\(\int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^2} \, dx\) [1316]

Optimal result

Integrand size = 22, antiderivative size = 55 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {49}{9 (2+3 x)^3}-\frac {77}{(2+3 x)^2}-\frac {1133}{2+3 x}-\frac {605}{3+5 x}+7480 \log (2+3 x)-7480 \log (3+5 x) \]

[Out]

-49/9/(2+3*x)^3-77/(2+3*x)^2-1133/(2+3*x)-605/(3+5*x)+7480*ln(2+3*x)-7480*ln(3+5*x)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 55, 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} \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {1133}{3 x+2}-\frac {605}{5 x+3}-\frac {77}{(3 x+2)^2}-\frac {49}{9 (3 x+2)^3}+7480 \log (3 x+2)-7480 \log (5 x+3) \]

[In]

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

[Out]

-49/(9*(2 + 3*x)^3) - 77/(2 + 3*x)^2 - 1133/(2 + 3*x) - 605/(3 + 5*x) + 7480*Log[2 + 3*x] - 7480*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 {49}{(2+3 x)^4}+\frac {462}{(2+3 x)^3}+\frac {3399}{(2+3 x)^2}+\frac {22440}{2+3 x}+\frac {3025}{(3+5 x)^2}-\frac {37400}{3+5 x}\right ) \, dx \\ & = -\frac {49}{9 (2+3 x)^3}-\frac {77}{(2+3 x)^2}-\frac {1133}{2+3 x}-\frac {605}{3+5 x}+7480 \log (2+3 x)-7480 \log (3+5 x) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.04 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {49}{9 (2+3 x)^3}-\frac {77}{(2+3 x)^2}-\frac {1133}{2+3 x}-\frac {605}{3+5 x}+7480 \log (5 (2+3 x))-7480 \log (3+5 x) \]

[In]

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

[Out]

-49/(9*(2 + 3*x)^3) - 77/(2 + 3*x)^2 - 1133/(2 + 3*x) - 605/(3 + 5*x) + 7480*Log[5*(2 + 3*x)] - 7480*Log[3 + 5
*x]

Maple [A] (verified)

Time = 2.35 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.87

method	result	size
norman	\(\frac {-132396 x^{2}-67320 x^{3}-\frac {780464}{9} x -\frac {56743}{3}}{\left (2+3 x \right )^{3} \left (3+5 x \right )}+7480 \ln \left (2+3 x \right )-7480 \ln \left (3+5 x \right )\)	\(48\)
risch	\(\frac {-132396 x^{2}-67320 x^{3}-\frac {780464}{9} x -\frac {56743}{3}}{\left (2+3 x \right )^{3} \left (3+5 x \right )}+7480 \ln \left (2+3 x \right )-7480 \ln \left (3+5 x \right )\)	\(49\)
default	\(-\frac {49}{9 \left (2+3 x \right )^{3}}-\frac {77}{\left (2+3 x \right )^{2}}-\frac {1133}{2+3 x}-\frac {605}{3+5 x}+7480 \ln \left (2+3 x \right )-7480 \ln \left (3+5 x \right )\)	\(54\)
parallelrisch	\(\frac {24235200 \ln \left (\frac {2}{3}+x \right ) x^{4}-24235200 \ln \left (x +\frac {3}{5}\right ) x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{3}-63011520 \ln \left (x +\frac {3}{5}\right ) x^{3}+2553435 x^{4}[phone] \ln \left (\frac {2}{3}+x \right ) x^{2}-61395840 \ln \left (x +\frac {3}{5}\right ) x^{2}+5023251 x^{3}[phone] \ln \left (\frac {2}{3}+x \right ) x -26568960 \ln \left (x +\frac {3}{5}\right ) x +3291198 x^{2}+4308480 \ln \left (\frac {2}{3}+x \right )-4308480 \ln \left (x +\frac {3}{5}\right )+718084 x}{24 \left (2+3 x \right )^{3} \left (3+5 x \right )}\)	\(116\)

[In]

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

[Out]

(-132396*x^2-67320*x^3-780464/9*x-56743/3)/(2+3*x)^3/(3+5*x)+7480*ln(2+3*x)-7480*ln(3+5*x)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.73 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {605880 \, x^{3} + 1191564 \, x^{2} + 67320 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (5 \, x + 3\right ) - 67320 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )} \log \left (3 \, x + 2\right ) + 780464 \, x + 170229}{9 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} \]

[In]

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

[Out]

-1/9*(605880*x^3 + 1191564*x^2 + 67320*(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(5*x + 3) - 67320*(135*x^
4 + 351*x^3 + 342*x^2 + 148*x + 24)*log(3*x + 2) + 780464*x + 170229)/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 2
4)

Sympy [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.96 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^2} \, dx=\frac {- 605880 x^{3} - 1191564 x^{2} - 780464 x - 170229}{1215 x^{4} + 3159 x^{3} + 3078 x^{2} + 1332 x + 216} - 7480 \log {\left (x + \frac {3}{5} \right )} + 7480 \log {\left (x + \frac {2}{3} \right )} \]

[In]

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

[Out]

(-605880*x**3 - 1191564*x**2 - 780464*x - 170229)/(1215*x**4 + 3159*x**3 + 3078*x**2 + 1332*x + 216) - 7480*lo
g(x + 3/5) + 7480*log(x + 2/3)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {605880 \, x^{3} + 1191564 \, x^{2} + 780464 \, x + 170229}{9 \, {\left (135 \, x^{4} + 351 \, x^{3} + 342 \, x^{2} + 148 \, x + 24\right )}} - 7480 \, \log \left (5 \, x + 3\right ) + 7480 \, \log \left (3 \, x + 2\right ) \]

[In]

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

[Out]

-1/9*(605880*x^3 + 1191564*x^2 + 780464*x + 170229)/(135*x^4 + 351*x^3 + 342*x^2 + 148*x + 24) - 7480*log(5*x
+ 3) + 7480*log(3*x + 2)

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.05 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^2} \, dx=-\frac {605}{5 \, x + 3} + \frac {5 \, {\left (\frac {34464}{5 \, x + 3} + \frac {6934}{{\left (5 \, x + 3\right )}^{2}} + 44661\right )}}{{\left (\frac {1}{5 \, x + 3} + 3\right )}^{3}} + 7480 \, \log \left ({\left | -\frac {1}{5 \, x + 3} - 3 \right |}\right ) \]

[In]

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

[Out]

-605/(5*x + 3) + 5*(34464/(5*x + 3) + 6934/(5*x + 3)^2 + 44661)/(1/(5*x + 3) + 3)^3 + 7480*log(abs(-1/(5*x + 3
) - 3))

Mupad [B] (verification not implemented)

Time = 1.19 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \frac {(1-2 x)^2}{(2+3 x)^4 (3+5 x)^2} \, dx=14960\,\mathrm {atanh}\left (30\,x+19\right )-\frac {\frac {1496\,x^3}{3}+\frac {44132\,x^2}{45}+\frac {780464\,x}{1215}+\frac {56743}{405}}{x^4+\frac {13\,x^3}{5}+\frac {38\,x^2}{15}+\frac {148\,x}{135}+\frac {8}{45}} \]

[In]

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

[Out]

14960*atanh(30*x + 19) - ((780464*x)/1215 + (44132*x^2)/45 + (1496*x^3)/3 + 56743/405)/((148*x)/135 + (38*x^2)
/15 + (13*x^3)/5 + x^4 + 8/45)