\(\int \frac {1-2 x}{(3+5 x)^2} \, dx\) [1214]

Optimal result

Integrand size = 13, antiderivative size = 22 \[ \int \frac {1-2 x}{(3+5 x)^2} \, dx=-\frac {11}{25 (3+5 x)}-\frac {2}{25} \log (3+5 x) \]

[Out]

-11/25/(3+5*x)-2/25*ln(3+5*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int \frac {1-2 x}{(3+5 x)^2} \, dx=-\frac {11}{25 (5 x+3)}-\frac {2}{25} \log (5 x+3) \]

[In]

Int[(1 - 2*x)/(3 + 5*x)^2,x]

[Out]

-11/(25*(3 + 5*x)) - (2*Log[3 + 5*x])/25

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {11}{5 (3+5 x)^2}-\frac {2}{5 (3+5 x)}\right ) \, dx \\ & = -\frac {11}{25 (3+5 x)}-\frac {2}{25} \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1-2 x}{(3+5 x)^2} \, dx=-\frac {11}{25 (3+5 x)}-\frac {2}{25} \log (3+5 x) \]

[In]

Integrate[(1 - 2*x)/(3 + 5*x)^2,x]

[Out]

-11/(25*(3 + 5*x)) - (2*Log[3 + 5*x])/25

Maple [A] (verified)

Time = 1.80 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77

method	result	size
risch	\(-\frac {11}{125 \left (x +\frac {3}{5}\right )}-\frac {2 \ln \left (3+5 x \right )}{25}\)	\(17\)
default	\(-\frac {11}{25 \left (3+5 x \right )}-\frac {2 \ln \left (3+5 x \right )}{25}\)	\(19\)
norman	\(\frac {11 x}{15 \left (3+5 x \right )}-\frac {2 \ln \left (3+5 x \right )}{25}\)	\(20\)
meijerg	\(\frac {11 x}{45 \left (1+\frac {5 x}{3}\right )}-\frac {2 \ln \left (1+\frac {5 x}{3}\right )}{25}\)	\(20\)
parallelrisch	\(-\frac {30 \ln \left (x +\frac {3}{5}\right ) x +18 \ln \left (x +\frac {3}{5}\right )-55 x}{75 \left (3+5 x \right )}\)	\(27\)

[In]

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

[Out]

-11/125/(x+3/5)-2/25*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1-2 x}{(3+5 x)^2} \, dx=-\frac {2 \, {\left (5 \, x + 3\right )} \log \left (5 \, x + 3\right ) + 11}{25 \, {\left (5 \, x + 3\right )}} \]

[In]

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

[Out]

-1/25*(2*(5*x + 3)*log(5*x + 3) + 11)/(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.77 \[ \int \frac {1-2 x}{(3+5 x)^2} \, dx=- \frac {2 \log {\left (5 x + 3 \right )}}{25} - \frac {11}{125 x + 75} \]

[In]

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

[Out]

-2*log(5*x + 3)/25 - 11/(125*x + 75)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82 \[ \int \frac {1-2 x}{(3+5 x)^2} \, dx=-\frac {11}{25 \, {\left (5 \, x + 3\right )}} - \frac {2}{25} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

-11/25/(5*x + 3) - 2/25*log(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {1-2 x}{(3+5 x)^2} \, dx=-\frac {11}{25 \, {\left (5 \, x + 3\right )}} + \frac {2}{25} \, \log \left (\frac {{\left | 5 \, x + 3 \right |}}{5 \, {\left (5 \, x + 3\right )}^{2}}\right ) \]

[In]

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

[Out]

-11/25/(5*x + 3) + 2/25*log(1/5*abs(5*x + 3)/(5*x + 3)^2)

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.73 \[ \int \frac {1-2 x}{(3+5 x)^2} \, dx=-\frac {2\,\ln \left (x+\frac {3}{5}\right )}{25}-\frac {11}{125\,\left (x+\frac {3}{5}\right )} \]

[In]

int(-(2*x - 1)/(5*x + 3)^2,x)

[Out]

- (2*log(x + 3/5))/25 - 11/(125*(x + 3/5))