\(\int \frac {(1-2 x)^2}{3+5 x} \, dx\) [1296]

Optimal result

Integrand size = 15, antiderivative size = 23 \[ \int \frac {(1-2 x)^2}{3+5 x} \, dx=-\frac {32 x}{25}+\frac {2 x^2}{5}+\frac {121}{125} \log (3+5 x) \]

[Out]

-32/25*x+2/5*x^2+121/125*ln(3+5*x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {45} \[ \int \frac {(1-2 x)^2}{3+5 x} \, dx=\frac {2 x^2}{5}-\frac {32 x}{25}+\frac {121}{125} \log (5 x+3) \]

[In]

Int[(1 - 2*x)^2/(3 + 5*x),x]

[Out]

(-32*x)/25 + (2*x^2)/5 + (121*Log[3 + 5*x])/125

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 {32}{25}+\frac {4 x}{5}+\frac {121}{25 (3+5 x)}\right ) \, dx \\ & = -\frac {32 x}{25}+\frac {2 x^2}{5}+\frac {121}{125} \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^2}{3+5 x} \, dx=\frac {1}{125} \left (-114-160 x+50 x^2+121 \log (3+5 x)\right ) \]

[In]

Integrate[(1 - 2*x)^2/(3 + 5*x),x]

[Out]

(-114 - 160*x + 50*x^2 + 121*Log[3 + 5*x])/125

Maple [A] (verified)

Time = 2.31 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.70

method	result	size
parallelrisch	\(\frac {2 x^{2}}{5}-\frac {32 x}{25}+\frac {121 \ln \left (x +\frac {3}{5}\right )}{125}\)	\(16\)
default	\(-\frac {32 x}{25}+\frac {2 x^{2}}{5}+\frac {121 \ln \left (3+5 x \right )}{125}\)	\(18\)
norman	\(-\frac {32 x}{25}+\frac {2 x^{2}}{5}+\frac {121 \ln \left (3+5 x \right )}{125}\)	\(18\)
risch	\(-\frac {32 x}{25}+\frac {2 x^{2}}{5}+\frac {121 \ln \left (3+5 x \right )}{125}\)	\(18\)
meijerg	\(\frac {121 \ln \left (1+\frac {5 x}{3}\right )}{125}-\frac {4 x}{5}-\frac {2 x \left (-5 x +6\right )}{25}\)	\(21\)

[In]

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

[Out]

2/5*x^2-32/25*x+121/125*ln(x+3/5)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^2}{3+5 x} \, dx=\frac {2}{5} \, x^{2} - \frac {32}{25} \, x + \frac {121}{125} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

2/5*x^2 - 32/25*x + 121/125*log(5*x + 3)

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {(1-2 x)^2}{3+5 x} \, dx=\frac {2 x^{2}}{5} - \frac {32 x}{25} + \frac {121 \log {\left (5 x + 3 \right )}}{125} \]

[In]

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

[Out]

2*x**2/5 - 32*x/25 + 121*log(5*x + 3)/125

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.74 \[ \int \frac {(1-2 x)^2}{3+5 x} \, dx=\frac {2}{5} \, x^{2} - \frac {32}{25} \, x + \frac {121}{125} \, \log \left (5 \, x + 3\right ) \]

[In]

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

[Out]

2/5*x^2 - 32/25*x + 121/125*log(5*x + 3)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.78 \[ \int \frac {(1-2 x)^2}{3+5 x} \, dx=\frac {2}{5} \, x^{2} - \frac {32}{25} \, x + \frac {121}{125} \, \log \left ({\left | 5 \, x + 3 \right |}\right ) \]

[In]

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

[Out]

2/5*x^2 - 32/25*x + 121/125*log(abs(5*x + 3))

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.65 \[ \int \frac {(1-2 x)^2}{3+5 x} \, dx=\frac {121\,\ln \left (x+\frac {3}{5}\right )}{125}-\frac {32\,x}{25}+\frac {2\,x^2}{5} \]

[In]

int((2*x - 1)^2/(5*x + 3),x)

[Out]

(121*log(x + 3/5))/125 - (32*x)/25 + (2*x^2)/5