, size = 165, normalized size = 4.23 \begin{align*} \frac{2450 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (5 \, x + 3\right ) - 2450 \,{\left (25 \, x^{2} + 30 \, x + 9\right )} \log \left (3 \, x + 2\right ) + 4070 \, x + 2321}{50 \,{\left (25 \, x^{2} + 30 \, x + 9\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/50*(2450*(25*x^2 + 30*x + 9)*log(5*x + 3) - 2450*(25*x^2 + 30*x + 9)*log(3*x + 2) + 4070*x + 2321)/(25*x^2 +
30*x + 9)

________________________________________________________________________________________

Sympy [A]  time = 0.135583, size = 31, normalized size = 0.79 \begin{align*} \frac{4070 x + 2321}{1250 x^{2} + 1500 x + 450} + 49 \log{\left (x + \frac{3}{5} \right )} - 49 \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)/(3+5*x)**3,x)

[Out]

(4070*x + 2321)/(1250*x**2 + 1500*x + 450) + 49*log(x + 3/5) - 49*log(x + 2/3)

________________________________________________________________________________________

Giac [A]  time = 2.71257, size = 45, normalized size = 1.15 \begin{align*} \frac{11 \,{\left (370 \, x + 211\right )}}{50 \,{\left (5 \, x + 3\right )}^{2}} + 49 \, \log \left ({\left | 5 \, x + 3 \right |}\right ) - 49 \, \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)/(3+5*x)^3,x, algorithm="giac")

[Out]

11/50*(370*x + 211)/(5*x + 3)^2 + 49*log(abs(5*x + 3)) - 49*log(abs(3*x + 2))