\(\int \frac {(1-2 x)^2}{(2+3 x)^2 (3+5 x)^3} \, dx\) [1331]

Optimal result

Integrand size = 22, antiderivative size = 46 \[ \int \frac {(1-2 x)^2}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {49}{2+3 x}-\frac {121}{10 (3+5 x)^2}+\frac {154}{3+5 x}-707 \log (2+3 x)+707 \log (3+5 x) \]

[Out]

49/(2+3*x)-121/10/(3+5*x)^2+154/(3+5*x)-707*ln(2+3*x)+707*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {90} \[ \int \frac {(1-2 x)^2}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {49}{3 x+2}+\frac {154}{5 x+3}-\frac {121}{10 (5 x+3)^2}-707 \log (3 x+2)+707 \log (5 x+3) \]

[In]

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

[Out]

49/(2 + 3*x) - 121/(10*(3 + 5*x)^2) + 154/(3 + 5*x) - 707*Log[2 + 3*x] + 707*Log[3 + 5*x]

Rule 90

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*} \text {integral}& = \int \left (-\frac {147}{(2+3 x)^2}-\frac {2121}{2+3 x}+\frac {121}{(3+5 x)^3}-\frac {770}{(3+5 x)^2}+\frac {3535}{3+5 x}\right ) \, dx \\ & = \frac {49}{2+3 x}-\frac {121}{10 (3+5 x)^2}+\frac {154}{3+5 x}-707 \log (2+3 x)+707 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^2}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {49}{2+3 x}-\frac {121}{10 (3+5 x)^2}+\frac {154}{3+5 x}-707 \log (5 (2+3 x))+707 \log (3+5 x) \]

[In]

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

[Out]

49/(2 + 3*x) - 121/(10*(3 + 5*x)^2) + 154/(3 + 5*x) - 707*Log[5*(2 + 3*x)] + 707*Log[3 + 5*x]

Maple [A] (verified)

Time = 2.36 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.96

method	result	size
risch	\(\frac {3535 x^{2}+\frac {43597}{10} x +\frac {6704}{5}}{\left (2+3 x \right ) \left (3+5 x \right )^{2}}-707 \ln \left (2+3 x \right )+707 \ln \left (3+5 x \right )\)	\(44\)
default	\(\frac {49}{2+3 x}-\frac {121}{10 \left (3+5 x \right )^{2}}+\frac {154}{3+5 x}-707 \ln \left (2+3 x \right )+707 \ln \left (3+5 x \right )\)	\(45\)
norman	\(\frac {-\frac {62041}{9} x^{2}-\frac {16760}{3} x^{3}-\frac {12725}{6} x}{\left (2+3 x \right ) \left (3+5 x \right )^{2}}-707 \ln \left (2+3 x \right )+707 \ln \left (3+5 x \right )\)	\(47\)
parallelrisch	\(-\frac {954450 \ln \left (\frac {2}{3}+x \right ) x^{3}-954450 \ln \left (x +\frac {3}{5}\right ) x^{3}+1781640 \ln \left (\frac {2}{3}+x \right ) x^{2}-1781640 \ln \left (x +\frac {3}{5}\right ) x^{2}+100560 x^{3}+1107162 \ln \left (\frac {2}{3}+x \right ) x -1107162 \ln \left (x +\frac {3}{5}\right ) x +124082 x^{2}+229068 \ln \left (\frac {2}{3}+x \right )-229068 \ln \left (x +\frac {3}{5}\right )+38175 x}{18 \left (2+3 x \right ) \left (3+5 x \right )^{2}}\)	\(93\)

[In]

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

[Out]

75*(707/15*x^2+43597/750*x+6704/375)/(2+3*x)/(3+5*x)^2-707*ln(2+3*x)+707*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.63 \[ \int \frac {(1-2 x)^2}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {35350 \, x^{2} + 7070 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (5 \, x + 3\right ) - 7070 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )} \log \left (3 \, x + 2\right ) + 43597 \, x + 13408}{10 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} \]

[In]

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

[Out]

1/10*(35350*x^2 + 7070*(75*x^3 + 140*x^2 + 87*x + 18)*log(5*x + 3) - 7070*(75*x^3 + 140*x^2 + 87*x + 18)*log(3
*x + 2) + 43597*x + 13408)/(75*x^3 + 140*x^2 + 87*x + 18)

Sympy [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.89 \[ \int \frac {(1-2 x)^2}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {35350 x^{2} + 43597 x + 13408}{750 x^{3} + 1400 x^{2} + 870 x + 180} + 707 \log {\left (x + \frac {3}{5} \right )} - 707 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

(35350*x**2 + 43597*x + 13408)/(750*x**3 + 1400*x**2 + 870*x + 180) + 707*log(x + 3/5) - 707*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {(1-2 x)^2}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {35350 \, x^{2} + 43597 \, x + 13408}{10 \, {\left (75 \, x^{3} + 140 \, x^{2} + 87 \, x + 18\right )}} + 707 \, \log \left (5 \, x + 3\right ) - 707 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

1/10*(35350*x^2 + 43597*x + 13408)/(75*x^3 + 140*x^2 + 87*x + 18) + 707*log(5*x + 3) - 707*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.07 \[ \int \frac {(1-2 x)^2}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {49}{3 \, x + 2} - \frac {33 \, {\left (\frac {206}{3 \, x + 2} - 865\right )}}{2 \, {\left (\frac {1}{3 \, x + 2} - 5\right )}^{2}} + 707 \, \log \left ({\left | -\frac {1}{3 \, x + 2} + 5 \right |}\right ) \]

[In]

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

[Out]

49/(3*x + 2) - 33/2*(206/(3*x + 2) - 865)/(1/(3*x + 2) - 5)^2 + 707*log(abs(-1/(3*x + 2) + 5))

Mupad [B] (verification not implemented)

Time = 1.17 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.76 \[ \int \frac {(1-2 x)^2}{(2+3 x)^2 (3+5 x)^3} \, dx=\frac {\frac {707\,x^2}{15}+\frac {43597\,x}{750}+\frac {6704}{375}}{x^3+\frac {28\,x^2}{15}+\frac {29\,x}{25}+\frac {6}{25}}-1414\,\mathrm {atanh}\left (30\,x+19\right ) \]

[In]

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

[Out]

((43597*x)/750 + (707*x^2)/15 + 6704/375)/((29*x)/25 + (28*x^2)/15 + x^3 + 6/25) - 1414*atanh(30*x + 19)